A Simple Differential Geometry for Complex Networks
AA Simple Differential Geometry for Complex Networks
Emil Saucan, ∗ Areejit Samal,
2, 3 and J¨urgen Jost
3, 4 Department of Applied Mathematics, ORT Braude College, Karmiel 2161002, Israel The Institute of Mathematical Sciences (IMSc),Homi Bhabha National Institute (HBNI), Chennai 600113 India Max Planck Institute for Mathematics in the Sciences, Leipzig 04103 Germany The Santa Fe Institute, Santa Fe, New Mexico 87501, USA
We introduce new definitions of sectional, Ricci and scalar curvature for networks and their higherdimensional counterparts, derived from two classical notions of curvature for curves in general metricspaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicableto unweighted or weighted and undirected or directed networks, and are more intuitive and easierto compute than other network curvatures. In particular, the proposed curvatures based on theinterpretation of Haantjes definition as geodesic curvature allow us to give a network analogue ofthe classical local Gauss-Bonnet theorem. Furthermore, we propose even simpler and more intuitiveproxies for the Haantjes curvature that allow for even faster and easier computations in large-scalenetworks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures.Lastly, we also investigate the behaviour, both on model and real-world networks, of the curvaturesintroduced herein with more established notions of Ricci curvature and other widely-used networkmeasures.
1. INTRODUCTION
Till recently, mathematical analysis of complex networks was largely based upon combinatorial invariants andmodels [1–5], at the detriment of the geometric approach. Especially, theoretical developments in discrete differentialgeometry leading to notions of curvature [6, 7] for graphs or networks remained largely unexplored until recentlyin network science [8–10]. Noteworthy, one of the widely-used combinatorial measure, the clustering coefficient [1],to characterize complex networks is in fact, a discretization of the classical Gauss curvature [11]. Lately, notions ofnetwork curvature have proven to be an important tool in the analysis of complex networks [8–10, 12]. In particular,Ollivier’s Ricci curvature [6] has been extensively used in the analysis of complex networks in its various avatars[8, 9, 13–15]. Based on Forman’s work [7], another approach towards the introduction of Ricci curvature in the studyof networks was also proposed [10, 16]. Moreover, the two above-mentioned notions of Ricci curvature in severalmodel and real-world networks were compared [12]. However, these two notions of Ricci curvature for networks havecertain drawbacks. Their common denominator is the theoretical advanced apparatus which stands at the base ofboth notions of discrete Ricci curvature for networks. While Ollivier-Ricci curvature is prohibitively hard to computein large networks, Forman-Ricci curvature is extremely simple to compute in large networks [12]. On the other hand,Forman-Ricci curvature is less intuitive than Ollivier-Ricci curvature, as it is based on a discretization of the socalled
Bochner-Weitzenb¨ock formula [17]. These issues may deter the interdisciplinary community of engineers, socialscientists and biologists active in network science from employing these two notions of discrete Ricci curvature in theirresearch.Therefore, it is worthwhile to ask whether it is possible to define more intuitive notions of curvature (specifically,Ricci curvature) for networks which are modelled using the simple framework of graphs. The answer to this quandaryis immediate after the realization that the simplest, more general, geometric notion at our disposal is that of metricspace. Furthermore, unweighted and weighted graphs can be easily endowed with a metric, be it the combinatorialweights, the path distance, the given weights, the Wasserstein metric (also called the Earth mover’s distance) [18], ora metric such as the path degree metric [19, 20] that can be easily obtained from the given weights. For metric spaces,there indeed exist a number of simple and intuitive notions of curvature, and in particular, two definitions of curvatureof curves which go back to Menger [21] and Haantjes [22], respectively. By partially extending ideas previously appliedin the context of imaging and graphics (
P L manifolds in general) [23–25], we show that the definitions of Menger [21]and Haantjes [22] allow us to naturally define Ricci and scalar curvatures for networks and hypernetworks includingunweighted, weighted, undirected and directed ones.The simplest and better known among the two notions of metric curvature is the Menger curvature. Notably, theMenger curvature defined for metric triangles naturally allows us to define scalar and Ricci curvatures for networks, ∗ Correspondence to: [email protected] a r X i v : . [ m a t h . M G ] A p r FIG. 1. The spherical network representing the 1-skeleton of the canonical triangulation with fundamental triangles of thespherical counterpart of the truncated icosidodecahedron (see also Figure 3). hypernetworks, simplicial complexes and clique complexes including unweighted, weighted, undirected and directedones. Furthermore, we show that this approach holds not only for the usual, Euclidean model geometry, but as wellfor hyperbolic and spherical background geometry. We believe this is an important feature of the Menger curvaturedue to the following reason. Currently, while network embedding is commonly considered, such embedding is eitherpurely combinatorial or, at most, equipped with an undescriptive Euclidean geometry which is decoupled from theone of the ambient space, and thus, a vital part of the expressiveness of the embedding is lost.In comparison to the Menger curvature, Haantjes curvature is far less familiar among researchers but is a muchmore flexible notion. This is because Haantjes curvature is not restricted solely to triangles, but rather holds forgeneral metric arcs. In consequence, Haantjes curvature presents two distinct advantages. Firstly, it is applicable toany 2-cell, not just to triangles. Secondly, it can be better used as discrete version of the classical geodesic curvature(of curves on smooth surfaces). In consequence, Haantjes curvature has a clear advantage over Menger curvature asit is applicable to networks or graphs, without any assumption on the background geometry . In fact, not only canit be employed in networks of variable curvature , it can be used to define the curvature of such a discrete space.In effect, this allows us to study the intrinsic , not just the extrinsic , geometry of networks. From a practical viewof applications, the definition of Haantjes curvature allows inference of geometric, highly descriptive properties fromits given characteristic, rather than presume them. From a more theoretical perspective but still with practicalimportance, for instance, while predicting the long time behavior, Haantjes curvature enables us to study networksas geometric spaces in their own right, not just as graphs realized in some largely arbitrary ambient familiar space.The reminder of the paper is structured as follows. In Section 2, we introduce the Menger curvature. In Section3, we introduce Haantjes curvature, its generalizations and extensions, as well as its use in the introduction of localGauss-Bonnet theorem for networks. In Section 4, we provide a brief overview of the embedding properties of themetric curvatures presented here. In Section 5, we present empirical results from analysis of the Menger curvatureand the Haantjes curvature in various model and real-world networks. In Section 6, we conclude with a summary andfuture outlook. In appendix A, we present the mutual relationship between the two metric curvatures, the Mengercurvature and the Haantjes curvature, and also their connections to the classical notion of curvature for curves. Someof the results reported in this manuscript were recently presented in a conference proceeding [26].
2. MENGER CURVATURE
The simplest, most elementary manner, of introducing curvature in metric spaces is due to Menger [21]. Here, onesimply defines the curvature K ( T ) of a triangle T , i.e., a metric triple of points with sides of lengths a, b, c as R ( T ) ,where R ( T ) is the radius of the circle circumscribed to the triangle. An elementary computation yields κ M,E ( T ) = 1 R ( T ) = abc (cid:112) p ( p − a )( p − b )( p − c ) , (1)where p = ( a + b + c )2 denotes the half-perimeter of T .However, there is conceptual problem with the above definition which utilizes the geometry of the Euclidean plane.In the general setting of networks, it is not natural to assume a Euclidean background. This is analogous to the (a) (b) w u e w u ev v v v v v v v v v FIG. 2. Sign convention for (a) directed triangles and (b) directed polygons. geometry of surfaces, where the metric need not be Euclidean, but could be spherical, hyperbolic, or of varying Gausscurvature. For example, embedding networks in hyperbolic plane and space is becoming quite common [27–29].Of course, one may formulate a spherical or hyperbolic analogue of Eq. 1, see e.g. [30]. The spherical version is κ M,S ( T ) = 1tan R ( T ) = (cid:112) sin p sin ( p − a ) sin ( p − b ) sin ( p − c )2 sin a sin b sin c , (2)whereas the hyperbolic version is κ M,H ( T ) = 1tanh R ( T ) = (cid:112) sinh p sinh ( p − a ) sinh ( p − b ) sinh ( p − c )2 sinh a sinh b sinh c . (3)Note that, in the setting of networks, the constant factors “4” and “2”, respectively, appearing in the denominatorsof the above equations are less relevant, and they can be discarded in this context. Remark:
The factors ( p − a ) , ( p − b ) , ( p − c ) in the definition above are, in fact, the so called Gromov products of thevertices of T , which is employed in the definition of Gromov δ -hyperbolicity [31]. Examples of computing Menger curvature with spherical background geometry
1. Let (cid:52)
ABC be a spherical triangle with angles α = β = γ = π on a sphere of radius 1. Then the length of thesides of the triangle, a = b = c = π , and the formula above renders κ M,S ( (cid:52) ABC ) = √ .2. The 1-skeleton of the triangulation of the sphere consisting of the fundamental triangles of a spherical counterpartof an Archimedean polyehdron (see Figure 1) represents a spherical network. The side of the spherical faces istaken to be 2 units. The fundamental triangle of an n -gonal face has angles ( π , π , πn ), n = 4 , ,
10. Hence, theremaining sides of the triangles, thence their Menger curvatures, can be computed using the classical formulasof spherical trigonometry, see e.g. [30].
Examples of computing Menger curvature with hyperbolic background geometry.
1. The hyperbolic plane admits a tessellation with triangles of angles ( π , π , π ). The dual graph represents also atessellation with regular (hyperbolic) squares, hexagons and 14-gones, symbolically denoted as (4 , , Cayley graph of the group of the symmetries of the tessellation above, as generated by thereflections in the sides of the triangles, see e.g. [32]. Assuming that the edge is opposite the angle of measure π , one can compute the remaining sides using the classical formulas of hyperbolic trigonometry, see e.g. [30].2. It is possible to tessellate the hyperbolic space with regular dodecahedra having all the faces with anglesequal to π , see e.g. [33]. Then the fundamental triangle of the faces of these dodecahedra has angles ( π , π , π ).Normalizing the sides of the dodecahedra, such that half of its length be equal to 1, the hyperbolic law of sinuses ,see e.g. [30], gives sinh a = √ · sinh 1sinh π/ for the side opposite the angle of measure π , and sinh a = sinh 1sinh π/ forthe side opposite the right angle. From these the κ M,H ( T ) of the triangle can be readily computed.Hyperbolic geometry is considered better suited to represent the background network geometry as it capturesthe qualitative aspects of networks of exponential growth such as the World Wide Web, and thus, it is used asthe setting for variety of purposes. However, spherical geometry is usually not considered as a model geometry fornetworks because that geometry has finite diameter, hence finite growth. However, spherical networks naturally arisein at least two instances. The first one is that of global communication, where the vertices represent relay stations,satellites, sensors or antennas that are distributed over the geo-sphere or over a thin spherical shell that can, andusually is, modeled as a sphere. The second one is that of brain networks, where the cortex neurons are envisioned,due to the spherical topology of the brain, as being distributed on a sphere or, in some cases, again on a very thin(only a few neurons deep) spherical shell, that can also be viewed as essentially spherical. One can also devise ananalogous, although less explicit formula in spaces of variable curvature, but in network analysis, it is not clear wherethat background curvature should come from. After all, the purpose here is to define curvature, and not take it asgiven.As defined, the Menger curvature is always positive. This may not be desirable, as in geometry, the distinctionbetween positive and negative curvature is important. For directed networks, however, a sign ε ( T ) ∈ {− , , +1 } is naturally attached to a directed triangle T (Figure 2), and the Menger curvature of the directed triangle is thendefined, in a straightforward manner as κ M,O ( T ) = ε ( T ) · κ M ( T ) , (4)where κ M could be the Euclidean, the spherical or the hyperbolic version, accordingly to the given setting. Notethat, since in simplicial complexes, triangles adjacent to an edge represent discrete ( P L ) analogues of 2-dimensionalsections, the Menger curvature of each such triangle T can be naturally interpreted as the sectional curvature of T .We can then define the Menger-Ricci curvature of an edge by averaging as in differential or piecewise linear geometryas κ M ( e ) not = Ric M ( e ) = (cid:88) T e ∼ e κ M ( T e ) , (5)where T e ∼ e denote the triangles adjacent to the edge e , and the Menger-scalar curvature of a vertex is given by κ M ( v ) not = scal M ( v ) = (cid:88) e k ∼ v Ric M ( e k ) = (cid:88) T ∼ v κ M ( T ) , (6)where e k ∼ v and T ∼ v stand for all the edges e k adjacent to the vertex v and all the triangles T having v as avertex, respectively. Remark:
Ric M ( e ) captures, in keeping with the intuition behind κ M ( T ), the geodesic dispersion rate aspect of Riccicurvature. See [12] for a succinct overview of the different aspects of Ricci curvature.As indicated above, we see two drawbacks for Menger curvature as a tool in network analysis. It depends on abackground geometry model, and it naturally applies only to triangles, but not to more general 2-cells. Therefore,we turn, in the next section, to Haantjes curvature, which we find more flexible in applications. (See, however, theremark below.) Remark:
It is possible to prescribe a Menger curvature for general paths as well, in the following manner. Let π = v , v , . . . , v n − , v n be a path, subtended by the chord ( v , v n ), and let v k , ≤ k ≤ n − v k divides the path π into two paths π = v , . . . , v k and π = v k , . . . , v n . Let us denote a = l ( π ) , b = l ( π ) , c = l ( v , v n ). Then one can consider the Menger curvature of the metric triangle (cid:52) v v k v n . Note that it does notdepend on the choice of the vertex v k . Again, the Menger curvature can be computed with the Euclidean, sphericalor hyperbolic flavor, according to the preferred model geometry for the given network. As already noted above, thisapproach has the limitation of prescribing a predefined curvature for the network. On the other hand, this approachallows for the passing to an Alexandrov comparison type of coarse geometry , see e.g. [34].
3. HAANTJES CURVATURE
Haantjes [22] defined metric curvature by comparing the ratio between the length of an arc of curve and that ofthe chord it subtends. More precisely, if c is a curve in a metric space ( X, d ), and p, q, r are points on c , p between q and r , the Haantjes curvature is defined as κ H ( p ) = 24 lim q,r → p l ( (cid:98) qr ) − d ( q, r ) (cid:0) d ( q, r ) (cid:1) , (7) FIG. 3. The truncated icosidodecahedron, another Archimedean polyhedron. This is an Euclidean counterpart of the sphericalpolyhedron in Figure 1. There are three types of edges in this object, and the Forman-Ricci curvature for the square-hexagon,square-decagon and hexagon-decagon edges are equal to √ √ √ √ π π π π u v FIG. 4. Haantjes-Ricci curvature in the direction uv , is defined as Ric H ( uv ) = (cid:80) m K iH,uv = (cid:80) m κ H,uv ( π i ) where κ H,uv ( π i ) = l ( π i ) − l ( π ) l ( π ) i = 1 , , π being the shortest path connecting the vertices u and v . where l ( (cid:98) qr ) denotes the length, in the intrinsic metric induced by d , of the arc (cid:98) qr (see also Appendix below). In thenetwork case, (cid:98) qr is replaced by a path π = v , v , . . . , v n , and the subtending chord by edge e = ( v , v n ). Clearly,the limiting process has no meaning in this discrete case. Furthermore, the normalizing constant “24” which ensuresthat, the limit in the case of smooth planar curves will coincide with the classical notion, is superfluous in this setting.This leads to the following definition of the Haantjes curvature of a simple path π : κ H ( π ) = l ( π ) − l ( v , v n ) l ( v , v n ) , (8)where, if the graph is a metric graph, l ( v , v n ) = d ( v , v n ). In particular, in the case of the combinatorial metric,we obtain that, for path π = v , v , . . . , v n as above, κ H ( π ) = √ n −
1. Note that considering simple paths is not a (a) (b) (c)
FIG. 5. Haantjes-Ricci curvature Ric H for directed networks, endowed with combinatorial weights: (a) Ric H = 6 π − (b) Ric H = 2 π −
1, and (c)
Ric H = 6 π − − √
2. Note that while Forman-Ricci curvature is a counter of triangles in simplicialcomplexes, Haantjes-Ricci curvature represents a counter of all n -gones, since each n -gone contributes a √ n term. restriction, given that a metric arc is, by definition, a simple curve. However, to capture in the discrete context thelocal nature of the Ricci (and scalar) curvature, we shall restrict to paths π such that π ∗ = v , v , . . . , v n , v is an elementary cycle .Clearly, one can extend the above definition of Haantjes curvature to directed paths in the same manner as donefor the Menger curvature in the preceding section, namely κ H,O ( π ) = ε ( π ) · κ H ( π ) , (9)for every directed path π , where ε ∈ {− , , +1 } denotes the direction of path π (see also Section 3.3 below).In a straightforward manner, we can define Haantjes-Ricci curvature and Haantjes-scalar curvature, similar to theMenger curvature, as κ H ( e ) not = Ric H ( e ) = (cid:88) π ∼ e κ H ( π ) , (10)where π ∼ e denote the paths that connect the vertices anchoring the edge e , and κ H ( v ) not = scal H ( v ) = (cid:88) e k ∼ v Ric H ( e k ) , (11)where e k ∼ v stands for all the edges e k adjacent to the vertex v . In the following, we shall however significantlystrengthen the somewhat simplistic definition of the Haantjes-Ricci curvature presented above. Examples of Archimedean (semi-regular) polyhedra. The truncated dodecahedron is a typical Archimedean convex body, hence of positive combinatorial curvatureconcentrated at vertices. However, the two types of edges, the pentagon-hexagon ones and the hexagon-hexagonones, in this object have Forman-Ricci curvature Ric F ( e ) equal to -1 and -2, respectively. In contrast, theHaantjes-Ricci curvature is, by definition, always positive: Ric H ( e ) = 9 for the pentagon-hexagon edges, andRic H ( e ) = 10, for the hexagon-hexagon edges.2. The truncated octahedron is another Archimedean polyhedron. The positivity condition for Forman-Ricci cur-vature for all edges does not hold here as well: Ric F ( e ) = 0 for square-hexagon edges, and Ric F ( e ) = − < H ( e ) = 8 forthe square-hexagon edges, and Ric H ( e ) = 10, for the hexagon-hexagon edges. Let us also note that the edgesof this object represent the Cayley graph of the symmetric group S (4) with respect to the set of generators { τ , τ , τ } , where τ , τ , τ are the transpositions τ = (12) , τ = (23) , τ = (34) (see [32]). Examples of non-convex uniform polyhedra. The tetrahemihexahedron (4 . . .
3) is a simple non-orientable polyhedron, representing a (minimalistic) modelof the real projective plane. As such, it’s Euler characteristic is equal to 1. The Forman-Ricci curvature of eachedge here is equal to 1, and thus, strictly positive, whereas the Haantjes-Ricci curvature is equal to 1 + √ The octahemoctahedron (6 . . .
3) is the only toroidal uniform poyhedron with Euler characteristic 0. TheForman-Ricci curvature of each triangle-hexahedron edge here is equal to -1, whereas the Haantjes-Ricci curva-ture is equal to 3.3.
The nonconvex great rhombicuboctahedron (or quasi-rhombicuboctahedron) (4 . . .
4) is composed of 8 regulartriangles and 18 squares. The triangles are retrograde , and thus, their Haantjes curvature should be taken with a negative sign. Therefore, the Haantjes-Ricci curvature of an triangle-square edge here is √ −
1, and not √ Examples of three-dimensional polyhedral complexes. The Seifert-Weber dodecahedral space is a three manifold obtained by gluing, via of a clockwise full twist,the opposite faces of a regular dodecahedron. Each edge is incident to 5 dodecahedra/pentagonal faces, andhence, the Forman-Ricci curvature of such edges is −
7, thus corresponding to the fact that it is possible to tilethe Hyperbolic space with such dodecahedra, whereas the Haantjes-Ricci curvature is equal to Ric H ( e ) = 5 √ TABLE I. Comparison of undirected curvatures for a number of standard grids (tessellations) of the Euclidean plane and space.Curvature Triangular Square Hexagonal EuclideanType Tessellation Tessellation Tessellation CubulationRic H ( e ) 4 π - 2 4 π - 4 4 π − √ π − √ F,r ( e ) -8 -2 -2 -4Ric F ( e ) -2 0 4 4Ric O ( e ) 1 -1 – - The Poincar´e dodecahedral space is a three manifold obtained by gluing, via of a clockwise full twist, theopposite faces of a regular dodecahedron. Each edge is incident to 3 dodecahedra/2-faces, and hence, theForman-Ricci curvature of such edges is equal to −
1, even though the Poincar´e is the classical homology 3-sphere, and thus, expected to have a positive curvature. On the other hand, the Haantjes-Ricci curvature ofsuch edges is positive, more precisely, Ric H ( e ) = 3 √ Remark:
Among metric curvatures, the Haantjes curvature has several advantanges from the perspective of networkscience applications. Aside from the simplicity of computation in networks, Haantjes-Ricci curvature for an edge inundirected and unweighted (combinatorial) simplicial complexes can be obtained by simply counting the triangles t containing an edge e , that is Ric H ( e ) = (cid:93) { t | t > e } . (12)Moreover, in k -regular undirected and unweighted simplicial complexes where each vertex is incident to precisely k edges, the (augmented) Forman-Ricci curvature of an edge e [12] is given byRic F ( e ) = 4 − (cid:93) { t | t > e } − k , (13)and thus, the formula also reduces to counting the triangles t containing an edge e . While the later formula above isalso dependent on vertex degree k , for any given k the two types of Ricci curvature, Haantjes and Forman, for edgesin undirected combinatorial simplicial complexes both reduce to the counting of triangles adjacent to a given edge.The computation of Haantjes-Ricci curvature above has an additional advantage of not being dependent on k . On theother hand, note that Haantjes-Ricci curvature is always positive while Forman-Ricci curvature is mostly negative.Still, the distributions of the two curvatures, over a network, are likely to strongly correlated. We should, however,note here that Ric F covers further aspects of network geometry that Ric H does not. (However, see also the discussionin the concluding section.) Due to its advantages over Menger curvature, we shall now use Haantjes curvature to provide stronger definitionsof scalar curvature and Ricci curvature of networks. Here, the basic idea is to adapt the local Gauss-Bonnet theorem to this discrete setting. Recall that, in the classical context of smooth surfaces, the theorem states that (cid:90) D KdA + p (cid:88) (cid:90) v i +1 v i k g dl + p (cid:88) ϕ i = 2 πχ ( D ) , (14)where D (cid:39) B is a (simple) region in the surface, having as boundary ∂D a piecewise-smooth curve π , of vertices(i.e., points where ∂D is not smooth) v i , i = 1 , . . . , n , ( v n = v ), ϕ i denotes the external angles of ∂D at the vertex v i , and, K and k g denote the Gaussian and geodesic curvatures, respectively.Let us first note that, in the absence of a background curvature, the very notion of angle is undefinable. Thus, forabstract (non-embedded) cells, there exists no honest notion of angle. Hence, the last term on the left side of aboveEq. 14 has no proper meaning, and thus, can be discarded. Moreover, the distances between non-adjacent verticeson the same cycle (apart from the path metric) are not defined, and thus, the third term on the left side of above Eq.14 also vanishes.We next concentrate on the case of combinatorial (unweighted) networks. For such networks endowed with thecombinatorial metric, the area of each cell is usually taken to be equal to 1. Moreover, one can naturally assume thatthe curvature is constant on each cell, and thus, the first term on the left side of Eq. 14 reduces simply to K . Inaddition, given that D is a 2-cell, we have χ ( D ) = 1. Therefore, in the absence of the definition of an angle, it isnaturally to define K = 2 π − (cid:90) ∂D k g dl . (15)It is tempting to next consider ∂D as being composed of segments (on which k g vanishes), except at the vertices, thusrendering the above expression as K = 2 π − n (cid:88) κ H ( v i ) . (16) Remark
An alternative approach to defining the curvature of cell would be the following: Since in a Euclidean polygon,the sum of the angles equals π ( n − n represents the number of vertices of the polygon, one could replacethe angle sum term in Eq. 14 simply by π ( n − v i − and v i +1 on the same cycle can beimplicitly assumed (apart from the one given by the path metric, which would produce trivial zero curvature at vertex v i ). In fact, in this general case, neither can the arc (path) π = v , v , . . . , v n be truly viewed as smooth. Therefore,we have no choice but to replace the second term on the right side of above Eq. 16 by κ H ( π ), where it should beremembered that π represents the path v , v , . . . , v n of chord e = ( v , v n ).We can now define the Haantjes-sectional curvature of a 2-cell c . Given an edge e = ( u, v ) and a 2-cell c , ∂ c = ( u = v , v , . . . , v n = v ) ( relative to the edge e ∈ ∂ c ), we have K H,e ( c ) = 2 π − κ H,e ( π ) , (17)where π denotes the path v , v , . . . , v n subtended by the chord e = ( v , v n ) (see Figure 4), and κ H,e ( π ) denotes itsrespective Haantjes curvature.Note that the definition above is much more general than the one based on Menger curvature. Indeed, not onlyis it applicable to cells whose boundary has (combinatorial) length greater than three, i.e., not just to triangles, italso does not presume any convexity condition for the cells, even in the case when they are realized in some modelspace, e.g. in R . However, for simplicial complexes endowed with the combinatorial metric, the two notions coincideup to a constant. More precisely, in this case, for any triangle T , κ M ( T ) /κ H ( T ) = √ /
3. In fact, for the case ofsmooth, planar curves, Menger and (unnormalized) Haantjes curvature coincide in the limit, and furthermore, theyagree with the classical concept. However, for networks there is no proper notion of convergence, a fact which allowedus to discard the factor 24 in the original definition (Eq. 7) of Haantjes curvature.We can now define, analogous to Eq. 6, the
Haantjes-Ricci curvature of an edge e asRic H ( e ) = (cid:88) c ∼ e K H,e ( c ) = (cid:88) c ∼ e (2 π − κ H,e ( π )) , (18)where the sum is taken over all the 2-cells c adjacent to e . See Figure 5 for examples of computation of Haantjes-Riccicurvature in networks, in the directed case. See Table 1 for the comparison on a number of (undirected) standardplanar and spatial grids of the various types of Ricci curvature at our disposal. In this subsection, we return to the general case of weighted graphs. Firstly, note that it is not reasonable to attacharea 1 to every 2-cell in such graphs. However, as discussed in [25, 35], it is possible to endow cells in an abstractweighted graph with weights that are both derived from the original ones and have a geometric content. For instance,in the case of unweighted social or biological networks, endowed with the combinatorial metric, one can designateto each face, instead of the canonical combinatorial weight equal to 1, a weight that penalizes the faces with more Note that the field assigned to the hexagonal tilling for Ollivier-Ricci curvature in Table 1 is marked as “–”, since in this case theOllivier-Ricci curvature is not applicable, see [55]. edges, and thus, reflecting the weaker mutual connections between the vertices of such a face. Thus, it is possible toderive a proper local Gauss-Bonnet formula for such general networks, in a manner that still retains the given data,yet captures the geometric meaning of area, volume, etc. Thus, when considering any such geometric weight w g ( c )of a 2-cell c , the appropriate form of the first term on the left side of Eq. 14 becomes Kw g ( c ) , and the fitting form of Eq. 17 K H,e ( c ) = − w g ( c ) (2 π − κ H,e ( π )) . (19)Before passing to the problem of extending the above definition to the case of general weights, let us note that theobservations above regarding Menger curvature for directed networks apply also to Haantjes curvature, after properlyextending the notion to directed 1-cycles of any length and not just to directed triangles (see Figure 2). Again, as forthe Menger curvature, considering directed networks actually simplifies the problem, in the sense that it allows forvariable curvature (and not just one with constant sign). For general edge weights, we have the problem that the totalweight w ( π ) of a path π = v , v , . . . , v n is not necessarily smaller than the weight of its subtending chord e = ( v , v n ) , thus Haantjes’ definition cannot be applied. However, we can turn this to our own advantage by reversing the rolesof w ( π ) and w ( v , v n ) in the definition of the Haantjes curvature and assigning a minus sign to the curvature of cyclesfor which this occurs. Thus, this approach actually allows us to define a variable sign Haantjes curvature of cycles(hence, a Ricci curvature as well), even if the given network is not a naturally directed one.Note that the case when w ( v , v , . . . , v n ) = w ( v , v n ), i.e., that of zero curvature of the 2-cell c with ∂ c = v , v , . . . , v n , v straightforwardly corresponds to the splitting case for the path metric induced by the weights w ( v i , v i +1 ).Indeed, the method suggested above reduces to the use of the path metric, in most of the cases. One can alwayspass to the path metric and apply to it the Haantjes curvature. Beyond the complications that this might induce incertain cases, it is, in our view, less general, at least from a theoretical viewpoint, since it necessitates the passage toa metric. However, in the case of most general weights, i.e. both vertex and edge weights, one has to pass to a metric.We find the path degree metric (see e.g. [20]) especially alluring as it is both simple and has the capacity to capture,in the discrete context, essential geometric properties of Riemannian metrics. However, we also refer the reader to[36] for an ad hoc metric devised precisely for use on graphs in tandem with Haantjes curvature. Note that Eqs. 7 and 8 are meaningful not only for a single edge, we can consider any two vertices u, v that canbe connected by a path π . Among the simple paths π , . . . , π m connecting the vertices, the shortest one, i.e., the onefor which l ( π i ) = min { l ( π ) , . . . , l ( π m ) } is attended represents the metric segment of ends u and v . Therefore, givenany two such vertices, we can define the Haantjes-Ricci curvature in the direction uv to beRic H ( uv ) = m (cid:88) K iH,uv = m (cid:88) κ H,uv ( π i ) (20)where K iH,uv denotes the Haantjes-Ricci curvature of the cell c i , where ∂ c i = π i π − , relative to the direction uv , andwhere κ H ( π i ) = l ( π i ) − l ( π ) l ( π ) , (21)and where the paths π , . . . , π m satisfy the condition that π i π − is an elementary cycle . This represents a localitycondition in the network setting. Remark:
Any of the paths π i above can be viewed, according to Stone [37] as a variation of the geodesic uv . Thus, inthe setting of metric measure spaces [38], and for P L manifolds [37], Eq. 20 connects in the network context
Jacobifields to Ricci curvature as in the classical case, see e.g. [17].We conclude this section by noting that both the version of the curvature for directed networks and weightednetworks, can be extended, mutatis mutandis , to this generalized definition. We suggest the name strong local metrics for those sets of positive weights that satisfy the generalized triangle inequality w ( v , v , . . . , v n , v n +1 ) < w ( v , v n ), for any elementary 1-cycle v , v , . . . , v n , v . Remark:
For the case of networks and simplicial complexes, it is useful to note that Eq. 8 for a triangle T = T ( uvw )reduces to κ H ( T ) = d ( u, v ) + d ( v, w ) − d ( u, w )( d ( u, w )) . (22)Thus, Haantjes curvature of triangles is closely related to two other measures, namely the excess exc( T ) and aspectratio ar( T ), that are defined as follows:exc( T ) = max v ∈{ u,v,w } ( d ( u, v ) + d ( v, w ) − d ( u, w )) , (23)and ar( T ) = exc ( T ) d ( T ) , (24)where d ( T ) denotes the diameter of a triangle T = T ( uvw ). There are strong connections between the excess, aspectratio, and curvature. In particular, for the normalized Haantjes curvature introduced above, we have the followingrelation between the three notions: κ H ( T ( v )) = e ( T ( v )) d , (25)that is κ H ( T ( v )) = (cid:112) ar( T ( v )) d . (26)Since the factor d has the role of ensuring that, in the limit, the curvature of a triangle will have the dimensionalityof the curvature at a point of a planar curve, the aspect ratio can be viewed as a (skewed), unnormalized version ofthe curvature, and Haantjes curvature can be viewed as a scaled version of excess. Thus, curvature can be replacedby these surrogates, as the notion of scale is not of true import in many aspects of network characterization. Also,for the global understanding of the shape of networks, it is useful to compute, as is common in the manifold context,the maximal excess and minimal aspect ratio over all triangles in the network. Remark:
Note that Eqs. 22 and 23 above again contain Gromov products [31], that already appeared in the definitionof Menger curvature.
4. EMBEDDING PROPERTIES
As noted in the Introduction, while studying complex networks, they are often considered for convenience to beequipped with the combinatorial metric, or they are considered to be topologically embedded in some model spaceand endowed with the induced ( extrinsic ) metric, e.g. the hyperbolic one. However, at least in many instances,a more realistic approach would be to consider the networks to be endowed with an intrinsic metric, obtained viasome expressive (i.e., essential properties preserving) manner, from the weights assigned to vertices and edges in thenetwork. In this case, a natural question to ask is whether a (topological) embedding of the network into an ambientspace will preserve the metric, or in other words, whether such an embedding is an isometric one.Evidently, Haantjes-Ricci curvature, due to its purely metric definition, is geometric embedding invariant : Anisometric embedding will also preserve the curvature as is the case for Menger-Ricci curvature. Moreover, in thecase where face weights are combinatorial, the sectional curvatures of faces, and hence, the Haantjes-Ricci curvaturesare dependent solely on the metric structure of the edges, that is, Ric H is a strong geometric embedding invariant therefore face areas or weights are also preserved by an isometric embedding. However, in the case where face weightsare general, the Haantjes-Ricci curvature is dependent on the face weights, as a consequence of Eq. 19 above, and wehave the following result for general weighted networks. Proposition 4.1.
Let ( N , W ) be a weighted network with general weights. Then, the Haantjes-Ricci curvature Ric H is a geometric embedding invariant, however, not a strong geometric embedding one. Remark:
Any topological embedding that preserves edge and 2-face cell weights is a strong geometric embedding.In particular, for combinatorial (unweighted) networks, the Haantjes-Ricci curvature Ric H is a strong geometricembedding invariant.1 F r e qu e n cy Menger-Ricci curvature
0 0.2 0.4 0.6 0.8 1 1.2 F r e qu e n cy F r e qu e n cy (a) ER network (b) WS network(c) BA network FIG. 6. Distribution of Menger-Ricci curvature in model networks. (a)
ER network, (b)
WS network, and (c)
BA networkwith 1000 vertices and average degree 4.
5. APPLICATION TO MODEL AND REAL-WORLD NETWORKS
In this section, we explore the two notions of metric curvature introduced herein, in different types of model networksand real-world complex networks. Furthermore, we compare the two notions of metric curvature with existing networkmeasures (including discrete notions of Ricci curvature) in both model and real-world complex networks.
For this empirical exploration, we considered 3 models of undirected and unweighted complex networks, 4 undirectedand unweighted real networks, and 2 directed and unweighted real networks.The 3 models of undirected and unweighted complex networks considered here include the Erd¨os-R´enyi (ER) model2 (a) ER network(b) WS network(c) BA network F r e qu e n cy Haantjes-Ricci curvature
200 400 600 800 1000 1200 1400 F r e qu e n cy
200 400 600 800 1000 1200 F r e qu e n cy
200 400 600 800 1000 1200 1400 1600 1800
FIG. 7. Distribution of Haantjes-Ricci curvature in model networks. (a)
ER network, (b)
WS network, and (c)
BA networkwith 1000 vertices and average degree 4. [39], the Watts-Strogatz (WS) model [1] and the Barab´asi-Albert (BA) model [2]. The ER model generates anensemble of random graphs G ( n, p ) where n is the number of vertices and p is the probability that any pair of verticesare connected via an edge in the network. The WS model generates small-world graphs with high clustering andsmall average path length. The parameters of the WS model are the number n of vertices, the number k of nearestneighbours to which each vertex is connected in the initial network, and the rewiring probability β of edges in theinitial network. The BA model generates scale-free graphs with power-law degree distribution. The parameters of theBA model are the number n of vertices in the final network, the number m of vertices in the initial network, andthe number m of existing vertices to which a new vertex is connected to at each step of this growing network model.Notably, the BA model uses a preferential attachment scheme whereby high-degree vertices have a higher chance ofacquiring new edges than low-degree vertices at each step of this growing network model.The 4 undirected and unweighted real-world networks considered here are as follows. The Karate club networkconsists of 34 vertices representing members of the club, and 78 edges representing ties between pairs of members ofthe club, and this dataset was collected by Zachary in 1977 [40]. The Euro road network [41] consists of 1174 verticesrepresenting cities in Europe, and 1417 edges representing roads in the international E-road network linking differentcities. The Yeast protein interaction network [42] consists of 1870 vertices representing proteins in Saccharomycescerevisiae , and 2277 edges representing interactions between pairs of proteins. The US Power Grid network [43]consists of 4941 vertices representing generators or transformers or substations in USA, and 6594 edges representingpower supply lines linking them. The 2 directed and unweighted real-world networks considered here are as follows.3 (a) Karate Club(b) Euro Road(c) Yeast Protein Interaction F r e qu e n cy F r e qu e n cy
100 200 300 400 500 600 700 800 900 1000 F r e qu e n cy
500 1000 1500 2000 (d) US Power Grid F r e qu e n cy Haantjes-Ricci curvature
FIG. 8. Distribution of Haantjes-Ricci curvature in undirected real networks. (a)
Karate club, (b)
Euro road, (c)
Yeast proteininteraction, and (d)
US Power Grid.
The Air traffic control network [44] consists of 1226 vertices representing airports and 2613 directed edges representingpreferred routes between airports. The
E. coli transcriptional regulatory network (TRN) [45] consists of 3073 verticesrepresenting genes and 7853 directed edges representing control of target gene expression through transcription factors.4 (a) Air traffic control network(b)
E. coli transcriptional regulatory network F r e qu e n cy Haantjes-Ricci curvature
200 400 600 800 1000 1200 1400 1600 1800 F r e qu e n cy
500 1000 1500 2000 2500 3000 3500 4000 4500
FIG. 9. Distribution of Haantjes-Ricci curvature in directed real networks. (a)
Air Traffic Control, and (b)
E. coli tran-scriptional regulatory network. Note that, in this case of directed networks, the Haantjes-Ricci curvature of edges can benegative.
We have investigated the distribution of the two notions of metric curvature, namely, Menger-Ricci curvatureand Haantjes-Ricci curvature in model networks considered here. Note that the Menger-Ricci curvature of edges inconsidered networks was computed using Eq. 6 with Euclidean background geometry. In Figure 6, we display thedistribution of Menger-Ricci curvature of edges in ER, WS and BA networks with 1000 vertices and average degree4. From the figure, it is seen that the distribution of Menger-Ricci curvature of edges is broader in BA networks incomparison to WS networks which in turn is broader in comparison to ER networks, and this trend is preserved evenwhen one compares ER, WS and BA networks with 1000 vertices and average degree 6 or 8 (data not shown).Note that the Haantjes-Ricci curvature of edges in considered networks was computed using Eq. 10. Due tocomputational constraints, we only consider simple paths π i of length ≤ We next investigated the correlation between the two notions of metric curvature, namely, Menger-Ricci curvatureand Haantjes-Ricci curvature in model and real networks considered here. Recall that, due to computational con-straints, the computation of Haantjes-Ricci curvature of edges in considered networks using Eq. 10 was restricted tosimple paths π i of maximum length ≤ TABLE II. Correlation between Menger-Ricci (MR) curvature and Haantjes-Ricci (HR) curvature in model and real networks.
Network MR versus HR curvature computed usingpaths of maximum length up to2 3 4 5Undirected model networks
ER with average degree 4 1 0.31 0.14 0.08ER with average degree 6 1 0.27 0.14 0.09ER with average degree 8 1 0.24 0.12 0.11WS with average degree 4 1 0.75 0.55 0.38WS with average degree 6 1 0.77 0.63 0.44WS with average degree 8 1 0.82 0.71 0.52BA with average degree 4 1 0.59 0.52 0.51BA with average degree 6 1 0.71 0.71 0.71BA with average degree 8 1 0.88 0.87 0.86
Undirected real networks
Karate Club 1 0.69 0.44 0.27Euro Road 1 0.51 0.37 0.32Yeast Protein Interaction 1 0.83 0.74 0.64US Power Grid 1 0.82 0.67 0.58
Directed real networks
Air Traffic Control 1 0.53 0.39 0.28
E. coli
TRN 1 0.83 0.75 0.71 more, we have also studied the correlation between Menger-Ricci curvature and Haantjes-Ricci curvature in modeland real networks as a function of the maximum length of simple paths π i that are included in Eq. 10 while computingthe Haantjes-Ricci curvature.In Table 2, we report the correlation between Menger-Ricci and Haantjes-Ricci curvature in model and real networksanalyzed here. As expected, the Menger-Ricci curvature is perfectly correlated with Haantjes-Ricci curvature when thecomputation is restricted to paths of maximum length up to 2 or triangles (Table 2). However, the positive correlationbetween Menger-Ricci and Haantjes-Ricci curvature decreases with increase in the maximum length of paths accountedin the computation (Table 2). Specifically, Menger-Ricci and Haantjes-Ricci curvatures have minimal correlation inER networks and moderate correlation in WS networks when paths of maximum length up to 5 are accounted inthe computation. In contrast, there is high positive correlation between Menger-Ricci and Haantjes-Ricci curvaturesin BA networks even when paths of maximum length up to 5 are accounted in the computation (Table 2). In realnetworks, we find moderate to high positive correlation between Menger-Ricci and Haantjes-Ricci curvatures evenwhen paths of maximum length up to 5 are accounted in the computation (Table 2). We next investigated the correlation of the two notions of metric curvature, Menger-Ricci and Haantjes-Ricci, withother network measures in model and real networks considered here. Specifically, we have studied the correlationwith two other notions of Ricci curvature, (augmented) Forman-Ricci curvature and Ollivier-Ricci curvature [12,46], in networks. In Table 3, we report the correlation between augmented Forman-Ricci curvature and Menger-Ricci or Haantjes-Ricci curvature in model and real networks analyzed here. We find no consistent trend in thecorrelation between augmented Forman-Ricci and Menger-Ricci curvature, or augmented Forman-Ricci and Haantjes-Ricci curvature in model and real networks analyzed here (Table 3). It is seen that there is high negative correlationbetween augmented Forman-Ricci and Haantjes-Ricci curvature in ER and BA networks, while there is high positivecorrelation between augmented Forman-Ricci and Menger-Ricci curvature in WS networks (Table 3). In Table 3, wealso report the correlation between Ollivier-Ricci curvature and Menger-Ricci or Haantjes-Ricci curvature in model andreal networks analyzed here. We again find no consistent trend in the correlation between Ollivier-Ricci and Menger-Ricci curvature, or Ollivier-Ricci and Haantjes-Ricci curvature in model and real networks analyzed here (Table 3).Importantly, in recent work [12, 46], it was shown that augmented Forman-Ricci and Ollivier-Ricci curvature have high6
TABLE III. Comparison of augmented Forman-Ricci (AFR) curvature and Ollivier-Ricci (OR) curvature with Menger-Ricci(MR) curvature and Haantjes-Ricci (HR) curvature of edges in model and real networks. Here, HR curvature was computedby accounting for paths of maximum length up to 5 in Eq. 10.
Network AFR curvature OR curvatureversus versusMR HR MR HRUndirected model networks
ER with average degree 4 0.04 -0.55 0.08 -0.30ER with average degree 6 0.05 -0.77 0.22 -0.31ER with average degree 8 0.06 -0.86 0.29 -0.06WS with average degree 4 0.52 0.02 0.67 0.32WS with average degree 6 0.65 0.04 0.78 0.46WS with average degree 8 0.70 0.06 0.80 0.52BA with average degree 4 -0.37 -0.60 -0.10 -0.36BA with average degree 6 -0.44 -0.69 0.03 -0.13BA with average degree 8 -0.56 -0.69 0.17 0.09
Undirected real networks
Karate Club 0.45 -0.22 0.13 -0.42Euro Road 0.02 -0.52 0.02 -0.25Yeast Protein Interaction 0.12 -0.04 0.07 -0.10US Power Grid 0.11 -0.07 0.16 0.03
Directed real networks
Air Traffic Control 0.10 0.13 -0.13 -0.02
E. coli
TRN 0.08 0.04 -0.06 -0.07TABLE IV. Comparison of edge betweenness centrality with Menger-Ricci (MR) curvature and Haantjes-Ricci (HR) curvatureof edges in model and real networks. Here, HR curvature was computed by accounting for paths of maximum length up to 5in Eq. 10.
Network Edge betweennessversusMR HRUndirected model networks
ER with average degree 4 -0.03 0.35ER with average degree 6 -0.05 0.51ER with average degree 8 -0.06 0.56WS with average degree 4 -0.40 -0.20WS with average degree 6 -0.56 -0.24WS with average degree 8 -0.64 -0.27BA with average degree 4 0.38 0.83BA with average degree 6 0.52 0.81BA with average degree 8 0.62 0.76
Undirected real networks
Karate Club -0.20 0.27Euro Road -0.02 0.04Yeast Protein Interaction -0.07 0.05US Power Grid -0.03 -0.02
Directed real networks
Air Traffic Control -0.06 -0.13
E. coli
TRN -0.08 -0.06
6. CONCLUSIONS AND FUTURE OUTLOOK
In previous work, notions of curvature for networks have been proposed, notably, Ollivier-Ricci curvature [6] andForman-Ricci curvature [7]. Ollivier-Ricci curvature is cumbersome to compute in large networks while Forman-Ricci curvature is much less intuitive. Therefore, in search of simpler and more intuitive notions of curvature fornetworks and their higher-dimensional generalizations including simplicial and clique complexes, we have adaptedhere classical metric curvatures of curves, namely, introduced by Menger [21] and Haantjes [22], for analysis ofcomplex networks. In particular, we are able to define expressive metric Ricci curvatures for networks, both weightedand unweighted. Moreover, the simple yet elegant definitions of the metric curvatures introduced here are bothcomputationally efficient (especially, those based on Menger curvature) and extremely versatile (especially, thosederived from Haantjes curvature) from the perspective of application to complex networks. In fact, for the analysis ofcombinatorial (unweighted) polyhedral or simplicial complexes, the definitions of Haantjes-Ricci curvature is shownhere to be more expressive than the (augmented) Forman-Ricci curvature as the Haantjes-Ricci curvature accountsfor general n -gones and not only triangles like in the case of (augmented) Forman-Ricci curvature.Previously proposed metric curvature [49] based on the Wald metric curvature enables easy derivation of convergenceresults as well as proofs of theoretical results, such as a
P L version of the classical Bonnet-Myers theorem [24].Unfortunately, the metric curvatures prosposed in [49] are computationally expensive, and thus, impractical foranalysis of large networks. This is in sharp contrast to the simplicity and efficiency of the metric curvatures fornetworks proposed in the present article.We have also investigated model and real-world networks using the two notions of metric curvature introducedherein. It is seen that the distribution of Menger-Ricci curvature and Haantjes-Ricci curvature of edges is broaderin scale-free BA networks in comparison to random ER networks or small-world WS networks. We also find apositive correlation between Menger-Ricci and Haantjes-Ricci curvature of edges in model and real-world networks,however, this correlation decreases with the increase in the maximum length of paths accounted in Eq. 10 forthe computation of Haantjes-Ricci curvature. Thereafter, we have compared the Menger-Ricci or Haantjes-Riccicurvature with augmented Forman-Ricci or Ollivier-Ricci curvature in model and real networks, and no consistenttrend in the correlation between Menger-Ricci or Haantjes-Ricci curvature with augmented Forman-Ricci or Ollivier-Ricci curvature was found in analyzed networks. Lastly, we also find no consistent trend in the correlation betweenedge betweenness centrality and Menger-Ricci or Haantjes-Ricci curvature in analyzed networks. Our empirical resultssuggest that the metric curvatures introduced herein capture different aspects of the network organization in contrastto previously proposed edge-centric measures, such as Forman-Ricci curvature, Ollivier-Ricci curvature and edgebetweenness centrality, in complex networks. For instance, for combinatorial networks of bounded vertex degree,and in particular for finite networks, large Haantjes-Ricci curvature implies the existence of (arbitrarily) long simplecycles. Thus, the Haantjes-Ricci curvature, even if defined as a local invariant, has the potential of shedding light onthe large scale topological structure of the network.From a network application perspective, a natural extension is to develop algorithms based on Ricci curvaturesintroduced herein to detect clusters, modules or communities in real-world networks, and thereafter, compare theresults obtained with algorithms for community detection in networks based on other notions of network curvature[14, 36, 50]. Furthermore, another interesting exploratrion would be to investigate the correlation between the metricnotions of curvature introduced herein and hyperbolic embeddings of networks. In particular, it is worthwhile toinvestigate the extent to which the curvatures can predict the values obtained using the inferred hyperbolic distancesamong vertices in embeddings considered in previous work [51], and also study the extent to which these values are inconcordance with the curvatures measured on the given network. Among the theoretical problems naturally arisingfrom this work, one would like to first and foremost prove such results as a fitting analogue of the fundamental globalGauss-Bonnet theorem (which has important consequences in the study of long time evolution of networks [52]), as8well as a fitting Bonnet-Myers theorem.
Acknowledgement
The authors would like to thank the anonymous reviewers of the conference proceeding [26] where some of theseresults were first reported. AS would like to thank Max Planck Society, Germany, for the award of a Max PlanckPartner Group in Mathematical Biology.
Appendix A: Metric curvatures and classical curvature of curves
Both Menger curvature [21] and Haantjes curvature [22] were devised as generalizations to metric spaces of theclassical notion of curvature for smooth plane curves. Therefore, it is only natural to ask whether they representgood approximations of the classical invariant. Indeed, as expected, this turns out to be the case for both metriccurvatures. To state these results more formally we need the following definition:
Definition A.1
Let (
M, d ) be a metric space and let p ∈ M be an accumulation point. We say that M has Mengercurvature κ M ( p ) at p iff for any ε >
0, there exists δ >
0, such that for any triple of points p , p , p , satisfying d ( p, p i ) < δ, i = 1 , ,
3, the following inequality holds: | K M ( p , p , p ) − κ M ( p ) | < ε .While the corresponding definition for Haantjes curvature was essentially introduced in Eq. 7, we again state thesame more formally here: Definition A.2
Let (
M, d ) be a metric space and let c : I = [0 , ∼ → M be a homeomorphism, and let p, q, r ∈ c ( I ) , q, r (cid:54) = p . Denote by (cid:98) qr the arc of c ( I ) between q and r , and by qr line segment from q to r . We say that thecurve c = c ( I ) has Haantjes (or
Finsler-Haantjes ) curvature κ H ( p ) at the point p iff: κ H ( p ) = 24 lim q,r → p l ( (cid:98) qr ) − d ( q, r ) (cid:0) l ( (cid:98) qr ) (cid:1) , (A1)where l ( (cid:98) qr ) denotes the length, in the intrinsic metric induced by d , of (cid:98) qr . Note that the above definition is not theprecise one used in this work.For points or arcs where Haantjes curvature exists, we have l ( (cid:98) qr ) d ( q,r ) → d ( q, r ) → κ H can be defined by (see e.g. [53]) κ H ( p ) = 24 lim q,r → p l ( (cid:98) qr ) − d ( q, r ) (cid:0) d ( q, r )) (cid:1) . (A2)In the setting of graphs or networks, we prefer the above alternative form of the definition of Haantjes curvaturewhich is even more intuitive and simpler to use.As expected, Haantjes curvature for smooth curves in the Euclidean plane (or space) coincides with the standard(differential) notion. More precisely, we have the following result, see [22]. Theorem A.3
Let c ∈ C be smooth curve in R and let p ∈ c be a regular point. Then the metric curvature κ H ( p )exists and equals the classical curvature of γ at p .A similar result also holds for Menger curvature. However, we will arrive at the result in an indirect fashion whichwill enable us to establish the connection between the two types of metric curvature used here. Firstly, note thatalthough the Haantjes curvature exhibits additional flexibility in the network context in comparison to the Mengercurvature, the formal definition of Haantjes curvature in the setting of metric spaces is a more restricted notion thanthe Menger curvature, since it can be employed only for rectifiable curves. However, the following theorem due to[54] holds: Theorem A.4
Let (
X, d ) be a metric continuum, and consider p ∈ X . If κ M ( p ) exists, then X is a rectifiable arc ina neighbourhood of p . Corollary A.5
Let (
X, d ) be a metric arc. If κ M ( p ) exists at all points p ∈ X , then X is rectifiable.In consequence, the existence of one of the considered metric curvatures implies the existence of the other. In fact,the two definitions coincide whenever they both are applicable:9 Theorem A.6 [22] Let c be a rectifiable arc in a metric space ( M, d ), and let p ∈ c . If κ M and κ H exist, then theyare equal. [1] D. J. Watts and S. H. Strogatz, Nature , 440 (1998).[2] A. L. Barab´asi and R. Albert, Science , 509 (1999).[3] R. Albert and A. L. Barab´asi, Reviews of Modern Physics , 47 (2002).[4] M. E. J. Newman, Networks: An Introduction (Oxford University Press, 2010).[5] S. N. Dorogovtsev and J. F. F. Mendes,
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