Multiscale network renormalization: scale-invariance without geometry
MMultiscale network renormalization: scale-invariance without geometry
Elena Garuccio, Margherita Lalli, and Diego Garlaschelli
2, 1 Instituut-Lorentz for Theoretical Physics, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, Netherlands IMT School for Advanced Studies, Piazza S. Francesco 19, 55100 Lucca, Italy
Systems with lattice geometry can be renormalized exploiting their embedding in metric space,which naturally defines the coarse-grained nodes. By contrast, complex networks defy the usual tech-niques because of their small-world character and lack of explicit metric embedding. Current networkrenormalization approaches require strong assumptions (e.g. community structure, hyperbolicity,scale-free topology), thus remaining incompatible with generic graphs and ordinary lattices. Herewe introduce a graph renormalization scheme valid for any hierarchy of coarse-grainings, therebyallowing for the definition of ‘block nodes’ across multiple scales. This approach reveals a necessaryand specific dependence of network topology on an additive hidden variable attached to nodes, plusoptional dyadic factors. Renormalizable networks turn out to be consistent with a unique specifica-tion of the fitness model, while they are incompatible with preferential attachment, the configurationmodel or the stochastic blockmodel. These results highlight a deep conceptual distinction betweenscale-free and scale-invariant networks, and provide a geometry-free route to renormalization. If thehidden variables are annealed, the model spontaneously leads to realistic scale-free networks withcut-off. If they are quenched, the model can be used to renormalize real-world networks with nodeattributes and distance-dependence or communities. As an example we derive an accurate multiscalemodel of the International Trade Network applicable across hierarchical geographic partitions.
Introduction.
Several societal challenges, including asafer regulation of financial markets, the containmentof infectious diseases, the optimization of transportationsystems and the preservation of biodiversity, require athorough understanding of the network structure con-necting the underlying units of the system [1–3]. One ofthe obstacles systematically encountered in the analysisand modelling of real-world networks is the simultaneouspresence of structures at multiple interacting scales. Forinstance, socioeconomic networks are organized hierar-chically across several levels, from single individuals up togroups, firms, countries and whole geographical regions.Besides the interactions taking place horizontally withinthe same hierarchical level (e.g. social ties among indi-viduals or international trade relationships among coun-tries), there are important cross-level (e.g. individual-firm, firm-country, country-region) interactions that re-quire a multiscale description. Establishing a consistentrepresentation of a graph at multiple scales is in fact along-standing problem whose solution would enable con-siderable progress in the description, modelling, and con-trol of real-world complex systems.In the language of statistical physics, achieving aproper multiscale description of a network requires theintroduction of a renormalization scheme whereby a net-work can be coarse-grained iteratively by partitioningnodes into ‘block-nodes’ either horizontally, i.e. at ho-mogeneous levels of the hierarchy, or across hierarchicallevels, thus allowing block-nodes to contain possibly verydifferent numbers of nodes. While the traditional block-renormalization scheme (whereby equally sized blocks ofneighbouring nodes in a regular lattice are replaced byidentical block-nodes leading to a reduced lattice withthe same geometry) is feasible for geometrically embed-ded networks where the coordinates of nodes naturally in-duce a definition of block-nodes of equal size, it becomes ill-defined in arbitrary graphs and especially in real-worldnetworks with broad degree distribution and small-worldproperty. Several renormalization schemes for complexnetworks have been proposed [4–13]. For instance, inanalogy with fractal analysis, a box-covering techniquedefining block-nodes as certain sets of neighbouring nodeshas been defined [4–6]. Other methods have designeda coarse-graining scheme based on the preservation ofcertain spectral properties of the original network [7, 8].Another notable approach is the geometrical embeddingof networks in a hidden hyperbolic metric space [9, 10],followed by the coarse-graining of hyperbolically nearbynodes [11].Despite progress has been made, the current ap-proaches suffer from a number of limitations, includ-ing lack of generality, assumption of specific topologicalproperties (e.g. community structure, hyperbolic geom-etry and/or scale-freeness), irreducibility to the ordinaryrenormalization scheme in the special case of lattices,and limited iterability in small-world networks with shortpath lengths. These limitations significantly diminish thescope of application and theoretical justification of theavailable techniques. Moreover, the requirement that therenormalization can act across hierarchical levels is notexplicitly enforced in any of the available methods.Here we propose a general renormalization schemebased on a random network model that remains invariantacross all scales, for any desired partition of nodes intoblock-nodes . In a certain ‘quenched’ setting, the modelcan guide the renormalization of graphs with arbitrarytopology, including regular lattices or real-world net-works with node attributes and (optionally, but notnecessarily) dyadic properties such as distances and/orcommunity structure. In a different ‘annealed’ setting, itcan generate realistic scale-free networks spontaneously,only as a result of the requirement of scale-invariance, a r X i v : . [ phy s i c s . s o c - ph ] S e p without fine-tuning and without geometry. The renormalizable network model.
Let us considera binary undirected graph with N ‘fundamental’ nodes(labeled as i = 1 , N ) and its N × N adjacency matrix A (0) with entries a (0) i ,j = 1 if the nodes i and j areconnected, and a (0) i ,j = 0 otherwise. Note that we allowfor self-loops, i.e. each diagonal entry can take values a (0) i ,i = 0 ,
1. We want to aggregate the N nodes into N ≤ N block-nodes (labeled as i = 1 , N ) forminga non-overlapping partition Ω of the original N nodes,and connect two block-nodes if at least one link is presentbetween the nodes across the two blocks, as illustrated inFig. 1. Therefore the coarse-grained graph is described bythe N × N adjacency matrix A (1) with entries a (1) i ,j =1 − (cid:81) i ∈ i (cid:81) j ∈ j (1 − a (0) i ,j ), where i ∈ i denotes thatthe chosen partition Ω maps the original node i onto 𝐺 (cid:3036)(cid:4666)(cid:2868)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036)(cid:2868) (cid:4669) ∈ 𝑀 (cid:2875)(cid:3400)(cid:2875) 𝐺 (cid:3036) (cid:4666)(cid:2869)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036) (cid:2869) (cid:4669) ∈ 𝑀 (cid:2871)(cid:3400)(cid:2871) Ω (cid:2868) arbitrary ⋯ ⋯ (cid:3533) (cid:4668)(cid:3002) (cid:4666)(cid:3116)(cid:4667) (cid:4669) 𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667) (cid:1089) 1
𝑃 𝐴 (cid:2869) (cid:2016) (cid:2869) (cid:3404) (cid:3533) (cid:3002) (cid:3116) Ω(cid:3116) (cid:3002) (cid:3117)
𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667)
This is a scale-invariantrequirement!
Building the Scale-Invariant Random Graph Model 𝐺 (cid:3036)(cid:4666)(cid:2868)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036)(cid:2868) (cid:4669) ∈ 𝑀 (cid:2875)(cid:3400)(cid:2875) 𝐺 (cid:3036) (cid:4666)(cid:2869)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036) (cid:2869) (cid:4669) ∈ 𝑀 (cid:2871)(cid:3400)(cid:2871) Ω (cid:2868) arbitrary ⋯ ⋯ (cid:3533) (cid:4668)(cid:3002) (cid:4666)(cid:3116)(cid:4667) (cid:4669) 𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667) (cid:1089) 1
𝑃 𝐴 (cid:2869) (cid:2016) (cid:2869) (cid:3404) (cid:3533) (cid:3002) (cid:3116) Ω(cid:3116) (cid:3002) (cid:3117)
𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667)
This is a scale-invariantrequirement!
Building the Scale-Invariant Random Graph Model 𝐺 (cid:3036)(cid:4666)(cid:2868)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036)(cid:2868) (cid:4669) ∈ 𝑀 (cid:2875)(cid:3400)(cid:2875) 𝐺 (cid:3036) (cid:4666)(cid:2869)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036) (cid:2869) (cid:4669) ∈ 𝑀 (cid:2871)(cid:3400)(cid:2871) Ω (cid:2868) arbitrary ⋯ ⋯ (cid:3533) (cid:4668)(cid:3002) (cid:4666)(cid:3116)(cid:4667) (cid:4669) 𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667) (cid:1089) 1
𝑃 𝐴 (cid:2869) (cid:2016) (cid:2869) (cid:3404) (cid:3533) (cid:3002) (cid:3116) Ω(cid:3116) (cid:3002) (cid:3117)
𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667)
This is a scale-invariantrequirement!
Building the Scale-Invariant Random Graph Model 𝐺 (cid:3036)(cid:4666)(cid:2868)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036)(cid:2868) (cid:4669) ∈ 𝑀 (cid:2875)(cid:3400)(cid:2875) 𝐺 (cid:3036) (cid:4666)(cid:2869)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036) (cid:2869) (cid:4669) ∈ 𝑀 (cid:2871)(cid:3400)(cid:2871) Ω (cid:2868) arbitrary ⋯ ⋯ (cid:3533) (cid:4668)(cid:3002) (cid:4666)(cid:3116)(cid:4667) (cid:4669) 𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667) (cid:1089) 1
𝑃 𝐴 (cid:2869) (cid:2016) (cid:2869) (cid:3404) (cid:3533) (cid:3002) (cid:3116) Ω(cid:3116) (cid:3002) (cid:3117)
𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667)
This is a scale-invariantrequirement!
Building the Scale-Invariant Random Graph Model 𝐺 (cid:3036)(cid:4666)(cid:2868)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036)(cid:2868) (cid:4669) ∈ 𝑀 (cid:2875)(cid:3400)(cid:2875) 𝐺 (cid:3036) (cid:4666)(cid:2869)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036) (cid:2869) (cid:4669) ∈ 𝑀 (cid:2871)(cid:3400)(cid:2871) Ω (cid:2868) arbitrary ⋯ ⋯ (cid:3533) (cid:4668)(cid:3002) (cid:4666)(cid:3116)(cid:4667) (cid:4669) 𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667) (cid:1089) 1
𝑃 𝐴 (cid:2869) (cid:2016) (cid:2869) (cid:3404) (cid:3533) (cid:3002) (cid:3116) Ω(cid:3116) (cid:3002) (cid:3117)
𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667)
This is a scale-invariantrequirement!
Building the Scale-Invariant Random Graph Model 𝐺 (cid:3036)(cid:4666)(cid:2868)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036)(cid:2868) (cid:4669) ∈ 𝑀 (cid:2875)(cid:3400)(cid:2875) 𝐺 (cid:3036) (cid:4666)(cid:2869)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036) (cid:2869) (cid:4669) ∈ 𝑀 (cid:2871)(cid:3400)(cid:2871) Ω (cid:2868) arbitrary ⋯ ⋯ (cid:3533) (cid:4668)(cid:3002) (cid:4666)(cid:3116)(cid:4667) (cid:4669) 𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667) (cid:1089) 1
𝑃 𝐴 (cid:2869) (cid:2016) (cid:2869) (cid:3404) (cid:3533) (cid:3002) (cid:3116) Ω(cid:3116) (cid:3002) (cid:3117)
𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667)
This is a scale-invariantrequirement!
Building the Scale-Invariant Random Graph Model 𝐺 (cid:3036)(cid:4666)(cid:2868)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036)(cid:2868) (cid:4669) ∈ 𝑀 (cid:2875)(cid:3400)(cid:2875) 𝐺 (cid:3036) (cid:4666)(cid:2869)(cid:4667) ∶ (cid:4668)𝐴 (cid:3036) (cid:2869) (cid:4669) ∈ 𝑀 (cid:2871)(cid:3400)(cid:2871) Ω (cid:2868) arbitrary ⋯ ⋯ (cid:3533) (cid:4668)(cid:3002) (cid:4666)(cid:3116)(cid:4667) (cid:4669) 𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667) (cid:1089) 1
𝑃 𝐴 (cid:2869) (cid:2016) (cid:2869) (cid:3404) (cid:3533) (cid:3002) (cid:3116) Ω(cid:3116) (cid:3002) (cid:3117)
𝑃 𝐴 (cid:4666)(cid:2868)(cid:4667) (cid:2016) (cid:2868) (cid:4667)
This is a scale-invariantrequirement!
Building the Scale-Invariant Random Graph Model
The renormalizable network model.
Let us considera binary undirected graph with N ‘fundamental’ nodes(labeled as i = 1 , N ) and its N ⇥ N adjacency matrix A (0) with entries a (0) i ,j = 1 if the nodes i and j areconnected, and a (0) i ,j = 0 otherwise. Note that we allowfor self-loops, i.e. each diagonal entry can take values a (0) i ,i = 0 ,
1. We want to aggregate the N nodes into N N block-nodes (labeled as i = 1 , N ) forminga non-overlapping partition ⌦ of the original N nodes,and connect two block-nodes if at least one link is presentbetween the nodes across the two blocks, as illustrated inFig. 1. Therefore the coarse-grained graph is described bythe N ⇥ N adjacency matrix A (1) with entries a (1) i ,j =1 Q i i Q j j (1 a (0) i ,j ), where i i denotes thatthe chosen partition ⌦ maps the original node i ontothe block-node i , i.e. i = ⌦ ( i ). Note that we havenot required i = j , as we keep allowing for self-loops.In general i is not the only node mapped to i , i.e. ⌦ is surjective . We call A (0) the 0 -graph and A (1) the 1 -graph .Similarly, we call the N nodes the 0 -nodes and the N block-nodes the 1 -nodes . Iterating the coarse-grainingproduces a hierarchy of ‘blocks of block-nodes’, wherebythe partition ⌦ l leads to an ( l +1)-graph with N l +1 ( l +1)-nodes and adjacency matrix A ( l +1) with entries a ( l +1) i l +1 ,j l +1 = 1 Y i l i l +1 Y j l j l +1 ⇣ a ( l ) i l ,j l ⌘ (1)where i l and j l are l -nodes, while i l +1 = ⌦ l ( i l ) and j l +1 = ⌦ l ( j l ) are ( l + 1)-nodes.The hierarchy of desired partitions { ⌦ l } l can beuniquely parametrized in terms of a dendrogram asshown in Fig. 2. Our first objective is the identifica-tion of a random graph model that can be renormal-ized under any partition obtained from { ⌦ l } l via ei-ther a ‘horizontal’ (left) or a ‘multi-scale’ (right) cutof the dendrogram. Note that, since any ‘multi-scale’coarse-graining is ultimately another partition of the FIG. 1.
Schematic example of the coarse-graining ofa graph . Nodes of the original network (left) are groupedtogether to form the block-nodes A , B and C (right). Ingeneral, block-nodes can contain di↵erent numbers of nodes. same 0-nodes, we can equivalently produce it ‘horizon-tally’ as well, but on a certain modified hierarchy { ⌦ l } l obtained from { ⌦ l } l . Therefore, requiring that themodel is scale-invariant for any specified hierarchy ofpartitions automatically allows for multi-scale coarse-grainings as well. To enforce this requirement, we fixsome { ⌦ l } l and regard the initial 0-graph A (0) notas deterministic, but as generated by a random pro-cess with some probability P A (0) ⇥ normalized sothat P A (0) N P A (0) ⇥ = 1, where ⇥ denotesall parameters of the model (including N ) and G N de-notes the set of all binary undirected graphs with N nodes. A given partition ⌦ will in general map mul-tiple 0-graphs { A (0) } onto the same coarse-grained 1-graph A (1) , and the notation { A (0) } ⌦ ! A (1) will de-note such surjective mapping. Therefore P A (0) ⇥ will induce a random process at the next level, gen-erating each possible 1-graph A (1) with probability P { A (0) } ⌦ ! A (1) P A (0) ⇥ , where the sum runs overall 0-graphs that are projected onto A (1) by ⌦ . Iterat-ing l times, we induce a process generating the l -graph A ( l ) with probability P { A (0) } ⌦ l ··· ⌦ ! A ( l ) P A (0) ⇥ ,where ⌦ l · · · ⌦ is the composition of the l partitions { ⌦ k } l k =0 , which is ultimately a partition of the 0-nodes.We now enforce a scale-invariant random graph modelthat, for any level l , can generate the possible l -graphs intwo equivalent ways: either hierarchically , i.e. by firstgenerating the 0-graphs with probability P A (0) ⇥ and then coarse-graining them l times via the parti-tions { ⌦ k } l k =0 , or directly , i.e with a certain probability P l A ( l ) ⇥ l that depends on l only through a set ⇥ l ofrenormalized parameters that should be obtained from ⇥ using ⌦ l · · · ⌦ . This scale-invariance is equivalent FIG. 2.
Horizontal and multiscale renormalization.
Left: the desired hierarchy of coarse-grainings can be repre-sented as a dendrogram where the 0-nodes are the bottom‘leaves’ and the l -nodes are the ‘branches’ cut out by a hor-izontal line placed at a suitable height. Right: if the den-drogram is cut at di↵erent heights, one obtains a multiscalerenormalization scheme with block-nodes defined across mul-tiple hierarchical levels. This is ultimately another partitionof the 0-nodes and is therefore readily implemented in ourapproach, which is designed to work for any partition. The renormalizable network model.
Let us considera binary undirected graph with N ‘fundamental’ nodes(labeled as i = 1 , N ) and its N ⇥ N adjacency matrix A (0) with entries a (0) i ,j = 1 if the nodes i and j areconnected, and a (0) i ,j = 0 otherwise. Note that we allowfor self-loops, i.e. each diagonal entry can take values a (0) i ,i = 0 ,
1. We want to aggregate the N nodes into N N block-nodes (labeled as i = 1 , N ) forminga non-overlapping partition ⌦ of the original N nodes,and connect two block-nodes if at least one link is presentbetween the nodes across the two blocks, as illustrated inFig. 1. Therefore the coarse-grained graph is described bythe N ⇥ N adjacency matrix A (1) with entries a (1) i ,j =1 Q i i Q j j (1 a (0) i ,j ), where i i denotes thatthe chosen partition ⌦ maps the original node i ontothe block-node i , i.e. i = ⌦ ( i ). Note that we havenot required i = j , as we keep allowing for self-loops.In general i is not the only node mapped to i , i.e. ⌦ is surjective . We call A (0) the 0 -graph and A (1) the 1 -graph .Similarly, we call the N nodes the 0 -nodes and the N block-nodes the 1 -nodes . Iterating the coarse-grainingproduces a hierarchy of ‘blocks of block-nodes’, wherebythe partition ⌦ l leads to an ( l +1)-graph with N l +1 ( l +1)-nodes and adjacency matrix A ( l +1) with entries a ( l +1) i l +1 ,j l +1 = 1 Y i l i l +1 Y j l j l +1 ⇣ a ( l ) i l ,j l ⌘ (1)where i l and j l are l -nodes, while i l +1 = ⌦ l ( i l ) and j l +1 = ⌦ l ( j l ) are ( l + 1)-nodes.The hierarchy of desired partitions { ⌦ l } l can beuniquely parametrized in terms of a dendrogram asshown in Fig. 2. Our first objective is the identifica-tion of a random graph model that can be renormal-ized under any partition obtained from { ⌦ l } l via ei-ther a ‘horizontal’ (left) or a ‘multi-scale’ (right) cutof the dendrogram. Note that, since any ‘multi-scale’coarse-graining is ultimately another partition of the FIG. 1.
Schematic example of the coarse-graining ofa graph . Nodes of the original network (left) are groupedtogether to form the block-nodes A , B and C (right). Ingeneral, block-nodes can contain di↵erent numbers of nodes. same 0-nodes, we can equivalently produce it ‘horizon-tally’ as well, but on a certain modified hierarchy { ⌦ l } l obtained from { ⌦ l } l . Therefore, requiring that themodel is scale-invariant for any specified hierarchy ofpartitions automatically allows for multi-scale coarse-grainings as well. To enforce this requirement, we fixsome { ⌦ l } l and regard the initial 0-graph A (0) notas deterministic, but as generated by a random pro-cess with some probability P A (0) ⇥ normalized sothat P A (0) N P A (0) ⇥ = 1, where ⇥ denotesall parameters of the model (including N ) and G N de-notes the set of all binary undirected graphs with N nodes. A given partition ⌦ will in general map mul-tiple 0-graphs { A (0) } onto the same coarse-grained 1-graph A (1) , and the notation { A (0) } ⌦ ! A (1) will de-note such surjective mapping. Therefore P A (0) ⇥ will induce a random process at the next level, gen-erating each possible 1-graph A (1) with probability P { A (0) } ⌦ ! A (1) P A (0) ⇥ , where the sum runs overall 0-graphs that are projected onto A (1) by ⌦ . Iterat-ing l times, we induce a process generating the l -graph A ( l ) with probability P { A (0) } ⌦ l ··· ⌦ ! A ( l ) P A (0) ⇥ ,where ⌦ l · · · ⌦ is the composition of the l partitions { ⌦ k } l k =0 , which is ultimately a partition of the 0-nodes.We now enforce a scale-invariant random graph modelthat, for any level l , can generate the possible l -graphs intwo equivalent ways: either hierarchically , i.e. by firstgenerating the 0-graphs with probability P A (0) ⇥ and then coarse-graining them l times via the parti-tions { ⌦ k } l k =0 , or directly , i.e with a certain probability P l A ( l ) ⇥ l that depends on l only through a set ⇥ l ofrenormalized parameters that should be obtained from ⇥ using ⌦ l · · · ⌦ . This scale-invariance is equivalent FIG. 2.
Horizontal and multiscale renormalization.
Left: the desired hierarchy of coarse-grainings can be repre-sented as a dendrogram where the 0-nodes are the bottom‘leaves’ and the l -nodes are the ‘branches’ cut out by a hor-izontal line placed at a suitable height. Right: if the den-drogram is cut at di↵erent heights, one obtains a multiscalerenormalization scheme with block-nodes defined across mul-tiple hierarchical levels. This is ultimately another partitionof the 0-nodes and is therefore readily implemented in ourapproach, which is designed to work for any partition. The renormalizable network model.
Let us considera binary undirected graph with N ‘fundamental’ nodes(labeled as i = 1 , N ) and its N ⇥ N adjacency matrix A (0) with entries a (0) i ,j = 1 if the nodes i and j areconnected, and a (0) i ,j = 0 otherwise. Note that we allowfor self-loops, i.e. each diagonal entry can take values a (0) i ,i = 0 ,
1. We want to aggregate the N nodes into N N block-nodes (labeled as i = 1 , N ) forminga non-overlapping partition ⌦ of the original N nodes,and connect two block-nodes if at least one link is presentbetween the nodes across the two blocks, as illustrated inFig. 1. Therefore the coarse-grained graph is described bythe N ⇥ N adjacency matrix A (1) with entries a (1) i ,j =1 Q i i Q j j (1 a (0) i ,j ), where i i denotes thatthe chosen partition ⌦ maps the original node i ontothe block-node i , i.e. i = ⌦ ( i ). Note that we havenot required i = j , as we keep allowing for self-loops.In general i is not the only node mapped to i , i.e. ⌦ is surjective . We call A (0) the 0 -graph and A (1) the 1 -graph .Similarly, we call the N nodes the 0 -nodes and the N block-nodes the 1 -nodes . Iterating the coarse-grainingproduces a hierarchy of ‘blocks of block-nodes’, wherebythe partition ⌦ l leads to an ( l +1)-graph with N l +1 ( l +1)-nodes and adjacency matrix A ( l +1) with entries a ( l +1) i l +1 ,j l +1 = 1 Y i l i l +1 Y j l j l +1 ⇣ a ( l ) i l ,j l ⌘ (1)where i l and j l are l -nodes, while i l +1 = ⌦ l ( i l ) and j l +1 = ⌦ l ( j l ) are ( l + 1)-nodes.The hierarchy of desired partitions { ⌦ l } l can beuniquely parametrized in terms of a dendrogram asshown in Fig. 2. Our first objective is the identifica-tion of a random graph model that can be renormal-ized under any partition obtained from { ⌦ l } l via ei-ther a ‘horizontal’ (left) or a ‘multi-scale’ (right) cutof the dendrogram. Note that, since any ‘multi-scale’coarse-graining is ultimately another partition of the FIG. 1.
Schematic example of the coarse-graining ofa graph . Nodes of the original network (left) are groupedtogether to form the block-nodes A , B and C (right). Ingeneral, block-nodes can contain di↵erent numbers of nodes. same 0-nodes, we can equivalently produce it ‘horizon-tally’ as well, but on a certain modified hierarchy { ⌦ l } l obtained from { ⌦ l } l . Therefore, requiring that themodel is scale-invariant for any specified hierarchy ofpartitions automatically allows for multi-scale coarse-grainings as well. To enforce this requirement, we fixsome { ⌦ l } l and regard the initial 0-graph A (0) notas deterministic, but as generated by a random pro-cess with some probability P A (0) ⇥ normalized sothat P A (0) N P A (0) ⇥ = 1, where ⇥ denotesall parameters of the model (including N ) and G N de-notes the set of all binary undirected graphs with N nodes. A given partition ⌦ will in general map mul-tiple 0-graphs { A (0) } onto the same coarse-grained 1-graph A (1) , and the notation { A (0) } ⌦ ! A (1) will de-note such surjective mapping. Therefore P A (0) ⇥ will induce a random process at the next level, gen-erating each possible 1-graph A (1) with probability P { A (0) } ⌦ ! A (1) P A (0) ⇥ , where the sum runs overall 0-graphs that are projected onto A (1) by ⌦ . Iterat-ing l times, we induce a process generating the l -graph A ( l ) with probability P { A (0) } ⌦ l ··· ⌦ ! A ( l ) P A (0) ⇥ ,where ⌦ l · · · ⌦ is the composition of the l partitions { ⌦ k } l k =0 , which is ultimately a partition of the 0-nodes.We now enforce a scale-invariant random graph modelthat, for any level l , can generate the possible l -graphs intwo equivalent ways: either hierarchically , i.e. by firstgenerating the 0-graphs with probability P A (0) ⇥ and then coarse-graining them l times via the parti-tions { ⌦ k } l k =0 , or directly , i.e with a certain probability P l A ( l ) ⇥ l that depends on l only through a set ⇥ l ofrenormalized parameters that should be obtained from ⇥ using ⌦ l · · · ⌦ . This scale-invariance is equivalent FIG. 2.
Horizontal and multiscale renormalization.
Left: the desired hierarchy of coarse-grainings can be repre-sented as a dendrogram where the 0-nodes are the bottom‘leaves’ and the l -nodes are the ‘branches’ cut out by a hor-izontal line placed at a suitable height. Right: if the den-drogram is cut at di↵erent heights, one obtains a multiscalerenormalization scheme with block-nodes defined across mul-tiple hierarchical levels. This is ultimately another partitionof the 0-nodes and is therefore readily implemented in ourapproach, which is designed to work for any partition. The renormalizable network model.
Let us considera binary undirected graph with N ‘fundamental’ nodes(labeled as i = 1 , N ) and its N ⇥ N adjacency matrix A (0) with entries a (0) i ,j = 1 if the nodes i and j areconnected, and a (0) i ,j = 0 otherwise. Note that we allowfor self-loops, i.e. each diagonal entry can take values a (0) i ,i = 0 ,
1. We want to aggregate the N nodes into N N block-nodes (labeled as i = 1 , N ) forminga non-overlapping partition ⌦ of the original N nodes,and connect two block-nodes if at least one link is presentbetween the nodes across the two blocks, as illustrated inFig. 1. Therefore the coarse-grained graph is described bythe N ⇥ N adjacency matrix A (1) with entries a (1) i ,j =1 Q i i Q j j (1 a (0) i ,j ), where i i denotes thatthe chosen partition ⌦ maps the original node i ontothe block-node i , i.e. i = ⌦ ( i ). Note that we havenot required i = j , as we keep allowing for self-loops.In general i is not the only node mapped to i , i.e. ⌦ is surjective . We call A (0) the 0 -graph and A (1) the 1 -graph .Similarly, we call the N nodes the 0 -nodes and the N block-nodes the 1 -nodes . Iterating the coarse-grainingproduces a hierarchy of ‘blocks of block-nodes’, wherebythe partition ⌦ l leads to an ( l +1)-graph with N l +1 ( l +1)-nodes and adjacency matrix A ( l +1) with entries a ( l +1) i l +1 ,j l +1 = 1 Y i l i l +1 Y j l j l +1 ⇣ a ( l ) i l ,j l ⌘ (1)where i l and j l are l -nodes, while i l +1 = ⌦ l ( i l ) and j l +1 = ⌦ l ( j l ) are ( l + 1)-nodes.The hierarchy of desired partitions { ⌦ l } l can beuniquely parametrized in terms of a dendrogram asshown in Fig. 2. Our first objective is the identifica-tion of a random graph model that can be renormal-ized under any partition obtained from { ⌦ l } l via ei-ther a ‘horizontal’ (left) or a ‘multi-scale’ (right) cutof the dendrogram. Note that, since any ‘multi-scale’coarse-graining is ultimately another partition of the FIG. 1.
Schematic example of the coarse-graining ofa graph . Nodes of the original network (left) are groupedtogether to form the block-nodes A , B and C (right). Ingeneral, block-nodes can contain di↵erent numbers of nodes. same 0-nodes, we can equivalently produce it ‘horizon-tally’ as well, but on a certain modified hierarchy { ⌦ l } l obtained from { ⌦ l } l . Therefore, requiring that themodel is scale-invariant for any specified hierarchy ofpartitions automatically allows for multi-scale coarse-grainings as well. To enforce this requirement, we fixsome { ⌦ l } l and regard the initial 0-graph A (0) notas deterministic, but as generated by a random pro-cess with some probability P A (0) ⇥ normalized sothat P A (0) N P A (0) ⇥ = 1, where ⇥ denotesall parameters of the model (including N ) and G N de-notes the set of all binary undirected graphs with N nodes. A given partition ⌦ will in general map mul-tiple 0-graphs { A (0) } onto the same coarse-grained 1-graph A (1) , and the notation { A (0) } ⌦ ! A (1) will de-note such surjective mapping. Therefore P A (0) ⇥ will induce a random process at the next level, gen-erating each possible 1-graph A (1) with probability P { A (0) } ⌦ ! A (1) P A (0) ⇥ , where the sum runs overall 0-graphs that are projected onto A (1) by ⌦ . Iterat-ing l times, we induce a process generating the l -graph A ( l ) with probability P { A (0) } ⌦ l ··· ⌦ ! A ( l ) P A (0) ⇥ ,where ⌦ l · · · ⌦ is the composition of the l partitions { ⌦ k } l k =0 , which is ultimately a partition of the 0-nodes.We now enforce a scale-invariant random graph modelthat, for any level l , can generate the possible l -graphs intwo equivalent ways: either hierarchically , i.e. by firstgenerating the 0-graphs with probability P A (0) ⇥ and then coarse-graining them l times via the parti-tions { ⌦ k } l k =0 , or directly , i.e with a certain probability P l A ( l ) ⇥ l that depends on l only through a set ⇥ l ofrenormalized parameters that should be obtained from ⇥ using ⌦ l · · · ⌦ . This scale-invariance is equivalent FIG. 2.
Horizontal and multiscale renormalization.
Left: the desired hierarchy of coarse-grainings can be repre-sented as a dendrogram where the 0-nodes are the bottom‘leaves’ and the l -nodes are the ‘branches’ cut out by a hor-izontal line placed at a suitable height. Right: if the den-drogram is cut at di↵erent heights, one obtains a multiscalerenormalization scheme with block-nodes defined across mul-tiple hierarchical levels. This is ultimately another partitionof the 0-nodes and is therefore readily implemented in ourapproach, which is designed to work for any partition. w i t h s o m e p r o b a b ili t y P A ( ) ⇥ n o r m a li ze d s o t h a t P A ( ) G N P A ( ) ⇥ = , w h e r e ⇥ d e n o t e s a ll p a - r a m e t e r s o f t h e m o d e l ( i n c l ud i n g N ) a nd G N d e n o t e s t h e s e t o f a ll b i n a r y und i r ec t e d g r a ph s w i t h N n o d e s . A g i v e n p a r t i t i o n ⌦ w illi n g e n e r a l m a p m u l t i p l e - g r a ph s { A ( ) } o n t o t h e s a m ec oa r s e - g r a i n e d - g r a ph A ( ) , a nd t h e n o - t a t i o n { A ( ) } ⌦ ! A ( ) w ill d e n o t e s u c h s u r j ec t i v e m a p - p i n g . T h e r e f o r e P A ( ) ⇥ w illi ndu ce a r a nd o m p r o - ce ss a tt h e n e x t l e v e l, g e n e r a t i n g e a c hp o ss i b l e - g r a ph A ( ) w i t hp r o b a b ili t y X { A ( l ) } ⌦ l ! A ( l + ) P l A ( l ) = P l + A ( l + ) P { A ( ) } ⌦ ! A ( ) P A ( ) ⇥ , w h e r e t h e s u m r un s o v e r a ll - g r a ph s t h a t a r e p r o j ec t e d o n t o A ( ) b y ⌦ . I t e r a t - i n g l t i m e s , w e i ndu ce a p r o ce ss g e n e r a t i n g t h e l - g r a ph A ( l ) w i t hp r o b a b ili t y P { A ( ) } ⌦ l ··· ⌦ ! A ( l ) P A ( ) ⇥ , w h e r e ⌦ l ··· ⌦ i s t h ec o m p o s i t i o n o f t h e l p a r t i t i o n s { ⌦ k } l k = , w h i c h i s u l t i m a t e l y a p a r t i t i o n o f t h e - n o d e s . W e n o w e n f o r ce a s c a l e - i n v a r i a n t r a nd o m g r a ph m o d e l t h a t , f o r a n y l e v e l l , c a n g e n e r a t e t h e p o ss i b l e l - g r a ph s i n t w o e q u i v a l e n t w a y s : e i t h e r h i e r a r c h i c a ll y ,i. e . b y fi r s t g e n e r a t i n g t h e - g r a ph s w i t hp r o b a b ili t y P A ( ) ⇥ a nd t h e n c oa r s e - g r a i n i n g t h e m l t i m e s v i a t h e p a r t i - t i o n s { ⌦ k } l k = , o r d i r ec t l y ,i. e w i t h a ce r t a i np r o b a b ili t y P l A ( l ) ⇥ l t h a t d e p e nd s o n l o n l y t h r o u g h a s e t ⇥ l o f r e n o r m a li ze dp a r a m e t e r s t h a t s h o u l db e o b t a i n e d f r o m ⇥ u s i n g ⌦ l ··· ⌦ . T h i ss c a l e - i n v a r i a n ce i s e q u i v a l e n t t o i m p o s i n g t h a t , f o r a n y p a i r l , m ( w i t h l m ) , P l A ( l ) ⇥ l = X { A ( m ) } ⌦ l ··· ⌦ m ! A ( l ) P m A ( m ) ⇥ m ( ) w h e r e t h e r e n o r m a li ze dp a r a m e t e r s ⇥ l a r e o b t a i n e d o n l y f r o m ⇥ m , g i v e n ⌦ l ··· ⌦ m . W e l oo k f o r t h e g e n e r a l s o l u t i o n i n t h ec a s e o f r a nd o m g r a ph s w i t h i nd e p e nd e n t li n k s , w h e r e P l A ( l ) | ⇥ l f a c t o r i ze s a s N l Y i l = i l Y j l = ⇥ p ( l ) i l , j l ⇥ l ⇤ a ( l ) i l , j l ⇥ p ( l ) i l , j l ⇥ l ⇤ a ( l ) i l , j l , ( ) w h e r e p ( l ) i l , j l ⇥ l i s t h e p r o b a b ili t y t h a tt w o l - n o d e s i l a nd j l a r e li n k e d . I n t h i s c a s e i t i s n a t u r a l t o r e q u i r e t h a t ⇥ l c o n t a i n s ( b e s i d e s N l ) a n o v e r a ll c o n s t a n t l ( w h i c h w ill s e tt h e g l o b a lli n k d e n s i t y ) , a s e t o f a dd i t i v e n o d e - s p ec i fi c p a r a m e t e r s { x i l } N l i l = ( w h i c h w ill d i s t r i bu t e t h e t o t a l nu m b e r o f li n k s h e t e r og e n e o u s l y a m o n g n o d e s ) , a nd a n ( o p t i o n a l ) s e t o f d y a d i c ( p a i r - s p ec i fi c ) p a r a m e - t e r s { d i l , j l } N l i l , j l = . W ec a n t h e r e f o r e u s e t h e n o t a t i o n p ( l ) i l , j l ⇥ l = p ( l ) i l , j l ( l ) w h e r e w e k ee p o n l y l e x p li c i t i n t h e a r g u m e n t o f p i l , j l , b ec a u s e t h e d e p e nd e n ce o n t h e o t h e r v a r i a b l e s x i l , x j l , d i l , j l i s a l r e a d y d e n o t e db y t h e s ub s c r i p t s i l , j l ( i nd ee d , p i l , j l d e p e nd s o n i l a nd j l o n l y t h r o u g h x i l , x j l , d i l , j l ) . N o w , t h e s c a l e - i n v a r i a n ce r e q u i r e m e n t c o n s t r a i n s p ( l ) i l , j l t o b e f un c t i o n a ll y i n v a r i a n t w i t h r e s p ec tt o l ,i. e . p ( l ) i l , j l ( l ) = p i l , j l ( l ) . A s w e s h o w i nSupp l e m e n t a r y I n - f o r m a t i o n ( S I ) , t h e r e i s a un i q u e s o l u t i o n g i v e nb y p i l , j l ( ) = e x i l x j l f ( d i l , j l ) , , x i l , x j l , f > , ( ) w h e r e f i s a n a r b i t r a r y p o s i t i v e f un c t i o n a nd t h e f o ll o w - i n g r e n o r m a li z a t i o n r u l e s a pp l y : l + ⌘ l ⌘ , ( ) x i l + ⌘ X i l i l + x i l , ( ) f d i l + , j l + ⌘ P i l i l + P j l j l + x i l x j l f d i l , j l P i l i l + x i l P j l j l + x j l ( ) ( i. e . i ss c a l e - i n v a r i a n t , x i s n o d e - a dd i t i v e a nd f r e n o r - m a li ze s a s a s p ec i fi c x - d e p e nd e n t w e i g h t e d a v e r ag e ) . P u tt i n g E q . ( ) b a c k i n t o E q . ( ) , w e fin a ll y a rr i v e a t t h e s c a l e - i n v a r i a n t g r a php r o b a b ili t y P A ( l ) | = N l Y i l = i l Y j l = ⇥ p i l , j l ( ) ⇤ a ( l ) i l , j l ⇥ p ( l ) i l , j l ( ) ⇤ a ( l ) i l , j l = N l Y i l = i l Y j l = ⇥ e x i l x j l f ( d i l , j l ) ⇤ a ( l ) i l , j l e x i l x j l f ( d i l , j l ) = Q N l Y i l = i l Y j l ⇥ e x i l x j l f ( d i l , j l ) ⇤ a ( l ) i l , j l , ( ) w h e r e w e h a v e d e fin e d Q ⌘ N l Y i l = i l Y j l = e x i l x j l f ( d i l , j l ) = N Y i = i Y j = e x i x j f ( d i , j ) , ( ) w h i c h i s a c o n s t a n tt e r m t h a t d e p e nd s n e i t h e r o n t h e r e a li ze d l - g r a ph A ( l ) n o r , o w i n g t o E q . ( ) , o n t h e h i - e r a r c h i c a ll e v e l l b e i n g c o n s i d e r e d . N o t e t h a t , a s d e - s i r e d , P A ( l ) | d e p e nd s o n l o n l y t h r o u g h t h e p a r a m - e t e r s { x i l } N l i l = a nd { d i l , j l } N l i l , j l = , w h i c h r e n o r m a li ze a s s t a t e d i n E q s . ( ) a nd ( ) . E q u a t i o n s ( ) - ( ) a r e o u r k e y r e s u l t . O n e o f t h e i r r e m a r k a b l ec o n s e q u e n ce s i s t h a t , w h il e t h e d e p e nd e n ce o f t h ec o nn ec t i o np r o b a b ili t y p i l , j l ( ) o n t h e d y a d i c f a c t o r d i l , j l c a nb e s w i t c h e d o ↵ e n t i r e l y w i t h o u t d e s t r o y i n g t h e s c a l e - i n v a r i a n t p r o p e r t i e s o f t h e m o d e l ( e . g . b y t a k i n g f t o b e a c o n s t a n t f un c t i o n , w h e r e b y E q . ( ) i s a u t o m a t i c a ll y f u l fi ll e d ) , t h e d e p e nd e n ce o n t h e n o d e - s p ec i fi c f a c t o r s x i l x j l c a nn o t b e s w i t c h e d o ↵ , un l e ss t h e m o d e li s m a d e d e t e r m i n i s t i c b y f o r m a ll y r e q u i r i n g t h a t f ( d i l , j l )t a k e s o n l y t h e t w o v a l u e s f = ( i m p l y i n g p i l , j l ( ) = ) o r f =+ ( i m p l y i n g p i l , j l ( ) = ) . W e c o n s i d e r e x a m p l e s o f b o t h s i t u a t i o n s b e l o w . T h e r e f o r e , t h e d e p e n d e n ce o n d y ad i c f a c t o r s( i n c l u d i n gge o m e t r i c d i s t a n ce s) i s op t i o n a l , w h i l e t ha t o nn od e - s p ec i fi c f a c t o r s i s n ece ss a r y . T h i s i s ag e n e r a l r e s u l t f o ll o w i n go n l y f r o m t h ee n f o r ce m e n t o f s c a l e - i n v a r i a n ce . M o r e s p ec i fi c r e s u l t s a r e d i s c u ss e db e l o w . w i t h s o m e p r o b a b ili t y P A ( ) ⇥ n o r m a li ze d s o t h a t P A ( ) G N P A ( ) ⇥ = , w h e r e ⇥ d e n o t e s a ll p a - r a m e t e r s o f t h e m o d e l ( i n c l ud i n g N ) a nd G N d e n o t e s t h e s e t o f a ll b i n a r y und i r ec t e d g r a ph s w i t h N n o d e s . A g i v e np a r t i t i o n ⌦ w illi n g e n e r a l m a p m u l t i p l e - g r a ph s { A ( ) } o n t o t h e s a m ec oa r s e - g r a i n e d - g r a ph A ( ) , a nd t h e n o - t a t i o n { A ( ) } ⌦ ! A ( ) w ill d e n o t e s u c h s u r j ec t i v e m a p - p i n g . T h e r e f o r e P A ( ) ⇥ w illi ndu ce a r a nd o m p r o - ce ss a tt h e n e x t l e v e l, g e n e r a t i n g e a c hp o ss i b l e - g r a ph A ( ) w i t hp r o b a b ili t y X { A ( l ) } ⌦ l ! A ( l + ) P l A ( l ) = P l + A ( l + ) P { A ( ) } ⌦ ! A ( ) P A ( ) ⇥ , w h e r e t h e s u m r un s o v e r a ll - g r a ph s t h a t a r e p r o j ec t e d o n t o A ( ) b y ⌦ . I t e r a t - i n g l t i m e s , w e i ndu ce a p r o ce ss g e n e r a t i n g t h e l - g r a ph A ( l ) w i t hp r o b a b ili t y P { A ( ) } ⌦ l ··· ⌦ ! A ( l ) P A ( ) ⇥ , w h e r e ⌦ l ··· ⌦ i s t h ec o m p o s i t i o n o f t h e l p a r t i t i o n s { ⌦ k } l k = , w h i c h i s u l t i m a t e l y a p a r t i t i o n o f t h e - n o d e s . W e n o w e n f o r ce a s c a l e - i n v a r i a n t r a nd o m g r a ph m o d e l t h a t , f o r a n y l e v e l l , c a n g e n e r a t e t h e p o ss i b l e l - g r a ph s i n t w o e q u i v a l e n t w a y s : e i t h e r h i e r a r c h i c a ll y ,i. e . b y fi r s t g e n e r a t i n g t h e - g r a ph s w i t hp r o b a b ili t y P A ( ) ⇥ a nd t h e n c oa r s e - g r a i n i n g t h e m l t i m e s v i a t h e p a r t i - t i o n s { ⌦ k } l k = , o r d i r ec t l y ,i. e w i t h a ce r t a i np r o b a b ili t y P l A ( l ) ⇥ l t h a t d e p e nd s o n l o n l y t h r o u g h a s e t ⇥ l o f r e n o r m a li ze dp a r a m e t e r s t h a t s h o u l db e o b t a i n e d f r o m ⇥ u s i n g ⌦ l ··· ⌦ . T h i ss c a l e - i n v a r i a n ce i s e q u i v a l e n tt o i m p o s i n g t h a t , f o r a n y p a i r l , m ( w i t h l m ) , P l A ( l ) ⇥ l = X { A ( m ) } ⌦ l ··· ⌦ m ! A ( l ) P m A ( m ) ⇥ m ( ) w h e r e t h e r e n o r m a li ze dp a r a m e t e r s ⇥ l a r e o b t a i n e d o n l y f r o m ⇥ m , g i v e n ⌦ l ··· ⌦ m . W e l oo k f o r t h e g e n e r a l s o l u t i o n i n t h ec a s e o f r a nd o m g r a ph s w i t h i nd e p e nd e n t li n k s , w h e r e P l A ( l ) | ⇥ l f a c t o r i ze s a s N l Y i l = i l Y j l = ⇥ p ( l ) i l , j l ⇥ l ⇤ a ( l ) i l , j l ⇥ p ( l ) i l , j l ⇥ l ⇤ a ( l ) i l , j l , ( ) w h e r e p ( l ) i l , j l ⇥ l i s t h e p r o b a b ili t y t h a tt w o l - n o d e s i l a nd j l a r e li n k e d . I n t h i s c a s e i t i s n a t u r a l t o r e q u i r e t h a t ⇥ l c o n t a i n s ( b e s i d e s N l ) a n o v e r a ll c o n s t a n t l ( w h i c h w ill s e tt h e g l o b a lli n k d e n s i t y ) , a s e t o f a dd i t i v e n o d e - s p ec i fi c p a r a m e t e r s { x i l } N l i l = ( w h i c h w ill d i s t r i bu t e t h e t o t a l nu m b e r o f li n k s h e t e r og e n e o u s l y a m o n g n o d e s ) , a nd a n ( o p t i o n a l ) s e t o f d y a d i c ( p a i r - s p ec i fi c ) p a r a m e - t e r s { d i l , j l } N l i l , j l = . W ec a n t h e r e f o r e u s e t h e n o t a t i o n p ( l ) i l , j l ⇥ l = p ( l ) i l , j l ( l ) w h e r e w e k ee p o n l y l e x p li c i t i n t h e a r g u m e n t o f p i l , j l , b ec a u s e t h e d e p e nd e n ce o n t h e o t h e r v a r i a b l e s x i l , x j l , d i l , j l i s a l r e a d y d e n o t e db y t h e s ub s c r i p t s i l , j l ( i nd ee d , p i l , j l d e p e nd s o n i l a nd j l o n l y t h r o u g h x i l , x j l , d i l , j l ) . N o w , t h e s c a l e - i n v a r i a n ce r e q u i r e m e n t c o n s t r a i n s p ( l ) i l , j l t o b e f un c t i o n a ll y i n v a r i a n t w i t h r e s p ec tt o l ,i. e . p ( l ) i l , j l ( l ) = p i l , j l ( l ) . A s w e s h o w i nSupp l e m e n t a r y I n - f o r m a t i o n ( S I ) , t h e r e i s a un i q u e s o l u t i o n g i v e nb y p i l , j l ( ) = e x i l x j l f ( d i l , j l ) , , x i l , x j l , f > , ( ) w h e r e f i s a n a r b i t r a r y p o s i t i v e f un c t i o n a nd t h e f o ll o w - i n g r e n o r m a li z a t i o n r u l e s a pp l y : l + ⌘ l ⌘ , ( ) x i l + ⌘ X i l i l + x i l , ( ) f d i l + , j l + ⌘ P i l i l + P j l j l + x i l x j l f d i l , j l P i l i l + x i l P j l j l + x j l ( )( i. e . i ss c a l e - i n v a r i a n t , x i s n o d e - a dd i t i v e a nd f r e n o r - m a li ze s a s a s p ec i fi c x - d e p e nd e n t w e i g h t e d a v e r ag e ) . P u tt i n g E q . ( ) b a c k i n t o E q . ( ) , w e fin a ll y a rr i v e a tt h e s c a l e - i n v a r i a n t g r a php r o b a b ili t y P A ( l ) | = N l Y i l = i l Y j l = ⇥ p i l , j l ( ) ⇤ a ( l ) i l , j l ⇥ p ( l ) i l , j l ( ) ⇤ a ( l ) i l , j l = N l Y i l = i l Y j l = ⇥ e x i l x j l f ( d i l , j l ) ⇤ a ( l ) i l , j l e x i l x j l f ( d i l , j l ) = Q N l Y i l = i l Y j l ⇥ e x i l x j l f ( d i l , j l ) ⇤ a ( l ) i l , j l , ( ) w h e r e w e h a v e d e fin e d Q ⌘ N l Y i l = i l Y j l = e x i l x j l f ( d i l , j l ) = N Y i = i Y j = e x i x j f ( d i , j ) , ( ) w h i c h i s a c o n s t a n tt e r m t h a t d e p e nd s n e i t h e r o n t h e r e a li ze d l - g r a ph A ( l ) n o r , o w i n g t o E q . ( ) , o n t h e h i - e r a r c h i c a ll e v e l l b e i n g c o n s i d e r e d . N o t e t h a t , a s d e - s i r e d , P A ( l ) | d e p e nd s o n l o n l y t h r o u g h t h e p a r a m - e t e r s { x i l } N l i l = a nd { d i l , j l } N l i l , j l = , w h i c h r e n o r m a li ze a ss t a t e d i n E q s . ( ) a nd ( ) . E q u a t i o n s ( ) - ( ) a r e o u r k e y r e s u l t . O n e o f t h e i rr e m a r k a b l ec o n s e q u e n ce s i s t h a t , w h il e t h e d e p e nd e n ce o f t h ec o nn ec t i o np r o b a b ili t y p i l , j l ( ) o n t h e d y a d i c f a c t o r d i l , j l c a nb e s w i t c h e d o ↵ e n t i r e l y w i t h o u t d e s t r o y i n g t h e s c a l e - i n v a r i a n t p r o p e r t i e s o f t h e m o d e l ( e . g . b y t a k i n g f t o b e a c o n s t a n t f un c t i o n , w h e r e b y E q . ( ) i s a u t o m a t i c a ll y f u l fi ll e d ) , t h e d e p e nd e n ce o n t h e n o d e - s p ec i fi c f a c t o r s x i l x j l c a nn o t b e s w i t c h e d o ↵ , un l e ss t h e m o d e li s m a d e d e t e r m i n i s t i c b y f o r m a ll y r e q u i r i n g t h a t f ( d i l , j l )t a k e s o n l y t h e t w o v a l u e s f = ( i m p l y i n g p i l , j l ( ) = ) o r f =+ ( i m p l y i n g p i l , j l ( ) = ) . W ec o n s i d e r e x a m p l e s o f b o t h s i t u a t i o n s b e l o w . T h e r e f o r e , t h e d e p e n d e n ce o n d y ad i c f a c t o r s( i n c l u d i n gge o m e t r i c d i s t a n ce s) i s op t i o n a l , w h i l e t ha t o nn od e - s p ec i fi c f a c t o r s i s n ece ss a r y . T h i s i s ag e n e r a l r e s u l t f o ll o w i n go n l y f r o m t h ee n f o r ce m e n t o f s c a l e - i n v a r i a n ce . M o r e s p ec i fi c r e s u l t s a r e d i s c u ss e db e l o w . FIG. 1.
Schematic example of the graph coarse-graining and induced ensembles . Nodes of an l -graph A ( l ) (left) are grouped together, via a given partition Ω l , toform the block-nodes of the coarse-grained ( l +1)-graph A ( l +1) (right). Note that, in general, block-nodes can contain differ-ent numbers of nodes. A link between two block-nodes (ora self-loop at a single block-node) is drawn whenever a linkis present between any pair of constituents nodes. Differentrealizations of the l -graph are mapped onto realizations of the( l + 1)-graph via Ω l . Multiple realizations of the l -graph mayend up in the same realization of the ( l + 1)-graph. The scale-invariant requirement is obtained by viewing the realizationsof the l -graph as the outcome of a random graph generatingprocess with probability P l (cid:0) A ( l ) (cid:1) , and imposing that the in-duced probability P l +1 (cid:0) A ( l +1) (cid:1) at the next level has the samefunctional form as P l (cid:0) A ( l ) (cid:1) , with renormalized parameters. the block-node i , i.e. i = Ω ( i ). Note that we havenot required i (cid:54) = j , as we keep allowing for self-loops.In general i is not the only node mapped to i , i.e. Ω is surjective . We call A (0) the 0 -graph and A (1) the 1 -graph .Similarly, we call the N nodes the 0 -nodes and the N block-nodes the 1 -nodes . Iterating the coarse-grainingproduces a hierarchy of ‘blocks of block-nodes’, wherebythe partition Ω l leads to an ( l +1)-graph with N l +1 ( l +1)-nodes and adjacency matrix A ( l +1) with entries a ( l +1) i l +1 ,j l +1 = 1 − (cid:89) i l ∈ i l +1 (cid:89) j l ∈ j l +1 (cid:16) − a ( l ) i l ,j l (cid:17) (1)where i l and j l are l -nodes, while i l +1 = Ω l ( i l ) and j l +1 = Ω l ( j l ) are ( l + 1)-nodes.The hierarchy of desired partitions { Ω l } l ≥ can beuniquely parametrized in terms of a dendrogram asshown in Fig. 2. Our first objective is the identifica-tion of a random graph model that can be renormal-ized under any partition obtained from { Ω l } l ≥ via ei-ther a ‘horizontal’ (left) or a ‘multi-scale’ (right) cutof the dendrogram. Note that, since any ‘multi-scale’coarse-graining is ultimately another partition of thesame 0-nodes, we can equivalently produce it ‘horizon-tally’ as well, but on a certain modified hierarchy { Ω (cid:48) l } l ≥ obtained from { Ω l } l ≥ . Therefore, requiring that themodel is scale-invariant for any specified hierarchy ofpartitions automatically allows for multi-scale coarse-grainings as well. To enforce this requirement, we fixsome { Ω l } l ≥ and regard the initial 0-graph A (0) notas deterministic, but as generated by a random pro-cess with some probability P (cid:0) A (0) (cid:12)(cid:12) Θ (cid:1) normalized sothat (cid:80) A (0) ∈G N P (cid:0) A (0) (cid:12)(cid:12) Θ (cid:1) = 1, where Θ denotesall parameters of the model (including N ) and G N de-notes the set of all binary undirected graphs with N nodes. A given partition Ω will in general map mul- FIG. 2.
Horizontal vs multiscale renormalization.
Left: the desired hierarchy of coarse-grainings can be rep-resented as a dendrogram where the 0-nodes are the bottom‘leaves’ and the l -nodes are the ‘branches’ cut out by a hor-izontal line placed at a suitable height. Right: if the den-drogram is cut at different heights, one obtains a multiscalerenormalization scheme with block-nodes defined across mul-tiple hierarchical levels. This is ultimately another partitionof the 0-nodes and is therefore readily implemented in ourapproach, which is designed to work for any partition. tiple 0-graphs { A (0) } onto the same coarse-grained 1-graph A (1) , and the notation { A (0) } Ω −−→ A (1) will de-note such surjective mapping. Therefore P (cid:0) A (0) (cid:12)(cid:12) Θ (cid:1) will induce a random process at the next level, gen-erating each possible 1-graph A (1) with probability (cid:80) { A (0) } Ω −−→ A (1) P (cid:0) A (0) (cid:12)(cid:12) Θ (cid:1) , where the sum runs overall 0-graphs that are projected onto A (1) by Ω . Iterat-ing l times, we induce a process generating the l -graph A ( l ) with probability (cid:80) { A (0) } Ω l − ··· Ω −−−−−−→ A ( l ) P (cid:0) A (0) (cid:12)(cid:12) Θ (cid:1) ,where Ω l − · · · Ω is the composition of the l partitions { Ω k } l − k =0 , which is ultimately a partition of the 0-nodes.We now enforce a scale-invariant random graph modelthat, for any level l , can generate the possible l -graphs intwo equivalent ways: either hierarchically , i.e. by firstgenerating the 0-graphs with probability P (cid:0) A (0) (cid:12)(cid:12) Θ (cid:1) and then coarse-graining them l times via the parti-tions { Ω k } l − k =0 , or directly , i.e with a certain probability P l (cid:0) A ( l ) (cid:12)(cid:12) Θ l (cid:1) that depends on l only through a set Θ l ofrenormalized parameters that should be obtained from Θ using Ω l − · · · Ω . This scale-invariance is equivalentto imposing that, for any pair l, m (with l ≥ m ), P l (cid:0) A ( l ) (cid:12)(cid:12) Θ l (cid:1) = (cid:88) { A ( m ) } Ω l − ··· Ω m −−−−−−−→ A ( l ) P m (cid:0) A ( m ) (cid:12)(cid:12) Θ m (cid:1) (2)where the renormalized parameters Θ l are obtained onlyfrom Θ m , given Ω l − · · · Ω m . We look for the generalsolution in the case of random graphs with independentlinks, where P l (cid:0) A ( l ) | Θ l (cid:1) factorizes as N l (cid:89) i l =1 i l (cid:89) j l =1 (cid:2) p ( l ) i l ,j l (cid:0) Θ l (cid:1)(cid:3) a ( l ) il,jl (cid:2) − p ( l ) i l ,j l (cid:0) Θ l (cid:1)(cid:3) − a ( l ) il,jl , (3)where p ( l ) i l ,j l (cid:0) Θ l (cid:1) is the probability that two l -nodes i l and j l are linked. In this case it is natural to require that Θ l contains (besides N l ) an overall constant δ l (whichwill set the global link density), a set of additive node-specific parameters { x i l } N l i l =1 (which will distribute thetotal number of links heterogeneously among nodes),and an (optional) set of dyadic (pair-specific) parame-ters { d i l ,j l } N l i l ,j l =1 . We can therefore use the notation p ( l ) i l ,j l (cid:0) Θ l (cid:1) = p ( l ) i l ,j l ( δ l ) where we keep only δ l explicit inthe argument of p i l ,j l , because the dependence on theother variables x i l , x j l , d i l ,j l is already denoted by thesubscripts i l , j l (indeed, p i l ,j l depends on i l and j l onlythrough x i l , x j l , d i l ,j l ).Now, the scale-invariance requirement constrains p ( l ) i l ,j l to be functionally invariant with respect to l , i.e. p ( l ) i l ,j l ( δ l ) = p i l ,j l ( δ l ). As we show in Supplementary In-formation (SI), there is a unique solution given by p i l ,j l ( δ ) = 1 − e − δ x il x jl f ( d il,jl ) , δ, x i l , x j l , f > , (4)where f is an arbitrary positive function and the follow- ing renormalization rules apply: δ l +1 ≡ δ l ≡ δ, (5) x i l +1 ≡ (cid:88) i l ∈ i l +1 x i l , (6) f (cid:0) d i l +1 ,j l +1 (cid:1) ≡ (cid:80) i l ∈ i l +1 (cid:80) j l ∈ j l +1 x i l x j l f (cid:0) d i l ,j l (cid:1)(cid:80) i l ∈ i l +1 x i l (cid:80) j l ∈ j l +1 x j l (7)(i.e. δ is scale-invariant, x is node-additive and f renor-malizes as a specific x -dependent weighted average).Putting Eq. (4) back into Eq. (3), we finally arrive atthe scale-invariant graph probability P (cid:0) A ( l ) | δ (cid:1) = N l (cid:89) i l =1 i l (cid:89) j l =1 (cid:2) p i l ,j l ( δ ) (cid:3) a ( l ) il,jl (cid:2) − p ( l ) i l ,j l ( δ ) (cid:3) − a ( l ) il,jl = N l (cid:89) i l =1 i l (cid:89) j l =1 (cid:2) e δ x il x jl f ( d il,jl ) − (cid:3) a ( l ) il,jl e δ x il x jl f ( d il,jl ) = Q − δ N l (cid:89) i l =1 i l (cid:89) j l (cid:2) e δ x il x jl f ( d il,jl ) − (cid:3) a ( l ) il,jl , (8)where we have defined Q ≡ N l (cid:89) i l =1 i l (cid:89) j l =1 e x il x jl f ( d il,jl ) = N (cid:89) i =1 i (cid:89) j =1 e x i x j f ( d i ,j ) , (9)which is a constant term that depends neither on therealized l -graph A ( l ) nor, owing to Eq. (7), on the hi-erarchical level l being considered. Note that, as de-sired, P (cid:0) A ( l ) | δ (cid:1) depends on l only through the param-eters { x i l } N l i l =1 and { d i l ,j l } N l i l ,j l =1 , which renormalize asstated in Eqs. (6) and (7).Equations (4)-(7) are our key result. One of theirremarkable consequences is that, while the dependence ofthe connection probability p i l ,j l ( δ ) on the dyadic factor d i l ,j l can be switched off entirely without destroyingthe scale-invariant properties of the model (e.g. bytaking f to be a constant function, whereby Eq. (7) isautomatically fulfilled), the dependence on the node-specific factors x i l x j l cannot be switched off, unlessthe model is made deterministic by formally requiringthat f ( d i l ,j l ) takes only the two values f = 0 (implying p i l ,j l ( δ ) = 0) or f = + ∞ (implying p i l ,j l ( δ ) = 1). Weconsider examples of both situations below. Therefore,the dependence on dyadic factors (including geometricdistances) is optional, while that on node-specific factorsis necessary.
This is a general result following onlyfrom the enforcement of scale-invariance. More specificresults are discussed below.
Node-specific fitness.
The connection probability p i l ,j l increases as x i l and/or x j l increase. Therefore,as in the Fitness Model (FM) [14], x i l can be viewedas a hidden variable or ‘fitness’ that characterizes theintrinsic tendency of the l -node i l to form connections.Here, the fitness is defined across multiple hierarchicallevels and renormalizability ensures that it is also anadditive quantity summing up to the value in Eq. (6)when l -nodes are merged onto an ( l + 1)-node. Thereforethe sum X ≡ (cid:80) N l i l =1 x i l = (cid:80) N i =1 x i is independent of l , i.e. the total fitness of all nodes is preserved by therenormalization. For instance, if one starts with x i = 1for all i , then x i l will simply count how many 0-nodesare found within the l -node i l , and X = N . More in-teresting outcomes are obtained by using heterogeneousdistributions of the fitness, as we illustrate in detail later. Dyadic properties.
Unlike the fitness, d i l ,j l is a dyadicfactor (such as distance, similarity, co-membership inthe same community, etc.) associated with the nodepair ( i l , j l ). Although we are free to do otherwise, wemay regard d i l ,j l as a distance, in which case it maymake sense to assume that f is a decreasing function,ensuring that more distant nodes are less likely tobe connected. It is easy to realize that, if d i ,j isan ultrametric distance (i.e. such that the ‘stronger’triangle inequality d i ,j ≤ max { d i ,k , d j ,k } holds forevery triple i , j , k of 0-nodes [15]) that respects thehierarchy of coarse-grainings (i.e. such that all distancescan be represented as the heights of the branching pointsof the dendrogram shown in Fig. 2), then d i l ,j l = d i ,j and hence f ( d i l ,j l ) = f ( d i ,j ) whenever the 0-nodes i and i map onto the l -nodes i l and j l respectively, i.e.whenever i l = Ω l − · · · Ω ( i ) and j l = Ω l − · · · Ω ( j ).In such a case, Eq. (7) reduces to f (cid:0) d i l +1 ,j l +1 (cid:1) = f (cid:0) d i l ,j l (cid:1) with i l +1 = Ω l ( i l ) and j l +1 = Ω l ( j l ), showing that if the distances among the -nodes are ultrametric onthe dendrogram induced by the hierarchy of partitions,they decouple from the hidden variables and remaininvariant across the entire coarse-graining process, justlike the global parameter δ . Reversing the point ofview, we may equivalently say that, given an ultrametricdistance among the -nodes, any hierarchy of partitionsinduced by the associated dendrogram keeps the distancesscale-invariant . In weaker form, this also means that onemay use d i ,j to specify the dendrogram parametrizingthe desired hierarchy of partitions that will keep thedistances scale-invariant. The hierarchy may coincidewith e.g. a nested community structure that one maywant to impose. In any case we stress that, althoughultrametricity is an attractive property (especially in theannealed scenario that we introduce later), we do notrequire it as a necessary condition in general. Recovering the lattice case.
We can now discuss asimple but important extreme case, where the graphis constructed only as a function of distance andour approach reduces to the traditional scheme forrenormalizing regular lattices. For instance, assumethat the 0-nodes have coordinates at the sites of a2-dimensional grid with lattice spacing τ and that d i ,j is the Euclidean distance between these coordinates. Ifwe set f ≡ + ∞ if d i l ,j l ≤ l τ and f ≡ l -graph will be the usual renormalized latticewith spacing τ l = 2 l τ obtained through an appropriatepartition Ω l − that maps each square block of 4 nearest( l − l -node sitting at the centerof the square. In this case, each vertical line of thedendrogram of hierarchical partitions branches regularlyinto 4 ‘daughter’ lines and τ l = 2 l τ is the height of thebranching points splitting ( l + 1)-nodes into l -nodes.The renormalized distances d i l ,j l can be mapped exactlyto this dendrogram, thereby retrieving the standardlattice renormalization scheme as a special case of ourapproach. Importantly, other network renormalizationschemes are incompatible with this key limiting case be-cause they require specific topologies such as communitystructure [7] or scale-free degree distributions [11] thatare obviously absent in regular grids. Relation to other network models.
In the opposite,more interesting extreme, the dependence on the dyadicfactors can switched off. For instance, if we set f ≡ p i l ,j l ( δ ) = 1 − e − δx il x jl , δ, x i l , x j l > . (10)It is important to notice that, in the ‘sparse’ and‘bounded’ case (i.e. for δ (cid:28) x − and x max < + ∞ ,where x max is the maximum realized value of the fit-ness), Eq. (10) reduces to p i l ,j l ( δ ) ≈ δx i l x j l , whichincludes the Chung-Lu [16] or ‘sparse’ ConfigurationModel (CM) ( p i,j ≈ δx i x j with x i = k i and δ =(2 L ) − , where k i is the degree of node i and L is thetotal number of links). Similarly, in the same limitEq. (4) reduces to p i l ,j l ( δ ) ≈ δx i l x j l f ( d i l ,j l ), whichincludes the sparse degree-corrected Stochastic Block-Model (SBM) [17] ( p i,j ≈ δx i x j B i,j where B is a blockmatrix). The CM and SBM are among the most popularnetwork models and find diverse applications includingcommunity detection [18], pattern recognition [19] andnetwork reconstruction [20]. However, even if we startfrom a sufficiently sparse 0-graph for which these modelsare consistent with Eq. (4), successive coarse-grainingswill unavoidably increase x max and bring the network tothe dense regime where the CM and SBM are describedby their ‘full’ probability p i,j = δx i x j B i,j / (1+ δx i x j B i,j )that eventually deviates from Eq. (4). This means thatthese models are not renormalizable.Similar considerations apply to the traditional (non-degree-corrected) SBM [21] (for which p i,j = B i,j ), tothe Erd˝os-R´enyi (ER) model [22] (for which p i,j = p for all i, j ) and to growing network models based onpreferential attachment (PA) [23]. In the latter, nodesenter sequentially into the network and the time atwhich a node enters determines its expected topologicalproperties. There is no straightforward way to coarse-grain these models by defining block-nodes (possiblyacross different hierarchical levels) that respect thedifferent expected properties of the nodes they contain.The above considerations show that renormalizablenetworks are consistent with a unique specification of the FM, possibly enhanced by dyadic factors, whilethey are incompatible with the CM, SBM and PAmodels. Also, it is important to realize that, sincedistance-dependence has been switched off, the modelin Eq. (10) is exactly renormalizable for any possiblechoice of coarse-grainings . This shows that networkrenormalization does not require any notion of geometry (whether hyperbolic or not) or spatial embedding.
Scale-free versus scale-invariant networks.
The abovediscussion sheds new light on the distinction between scale-free networks (i.e. graphs with power-law tailsin the degree distribution, as usually appearing in theCM, degree-corrected SBM and PA models) and scale-invariant networks (i.e. graphs designed to be renormal-izable as defined here). The early renormalization ap-proaches reminiscent of fractal analysis [4–6] relied on theidea that real-world networks can be interpreted as scale-invariant, precisely because of their scale-free property.However the degrees, even when power-law distributed,cannot be renormalized exactly because they are neitherpreserved nor additively transformed upon renormaliza-tion. The non-renormalizability of the CM, SBM, ERand PA models originates precisely from the fact thattheir defining quantities are the node degrees.
Unlikefractals, the self-similarity of scale-free networks appliesto a topological property (the degree), not to a metric one.
The absence of an embedding metric space, which wouldprovide an ‘ambient’ dimensionality to harbour fractal-ity in the first place (e.g. to allow for the Hausdorff-Besicovitch dimension to be strictly larger than the in-trinsic topological dimension of the fractal), is also thereason why arbitrary networks cannot be easily renor-malized using metric coordinates. And even when theycan (as in the cases discussed here), renormalizability isnot due to their scale-free property, but because of theircompatibility with Eq. (4). As metioned above, only if δ is small enough and the fitness is not too broadly dis-tributed (so that x max < + ∞ ) there may be a sparseregime where Eq. (10) reduces to p i l ,j l ≈ δx i l , x j l with k i l = x i l , so that degrees are approximately additive.However it should be noted that, even in the latter case,degrees would be rigorously additive only if each ( l + 1)-node were obtained as a set of l -nodes that have noconnection among themselves. This prescription is com-pletely opposite to the usual scheme of merging nodesthat are tightly connected, e.g. because they are inthe same community. If mutually connected nodes aremapped onto the same block-node, the degree of the lat-ter is strictly smaller than the sum of the degrees of theoriginal nodes. We may say that the coarse-graining ofa network is usually designed in such a way that the ad-ditivity of degrees is maximally violated . In fact, thisproblem affects by construction all renormalization ap-proaches based on community structure or dense motifs.In any case, the sparse regime is destined to vanishinto the dense one through the action of renormalizationitself, eventually breaking the approximate additivity of degrees and producing an unavoidable upper cut-offin the degree distribution. Indeed, previous renormal-ization approaches based on the scale-free property ledto various contradictions, including lack of generality,irreducibility to the ordinary renormalization schemein the special case of lattices, and limited iterabilityin small-world networks with short path lengths. Bycontrast, the model proposed here is renormalizablethroughout the entire spectrum of network densitybecause it is designed via a fitness that remains additive(and globally conserved at any hierarchical level) uponcoarse grainings of nodes.Bearing in mind the aforementioned important distinc-tions i) between scale-free and scale-invariant networksand ii) between geometric and non-geometric renormal-ization, in what follows we focus first on the ‘quenched’case where the fitness (and distance if applicable) is fixedand possibly identified with some empirical quantity(thereby allowing for the renormalization of real-worldnetworks irrespective of their scale-free behaviour), andthen on an opposite ‘annealed’ scenario that sponta-neously leads to scale-invariant and scale-free networkswith a density-dependent cut-off (thereby providinga generic mechanism for the emergence of scale-freenetworks from scale-invariance, without geometry ). Case I: quenched fitness.
In the quenched case, thefitness of each 0-node i is assigned a fixed value x i . Forinstance, when modelling real-world networks, the ob-served nodes can be identified with the 0-nodes and x i can be taken to be the value of some measurable additiveempirical quantity attached to the 0-node i . Then, afterchoosing a hierarchy of partitions and consistently withEq. (6), the fitness x i l +1 of each ( l + 1)-node i l +1 (with l >
0) is calculated iteratively by summing the fitness ofall the l -nodes mapped onto i l +1 . For each pair ( i , j ) of0-nodes, a distance d i ,j may also be specified (and possi-bly measured from empirical data as well) and used to de-termine f ( d i ,j ). Consistently, the quantity f ( d i l +1 ,j l +1 )between each pair ( i l +1 , j l +1 ) of ( l + 1)-nodes is calcu-lated via Eq. (7). Together, fitness and distance deter-mine the probability (4) of connection between nodes atall scales. Clearly, once f is specified, the only free pa-rameter is δ , controlling the overall density of the ran-dom network. When considering real-world networks forwhich fitness and distance can be measured from em-pirical data separately from the network structure, wemay use the quenched model in order to check whetherEq. (4) reproduces the observed topological properties ofthe 0-graph itself and, if this is the case, to provide atestable multi-scale model of the renormalized networkat any higher level of aggregation.To illustrate this procedure, we consider the empiri-cal International Trade Network (ITN), using the BACI-Comtrade dataset [24] which reports the internationaltrade flows (imports and exports) between all pairs ofworld countries. We show the results for the year 2011;we have obtained similar results for the other years avail-able in the database. We select this particular networkbecause previous research has clarified that the topologyof the ITN is strongly dictated by the GDP of coun-tries [25–28]. Moreover, the economics literature has ex-tensively shown that both GDP and geographical dis-tance are key determinants of international trade, lead-ing to the so-called ‘Gravity Model’ of trade [29, 30].The additivity of the GDP (i.e. the aggregate GDP oftwo countries is the sum of their GDPs) makes the ITN aperfect candidate for our analysis, and allows us to intro-duce a novel renormalization scheme for this importanteconomic network across arbitrary levels of geographicalaggregation.Our aim is twofold. On the one hand, we want tointroduce a multiscale model of the ITN derived fromfirst principles, i.e. using the unique combination of GDPand geographical distances dictated by Eq. (4), ratherthan arbitrary or data-driven combinations. On the otherhand, we want to check whether the empirical topologyof the ITN is consistent with the multiscale model notonly at the country level at which it is usually studied(here, the 0-graph), but also across different hiearchicallevels using the renormalization rules in Eqs. (6) and (7).First, we define the multiscale model of the ITN. Weidentify each 0-node i with a specific country for whichthere are GDP data available from the World Bank [31]in the considered year. This results in N = 183 0-nodes (see SI). Then, we set the fitness x i of each 0-nodeequal to the empirical value of the GDP: x i = GDP i , i = 1 , N . For each pair ( i , j ) of countries, we also setthe distance d i ,j equal to the empirical geographicaldistance between the corresponding countries, using theBACI-CEPII GeoDist [32, 33] database (see SI). Next,we use these distances to induce a hierarchy of partitions { Ω l } l ≥ that define the possible coarse-grainings of theITN. Technically, this is done by merging geographicallyclose countries into ‘block-countries’ following a single-linkage hierarchical clustering algorithm based on theoriginal distances { d i ,j } N i ,j =1 . The output of this al-gorithm is a dendrogram like the one shown in Fig. 2,where the leaves are the original countries (0-nodes), thebranching points are the block-countries, and the heightof each branching point represents the geographical dis-tance between pairs of countries across the correspondingtwo branches. Cutting the dendrogram at a fixed height h l defines the hierarchical level l and identifies a uniquepartition Ω l of countries into a certain number N l of‘ l -countries’. This partition can be regarded as a multi-scale aggregation of countries into groups of varying size,following from actual geographical closeness rather thanpre-imposed regional or political criteria. Cutting thedendoram at multiple heights { h l } l ≥ (with h = 0) iden-tifies a set of hierarchical levels { l } , a geography-inducedhierarchy of partitions { Ω l } l ≥ , and a corresponding se-quence of { N l } l ≥ block-countries. We considered 18 hi-erarchical levels (from l = 0 to l = 17), such that thenumber of block-countries is N l = 183 for l = 0 and N l = 180 − l for l = 1 ,
17. For each of these levels, theadditivity of GDP ensures that Eq. (6) holds as a defini-tion for the empirical aggregate GDP of block-countries:GDP i l +1 ≡ (cid:88) i l ∈ i l +1 GDP i l . (11)We then fix the function f in Eq. (4) as f ( d ) = d − , sothat the renormalized geographical distances equal d − i l +1 ,j l +1 ≡ (cid:80) i l ∈ i l +1 (cid:80) j l ∈ j l +1 GDP i l GDP j l d − i l ,j l (cid:80) i l ∈ i l +1 GDP i l (cid:80) j l ∈ j l +1 GDP j l , (12)which is the GDP-averaged equivalent of certainpopulation-averaged distances commonly used in geog-raphy, e.g. in the GeoDist database itself [32] (see SI).In this way, d i l +1 ,j l +1 represents a sort of distance be-tween the ‘barycenters’ of the block-countries i l +1 and j l +1 , where the barycenter of each ( l + 1)-country is de-fined via the internal GDP distribution across the con-stituent l -countries. Putting all the above ingredientstogether, we arrive at the following multiscale model forthe ITN: p i l ,j l ( δ ) = 1 − e − δ GDP il GDP jl /d il,jl , (13)where δ is the only free parameter and where the renor-malization rules are given by Eqs. (11) and (12).Now that we have defined our multiscale model of theITN, we build the corresponding instances of the realnetwork at the chosen 18 levels of aggregation. To thisend, we construct the empirical 0-graph ˜ A (0) by drawingan undirected link between each pair of countries thathave a positive trade relationship in either direction inthe BACI-Comtrade dataset (see SI). Then, we use thedistance-induced partitions { Ω l } l ≥ defined above in or-der to construct the l -graph according to Eq. (1) for eachlevel l . This procedure creates a sequence { ˜ A ( l ) } l ≥ ofempirical coarse-grained versions of the ITN, each onerepresenting the existence of trade among l -countries.We can now test the multiscale model defined byEq. (13) against the real data { ˜ A ( l ) } l ≥ . Preliminarily,we calibrate the model by setting δ to the unique value ˜ δ that produces the same link density as the real ITN, i.e.such that the expected number of links in the 0-graph(that is a monotonically increasing function of δ ) equalsthe empirical value observed in ˜ A (0) (see SI). After thissingle parameter choice, all the probabilities in Eq. (13)are uniquely determined at all hierarchial levels andwe can test the model by comparing the empirical andexpected value of various topological properties of theITN at different coarse-grainings. In particular, for eachlevel l and for each l -node i l , we consider the degree k i l , the average nearest neighbour degree k nni l and the clustering coefficient c i l (see SI for all definitions). Wecarry out a first test of the model by considering theaverage of these quantities over all l -nodes, therebyobtaining (using bars to denote node averages) an D , D observed dataexpected data P > ( k ) , P > ( k ) observed dataexpected data M c t o t , c t o t observed dataexpected data D , < D > observed dataexpected data K nn , < K nn > observed dataexpected data N l c t o t , < c t o t > observed dataexpected data . . . D , < D > ob s e r v ed da t ae x pe c t ed da t a K nn , < K nn > ob s e r v ed da t ae x pe c t ed da t a N l . . c t o t , < c t o t > ob s e r v ed da t ae x pe c t ed da t a D , < D > observed dataexpected data K nn , < K nn > observed dataexpected data N l c t o t , < c t o t > observed dataexpected data D , < D > observed dataexpected data K nn , < K nn > observed dataexpected data N l c t o t , < c t o t > observed dataexpected data D , < D > observed dataexpected data K nn , < K nn > observed dataexpected data N l c t o t , < c t o t > observed dataexpected data D , < D > observed dataexpected data K nn , < K nn > observed dataexpected data N l c t o t , < c t o t > observed dataexpected data D , < D > observed dataexpected data K nn , < K nn > observed dataexpected data N l c t o t , < c t o t > observed dataexpected data D , < D > observed dataexpected data K nn , < K nn > observed dataexpected data N l c t o t , < c t o t > observed dataexpected data D , < D > observed dataexpected data K nn , < K nn > observed dataexpected data N l c t o t , < c t o t > observed dataexpected data . . . D,
1] irrespective of l , i.e.we consider the ratios ¯ k l /N l (which coincides with thelink density D l , see SI) and ¯ k nnl /N l , while ¯ c l is alreadynormalized.In Fig. 3 we plot, for all our 18 hierarchical levels, thethree normalized quantities as a function of the number N l of l -nodes. We see that the model replicates theempirical values accurately. This is remarkable, giventhat the only model parameter was calibrated in orderto match uniquely the density D of the 0-graph, whilethe agreement holds for the other quantities as well, andacross all levels. As expected, the link density increasesas the level increases (i.e. as N l decreases), neverthelessthe model always reproduces the properties of the realnetwork (no assumption of sparsity is needed). As aneven more stringent test of the model, we confirm theprediction that the local topological properties of theindividual (block-)countries, and in particular k i l , k nni l and c i l , should depend strongly on the empirical valueof GDP i l , in a way that is governed by Eq. (13) atall levels. As shown in Fig. 4, the model predictionsare confirmed by the empirical data. It is remarkablethat the agreement between observations and modelexpectations holds locally at the level of individual F i r s t , w e d e fin e t h e m u l t i s c a l e m o d e l o f t h e I T N a s f o ll o w s . W e i d e n t i f y e a c h - n o d e i w i t h a s p ec i fi cc o un - t r y f o r w h i c h t h e r e a r e G D P d a t aa v a il a b l e f r o m t h e W o r l d B a n k [ ? ]i n t h ec o n s i d e r e d y e a r . T h i s r e s u l t s i n N = - n o d e s ( s ee S I ) . T h e n , w e s e tt h e fi t n e ss x i o f e a c h - n o d ee q u a l t o t h ee m p i r i c a l v a l u e o f t h e G D P : x i = G D P i , i = , N . F o r e a c hp a i r ( i , j ) o f c o un t r i e s , w e a l s o s e tt h e d i s t a n ce d i , j e q u a l t o t h ee m - p i r i c a l g e og r a ph i c a l d i s t a n ce b e t w ee n t h ec o rr e s p o nd i n g c o un t r i e s , u s i n g t h e B A C I - C EP II G e o D i s t [ ] d a t a b a s e ( s ee S I ) . N e x t , w e u s e t h e s e d i s t a n ce s t o i ndu ce a h i e r a r - c h y o f p a r t i t i o n s { ⌦ l } l t h a t d e fin e t h e p o ss i b l ec oa r s e - g r a i n i n g s o f t h e I T N . T ec hn i c a ll y , t h i s i s d o n e b y m e r g i n g g e og r a ph i c a ll y c l o s ec o un t r i e s i n t o ‘ b l o c k - c o un t r i e s ’ f o l - l o w i n ga s i n g l e - li n k ag e h i e r a r c h i c a l c l u s t e r i n ga l go r i t h m b a s e d o n t h e o r i g i n a l d i s t a n ce s { d i , j } N i , j = ( s ee S I ) . T h e o u t pu t o f t h i s a l go r i t h m i s a d e nd r og r a m li k e t h e o n e s h o w n i n F i g . , w h e r e t h e l e a v e s a r e t h e o r i g i n a l c o un t r i e s ( - n o d e s ) , t h e b r a n c h i n g p o i n t s a r e t h e b l o c k - c o un t r i e s , a nd t h e h e i g h t o f e a c hb r a n c h i n g p o i n t r e p r e - s e n t s t h e g e og r a ph i c a l d i s t a n ce b e t w ee np a i r s o f c o un t r i e s a c r o ss t h ec o rr e s p o nd i n g t w o b r a n c h e s . C u tt i n g t h e d e n - d r og r a m a t a fi x e dh e i g h t h l d e fin e s t h e h i e r a r c h i c a ll e v e l l a nd i d e n t i fi e s a un i q u e p a r t i t i o n ⌦ l o f c o un t r i e s i n t o a ce r t a i nnu m b e r N l o f ‘ l - c o un t r i e s ’. T h i s p a r t i t i o n c a n b e r e ga r d e d a s a m u l t i s c a l e agg r e ga t i o n o f c o un t r i e s i n t o g r o up s o f v a r y i n g s i ze , f o ll o w i n g f r o m a c t u a l g e og r a ph i - c a l c l o s e n e ss r a t h e r t h a np r e - i m p o s e d r e g i o n a l o r p o li t - i c a l c r i t e r i a . C u tt i n g t h e d e nd o r a m a t m u l t i p l e h e i g h t s { h l } l ( w i t h h = ) i d e n t i fi e s a s e t o f h i e r a r c h i c a ll e v e l s { l } , ag e og r a ph y - i ndu ce dh i e r a r c h y o f p a r t i t i o n s { ⌦ l } l , a nd a c o rr e s p o nd i n g s e q u e n ce o f { N l } l b l o c k - c o un t r i e s . W ec o n s i d e r h i e r a r c h i c a ll e v e l s r a n g i n g f r o m N = t o N = ( s ee S I ) . F o r e a c h o f t h e s e l e v e l s , t h e a dd i - t i v i t y o f G D P e n s u r e s t h a t E q . ( ) h o l d s a s a d e fin i t i o n f o r t h ee m p i r i c a l agg r e ga t e G D P o f b l o c k - c o un t r i e s : G D P i l + ⌘ X i l i l + G D P i l . ( ) W e t h e nfi x t h e f un c t i o n f i n E q . ( ) a s f ( d ) = d , s o t h a tt h e r e n o r m a li ze d g e og r a ph i c a l d i s t a n ce s e q u a l d i l + , j l + ⌘ P i l i l + P j l j l + G D P i l G D P j l d i l , j l P i l i l + G D P i l P j l j l + G D P j l , ( ) w h i c h i s t h e G D P - a v e r ag e d e q u i v a l e n t o f ce r t a i n p o pu l a t i o n - a v e r ag e dd i s t a n ce s c o mm o n l y u s e d i n g e og - r a ph y , e . g .i n t h e G e o D i s t d a t a b a s e i t s e l f [ ]. I n t h i s w a y , d i l + , j l + r e p r e s e n t s a s o r t o f d i s t a n ce b e t w ee n t h e ‘ b a r y ce n t e r s ’ o f t h e b l o c k - c o un t r i e s i l + a nd j l + , w h e r e t h e b a r y ce n t e r o f e a c h ( l + ) - c o un t r y i s l o c a t e du s i n g t h e i n t e r n a l G D P d i s t r i bu t i o n a c r o ss i t s c o n s t i t u e n t l - c o un t r i e s . P u tt i n ga ll t h e a b o v e i n g r e d i e n t s t og e t h e r , w e a rr i v e a tt h e f o ll o w i n g m u l t i s c a l e m o d e l f o r t h e I T N : p i l , j l ( ) = e G D P i l G D P j l / d i l , j l , ( ) w h e r e i s t h e o n l y f r ee p a r a m e t e r a nd w h e r e t h e r e n o r - m a li z a t i o n r u l e s a r e g i v e nb y E q s . ( ) a nd ( ) . N o w t h a t w e h a v e d e fin e d o u r m u l t i s c a l e m o d e l o f t h e I T N , w e bu il d t h ec o rr e s p o nd i n g i n s t a n ce s o f t h e r e a l n e t w o r k a tt h ec h o s e n l e v e l s o f agg r e ga t i o n . T o t h i s e nd , w ec o n s t r u c tt h ee m p i r i c a l - g r a ph ˜ A ( ) b y d r a w i n g a nund i r ec t e d li n k b e t w ee n e a c hp a i r o f c o un t r i e s t h a t h a v e a p o s i t i v e t r a d e r e l a t i o n s h i p i n e i t h e r d i r ec t i o n i n t h e B A C I - C o m t r a d e d a t a s e t( s ee S I ) . T h e n , w e u s e t h e d i s t a n ce - i ndu ce dp a r t i t i o n s { ⌦ l } l d e fin e d a b o v e i n o r - d e r t o c o n s t r u c tt h e l - g r a ph a cc o r d i n g t o E q . ( ) f o r e a c h l e v e l l . T h i s p r o ce du r ec r e a t e s a s e q u e n ce { ˜ A ( l ) } l o f e m p i r i c a l c oa r s e - g r a i n e d v e r s i o n s o f t h e I T N , e a c h o n e r e p r e s e n t i n g t h ee x i s t e n ce o f t r a d e a m o n g l - c o un t r i e s . W ec a nn o w t e s tt h e m u l t i s c a l e m o d e l d e fin e db y E q . ( ) aga i n s tt h e r e a l d a t a { ˜ A ( l ) } l . P r e li m i n a r il y , w ec a li b r a t e t h e m o d e l b y s e tt i n g t o t h e un i q u e v a l u e ˜ t h a t p r o du ce s t h e s a m e li n k d e n s i t y a s t h e r e a l I T N ,i. e . s u c h t h a tt h ee x p ec t e dnu m b e r o f li n k s i n t h e - g r a ph (t h a t i s a m o n o t o n i c a ll y i n c r e a s i n g f un c t i o n o f ) e q u a l s t h ee m p i r i c a l v a l u e o b s e r v e d i n ˜ A ( ) ( s ee S I ) . A f t e r t h i s s i n g l e p a r a m e t e r c h o i ce , a ll t h e p r o b a b ili t i e s i n E q . ( ) a r e un i q u e l y d e t e r m i n e d a t a ll h i e r a r c h i a ll e v e l s a nd w e c a n t e s tt h e m o d e l b y c o m p a r i n g t h ee m p i r i c a l a nd e x - p ec t e d v a l u e o f v a r i o u s t o p o l og i c a l p r o p e r t i e s o f t h e I T N a t d i ↵ e r e n t c oa r s e - g r a i n i n g s . I np a r t i c u l a r , f o r e a c h l e v e l l a nd f o r e a c h l - n o d e i l , w ec o n s i d e r t h e d eg r ee k i l , t h e a ve r a ge n e a r e s t n e i g h b o u r d eg r ee k nn i l a nd t h e c l u s t e r i n g c o e c i e n t c i l ( s ee S I f o r a ll d e fin i t i o n s ) . W ec a rr y o u t a fi r s tt e s t o f t h e m o d e l b y c o n s i d e r i n g t h e a v e r ag e o f t h e s e q u a n t i t i e s o v e r a ll l - n o d e s , t h e r e b y o b t a i n i n g ( u s i n g b a r s t o d e n o t e n o d e a v e r ag e s ) a n o v e r a ll n o d e - a v e r ag e dd e - g r ee ¯ k l , a n o v e r a ll a v e r ag e n e a r e s t n e i g hb o u r d e g r ee ¯ k nn l a nd a n o v e r a ll c l u s t e r i n g c o e c i e n t ¯ c l f o r e a c h l e v e l l . T o c h ec k w h e t h e r t h e m o d e l r e p li c a t e s t h e s e p r o p e r t i e s a t a ll h i e r a r c h i c a ll e v e l s , w e fi r s t n o r m a li ze t h e s e q u a n t i t i e s t o pu tt h e m o n t h e s a m e i n t e r v a l[ , ]i rr e s p ec t i v e o f l ,i. e . w ec o n s i d e r t h e r a t i o s ¯ k l / N l ( w h i c h c o i n c i d e s w i t h t h e li n k d e n s i t y D l , s ee S I ) a nd ¯ k nn l / N l , w h il e ¯ c l i s a l r e a d y n o r m a li ze d . I n F i g . w e p l o t , f o r a ll o u r h i e r a r c h i c a ll e v e l s , t h e t h r ee n o r m a li ze d q u a n t i t i e s a s a f un c t i o n o f t h e nu m b e r N l o f l - n o d e s . W e s ee t h a tt h e m o d e l r e p li c a t e s t h e e m p i r i c a l v a l u e s a cc u r a t e l y . T h i s i s r e m a r k a b l e , g i v e n t h a tt h e o n l y m o d e l p a r a m e t e r w a s c a li b r a t e d i n o r d e r t o m a t c hun i q u e l y t h e d e n s i t y D o f t h e - g r a ph , w h il e t h e ag r ee m e n t h o l d s f o r t h e o t h e r q u a n t i t i e s a s w e ll, a nd a c r o ss a lll e v e l s . A s e x p ec t e d , t h e li n k d e n s i t y i n c r e a s e s a s t h e l e v e li n c r e a s e s ( i. e . a s N l d ec r e a s e s ) , n e v e r t h e l e ss t h e m o d e l a l w a y s r e p r o du ce s t h e p r o p e r t i e s o f t h e r e a l n e t w o r k ( n oa ss u m p t i o n o f s p a r s i t y i s n ee d e d ) . A s a n e v e n m o r e s t r i n g e n tt e s t o f t h e m o d e l, w ec o nfi r m t h e p r e d i c t i o n t h a tt h e l o c a l t o p o l og i c a l p r o p e r t i e s o f t h e i nd i v i du a l ( b l o c k - ) c o un t r i e s , a nd i np a r t i c u l a r k i l , k nn i l a nd c i l , s h o u l dd e p e nd s t r o n g l y o n t h ee m p i r i c a l v a l u e o f G D P i l ,i n a w a y t h a t i s go v e r n e db y E q . ( ) a t a lll e v e l s . A ss h o w n i n F i g . , t h e m o d e l p r e d i c t i o n s a r ec o nfi r m e db y t h ee m p i r i c a l d a t a . I t i s r e m a r k a b l e t h a tt h e ag r ee m e n t b e t w ee n o b s e r v a t i o n s a nd m o d e l e x p ec t a t i o n s h o l d s l o c a ll y a tt h e l e v e l o f i nd i v i du a l F i r s t , w e d e fin e t h e m u l t i s c a l e m o d e l o f t h e I T N a s f o ll o w s . W e i d e n t i f y e a c h - n o d e i w i t h a s p ec i fi cc o un - t r y f o r w h i c h t h e r e a r e G D P d a t aa v a il a b l e f r o m t h e W o r l d B a n k [ ? ]i n t h ec o n s i d e r e d y e a r . T h i s r e s u l t s i n N = - n o d e s ( s ee S I ) . T h e n , w e s e tt h e fi t n e ss x i o f e a c h - n o d ee q u a l t o t h ee m p i r i c a l v a l u e o f t h e G D P : x i = G D P i , i = , N . F o r e a c hp a i r ( i , j ) o f c o un t r i e s , w e a l s o s e tt h e d i s t a n ce d i , j e q u a l t o t h ee m - p i r i c a l g e og r a ph i c a l d i s t a n ce b e t w ee n t h ec o rr e s p o nd i n g c o un t r i e s , u s i n g t h e B A C I - C EP II G e o D i s t [ ] d a t a b a s e ( s ee S I ) . N e x t , w e u s e t h e s e d i s t a n ce s t o i ndu ce a h i e r a r - c h y o f p a r t i t i o n s { ⌦ l } l t h a t d e fin e t h e p o ss i b l ec oa r s e - g r a i n i n g s o f t h e I T N . T ec hn i c a ll y , t h i s i s d o n e b y m e r g i n g g e og r a ph i c a ll y c l o s ec o un t r i e s i n t o ‘ b l o c k - c o un t r i e s ’ f o l - l o w i n ga s i n g l e - li n k ag e h i e r a r c h i c a l c l u s t e r i n ga l go r i t h m b a s e d o n t h e o r i g i n a l d i s t a n ce s { d i , j } N i , j = ( s ee S I ) . T h e o u t pu t o f t h i s a l go r i t h m i s a d e nd r og r a m li k e t h e o n e s h o w n i n F i g . , w h e r e t h e l e a v e s a r e t h e o r i g i n a l c o un t r i e s ( - n o d e s ) , t h e b r a n c h i n g p o i n t s a r e t h e b l o c k - c o un t r i e s , a nd t h e h e i g h t o f e a c hb r a n c h i n g p o i n t r e p r e - s e n t s t h e g e og r a ph i c a l d i s t a n ce b e t w ee np a i r s o f c o un t r i e s a c r o ss t h ec o rr e s p o nd i n g t w o b r a n c h e s . C u tt i n g t h e d e n - d r og r a m a t a fi x e dh e i g h t h l d e fin e s t h e h i e r a r c h i c a ll e v e l l a nd i d e n t i fi e s a un i q u e p a r t i t i o n ⌦ l o f c o un t r i e s i n t o a ce r t a i nnu m b e r N l o f ‘ l - c o un t r i e s ’. T h i s p a r t i t i o n c a n b e r e ga r d e d a s a m u l t i s c a l e agg r e ga t i o n o f c o un t r i e s i n t o g r o up s o f v a r y i n g s i ze , f o ll o w i n g f r o m a c t u a l g e og r a ph i - c a l c l o s e n e ss r a t h e r t h a np r e - i m p o s e d r e g i o n a l o r p o li t - i c a l c r i t e r i a . C u tt i n g t h e d e nd o r a m a t m u l t i p l e h e i g h t s { h l } l ( w i t h h = ) i d e n t i fi e s a s e t o f h i e r a r c h i c a ll e v e l s { l } , ag e og r a ph y - i ndu ce dh i e r a r c h y o f p a r t i t i o n s { ⌦ l } l , a nd a c o rr e s p o nd i n g s e q u e n ce o f { N l } l b l o c k - c o un t r i e s . W ec o n s i d e r h i e r a r c h i c a ll e v e l s r a n g i n g f r o m N = t o N = ( s ee S I ) . F o r e a c h o f t h e s e l e v e l s , t h e a dd i - t i v i t y o f G D P e n s u r e s t h a t E q . ( ) h o l d s a s a d e fin i t i o n f o r t h ee m p i r i c a l agg r e ga t e G D P o f b l o c k - c o un t r i e s : G D P i l + ⌘ X i l i l + G D P i l . ( ) W e t h e nfi x t h e f un c t i o n f i n E q . ( ) a s f ( d ) = d , s o t h a tt h e r e n o r m a li ze d g e og r a ph i c a l d i s t a n ce s e q u a l d i l + , j l + ⌘ P i l i l + P j l j l + G D P i l G D P j l d i l , j l P i l i l + G D P i l P j l j l + G D P j l , ( ) w h i c h i s t h e G D P - a v e r ag e d e q u i v a l e n t o f ce r t a i n p o pu l a t i o n - a v e r ag e dd i s t a n ce s c o mm o n l y u s e d i n g e og - r a ph y , e . g .i n t h e G e o D i s t d a t a b a s e i t s e l f [ ]. I n t h i s w a y , d i l + , j l + r e p r e s e n t s a s o r t o f d i s t a n ce b e t w ee n t h e ‘ b a r y ce n t e r s ’ o f t h e b l o c k - c o un t r i e s i l + a nd j l + , w h e r e t h e b a r y ce n t e r o f e a c h ( l + ) - c o un t r y i s l o c a t e du s i n g t h e i n t e r n a l G D P d i s t r i bu t i o n a c r o ss i t s c o n s t i t u e n t l - c o un t r i e s . P u tt i n ga ll t h e a b o v e i n g r e d i e n t s t og e t h e r , w e a rr i v e a tt h e f o ll o w i n g m u l t i s c a l e m o d e l f o r t h e I T N : p i l , j l ( ) = e G D P i l G D P j l / d i l , j l , ( ) w h e r e i s t h e o n l y f r ee p a r a m e t e r a nd w h e r e t h e r e n o r - m a li z a t i o n r u l e s a r e g i v e nb y E q s . ( ) a nd ( ) . N o w t h a t w e h a v e d e fin e d o u r m u l t i s c a l e m o d e l o f t h e I T N , w e bu il d t h ec o rr e s p o nd i n g i n s t a n ce s o f t h e r e a l n e t w o r k a tt h ec h o s e n l e v e l s o f agg r e ga t i o n . T o t h i s e nd , w ec o n s t r u c tt h ee m p i r i c a l - g r a ph ˜ A ( ) b y d r a w i n g a nund i r ec t e d li n k b e t w ee n e a c hp a i r o f c o un t r i e s t h a t h a v e a p o s i t i v e t r a d e r e l a t i o n s h i p i n e i t h e r d i r ec t i o n i n t h e B A C I - C o m t r a d e d a t a s e t( s ee S I ) . T h e n , w e u s e t h e d i s t a n ce - i ndu ce dp a r t i t i o n s { ⌦ l } l d e fin e d a b o v e i n o r - d e r t o c o n s t r u c tt h e l - g r a ph a cc o r d i n g t o E q . ( ) f o r e a c h l e v e l l . T h i s p r o ce du r ec r e a t e s a s e q u e n ce { ˜ A ( l ) } l o f e m p i r i c a l c oa r s e - g r a i n e d v e r s i o n s o f t h e I T N , e a c h o n e r e p r e s e n t i n g t h ee x i s t e n ce o f t r a d e a m o n g l - c o un t r i e s . W ec a nn o w t e s tt h e m u l t i s c a l e m o d e l d e fin e db y E q . ( ) aga i n s tt h e r e a l d a t a { ˜ A ( l ) } l . P r e li m i n a r il y , w ec a li b r a t e t h e m o d e l b y s e tt i n g t o t h e un i q u e v a l u e ˜ t h a t p r o du ce s t h e s a m e li n k d e n s i t y a s t h e r e a l I T N ,i. e . s u c h t h a tt h ee x p ec t e dnu m b e r o f li n k s i n t h e - g r a ph (t h a t i s a m o n o t o n i c a ll y i n c r e a s i n g f un c t i o n o f ) e q u a l s t h ee m p i r i c a l v a l u e o b s e r v e d i n ˜ A ( ) ( s ee S I ) . A f t e r t h i s s i n g l e p a r a m e t e r c h o i ce , a ll t h e p r o b a b ili t i e s i n E q . ( ) a r e un i q u e l y d e t e r m i n e d a t a ll h i e r a r c h i a ll e v e l s a nd w e c a n t e s tt h e m o d e l b y c o m p a r i n g t h ee m p i r i c a l a nd e x - p ec t e d v a l u e o f v a r i o u s t o p o l og i c a l p r o p e r t i e s o f t h e I T N a t d i ↵ e r e n t c oa r s e - g r a i n i n g s . I np a r t i c u l a r , f o r e a c h l e v e l l a nd f o r e a c h l - n o d e i l , w ec o n s i d e r t h e d eg r ee k i l , t h e a ve r a ge n e a r e s t n e i g h b o u r d eg r ee k nn i l a nd t h e c l u s t e r i n g c o e c i e n t c i l ( s ee S I f o r a ll d e fin i t i o n s ) . W ec a rr y o u t a fi r s tt e s t o f t h e m o d e l b y c o n s i d e r i n g t h e a v e r ag e o f t h e s e q u a n t i t i e s o v e r a ll l - n o d e s , t h e r e b y o b t a i n i n g ( u s i n g b a r s t o d e n o t e n o d e a v e r ag e s ) a n o v e r a ll n o d e - a v e r ag e dd e - g r ee ¯ k l , a n o v e r a ll a v e r ag e n e a r e s t n e i g hb o u r d e g r ee ¯ k nn l a nd a n o v e r a ll c l u s t e r i n g c o e c i e n t ¯ c l f o r e a c h l e v e l l . T o c h ec k w h e t h e r t h e m o d e l r e p li c a t e s t h e s e p r o p e r t i e s a t a ll h i e r a r c h i c a ll e v e l s , w e fi r s t n o r m a li ze t h e s e q u a n t i t i e s t o pu tt h e m o n t h e s a m e i n t e r v a l[ , ]i rr e s p ec t i v e o f l ,i. e . w ec o n s i d e r t h e r a t i o s ¯ k l / N l ( w h i c h c o i n c i d e s w i t h t h e li n k d e n s i t y D l , s ee S I ) a nd ¯ k nn l / N l , w h il e ¯ c l i s a l r e a d y n o r m a li ze d . I n F i g . w e p l o t , f o r a ll o u r h i e r a r c h i c a ll e v e l s , t h e t h r ee n o r m a li ze d q u a n t i t i e s a s a f un c t i o n o f t h e nu m b e r N l o f l - n o d e s . W e s ee t h a tt h e m o d e l r e p li c a t e s t h e e m p i r i c a l v a l u e s a cc u r a t e l y . T h i s i s r e m a r k a b l e , g i v e n t h a tt h e o n l y m o d e l p a r a m e t e r w a s c a li b r a t e d i n o r d e r t o m a t c hun i q u e l y t h e d e n s i t y D o f t h e - g r a ph , w h il e t h e ag r ee m e n t h o l d s f o r t h e o t h e r q u a n t i t i e s a s w e ll, a nd a c r o ss a lll e v e l s . A s e x p ec t e d , t h e li n k d e n s i t y i n c r e a s e s a s t h e l e v e li n c r e a s e s ( i. e . a s N l d ec r e a s e s ) , n e v e r t h e l e ss t h e m o d e l a l w a y s r e p r o du ce s t h e p r o p e r t i e s o f t h e r e a l n e t w o r k ( n oa ss u m p t i o n o f s p a r s i t y i s n ee d e d ) . A s a n e v e n m o r e s t r i n g e n tt e s t o f t h e m o d e l, w ec o nfi r m t h e p r e d i c t i o n t h a tt h e l o c a l t o p o l og i c a l p r o p e r t i e s o f t h e i nd i v i du a l ( b l o c k - ) c o un t r i e s , a nd i np a r t i c u l a r k i l , k nn i l a nd c i l , s h o u l dd e p e nd s t r o n g l y o n t h ee m p i r i c a l v a l u e o f G D P i l ,i n a w a y t h a t i s go v e r n e db y E q . ( ) a t a lll e v e l s . A ss h o w n i n F i g . , t h e m o d e l p r e d i c t i o n s a r ec o nfi r m e db y t h ee m p i r i c a l d a t a . I t i s r e m a r k a b l e t h a tt h e ag r ee m e n t b e t w ee n o b s e r v a t i o n s a nd m o d e l e x p ec t a t i o n s h o l d s l o c a ll y a tt h e l e v e l o f i nd i v i du a l F i r s t , w e d e fin e t h e m u l t i s c a l e m o d e l o f t h e I T N a s f o ll o w s . W e i d e n t i f y e a c h - n o d e i w i t h a s p ec i fi cc o un - t r y f o r w h i c h t h e r e a r e G D P d a t aa v a il a b l e f r o m t h e W o r l d B a n k [ ? ]i n t h ec o n s i d e r e d y e a r . T h i s r e s u l t s i n N = - n o d e s ( s ee S I ) . T h e n , w e s e tt h e fi t n e ss x i o f e a c h - n o d ee q u a l t o t h ee m p i r i c a l v a l u e o f t h e G D P : x i = G D P i , i = , N . F o r e a c hp a i r ( i , j ) o f c o un t r i e s , w e a l s o s e tt h e d i s t a n ce d i , j e q u a l t o t h ee m - p i r i c a l g e og r a ph i c a l d i s t a n ce b e t w ee n t h ec o rr e s p o nd i n g c o un t r i e s , u s i n g t h e B A C I - C EP II G e o D i s t [ ] d a t a b a s e ( s ee S I ) . N e x t , w e u s e t h e s e d i s t a n ce s t o i ndu ce a h i e r a r - c h y o f p a r t i t i o n s { ⌦ l } l t h a t d e fin e t h e p o ss i b l ec oa r s e - g r a i n i n g s o f t h e I T N . T ec hn i c a ll y , t h i s i s d o n e b y m e r g i n g g e og r a ph i c a ll y c l o s ec o un t r i e s i n t o ‘ b l o c k - c o un t r i e s ’ f o l - l o w i n ga s i n g l e - li n k ag e h i e r a r c h i c a l c l u s t e r i n ga l go r i t h m b a s e d o n t h e o r i g i n a l d i s t a n ce s { d i , j } N i , j = ( s ee S I ) . T h e o u t pu t o f t h i s a l go r i t h m i s a d e nd r og r a m li k e t h e o n e s h o w n i n F i g . , w h e r e t h e l e a v e s a r e t h e o r i g i n a l c o un t r i e s ( - n o d e s ) , t h e b r a n c h i n g p o i n t s a r e t h e b l o c k - c o un t r i e s , a nd t h e h e i g h t o f e a c hb r a n c h i n g p o i n t r e p r e - s e n t s t h e g e og r a ph i c a l d i s t a n ce b e t w ee np a i r s o f c o un t r i e s a c r o ss t h ec o rr e s p o nd i n g t w o b r a n c h e s . C u tt i n g t h e d e n - d r og r a m a t a fi x e dh e i g h t h l d e fin e s t h e h i e r a r c h i c a ll e v e l l a nd i d e n t i fi e s a un i q u e p a r t i t i o n ⌦ l o f c o un t r i e s i n t o a ce r t a i nnu m b e r N l o f ‘ l - c o un t r i e s ’. T h i s p a r t i t i o n c a n b e r e ga r d e d a s a m u l t i s c a l e agg r e ga t i o n o f c o un t r i e s i n t o g r o up s o f v a r y i n g s i ze , f o ll o w i n g f r o m a c t u a l g e og r a ph i - c a l c l o s e n e ss r a t h e r t h a np r e - i m p o s e d r e g i o n a l o r p o li t - i c a l c r i t e r i a . C u tt i n g t h e d e nd o r a m a t m u l t i p l e h e i g h t s { h l } l ( w i t h h = ) i d e n t i fi e s a s e t o f h i e r a r c h i c a ll e v e l s { l } , ag e og r a ph y - i ndu ce dh i e r a r c h y o f p a r t i t i o n s { ⌦ l } l , a nd a c o rr e s p o nd i n g s e q u e n ce o f { N l } l b l o c k - c o un t r i e s . W ec o n s i d e r h i e r a r c h i c a ll e v e l s r a n g i n g f r o m N = t o N = ( s ee S I ) . F o r e a c h o f t h e s e l e v e l s , t h e a dd i - t i v i t y o f G D P e n s u r e s t h a t E q . ( ) h o l d s a s a d e fin i t i o n f o r t h ee m p i r i c a l agg r e ga t e G D P o f b l o c k - c o un t r i e s : G D P i l + ⌘ X i l i l + G D P i l . ( ) W e t h e nfi x t h e f un c t i o n f i n E q . ( ) a s f ( d ) = d , s o t h a tt h e r e n o r m a li ze d g e og r a ph i c a l d i s t a n ce s e q u a l d i l + , j l + ⌘ P i l i l + P j l j l + G D P i l G D P j l d i l , j l P i l i l + G D P i l P j l j l + G D P j l , ( ) w h i c h i s t h e G D P - a v e r ag e d e q u i v a l e n t o f ce r t a i n p o pu l a t i o n - a v e r ag e dd i s t a n ce s c o mm o n l y u s e d i n g e og - r a ph y , e . g .i n t h e G e o D i s t d a t a b a s e i t s e l f [ ]. I n t h i s w a y , d i l + , j l + r e p r e s e n t s a s o r t o f d i s t a n ce b e t w ee n t h e ‘ b a r y ce n t e r s ’ o f t h e b l o c k - c o un t r i e s i l + a nd j l + , w h e r e t h e b a r y ce n t e r o f e a c h ( l + ) - c o un t r y i s l o c a t e du s i n g t h e i n t e r n a l G D P d i s t r i bu t i o n a c r o ss i t s c o n s t i t u e n t l - c o un t r i e s . P u tt i n ga ll t h e a b o v e i n g r e d i e n t s t og e t h e r , w e a rr i v e a tt h e f o ll o w i n g m u l t i s c a l e m o d e l f o r t h e I T N : p i l , j l ( ) = e G D P i l G D P j l / d i l , j l , ( ) w h e r e i s t h e o n l y f r ee p a r a m e t e r a nd w h e r e t h e r e n o r - m a li z a t i o n r u l e s a r e g i v e nb y E q s . ( ) a nd ( ) . N o w t h a t w e h a v e d e fin e d o u r m u l t i s c a l e m o d e l o f t h e I T N , w e bu il d t h ec o rr e s p o nd i n g i n s t a n ce s o f t h e r e a l n e t w o r k a tt h ec h o s e n l e v e l s o f agg r e ga t i o n . T o t h i s e nd , w ec o n s t r u c tt h ee m p i r i c a l - g r a ph ˜ A ( ) b y d r a w i n g a nund i r ec t e d li n k b e t w ee n e a c hp a i r o f c o un t r i e s t h a t h a v e a p o s i t i v e t r a d e r e l a t i o n s h i p i n e i t h e r d i r ec t i o n i n t h e B A C I - C o m t r a d e d a t a s e t( s ee S I ) . T h e n , w e u s e t h e d i s t a n ce - i ndu ce dp a r t i t i o n s { ⌦ l } l d e fin e d a b o v e i n o r - d e r t o c o n s t r u c tt h e l - g r a ph a cc o r d i n g t o E q . ( ) f o r e a c h l e v e l l . T h i s p r o ce du r ec r e a t e s a s e q u e n ce { ˜ A ( l ) } l o f e m p i r i c a l c oa r s e - g r a i n e d v e r s i o n s o f t h e I T N , e a c h o n e r e p r e s e n t i n g t h ee x i s t e n ce o f t r a d e a m o n g l - c o un t r i e s . W ec a nn o w t e s tt h e m u l t i s c a l e m o d e l d e fin e db y E q . ( ) aga i n s tt h e r e a l d a t a { ˜ A ( l ) } l . P r e li m i n a r il y , w ec a li b r a t e t h e m o d e l b y s e tt i n g t o t h e un i q u e v a l u e ˜ t h a t p r o du ce s t h e s a m e li n k d e n s i t y a s t h e r e a l I T N ,i. e . s u c h t h a tt h ee x p ec t e dnu m b e r o f li n k s i n t h e - g r a ph (t h a t i s a m o n o t o n i c a ll y i n c r e a s i n g f un c t i o n o f ) e q u a l s t h ee m p i r i c a l v a l u e o b s e r v e d i n ˜ A ( ) ( s ee S I ) . A f t e r t h i s s i n g l e p a r a m e t e r c h o i ce , a ll t h e p r o b a b ili t i e s i n E q . ( ) a r e un i q u e l y d e t e r m i n e d a t a ll h i e r a r c h i a ll e v e l s a nd w e c a n t e s tt h e m o d e l b y c o m p a r i n g t h ee m p i r i c a l a nd e x - p ec t e d v a l u e o f v a r i o u s t o p o l og i c a l p r o p e r t i e s o f t h e I T N a t d i ↵ e r e n t c oa r s e - g r a i n i n g s . I np a r t i c u l a r , f o r e a c h l e v e l l a nd f o r e a c h l - n o d e i l , w ec o n s i d e r t h e d eg r ee k i l , t h e a ve r a ge n e a r e s t n e i g h b o u r d eg r ee k nn i l a nd t h e c l u s t e r i n g c o e c i e n t c i l ( s ee S I f o r a ll d e fin i t i o n s ) . W ec a rr y o u t a fi r s tt e s t o f t h e m o d e l b y c o n s i d e r i n g t h e a v e r ag e o f t h e s e q u a n t i t i e s o v e r a ll l - n o d e s , t h e r e b y o b t a i n i n g ( u s i n g b a r s t o d e n o t e n o d e a v e r ag e s ) a n o v e r a ll n o d e - a v e r ag e dd e - g r ee ¯ k l , a n o v e r a ll a v e r ag e n e a r e s t n e i g hb o u r d e g r ee ¯ k nn l a nd a n o v e r a ll c l u s t e r i n g c o e c i e n t ¯ c l f o r e a c h l e v e l l . T o c h ec k w h e t h e r t h e m o d e l r e p li c a t e s t h e s e p r o p e r t i e s a t a ll h i e r a r c h i c a ll e v e l s , w e fi r s t n o r m a li ze t h e s e q u a n t i t i e s t o pu tt h e m o n t h e s a m e i n t e r v a l[ , ]i rr e s p ec t i v e o f l ,i. e . w ec o n s i d e r t h e r a t i o s ¯ k l / N l ( w h i c h c o i n c i d e s w i t h t h e li n k d e n s i t y D l , s ee S I ) a nd ¯ k nn l / N l , w h il e ¯ c l i s a l r e a d y n o r m a li ze d . I n F i g . w e p l o t , f o r a ll o u r h i e r a r c h i c a ll e v e l s , t h e t h r ee n o r m a li ze d q u a n t i t i e s a s a f un c t i o n o f t h e nu m b e r N l o f l - n o d e s . W e s ee t h a tt h e m o d e l r e p li c a t e s t h e e m p i r i c a l v a l u e s a cc u r a t e l y . T h i s i s r e m a r k a b l e , g i v e n t h a tt h e o n l y m o d e l p a r a m e t e r w a s c a li b r a t e d i n o r d e r t o m a t c hun i q u e l y t h e d e n s i t y D o f t h e - g r a ph , w h il e t h e ag r ee m e n t h o l d s f o r t h e o t h e r q u a n t i t i e s a s w e ll, a nd a c r o ss a lll e v e l s . A s e x p ec t e d , t h e li n k d e n s i t y i n c r e a s e s a s t h e l e v e li n c r e a s e s ( i. e . a s N l d ec r e a s e s ) , n e v e r t h e l e ss t h e m o d e l a l w a y s r e p r o du ce s t h e p r o p e r t i e s o f t h e r e a l n e t w o r k ( n oa ss u m p t i o n o f s p a r s i t y i s n ee d e d ) . A s a n e v e n m o r e s t r i n g e n tt e s t o f t h e m o d e l, w ec o nfi r m t h e p r e d i c t i o n t h a tt h e l o c a l t o p o l og i c a l p r o p e r t i e s o f t h e i nd i v i du a l ( b l o c k - ) c o un t r i e s , a nd i np a r t i c u l a r k i l , k nn i l a nd c i l , s h o u l dd e p e nd s t r o n g l y o n t h ee m p i r i c a l v a l u e o f G D P i l ,i n a w a y t h a t i s go v e r n e db y E q . ( ) a t a lll e v e l s . A ss h o w n i n F i g . , t h e m o d e l p r e d i c t i o n s a r ec o nfi r m e db y t h ee m p i r i c a l d a t a . I t i s r e m a r k a b l e t h a tt h e ag r ee m e n t b e t w ee n o b s e r v a t i o n s a nd m o d e l e x p ec t a t i o n s h o l d s l o c a ll y a tt h e l e v e l o f i nd i v i du a l . . . D,
Prediction of the topological properties of the renormalized ITN across the full spectrum of geograph-ical aggregation using the multiscale model.
The panels show the agreement between the empirical and expected valuesof the link density D l (top), node-averaged rescaled average nearest neighbour degree ¯ k nnl / ( N l −
1) (middle) and node-averagedclustering coefficient ¯ c l (bottom) as functions of the number N l of l -countries, for all the 18 hierarchical levels considered( l = 0 , overall node-averaged degree ¯ k l , an overall averagenearest neighbour degree ¯ k nnl and an overall clusteringcoefficient ¯ c l for each level l . To check whether the modelreplicates these properties at all hierarchical levels, wefirst normalize these quantities to put them on the sameinterval [0 ,
1] irrespective of l , i.e. we consider the ratios¯ k l / ( N l −
1) (which coincides with the link density D l ,see SI) and ¯ k nnl / ( N l − c l is already normalized.In Fig. 3 we plot, for all our 18 hierarchical levels, thethree normalized quantities D l , ¯ k nnl / ( N l − c l as afunction of the number N l of l -nodes. We see that themodel replicates the empirical values accurately. Thisis remarkable, given that the model has only one freeparameter ( δ ), which was calibrated uniquely to matchthe density D of the 0-graph, while the agreement holdsfor the other quantities as well, and across all levels. Asexpected, the link density increases as the level increases(i.e. as N l decreases), nevertheless the model alwaysreproduces the properties of the real network. As aneven more stringent test of the model, we confirm the prediction that the local topological properties of theindividual (block-)countries, and in particular k i l , k nni l and c i l , should depend strongly on the empirical valueof GDP i l , in a way that is governed by Eq. (13) atall levels. As shown in Fig. 4, the model predictionsare confirmed by the empirical data. It is remarkablethat the agreement between observations and modelexpectations holds locally at the level of individualnodes and across all hierarchical levels, despite the factthat the single parameter δ was used to match only thedensity of the 0-graph, which is a global property definedat a single hierarchical level. As a final consistencycheck, we also confirmed that results similar to thoseshown in Figs. 3 and 4 are retrieved if δ is fixed in orderto match the empirical density of A ( l ) for any othergiven level l > Case II: annealed fitness.
In the annealed case we re-gard not only the graph structure, but also the fitness asa random variable. At the 0-th level, this means that, for
FIG. 4.
Comparison of empirical and expected node-specific properties across multiple levels.
Top panels (a,b,c):empirical (blue) and expected (red) degree k i l vs log(GDP i l ) for all N l nodes, for three representative hierarchical levels ( l = 0, l = 8, l = 13) such that N l = 183 (left), N l = 100 (centre) and N l = 50 (right). Middle panels (d,e,f): empirical (blue)and expected (red) average nearest-neighbour degree k nni l vs log(GDP i l ) for all N l nodes, for the same three hierarchical levels.Bottom panels (g,h,i): empirical (blue) and expected (red) clustering coefficient c i l vs log(GDP i l ) for all N l nodes, for the samethree hierarchical levels. all i = 1 , N , the value x i is drawn from from a certainprobability density function (PDF) ρ i ( x | Γ i ) with pos-itive support, where Γ i denotes all parameters of thePDF. As for the randomness in the topology, we im-pose that the randomness in the fitness, induced from { x i } N i =1 to { x i l } N l i l =1 at all higher levels l > x i l with exactly the same probability byproceeding along two equivalent ways: hierchically bysampling each value x i from its PDF ρ i ( x | Γ i ) and sum-ming up these values for all the 0-nodes that are mappedonto i l by the partition Ω l − · · · Ω , or directly by draw-ing x i l from a certain PDF ρ i l ( x | Γ i l ) that should havethe same functional form of ρ i ( x | Γ i ) and a set of renor- malized parameters Γ i l obtainable from { Γ i } N i =1 onlythrough the kwnoledge of Ω l − · · · Ω . In other words,the fitness values can be virtually resampled at each scale l from a universal distribution with scale-invariant func-tional form and possibly scale-dependent parameters.The above requirement is equivalent to imposing that ρ i l ( x | Γ i l ) belongs to the family of α - stable distribu-tions [34], which are characterized by the four param-eters Γ i l ≡ ( α i l , β i l , γ i l , µ i l ) where β i l ∈ [ − , µ i l ∈ R and γ i l > ρ i l ( x | Γ i l ) respectively, while α i l ≡ α ∈ (0 ,
2] is the(invariant) stability parameter, equal to the exponentasymptotically characterizing (if α <
2) the power-lawtails of the distribution, i.e. ρ i l ( x | Γ i l ) ∼ | x | − α − for x large. For α = 2, ρ i l ( x | Γ i l ) is instead Gaussian. TheGaussian ( α = 2), Cauchy ( α = 1, β i l = 0) and L´evy( α = 1 / β i l = 1) distributions are the only α -stabledistributions known in closed form. Despite this limita-tion, the characteristic function (CF) ϕ i l ( t | Γ i l ) ≡ (cid:90) e itx ρ i l ( x | Γ i l ) dx (14)of a general α -stable distribution is completely known: ϕ i l ( t | Γ i l ) = (cid:40) e itµ il −| γ il t | α (cid:2) − iβ il sign(t) tan πα (cid:3) if α (cid:54) = 1 ,e itµ il −| γ il t | (cid:2) iβ il π sign(t) log | t | (cid:3) if α = 1 . A key feature of α -stable distributions is that, underthe additive rule stated in Eq. (6), the parameters renor-malize as α i l +1 ≡ α, (15) β i l +1 ≡ (cid:80) i l ∈ i l +1 β i l γ αi l (cid:80) i l ∈ i l +1 γ αi l , (16) γ αi l +1 ≡ (cid:88) i l ∈ i l +1 γ αi l , (17) µ i l +1 ≡ (cid:88) i l ∈ i l +1 µ i l . (18)When 0 < α < β i l = 1, the support of α -stable dis-tributions is [ µ i l , + ∞ ). In order to ensure non-negativefitness values at all scales l ≥ p i l ,j l ), we therefore start from l = 0and set 0 < α < β i = 1, µ i = 0 for all i = 1 , N (note that we might set µ i > µ i l with l , while wedo not want to progressively restrict the possible valuesof the fitness as l increases). With this choice, Eqs. (15)-(18) imply that, at all higher levels, α i l +1 ≡ α ∈ (0 , , β i l +1 ≡ , γ αi l +1 ≡ (cid:88) i l ∈ i l +1 γ αi l , µ i l +1 ≡ , showing that α , β and µ are scale-invariant, while γ α isnode-additive. The above scaling rules, combined withthe form of ϕ i l ( t | Γ i l ) given above, finally lead to thescale-invariant CF of the fitness, for all α ∈ (0 ,
1) andfor all γ i l > ϕ i l ( t ) = e −| γ il t | α (cid:2) − i sign(t) tan πα (cid:3) . (19)In order to obtain also an explicit scale-invariant PDF ofthe fitness, we can use the only stable distribution knownin closed form within the above constraints, i.e. the L´evydistribution for which α = 1 / ρ i l ( x ) = (cid:114) γ i l π e − γ il / (2 x ) x / , x > . (20)(where we have kept µ i l = 0). In this case, the only re-maining free parameter is γ i l and the only relevant renor-malization rule is given by Eq. (17). Note that, as we did for P (cid:0) A ( l ) | δ (cid:1) previously, we have omitted the dependenceof ϕ i l ( t ) and ρ i l ( x ) on their parameters.In summary, in the annealed scenario at any hierarchi-cal level l the fitness of each l -node is a random variabledescribed by the CF ϕ i l ( t ) in Eq. (19) or (if α = 1 / ρ i l ( x ) in Eq. (20). Given a realization ofthese fitness values, the network is generated with prob-ability P (cid:0) A ( l ) | δ (cid:1) given by Eq. (8), i.e. by connectingpairs of l -nodes with connection probability p i l ,j l ( δ ) givenby Eq. (4). This construction is entirely self-consistentacross all hierarchical levels, i.e. the l -graph can be ei-ther be built bottom-up, starting from level 0 and coarse-graining the 0-graph up to level l , or directly at the l -thlevel, by sampling the fitness at that level and generatingthe resulting l -graph immediately. Note that, up to thispoint, the connection probability p i l ,j l can still depend onthe distances d i l ,j l as long as the latter are ultrametricon the histogram of desired coarse grainings and there-fore decoupled from the fitness, as discussed previously(if the distances between 0-nodes are not ultrametric,Eq. (7) would make the renormalized distances fitness-dependent and hence random in the annealed case).Notably, a unique property of the annealed case isthat the renormalization defines not only a semi-group proceeding bottom-up from the 0-graph to higher levelsas in usual schemes, but also a group : it can also proceedtop-down by resolving the 0-graph into a graph withany number of nodes bigger than N , indefinitely andin a scale-invariant manner. This possibility is ensuredby the fact that stable distributions are infinitelydivisible , i.e. they can be expressed as the probabilitydistribution of the sum of an arbitrary number of i.id.random variables from the same family. This propertyimplies that we can disaggregate each l -node (including l = 0) with fitness x i l into any desired number of( l − upscaling of the network,in a way that is conceptually similar, but physicallydifferent from the approach in Ref. [35] (which assumesa geometric embedding of nodes). We can thereforeattach no particular meaning to the level l = 0 andconsider any ‘negative’ level m < m -nodes are given and consistent with the higher-levelones, i.e. such that f ( d i l ,j l ) = f ( d i m ,j m ) whenever i l = Ω l − · · · Ω m ( i m ) and j l = Ω l − · · · Ω m ( j m ) for all l > m . Clearly, this requirement is always ensured intwo notable cases: i) if distances are ultrametric and theassociated dendrogram is used to define which m -nodesbranch into which ( m − ii) in the distance-free case f ≡
1. We consider the latter an instructive exampleand discuss it below.
Scale-free networks from scale-invariance without ge-ometry.
We have clarified that scale-free and scale-invariant networks are distinct concepts. Here we con-sider a special case of our annealed scale-invariant model0that spontaneously leads to scale-free networks, thusconnecting the two concepts and providing a nontrivialrecipe for generating scale-freeness purely from scale in-variance. To this end, we use Eq. (20) to provide a com-plete analytical characterization of the annealed model,although similar results can be obtained through numer-ical sampling of the fitness for all α ∈ (0 ,
1) [36–38]. Ingeneral, we may start from l = 0 and assign each 0-node i a different value of γ i , then specify a hierarchy ofcoarse-grainings and calculate the corresponding valuesof γ i l for all l -nodes and the resulting properties of thenetwork, for all l >
0. This leaves a lot of flexibility, inprinciple allowing us to taylor the resulting properties ofthe network to any degree of heterogeneity.However, to avoid making ad hoc assumptions, we putourselves in the simplest situation where distances areswitched off (i.e. f ≡
1, so that the model is governed byEq. (10) and is entirely non-geometric), all 0-nodes arestatistically equivalent (i.e. γ i ≡ γ for all i = 1 , N ),and the dendrogram of coarse grainings is m l -regular: ateach level l , the N l l -nodes are merged into N l +1 = N l m l = · · · = N (cid:81) ln =0 m n (21)( l + 1)-nodes, each formed by exactly m l l -nodes. Notethat this is the most homogeneous choice, as it preservesthe statistical equivalence of all the N l l -nodes at everyhierarchical level, i.e. γ i l ≡ γ l for all l where γ l +1 = m l /α γ l = · · · = l (cid:89) n =0 m /αn γ = (cid:18) N N l +1 (cid:19) /α γ (22)(with α = 1 / l , the fitness values { x i l } N l l =1 are i.i.d. withcommon distribution ρ l ( x ) = (cid:114) γ l π e − γ l / (2 x ) x / , x > , ρ l ( x ) ∝ x − / (23)effectively reducing a multivariate problem to a univari-ate one. This makes the model similar to the FM [14],with two special prescriptions: i) here the fitness is de-fined at all hierarchical levels simultaneously and ii) theconnection probability can only take the scale-invariantform given by Eq. (10). Note that the fitness distri-bution depends on the hierarchical level l through theparameter γ l , which, as clear from Eq. (22), cannot de-crease since N l cannot increase. This is just an overallshift towards larger values of the fitness as nodes arecoarse-grained. For instance, if we take m l = m (thebranching ratio is level-independent), then Eq. (22) im-plies γ l = m l/α γ = m l γ and the corresponding be-haviour of the fitness distribution is illustrated in the toppanel of Fig. 5. Irrespective of the rightward shift, thetail of the fitness distribution is always a pure power-law x − − α , independently of l .We can now discuss the topological properties of theresulting network. As we show in SI, the expected degree Annealed case: fitness variables drawn from a probability distribution (cid:4632)𝐴 (cid:4666)(cid:2868)(cid:4667) ~ 𝑃 (cid:4632)𝐴 (cid:4666)(cid:2868)(cid:4667) | (cid:1876) (cid:3036) (cid:4666)(cid:2868)(cid:4667) (cid:3015) (cid:3404) (cid:962) (cid:3036),(cid:3037) (cid:2879)(cid:3051) (cid:3284) (cid:3116) (cid:3051) (cid:3285) (cid:4666)(cid:3116)(cid:4667) • Scale-invariance of P • Additivity of (cid:1876) (cid:3036)(cid:4666) l (cid:4667) across different scales (cid:1876) (cid:3036) ~ (cid:2009) (cid:3398) 𝑠(cid:1872)𝑎𝑏𝑙𝑒 𝑑𝑖𝑠(cid:1872)𝑟𝑖𝑏(cid:1873)(cid:1872)𝑖𝑜𝑛 𝑃 𝑘 (cid:3406) 𝑘 (cid:2879)(cid:2870) , size-dependent cut-off 𝑃 (cid:1876) (cid:3406) (cid:1876) (cid:2879) (cid:3119)(cid:3118) , No cut-off
Fitness x Reduced degree k /( N -1) C u m u l a t i v e d i s t r i bu t i o n C u m u l a t i v e d i s t r i bu t i o n Annealed case: fitness variables drawn from a probability distribution (cid:4632)𝐴 (cid:4666)(cid:2868)(cid:4667) ~ 𝑃 (cid:4632)𝐴 (cid:4666)(cid:2868)(cid:4667) | (cid:1876) (cid:3036) (cid:4666)(cid:2868)(cid:4667) (cid:3015) (cid:3404) (cid:962) (cid:3036),(cid:3037) (cid:2879)(cid:3051) (cid:3284) (cid:3116) (cid:3051) (cid:3285) (cid:4666)(cid:3116)(cid:4667) • Scale-invariance of P • Additivity of (cid:1876) (cid:3036)(cid:4666) l (cid:4667) across different scales (cid:1876) (cid:3036) ~ (cid:2009) (cid:3398) 𝑠(cid:1872)𝑎𝑏𝑙𝑒 𝑑𝑖𝑠(cid:1872)𝑟𝑖𝑏(cid:1873)(cid:1872)𝑖𝑜𝑛 𝑃 𝑘 (cid:3406) 𝑘 (cid:2879)(cid:2870) , size-dependent cut-off 𝑃 (cid:1876) (cid:3406) (cid:1876) (cid:2879) (cid:3119)(cid:3118) , No cut-off -2 -1 a -6 -5 -4 -3 -2 -1 P ( a ) , a - P2(a)a -2 FIG. S2. Representation of the term P ( ) as a function of (blue line) and corresponding Taylor approximation ⇠ (red dashed line). expansion of log(1 y ) = P n =1 ( y ) n /n for | y | < B l ( ) = p l (1 ) ln (1 ) ⇡ p l (1 )( ) ⇡ p l (1 )( + O [ ]) ⇡ p l ( + O [ ]) ⇡ C l ( ) where C l ( ) is a cut-o↵ function with a peak at values of that increase towards 1 as l increases (see Fig. S2).Putting the pieces together, the right tail of the re-duced degree distribution behaves as Q l ( ) ⇡ C l ( ) (S36)where C l ( ) is the cut-o↵ function. This proves our state-ment in the main text, and is confirmed by the numericalsimulations in the bottom panel of Fig. 5. ↵ (0 ,
1) [ ? ][ ? ][ ? ]. In general, we may start from l = 0 and assign each0-node i a di↵erent value of i , then specify a hierarchyof coarse-grainings and calculate the corresponding val-ues of i l for all l -nodes and the resulting properties ofthe network, for all l >
0. This leaves a lot of flexibility,in principle allowing us to taylor the resulting propertiesof the network to any degree of heterogeneity.However, to avoid making ad hoc assumptions, we putourselves in the simplest situation where distances areswitched o↵ (i.e. f ⌘
1, so that the model is governed byEq. (10) and is entirely non-geometric), all 0-nodes arestatistically equivalent (i.e. i ⌘ for all i = 1 , N ),and the dendrogram of coarse grainings is m l -regular: ateach level l , the N l l -nodes are merged into N l +1 = N l m l = · · · = N Q ln =0 m n (21)( l + 1)-nodes, each formed by exactly m l l -nodes. Notethat this is the most homogeneous choice, as it preservesthe statistical equivalence of all the N l l -nodes at everyhierarchical level, i.e. i l ⌘ l for all l where l +1 = m l /↵ l = · · · = l Y n =0 m /↵n = ✓ N N l +1 ◆ /↵ (22)(with ↵ = 1 / l , the fitness values { x i l } N l l =1 are i.i.d. withcommon distribution ⇢ l ( x ) = r l ⇡ e l / (2 x ) x / , x > , ⇢ l ( x ) / x / (23)e↵ectively reducing a multivariate problem to a univari-ate one. This makes the model similar to the FM [ ? ],with two special prescriptions: i) here the fitness is de-fined at all hierarchical levels simultaneously and ii) theconnection probability can only take the scale-invariantform given by Eq. (10). Note that the fitness distri-bution depends on the hierarchical level l through theparameter l , which, as clear from Eq. (22), cannot de-crease since N l cannot increase. This is just an overallshift towards larger values of the fitness as nodes arecoarse-grained. For instance, if we take m l = m (thebranching ratio is level-independent), then Eq. (22) im-plies l = m l/↵ = m l and the corresponding be-haviour of the fitness distribution is illustrated in the toppanel of Fig. 5. Irrespective of the rightward shift, thetail of the fitness distribution is always a pure power-law x ↵ , independently of l .We can now discuss the topological properties of theresulting network. As we show in SI, the expected degree Annealed case: fitness variables drawn from a probability distribution (cid:4632)𝐴 (cid:4666)(cid:2868)(cid:4667) ~ 𝑃 (cid:4632)𝐴 (cid:4666)(cid:2868)(cid:4667) | (cid:1876) (cid:3036) (cid:4666)(cid:2868)(cid:4667) (cid:3015) (cid:3404) (cid:962) (cid:3036),(cid:3037) (cid:2879)(cid:3051) (cid:3284) (cid:3116) (cid:3051) (cid:3285) (cid:4666)(cid:3116)(cid:4667) • Scale-invariance of P • Additivity of (cid:1876) (cid:3036)(cid:4666) l (cid:4667) across different scales (cid:1876) (cid:3036) ~ (cid:2009) (cid:3398) 𝑠(cid:1872)𝑎𝑏𝑙𝑒 𝑑𝑖𝑠(cid:1872)𝑟𝑖𝑏(cid:1873)(cid:1872)𝑖𝑜𝑛 𝑃 𝑘 (cid:3406) 𝑘 (cid:2879)(cid:2870) , size-dependent cut-off 𝑃 (cid:1876) (cid:3406) (cid:1876) (cid:2879) (cid:3119)(cid:3118) , No cut-off
Fitness x Reduced degree k /( N -1) C u m u l a t i v e d i s t r i bu t i o n C u m u l a t i v e d i s t r i bu t i o n Annealed case: fitness variables drawn from a probability distribution (cid:4632)𝐴 (cid:4666)(cid:2868)(cid:4667) ~ 𝑃 (cid:4632)𝐴 (cid:4666)(cid:2868)(cid:4667) | (cid:1876) (cid:3036) (cid:4666)(cid:2868)(cid:4667) (cid:3015) (cid:3404) (cid:962) (cid:3036),(cid:3037) (cid:2879)(cid:3051) (cid:3284) (cid:3116) (cid:3051) (cid:3285) (cid:4666)(cid:3116)(cid:4667) • Scale-invariance of P • Additivity of (cid:1876) (cid:3036)(cid:4666) l (cid:4667) across different scales (cid:1876) (cid:3036) ~ (cid:2009) (cid:3398) 𝑠(cid:1872)𝑎𝑏𝑙𝑒 𝑑𝑖𝑠(cid:1872)𝑟𝑖𝑏(cid:1873)(cid:1872)𝑖𝑜𝑛 𝑃 𝑘 (cid:3406) 𝑘 (cid:2879)(cid:2870) , size-dependent cut-off 𝑃 (cid:1876) (cid:3406) (cid:1876) (cid:2879) (cid:3119)(cid:3118) , No cut-off
FIG. 5.
Top: cumulative distribution of the node fitness x across di↵erent hierarchical levels, for the parameter choice ↵ = 1 / N = 10 and m l = m = 2 (constant number of l -nodes coarse grained into ( l + 1)-nodes). The dashed lineis a power-law with exponent /
2, confirming that the non-cumulative fitness distribution has power-law tails with expo-nent /
2. Note that there is no upper cut-o↵ to this tail,despite the increasing network density for higher hierarchicallevels, because the fitness of a node has no bounds.
Bottom: corresponding cumulative distributions of the reduced degree , for the same parameter choice. The dashed line is a power-law with exponent
1, indicating the presence of a power-lawregime in the non-cumulative degree distribution with expo-nent distribution induced by Eqs. (10) and (23) is exactly cal-culated as P l ( k ) = 2 q l ⇡ exp l ln ⇣ kNl ⌘ ( N l ⇣ kN l ⌘ ln ⇣ kN l ⌘ (24) FIG. 5.
Fitness and degree distributions in the an-nealed scale-invariant model.
Top: cumulative distribu-tion of the node fitness x across different hierarchical levels,for the parameter choice α = 1 / N = 10 and m l = m = 2(constant number of l -nodes coarse grained into ( l +1)-nodes).The dashed line is a power-law with exponent − /
2, confirm-ing that the non-cumulative fitness distribution has power-law tails with exponent − /
2. Note that there is no uppercut-off to this tail, despite the increasing network density forhigher hierarchical levels, because the fitness of a node hasno bounds.
Bottom: corresponding cumulative distributionsof the reduced degree κ , for the same parameter choice. Thedashed line is a power-law with exponent −
1, indicating apower-law regime in the non-cumulative degree distributionwith exponent − C l ( κ ) becoming strongeras the hierarchical level (and density) increases. distribution induced by Eqs. (10) and (23) is exactly cal-culated as P l ( k ) = 2 (cid:113) δγ l π exp (cid:20) − δγ l ln (cid:16) − kNl − (cid:17) (cid:21) ( N l − (cid:16) − kN l − (cid:17) ln (cid:16) − kN l − (cid:17) (24)1for k ≥
0, and P l ( k ) = 0 otherwise. The degree distribu-tion above shows a more complicated dependence on thehierarchical level l . In this case, there are two contrastingtendencies as l increases. On the one hand, the numberof nodes N l decreases, hence the possible range of values[1 , N l −
1] for the degree k shrinks: this implies that thedegree tends to decrease. On the other hand, the ongoingcoarse-graining implies that, on average, l -nodes acquiremore and more links as l increases: this implies that thedegree tends to increase. To single out which of the twotendencies is the dominant one, we rescale the degree k by N l −
1, thereby defining the reduced degree κ ≡ kN l − ∈ [0 , . (25)This rescaling removes the effect of the first tendency, asthe possible range for κ is now independent of l . Theprobability distribution Q l ( κ ) for the reduced degree canbe easily calculated from P l ( k ) as Q l ( κ ) = 2 (cid:113) δγ l π exp (cid:104) − δγ l ln (1 − κ ) (cid:105) (1 − κ ) ln (1 − κ ) . (26)We now see that the distribution still has a residual de-pendence on the level l through the parameter γ l . As aconsequence, the reduced degree distributions obtainedfor different hierarchical levels do not collapse upon eachother, as confirmed in the bottom panel in Fig. 5 usingthe same parameter choice as above. This is the effect ofthe second tendency, which turns out to be the dominantone. Indeed we see that, as l increases, there is a moreand more pronounced accumulation of values of the re-duced degree κ close to the maximum value 1. This isa saturation effect cutting off the tail of the (reduced)degree distribution. For values of the degree that aresufficiently lower than the upper cut-off, the distributionhas a power-law trend proportional to κ − . Indeed, onecan prove analytically (see SI) that the right tail of thereduced degree distribution behaves as Q l ( κ ) ≈ κ − C l ( κ ) , (27)where C l ( κ ) is a cut-off function with a peak at valuesof κ that increase towards 1 as l increases. The cut-offfunction captures stronger and stronger finite-size effectsas the network size shrinks under the effect of coarse-graining. Indeed, the expected network can be calculatedexactly as (cid:104) D l (cid:105) = 1 − G , , (cid:0) ·− / , , (cid:12)(cid:12) δγ l / (cid:1) (28)where G m,np, q (cid:0) a ,...,a p b ,...,b q (cid:12)(cid:12) z (cid:1) denotes the Meijer- G function(see SI), which is an increasing function of the combina-tion δγ l , and therefore increases with l . Notably, the fact that the degree distribution has aninverse square power-law decay (with a cut off) is gener-ated only from the requirement of scale-invariance, with-out any other assumption. Moreover, it should be notedthat the cut-off is controlled by the parameter γ l , whichin turn is controlled by the particular choice m l = m .One may in principle make an alternative choice for thesequence m l , e.g. in order to keep an asymptotically con-stant density, i.e. an asymptotically constant γ l ≡ γ ∞ , as l increases. This would be accomplished by requiring that m l decreases with an appropriate speed as l increases. Insuch a way, the fitness and degree distributions, and ul-timately all the properties of the network, would remaininvariant across coarse-graining. Conclusions.
Our approach relied on the introductionof a random graph model whose mathematical formaliza-tion is invariant with respect to the scale of resolution.This means that the functional form of the probabilityfor two nodes of the network to be connected is inde-pendent on the resolution: at each scale, the model cangenerate the network either hierarchically, by generatingthe finest grained network and then coarse-graining itvia progressive non-overlapping (but otherwise arbitrary)partitions, or directly, with a given probability that de-pend on the scale only through a set of specific parame-ters. These parameters include a set of hidden variablesattached to each (block-)node, a global density param-eter and, if useful, a set of dyadic factors such as dis-tances or communities. These parameters transform (orrenormalize) following a rule that can be made explicit inboth directions of the hierarchy: from higher resolutionsto lower (via disaggregation of nodes) and, vice versa,bottom-up (via merging of nodes). The model can ei-ther generate scale-free networks spontaneously, withoutfine-tuning and without geometry, or guide the renor-malization of real graphs with arbitrary topology. Inthe first (annealed) case, the fitness variables must berandomly drawn according to a L´evy-stable distribution,whose properties of infinitely divisibility and additivityare strongly used in order to provide a scheme for thegeneration, at any hierarchical level, of a scale-free net-work which is also self-similar in the above mentionedway. In the latter (quenched) case, the parameters of themodel can be taken from empirical observations, withoutany assumption on the scale or on the existence of anunderlying metric space. In our application to the ITN,we found that a one-parameter fit of the model to theobserved density is enough to accurately replicate manytopological properties of individual nodes, even acrossseveral hierarchical levels of resolution.2 [1] Schweitzer, F., Fagiolo, G., Sornette, D., Vega-Redondo,F., Vespignani, A., and White, D. R., “Economic net-works: The new challenges”,
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SUPPLEMENTARY INFORMATION accompanying the paper “Multiscale network renormalization: scale-invariance without geometry” by E. Garuccio, M. Lalli, and D. Garlaschelli
S.I. FINDING THE SCALE-INVARIANTCONNECTION PROBABILITY
Here we show how the scale-invariance requirementstated in Eq. (2), for any model with independent linksas formulated in Eq. (3), leads to the unique form of theconnection probability given by Eq. (4).Let us consider a partition Ω l that maps an l -graphwith N l l -nodes and adjacency matrix A ( l ) to an ( l + 1)-graph with N l +1 ( l + 1)-nodes and adjacency matrix A ( l +1) . The relation between the entries of the matri-ces A ( l ) and A ( l +1) is given by Eq. (1) in the main text.Now, for any random graph model with independent linksas stated in Eq. (3), a ( l ) i l ,j l is a Bernoulli random variableequal to 1 with probability p ( l ) i l ,j l and equal to 0 withprobability 1 − p ( l ) i l ,j l . Similarly, a ( l +1) i l +1 ,j l +1 is a Bernoullirandom variable equal to 1 with probability p ( l +1) i l +1 ,j l +1 and equal to 0 with probability 1 − p ( l +1) i l +1 ,j l +1 . Now, thescale-invariance requirement in Eq. (2) demands that weshould create, with equal probability , any of the possi-ble realizations of the adjacency matrix A ( l +1) eitherby: i) generating the possible realizations of the ma-trix A ( l ) (using the associated probabilities { p ( l ) i l ,j l } ) andthen aggregating the corresponding l -graphs into ( l + 1)-graphs, or ii) directly generating all the possible realiza-tions of the matrix A ( l +1) (using the associated proba-bilities { p ( l +1) i l +1 ,j l +1 } ). Scale-invariance also demands that p ( l ) i l ,j l depends on l only through its parameters. Assum-ing that these parameters are a combination of global( δ l ), node-specific ( x i l , x j l ) and dyadic ( d i l ,j l ) factors, wecan write p ( l ) i l ,j l (cid:0) δ l (cid:1) = p i l ,j l (cid:0) δ l (cid:1) (see main text). Enforc-ing scale-invariance means finding not only the functionalform of p i l ,j l , but also the renormalization rules con-necting δ l , x i l , x j l , d i l ,j l to their next-level counterparts δ l +1 , x i l +1 , x j l +1 , d i l +1 ,j l +1 .To enforce the scale-invariance requirement we notethat, since a link between the pair ( i l +1 , j l +1 ) of ( l + 1)-nodes is present if and only if there is at least onelink present between any pair ( i l , j l ) of l -nodes suchthat i l ∈ i l +1 and j l ∈ j l +1 , the probability that i l +1 and j l +1 are not connected is equal, accordingto the procedure ii) described above, to the proba-bility that none of the pairs of underlying l -nodes isconnected. Since links are independent, this probabil-ity equals (cid:81) i l ∈ i l +1 (cid:81) j l ∈ j l +1 [1 − p i l ,j l ( δ )]. On the otherhand, according to the procedure i) , the same eventoccurs with probability 1 − p i l +1 ,j l +1 ( δ ). Enforcing theequality between the two probabilities leads to the con- dition1 − p i l +1 ,j l +1 ( δ ) = (cid:89) i l ∈ i l +1 (cid:89) j l ∈ j l +1 [1 − p i l ,j l ( δ )] . (S1)Taking the logarithm of both sides of Eq. (S1), we obtainln (cid:2) − p i l +1 ,j l +1 ( δ ) (cid:3) = (cid:88) i l ∈ i l +1 (cid:88) j l ∈ j l +1 ln [1 − p i l ,j l ( δ )] , (S2)from which we now derive the scale-invariant form ofthe connection probability. Note that Eq. (S1) is con-sistent with taking the expected values of both sides ofEq. (1) in the main text. However, it cannot be deriveddirectly in that way, because the two expected values aretaken with respect to different probability distributions,i.e. P l +1 (cid:0) A ( l +1) (cid:12)(cid:12) Θ l (cid:1) and P l (cid:0) A ( l ) (cid:12)(cid:12) Θ l (cid:1) respectively.Now, let us first consider the case where the connectionprobability p i l ,j l does not depend on any dyadic factor d i l ,j l . In this case, the only functional form of p i l +1 ,j l +1 compatible with Eq. (S2) for every pair of ( l + 1)-nodesis such thatln (cid:2) − p i l +1 ,j l +1 ( δ ) (cid:3) = − δ g ( x i l +1 ) g ( x j l +1 ) (S3)where g ( x ) is a positive function such that g ( x i l +1 ) = (cid:88) i l ∈ i l +1 g ( x i l ) (S4)and δ is positive and l -independent. Note that the pos-itivity of δ and g ( x ) follows from the fact that, since0 ≤ p i l +1 ,j l +1 ( δ ) ≤
1, ln (cid:2) − p i l +1 ,j l +1 ( δ ) (cid:3) has to be non-positive. On the other hand, g ( x ) has to have the samesign for all nodes, otherwise for some pair of nodes theproduct g ( x i l +1 ) g ( x j l +1 ) will be negative. Interpreting g ( x ) as the impact of the fitness x on the connectionprobability, it makes sense to choose the positive signfor g ( x ) (and, incidentally, that g ( x ) is monotonically in-creasing with x ). Similarly, δ has to be positive as well.Now, if the quantity x is node-additive (e.g. because itis identified with some empirical additive quantity, likethe GDP in our model of the ITN), then the fitness ofeach ( l + 1)-node x i l +1 should be consistently obtainedas a sum (cid:80) i l ∈ i l +1 x i l over the underlying l -nodes. Thisimplies that, after reabsorbing any (positive) proportion-ality factor into δ , the only possible choice for g ( x ) in theadditive case is g ( x ) = x . By constrast, if we do notrequire x to be node-additive, we can always invoke thedesired monotonicity of g ( x ) and redefine x ← g ( x ) (in-deed, there is no a priori reason why x i l , rather than g ( x i l ), should be regarded as the ‘natural’ node-specificfactor affecting the connection probabilities involving i l ).This makes the redefined fitness x additive by construc-tion. In summary, by redefining the node-specific factor x in a way that makes it node-additive, and reabsorbingany global constant into δ , the only possible functionalform for p i l ,j l under the requirement of scale-invariance(and in absence of dyadic factors) is such thatln (cid:2) − p i l +1 ,j l +1 ( δ ) (cid:3) = − δ x i l +1 x j l +1 , (S5)or equivalently p i l ,j l ( δ ) = 1 − e − δx il x jl , δ, x i l , x j l > , (S6)where δ is scale-invariant and x i l +1 = (cid:80) i l ∈ i l +1 x i l . Thisresult coincides with what stated in the main text inEq. (4) when f ≡
1, i.e. with Eq. (10), and with Eq. (6).If we add dyadic factors, i.e. we allow p i l ,j l to addition-ally depend on some non-constant positive function f ( d )of the dyadic quantity d , while at the same time preserv-ing the bilinear dependence of ln (cid:2) − p i l +1 ,j l +1 ( δ ) (cid:3) on x i l and x j l (i.e. preserving the additivity of the fitness), thenEq. (S5) has to be generalized toln (cid:2) − p i l +1 ,j l +1 ( δ ) (cid:3) = − δ x i l +1 x j l +1 f ( d i l +1 ,j l +1 ) , (S7)where Eq. (S2) demands that f ( d i l ,j l ) renormalizes as x i l +1 x j l +1 f ( d i l +1 ,j l +1 ) = (cid:88) i l ∈ i l +1 (cid:88) j l ∈ j l +1 x i l x j l f (cid:0) d i l ,j l (cid:1) . (S8)Equations (S7) and (S8) coincide with Eqs. (4) and (7),thus completing our proof. Note that in principle theconstant δ may be entirely reabsorbed into the function f ( d ), however it is useful to keep it separate as a singleparameter controlling the overall density of the graph.Also note that if the dyadic factor d is interpreted as afeature enhancing the connection probability (e.g. be-cause it represents similarity, correlation, co-affiliation,etc.), then f ( d ) has to be an increasing function. Bycontrast, if d diminishes the connection probability (e.g.because it represents distance or dissimilarity), then f ( d )has to be a decreasing function, as in our model of theITN. S.II. GDP, DISTANCE, AND TRADE DATA
GDP data are taken from the World Bank dataset [31]and are expressed in US Dollars. The results reported inthe main body of the paper use data for year 2011. Thenumber of countries for which GDP data are availablein that year is 183. Note that, unlike the internationaltrade data (see below), the World Bank GDP datasetcovers a slightly smaller number of countries as it doesnot include very small ones (typically islands).Geographic distance data are taken from the BACI-CEPII GeoDist database [32]. It reports bilateral inter-country distances measured as population-based averages among the most populated pairs of cities across each pairof countries. The database uses the general formula d i,j = (cid:32) (cid:80) k ∈ i (cid:80) l ∈ j POP k POP l d θk.l (cid:80) k ∈ i (cid:80) l ∈ j POP k POP l (cid:33) /θ (S9)developed by Head and Mayer [33] for calculating thedistance d i,j between country i and country j as apopulation-based average of the distances d k,l betweenpairs of agglomerations (cities, towns and places) across i and j . The symbol k ∈ i denotes that k runs overthe agglomerations in country i , and POP k denotesthe demographic population of agglomeration k . In theGeoDist database, population data were taken from theWorld Gazetteer ( )website. Note that d i,i >
0, i.e. the ‘distance’ of acountry to itself is non-zero (therefore it is not a propermetric distance). This is consistent with the fact that,at higher hierarchical levels, the distance between ablock-node to itself is necessarily positive as a resultof the renormalization rule. The exponent θ measuresthe sensitivity of trade flows to bilateral distance. Asnoted in the BACI-CEPII GeoDist documentation,selecting θ = − d i,j as apopulation-based average analogous to the GDP-basedaverage used later in our own renormalization procedurewhen coarse-graining the network. The agreementbetween our model and the ITN data actually suggeststhat, for the purpose of the analysis of internationaltrade, a better definition of inter-country distances couldpresumably be obtained by replacing POP with GDPin the above formula, to make inter-country distancesfully consistent with our GDP-averaged renormalizeddistances at higher levels. Unfortunately, GDP dataat the agglomeration level are much more difficult toobtain than the corresponding population data. For thisreason, we had to use population-averaged distancesin our analysis at level l = 0, and their GDP-averagedrenormalized values at higher levels l > S.III. NETWORK PROPERTIES: EMPIRICALAND EXPECTED VALUES
Here we define the key topological properties consid-ered in our analysis and modelling of the ITN. Each suchproperty is a function Y ( A ( l ) ) of the N l × N l adjacencymatrix A ( l ) (with entries a ( l ) i l ,j l = 0 ,
1) of the generic l -graph. Note that this matrix is symmetric and cancontain non-zero entries along the diagonal, representingself-loops. These self-loops may or may not be present inthe 0-graph, but are in any case eventually generated bythe coarse graining procedure if the nodes mapped ontothe same block-node are connected among themselves.When analysing the ITN, the relevant matrix A ( l ) is theempirical matrix ˜ A ( l ) obtained from the BACI-Comtradedata in year 2011 as described above. The correspondingempirical value of each topological property Y of inter-est will be denoted as ˜ Y ≡ Y ( ˜ A ( l ) ). When consideringthe multiscale model, A ( l ) is instead a random (symmet-ric) matrix whose entries { a ( l ) i l ,j l } are Bernoulli randomvariables with expected value (cid:104) a ( l ) i l ,j l (cid:105) = p i l ,j l ( δ ) = 1 − e − δ GDP il GDP jl /d il,jl , (S10)where, consistently with the possible presence of self-loops, we allow for i l = j l . Equation (S10) allows us tocalculate the expected value of several topological prop-erties. In particular, we use the total number of l -links(excluding possible self-loops) L l ( A ( l ) ) = N l (cid:88) i l =1 i l − (cid:88) j l =1 a ( l ) i l ,j l (S11)in order to fix the only free parameter δ to the uniquevalue ˜ δ such that the expected number of 0-links (ex-cluding possible self-loops, which are in any case absentin our data) (cid:104) L (cid:105) = N (cid:88) i =1 i − (cid:88) j =1 p i ,j ( δ ) (S12)equals the empirical value˜ L = N (cid:88) i =1 i − (cid:88) j =1 ˜ a (0) i ,j = 12018 (S13)observed in the ITN in year 2011. This selects the value˜ δ = 4 . · − (USD) − , where USD stands for US dol-lars (the unit of measure used in GDP data). Having fixed ˜ δ , we can generate unbiased realisations { A ( l ) } of the l -graphs from the multiscale model at anydesired hierarchical level l by sampling l -links indepen-dently with probability ˜ p i l ,j l ≡ p i l ,j l (˜ δ ). By averagingthe value Y ( A ( l ) ) of any topological property of inter-est over such realizations, we can efficiently estimate thecorresponding expected value (cid:104) ˜ Y (cid:105) ≡ (cid:88) A ( l ) ∈G Nl P (cid:0) A ( l ) | ˜ δ (cid:1) Y ( A ( l ) ) , (S14)where P (cid:0) A ( l ) | δ (cid:1) is given by Eq. (8), without actuallycalculating the above sum explicitly. If Y ( A ( l ) ) is linearin A ( l ) , we can even calculate (cid:104) ˜ Y (cid:105) exactly by directlyreplacing a ( l ) i l ,j l with ˜ p i l ,j l in the definition of Y ( A ( l ) ),without sampling any graph at all. This is indeed thecase for the number of links in Eq. (S11).Given any l -graph A ( l ) (be it the empirical l -graph ora random realization from the model), the topologicalproperties of interest to us are: the link density D l ( A ( l ) ) ≡ L l ( A ( l ) ) N l ( N l −
1) = 2 N l ( N l − N l (cid:88) i l =1 i l − (cid:88) j l =1 a ( l ) i l ,j l (S15)(representing the ratio of realized to maximum numberof links, excluding self-loops), the degree k i l ( A ( l ) ) ≡ (cid:88) j l (cid:54) = i l a ( l ) i l ,j l (S16)(counting the number of links of the l -node i l , excludingself-loops), the average nearest neighbour degree k nni l ( A ( l ) ) ≡ (cid:80) j l (cid:54) = i l (cid:80) k l (cid:54) = j l a ( l ) i l ,j l a ( l ) j l ,k l (cid:80) j l (cid:54) = i l a ( l ) i l ,j l (S17)(representing the average degree of the neighbours of i l ),and finally the clustering coefficient c i l ( A ( l ) ) ≡ (cid:80) j l (cid:54) = i l (cid:80) k l (cid:54) = i l ,j l a ( l ) i l ,j l a ( l ) j l ,k l a ( l ) k l ,i l (cid:80) j l (cid:54) = i l (cid:80) k l (cid:54) = i l ,j l a ( l ) i l ,j l a ( l ) k l ,i l (S18)(representing the number of triangles into which i l par-tipates, divided by the maximum realizable number oftriangles, given the value of k i l ). All the above quanti-ties can be averaged over nodes to obtain the followingoverall properties: ¯ k l ≡ N l N l (cid:88) i l =1 k i l , (S19)¯ k nnl ≡ N l N l (cid:88) i l =1 k nni l , (S20)¯ c l ≡ N l N l (cid:88) i l =1 c i l . (S21)It is important to stress that, of all the quantities de-fined in Eqs. (S15)-(S21) for each l -node ( i l = 1 , N l )and/or all levels ( l = 0 , D ofthe 0-graph is replicated by construction via the param-eter choice δ = ˜ δ : indeed, having enforced (cid:104) ˜ L (cid:105) = ˜ L byequating Eqs. (S12) and (S13) coincides with having re-quired (cid:104) ˜ D (cid:105) = ˜ D . For all the other properties, including D l for all l >
0, the agreement (shown in the main text)between the model and the empirical network is highlynontrivial and hence remarkable.
S.IV. ANALYTICAL FORM OF THE DEGREEDISTRIBUTION FOR α = 1 / Here we derive the functional form of the expecteddegree distribution in the annealed model with L´evy-distributed fitness (i.e. α = 1 /
2) as specified in Eq. (23)and distance-independent connection probability (i.e. f ≡
1) as given by Eq. (10). To this end, for anyfixed hierarchical level l we adapt the procedure out-lined in Ref. ?? to compute the distribution P l ( k ) ofexpected degrees from the PDF of the fitness ρ l ( x ) andthe connection probability p i l ,j l , written as a function p i l ,j l = f ( x i l , x j l ) of the fitness of the nodes involved,where in our case f ( x, y ) = 1 − e − δ x y . (S22)We first notice that, since f ( x, y ) is an increasing func-tion of both its arguments, the expected degree (cid:104) k i l (cid:105)(cid:104) k i l (cid:105) = (cid:88) j l (cid:54) = i l p i l ,j l = (cid:88) j l (cid:54) = i l f ( x i l , x j l ) (S23)is an increasing function of the fitness x i l . Indeed, anytwo l -nodes with the same fitness have the same expecteddegree, and l -nodes with higher fitness have larger ex-pected degree. For a large number N l of l -nodes, theabove discrete sum can be approximated as an integralover the number ( N l − ρ l ( y ) of l -nodes (except i l itself)with fitness in a neighbourhood of y : k ( x ) = ( N l − (cid:90) ∞ f ( x, y ) ρ l ( y )d y = ( N l − (cid:90) ∞ (cid:0) − e − δxy (cid:1) (cid:114) γ l π e − γ l / (2 y ) y / d y = ( N l − (cid:16) − e −√ δγ l x (cid:17) , (S24)where k ( x ) denotes the expected degree of a node withfitness x . Inverting, we find that the fitness of an l -nodewith expected degree k is x ( k ) = 12 δγ l ln (cid:18) N l − N l − − k (cid:19) , (S25)which implies dd k x ( k ) = ln (cid:16) − N l − N l − − k (cid:17) δγ l ( N l − − k ) . (S26) We can use the above expressions in order to obtainthe distribution P l ( k ) of the expected degrees from thedistribution ρ l [ x ( k )] of the corresponding fitness. Indeed,starting from the fundamental equation P l ( k )d k = ρ l [ x ( k )]d x ( k ) (S27)relating the probability distributions of the two randomvariables k and x , and using Eqs. (23), (S25) and (S26),we arrive at the explicit form of the distribution of ex-pected degrees: P l ( k ) = ρ l [ x ( k )] dd k x ( k )= 2 (cid:113) δγ l π exp (cid:20) − δγ l ln (cid:16) Nl − Nl − − k (cid:17) (cid:21) ( N l − − k ) ln (cid:16) N l − N l − − k (cid:17) = 2 (cid:113) δγ l π exp (cid:20) − δγ l ln (cid:16) − kNl − (cid:17) (cid:21) ( N l − (cid:16) − kN l − (cid:17) ln (cid:16) − kN l − (cid:17) (S28)for k ≥
0, and P l ( k ) = 0 otherwise.We can obtain the expected link density (cid:104) D l (cid:105) via thecalculation of the total expected degree, i.e. twice theexpected number of links, as (cid:104) L l (cid:105) = N l (cid:90) N l − P l ( k ) k d k = N l ( N l − (cid:18) − π (cid:90) ∞ e t − γ l √ δ/t d t (cid:19) (S29)where we have changed variables by introducing t = γ l √ δ log N l − N l − − k . (S30)The integral in Eq. (S29) can be expressed in terms ofone of the Meijer- G functions G m,np, q (cid:0) a ,...,a p b ,...,b q (cid:12)(cid:12) z (cid:1) . Theresulting expected density can be written as (cid:104) D l (cid:105) = (cid:104) L l (cid:105) N l ( N l −
1) = 1 − G , , (cid:0) ·− / , , (cid:12)(cid:12) δγ l / (cid:1) . (S31) S.V. THE SCALE-FREE RANGE WITHINVERSE SQUARE EXPONENT
We rewrite the distribution of the reduced degree κ = k/N l (see main text) shown in Eq. (26) as Q l ( κ ) = A l ( κ ) B l ( κ ) (S32)where A l ( κ ) ≡ exp (cid:20) − δγ l ln (1 − κ ) (cid:21) , (S33) B l ( κ ) ≡ (cid:112) δγ l /π (1 − κ ) ln (1 − κ ) . (S34)The term A l ( κ ) is a lower cut-off that rapidly saturatesto one as κ increases (see Fig. S1). On the other hand, B l ( κ ) has an intermediate power-law regime (for valuesof κ not too close to 1) and an upper cut-off (for κ closerto 1). This behaviour can be understood by using theexpansion of log(1 − y ) = − (cid:80) ∞ n =1 ( − y ) n /n for | y | < B l ( κ ) = √ δγ l (1 − κ ) ln (1 − κ ) ≈ √ δγ l (1 − κ )( − κ − κ ) ≈ √ δγ l (1 − κ )( κ + O [ κ ]) ≈ √ δγ l ( κ + O [ κ ]) ≈ κ − C l ( κ ) where C l ( κ ) is a cut-off function with a peak at values of κ that increase towards 1 as l increases (see Fig. S1).Putting the pieces together, the right tail of the re-duced degree distribution behaves as Q l ( κ ) ≈ κ − C l ( κ ) (S35)where C l ( κ ) is the cut-off function. This proves our state-ment in the main text, and is confirmed by the numericalsimulations in the bottom panel of Fig. 5. -2 -1 a P ( a ) -2 -1 a -6 -5 -4 -3 -2 -1 P ( a ) , a - P2(a)a -2 w i t hfi t n e ss i n a n e i g hb o u r h oo d o f y : k ( x ) = ( N l ) Z f ( x , y ) ⇢ l ( y ) d y = ( N l ) Z e x y r l ⇡ e l / ( y ) y / d y = ( N l ) ⇣ e p l x ⌘ , ( S ) w h e r e k ( x ) d e n o t e s t h ee x p ec t e dd e g r ee o f a n o d e w i t h fi t n e ss x . I n v e r t i n g , w e find t h a tt h e fi t n e ss o f a n l - n o d e w i t h e x p ec t e dd e g r ee k i s x ( k ) = l l n ✓ N l N l k ◆ , ( S ) w h i c h i m p li e s d d k x ( k ) = l n ⇣ N l N l k ⌘ l ( N l k ) . ( S ) W ec a nu s e t h e a b o v ee x p r e ss i o n s i n o r d e r t oo b t a i n t h e d i s t r i bu t i o n P l ( k ) o f t h ee x p ec t e dd e g r ee s f r o m t h e d i s t r i bu t i o n ⇢ l [ x ( k ) ] o f t h ec o rr e s p o nd i n g fi t n e ss . I nd ee d , s t a r t i n g f r o m t h e f und a m e n t a l e q u a t i o n P l ( k ) d k = ⇢ l [ x ( k ) ] d x ( k )( S ) r e l a t i n g t h e p r o b a b ili t y d i s t r i bu t i o n s o f t h e t w o r a nd o m v a r i a b l e s k a nd x , a ndu s i n g E q s . ( ) , ( S ) a nd ( S ) , w e a rr i v e a tt h ee x p li c i t f o r m o f t h e d i s t r i bu t i o n o f e x - p ec t e dd e g r ee s : P l ( k ) = ⇢ l [ x ( k ) ] d d k x ( k ) = q l ⇡ e x p l l n ⇣ N l N l k ⌘ ( N l k ) l n ⇣ N l N l k ⌘ = q l ⇡ e x p l l n ⇣ k N l ⌘ ( N l ) ⇣ k N l ⌘ l n ⇣ k N l ⌘ ( S ) f o r k , a nd P l ( k ) = t h e r w i s e . W ec a n o b t a i n t h ee x p ec t e d li n k d e n s i t y h D l i v i a t h e c a l c u l a t i o n o f t h e t o t a l e x p ec t e dd e g r ee ,i. e . t w i ce t h e e x p ec t e dnu m b e r o f li n k s , a s h L l i = N l Z N l P l ( k ) k d k = N l ( N l ) ✓ ⇡ Z e t l p / t d t ◆ ( S ) w h e r e w e h a v ec h a n g e d v a r i a b l e s b y i n t r o du c i n g t = l p l og N l N l k . ( S ) - - a . . . . . . P1(a) F I G . S . W e s h o w h e r e h o w t h ee x p o n e n t i a l t e r m i n P ( ) s a t u r a t e s a l r e a d y f o r s m a ll v a l u e o f . T h e i n t e g r a li n E q . ( S ) c a nb ee x p r e ss e d i n t e r m s o f o n e o f t h e M e i j e r - G f un c t i o n s G m , n p , q a ,..., a p b ,..., b q z . T h e r e s u l t i n g e x p ec t e dd e n s i t y c a nb e w r i tt e n a s h D l i = h L l i N l ( N l ) = G , , · / , , l / . ( S ) S . V . T H E S C A L E - F R EE R A N G E W I T H I N V E R S E S Q U A R EE X P O N E N T W e r e w r i t e t h e d i s t r i bu t i o n o f t h e r e du ce dd e g r ee = k / N l ( s ee m a i n t e x t) s h o w n i n E q . ( ) a s Q l ( ) = A l ( ) B l ( )( S ) w h e r e A l ( ) ⌘ e x p l l n ( ) , ( S ) B l ( ) ⌘ p l / ⇡ ( ) l n ( ) . ( S ) T h e t e r m A l ( ) i s a l o w e r c u t - o ↵ t h a t r a p i d l y s a t u r a t e s t oo n e a s i n c r e a s e s ( s ee F i g . S ) . O n t h e o t h e r h a nd , B l ( ) h a s a n i n t e r m e d i a t e p o w e r - l a w r e g i m e ( f o r v a l u e s o f n o tt oo c l o s e t o1 ) a nd a nupp e r c u t - o ↵ ( f o r c l o s e r t o1 ) . T h i s b e h a v i o u r c a nb e und e r s t oo db y u s i n g t h e e x p a n s i o n o f l og ( y ) = P n = ( y ) n / n f o r | y | < : B l ( ) = p l ( ) l n ( ) ⇡ p l ( )( ) ⇡ p l ( )( + O [ ] ) ⇡ p l ( + O [ ] ) ⇡ C l ( ) w h e r e C l ( ) i s a c u t - o ↵ f un c t i o n w i t h a p e a k a t v a l u e s o f t h a t i n c r e a s e t o w a r d s s l i n c r e a s e s ( s ee F i g . S ) . w i t hfi t n e ss i n a n e i g hb o u r h oo d o f y : k ( x ) = ( N l ) Z f ( x , y ) ⇢ l ( y ) d y = ( N l ) Z e x y r l ⇡ e l / ( y ) y / d y = ( N l ) ⇣ e p l x ⌘ , ( S ) w h e r e k ( x ) d e n o t e s t h ee x p ec t e dd e g r ee o f a n o d e w i t h fi t n e ss x . I n v e r t i n g , w e find t h a tt h e fi t n e ss o f a n l - n o d e w i t h e x p ec t e dd e g r ee k i s x ( k ) = l l n ✓ N l N l k ◆ , ( S ) w h i c h i m p li e s d d k x ( k ) = l n ⇣ N l N l k ⌘ l ( N l k ) . ( S ) W ec a nu s e t h e a b o v ee x p r e ss i o n s i n o r d e r t oo b t a i n t h e d i s t r i bu t i o n P l ( k ) o f t h ee x p ec t e dd e g r ee s f r o m t h e d i s t r i bu t i o n ⇢ l [ x ( k ) ] o f t h ec o rr e s p o nd i n g fi t n e ss . I nd ee d , s t a r t i n g f r o m t h e f und a m e n t a l e q u a t i o n P l ( k ) d k = ⇢ l [ x ( k ) ] d x ( k )( S ) r e l a t i n g t h e p r o b a b ili t y d i s t r i bu t i o n s o f t h e t w o r a nd o m v a r i a b l e s k a nd x , a ndu s i n g E q s . ( ) , ( S ) a nd ( S ) , w e a rr i v e a tt h ee x p li c i t f o r m o f t h e d i s t r i bu t i o n o f e x - p ec t e dd e g r ee s : P l ( k ) = ⇢ l [ x ( k ) ] d d k x ( k ) = q l ⇡ e x p l l n ⇣ N l N l k ⌘ ( N l k ) l n ⇣ N l N l k ⌘ = q l ⇡ e x p l l n ⇣ k N l ⌘ ( N l ) ⇣ k N l ⌘ l n ⇣ k N l ⌘ ( S ) f o r k , a nd P l ( k ) = t h e r w i s e . W ec a n o b t a i n t h ee x p ec t e d li n k d e n s i t y h D l i v i a t h e c a l c u l a t i o n o f t h e t o t a l e x p ec t e dd e g r ee ,i. e . t w i ce t h e e x p ec t e dnu m b e r o f li n k s , a s h L l i = N l Z N l P l ( k ) k d k = N l ( N l ) ✓ ⇡ Z e t l p / t d t ◆ ( S ) w h e r e w e h a v ec h a n g e d v a r i a b l e s b y i n t r o du c i n g t = l p l og N l N l k . ( S ) - - a . . . . . . P1(a) F I G . S . W e s h o w h e r e h o w t h ee x p o n e n t i a l t e r m i n P ( ) s a t u r a t e s a l r e a d y f o r s m a ll v a l u e o f . T h e i n t e g r a li n E q . ( S ) c a nb ee x p r e ss e d i n t e r m s o f o n e o f t h e M e i j e r - G f un c t i o n s G m , n p , q a ,..., a p b ,..., b q z . T h e r e s u l t i n g e x p ec t e dd e n s i t y c a nb e w r i tt e n a s h D l i = h L l i N l ( N l ) = G , , · / , , l / . ( S ) S . V . T H E S C A L E - F R EE R A N G E W I T H I N V E R S E S Q U A R EE X P O N E N T W e r e w r i t e t h e d i s t r i bu t i o n o f t h e r e du ce dd e g r ee = k / N l ( s ee m a i n t e x t) s h o w n i n E q . ( ) a s Q l ( ) = A l ( ) B l ( )( S ) w h e r e A l ( ) ⌘ e x p l l n ( ) , ( S ) B l ( ) ⌘ p l / ⇡ ( ) l n ( ) . ( S ) T h e t e r m A l ( ) i s a l o w e r c u t - o ↵ t h a t r a p i d l y s a t u r a t e s t oo n e a s i n c r e a s e s ( s ee F i g . S ) . O n t h e o t h e r h a nd , B l ( ) h a s a n i n t e r m e d i a t e p o w e r - l a w r e g i m e ( f o r v a l u e s o f n o tt oo c l o s e t o1 ) a nd a nupp e r c u t - o ↵ ( f o r c l o s e r t o1 ) . T h i s b e h a v i o u r c a nb e und e r s t oo db y u s i n g t h e e x p a n s i o n o f l og ( y ) = P n = ( y ) n / n f o r | y | < : B l ( ) = p l ( ) l n ( ) ⇡ p l ( )( ) ⇡ p l ( )( + O [ ] ) ⇡ p l ( + O [ ] ) ⇡ C l ( ) w h e r e C l ( ) i s a c u t - o ↵ f un c t i o n w i t h a p e a k a t v a l u e s o f t h a t i n c r e a s e t o w a r d s s l i n c r e a s e s ( s ee F i g . S ) . -2 -1 a -6 -5 -4 -3 -2 -1 P ( a ) , a - P2(a)a -2 FIG. S2. Representation of the term P ( ) as a function of (blue line) and corresponding Taylor approximation ⇠ (red dashed line). Putting the pieces together, the right tail of the re-duced degree distribution behaves as Q l ( ) ⇡ C l ( ) (S35)where C l ( ) is the cut-o↵ function. This proves our state-ment in the main text, and is confirmed by the numericalsimulations in the bottom panel of Fig. 5. -2 -1 a -6 -5 -4 -3 -2 -1 P ( a ) , a - P2(a)a -2 -2 -1 a -6 -5 -4 -3 -2 -1 P ( a ) , a - P2(a)a -2 FIG. S2. Representation of the term P ( ) as a function of (blue line) and corresponding Taylor approximation ⇠ (red dashed line). Putting the pieces together, the right tail of the re-duced degree distribution behaves as Q l ( ) ⇡ C l ( ) (S35)where C l ( ) is the cut-o↵ function. This proves our state-ment in the main text, and is confirmed by the numericalsimulations in the bottom panel of Fig. 5. FIG. S1. The two factors contributing to the cumulative dis-tribution of the rescaled degree κ . Top: lower cut-off function A l ( κ ) defined in Eq. (S33). The function rapidly saturates to A l ( κ ) ≈ κ increases. Bottom: tail function B l ( κ ) de-fined in Eq. (S34). The function behaves as a power law B l ( κ ) ≈ κ − (red dashed line) for a wide range of values for κ and has an ll