Edge-based analysis of networks: Curvatures of graphs and hypergraphs
Marzieh Eidi, Amirhossein Farzam, Wilmer Leal, Areejit Samal, Jürgen Jost
EEdge-based analysis of networks:Curvatures of graphs and hypergraphs
Marzieh Eidi, Amirhossein Farzam, Wilmer Leal,
1, 2
Areejit Samal,
3, 1 and J¨urgen Jost
1, 4, ∗ Max Planck Institute for Mathematics in the Sciences, Leipzig 04103 Germany Bioinformatics Group, Department of Computer Science, Universit¨at Leipzig, 04107 Leipzig, Germany The Institute of Mathematical Sciences (IMSc),Homi Bhabha National Institute (HBNI), Chennai 600113 India The Santa Fe Institute, Santa Fe, New Mexico 87501 USA
The relations, rather than the elements, constitute the structure of networks. We therefore developa systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is basedon structural properties of (hyper)edges, instead of vertices. For that purpose, we utilize so-callednetwork curvatures. These curvatures quantify the local structural properties of (hyper)edges, thatis, how, and how well, they are connected to others. In the case of directed networks, they assessthe input they receive and the output they produce, and relations between them. With those tools,we can investigate biological networks. As examples, we apply our methods here to protein-proteininteraction, transcriptional regulatory and metabolic networks.
1. INTRODUCTION
A central paradigm of structuralism [1, 2] is the analysis of structural relations regardless of the identity of theelements involved. That is, a structure is conceived in terms of the relations between elements. One wants tounderstand the types of relations, rather than the nature of the elements. This paradigm is obviously also fundamentalfor the analysis of empirical networks, be they from the biological sciences or other domains. Such an analysis thenagain abstracts from the specific content of the elements and concentrates on the formal relations between them. Inthat manner, one can both find universal features that hold across a wide range of networks from different domains,and properties that are specific to particular empirical domains.For that purpose, many different measures have been developed. Some of these measures, like for instance theassortativity (see for instance [3, 4]), are of a global nature, that is, associate some number to the entire network.Such a number is usually an average or perhaps, like the diameter of a network, an extremum of locally measuredquantities. In any case, the basis for such global measures is to first develop local measures. For a more refinedanalysis, one can then also look at the statistics of those local measures, instead of lumping them together in a singlenumber (for instance [5, 6] for assortativity).Some of these local measures require global computations in the network; for instance, for computing the diameter,one needs to evaluate the distances between any two elements. Therefore, some of these measures are difficult toevaluate in practice for networks of more than a moderate size. Others, including those that we shall concentrate onin this contribution, require only local computations and can be very easy to evaluate.Now, somewhat surprisingly in view of the above structuralist paradigm, many of the local measures assign numbersto the elements of the network, rather than directly to its relations. The most basic one here is the degree of anelement, the number of relations that it participates in. More global measures for instance evaluate the robustness ofthe network in terms of how many or which elements need to be eliminated in order to disconnect the network. Seefor instance [7, 8].In this situation, we and our collaborators have developed the research paradigm of a relation based analysis ofnetworks (for instance [6, 9–17]. That is, we evaluate relations and associate measures to them whose statistics acrossthe network then can provide structural insight.There is another shortcoming of much of traditional network analysis. It tries to represent all structures as graphs,that is, considers only pairwise relations. For instance, a relation between three elements is simply broken up intothree pairwise relations. That may, however, suppress some important structural insight. Take the example ofscientific collaborations. From preprint repositories in the internet, it is easy to extract patterns of collaborationsfrom coauthorships between authors. There are some single author papers, but of more interest are those written byseveral authors. There may be more than two authors involved in some paper, say
A, B , and C . Of course, one couldreduce it to pairwise relations and say that any two of them are coauthors. But there may be more structure. For ∗ [email protected] a r X i v : . [ c s . D M ] O c t instance, there may also exist a two-author paper by A and B , no such paper between A and C , and a paper of B and C with two other authors D and E . This is obviously not captured by the pairwise relations, and for a moreadequate model of the structure of scientific collaborations, we should rather use a hypergraph instead of a simplegraph. In a hypergraph, a hyperedge can connect any number of elements. See for instance [18–22]. In computerscience, directed hypergraphs are also known as Petri nets [23, 24]. They were originally proposed by Petri as modelsof chemical reactions. Over the years, while not as widely employed as graphs, they have found applications in manyfields, for instance recently as models of coupled dynamics in statistical physics [25–27], of social contagion [28] andfor knowledge representation in natural language processing [29].In this contribution, we shall summarize relation-based measures both for graphs, that being the simplest case, andfor hypergraphs.
2. THE IDEA OF CURVATURES
The name curvature derives from its origin in differential geometry. Originally, curvature was an infinitesimalquantity, obtained by taking second derivatives of functions describing shapes of smooth objects, like curves orsurfaces. In Riemannian geometry, curvatures obtained a deeper conceptual significance, as tensors encoding thegeometric invariants of Riemannian metrics of smooth manifolds [30]. In particular, the Ricci tensor is fundamentalnot only in Einstein’s theory of general relativity and in elementary particle physics (the Calabi-Yau manifolds of stringtheory, for instance, are characterized by the vanishing of the Ricci tensor), but it also permeates much of modernresearch in Riemannian geometry. While Ricci curvature in Riemannian geometry again is computed infinitesimally,by taking second derivatives of the metric tensor, it essentially encodes local property, like the average divergence ofgeodesics or the growth of the volume of balls as a function of their radii. Moreover, Bochner type identities link itto other important geometric quantities, like the first eigenvalue of the Laplace operator. See for instance [31] for asurvey.Since such objects and properties are also meaningful and important in metric spaces that are more general thanRiemannian manifolds, alternative definitions of Ricci curvatures have been proposed that are formulated in terms oflocal quantities and no longer depend on taking derivatives. Several of these definitions turned to be also meaning-and useful for graphs, and we have extended them to hypergraphs and are exploring their properties. Here, we shallnot recount the history in detail, but rather systematically develop a conceptual approach that is in line with theparadigm of structuralism described at the beginning. We only note the curious fact that these concepts, althoughextremely natural from a structuralist perspective, were not developed directly, but inspired by concepts in a different,and more highly developed branch of mathematics, Riemannian geometry.
3. HOW RELATIONS CONNECT
Abstractly, there are different types of relations. They can vary with respect to the number of elements involved,they can be symmetric or directed, that is, distinguish between inputs and outputs, and they may also carry weights.The simplest case are binary, symmetric and unweighted relations. Such a web of relations is then modelled by anundirected and unweighted graph whose vertices stand for the elements in question and whose edges represent thepresence of a relation between the two vertices they connect. For simplicity, we also assume that the graph is simple,that is, there is at most one edge between any two vertices, and that it is connected, that is, by passing from edge toedge we can reach any vertex from any other one, although these assumptions are not essential for any of the sequel.So, we start with that case.We want to assess how a relation, that is, an edge of such a graph, sits in the web of relations, that is, how itrelates to other relations. Two edges are called neighbors when they share a vertex. We can then already definethe simplest concept, called
Forman-Ricci curvature , because it was introduced by Forman [32] as an analogy withthe Ricci curvature of Riemannian geometry (the analogy relates to the role it plays in Bochner type identities). Wedefine the degree of an edge e as deg( e ) := e ) , (1)and define its Forman-Ricci curvature as F ( e ) := 2 − deg( e ) . (2)The 2 and the minus sign are somewhat unfortunate for our purposes, but they are there because of the analogywith the well-established Ricci curvature of Riemannian geometry, and they are useful from an abstract geometricperspective.When the edge e connects the vertices v, w , we can also assess their contribution to the number of neighbors of e .We let deg v ( e ) be the number of edges that share with e the vertex v . Then, obviously, F ( e ) = 2 − (deg v ( e ) + deg w ( e )) . (3)Instead of the sum of the degrees, we may also consider their difference. When the edge is not directed, there is nointrinsic structural difference between the two vertices that it connects, and so, it is natural to take the absolute valueof the difference and define the degree difference [6] as (cid:107) ( e ) := | deg v ( e ) − deg w ( e ) | . (4)Let us interpret the geometric significance of these quantities. (cid:107) ( e ) is large when e connects vertices of differenttypes, a well-connected one from which many further edges emanate, and a less well-connected one from which onlyfewer edges originate. The statistics of this quantity therefore quantify to what extent the network is assortative,that is, typically connect similar vertices (small (cid:107) ( e )), or disassortative, that is, typically connect dissimilar vertices(large (cid:107) ( e )). This is important, for instance, because social networks tend to be assortative [33] (well connectedpeople like to link with other well connected people, and this further improves their position in social networks). Incontrast, F ( e ) is very negative, that is, has a particularly large absolute value when both ends of an edge are wellconnected. Such edges may play a very important role in the network. In fact, we have found [13] that a quantitythat needs a global computation, edge-betweenness centrality (see [7]), is statistically well correlated with F ( e ). Thisedge-betweenness centrality measures how many shortest connections between pairs of vertices in the network passthrough that particular edge. The computation of that quantity is expensive because all shortest connections betweenany two vertices have to be evaluated. In contrast, the computation of F ( e ) is very quick and easy, because only localneighborhoods have to be evaluated.Edges with large | F ( e ) | also play an important role for spreading in the network because from its vertices manyother vertices in the network can be reached in a single step. There is one issue here, however. Edges from the twovertices v, w of e may end at the same vertex z , that is, v, w, z may form a triangle. In that case, they would notcontribute to spreading into different directions. Or the endpoint of an edge from v and that of an edge from w maybe connected themselves by an edge. That is, they form a quadrangle together with v and w . Again, that does notreally constitute spreading into different directions. It is possible to address this issue by inserting two-dimensionalfaces into such triangles and perhaps also into quadrangles, and then to evaluate the Forman curvature of the resultingsimplicial or polyhedral complex. Those faces would then increase the Forman curvature and make it less negative oreven positive. See for instance [14].This aspect is taken care of in a different way by a more refined concept of Ricci curvature, the Ollivier-Ricci curvature introduced in [34]. For that purpose, consider the edge e = ( v, w ) and let e v = ( v, v ) and e w = ( w, w ) beedges emanating from v and w , respectively. We then define their distance w.r.t. e as d e ( e v , e w ) := d ( v , w ) (5)where d ( v , w ) denotes the distance between v and w in the network, that is, the minimal edges that have to betraversed for getting from v to w . Let E v be the set of edges that have v as a vertex, and let | E v | be its cardinality.We then define a probability measure µ v on the set of all edges E by giving each edge e v ∈ E v the weight | E v | andall edges not in E v the weight 0. We then define the Ollivier-Ricci curvature [34] of the edge e = ( v, w ) as O ( e ) := 1 − W ( µ v , µ w ) (6)where W is the 1-Wasserstein distance between µ v and µ w , W ( µ v , µ w ) := inf p ∈ Π( µ v ,µ w ) (cid:88) ( e ,e ) ∈ E × E d e ( e , e ) p ( e , e ) (7)and Π( µ v , µ w ) is the set of measures on E × E that project to µ v and µ w , resp. We thus try to arrange the twocollections E v , E w of edges sharing one of their endpoints with e in an optimal manner, that is, that the averagedistances of the arranged pairs become as small as possible. We note that the sets E v and E w both include the edge e = ( v, w ) that we are evaluating. This convention is only needed to let our definition agree with that originallyproposed in the literature, but could otherwise be abandoned, to make the definition more natural in the presentcontext.In order to evaluate (6), we have to optimize the arrangement between the edges in E v and E w , to make thetransportation cost as small as possible. Since this is a quantity all edges in those two edge sets, it is not necessarilythe case that an optimal transport plan arranges each edge e in E v with the edge e in E w closest to it. Theremight be some competition, as there might be other edges e , e , . . . for which e is closest. But even if there is nosuch competition, it might be overall more beneficial to arrange e with an edge different from e . Also, because ofthe normalization, the edges in E v and E w have fractional weights, and if the cardinalities of the two edge sets aredifferent, also the corresponding weights are different, necessitating an arrangement where some part of an edge in E v is arranged with some part of an edge in E w , and other parts with other ones.Notwithstanding these complications, let m i be the fraction of edges in E v that are moved a distance i in someoptimal transport plan (such an optimal arrangement need not be unique, but that does not matter for our discussion).Then [35] O ( e ) = m − m − m . (8)In particular, moving an edge a distance 1 does not contribute at all to O ( e ). (While m itself does not appear in (8),its computation is nevertheless needed as an intermediate step for computing m and m .) Distance 0, that is, when e participates in a triangle, has a positive contribution. A pentagon, that is, distance 2, has a negative contribution, butnot as a negative as the maximal distance, that can occur in a transportation plan, which is 3. This simple formulathus encodes the essential features of Ollivier-Ricci curvature. In fact, we could simply take (8) as the definition of O ( e ), instead of utilizing the more complicated formula (7).More generally, the Ollivier-Ricci curvature is related to the clustering coefficient, that is, the relative frequency oftriangles in the network [36]. Protein-protein interaction networks
To illustrate an application of these structural measures to empirical data, we have studied the protein-proteininteraction (PPI) networks in human [37], with 8275 nodes and 52569 edges, and fission yeast
S. pombe [38], with1306 nodes and 2278 edges. The edges in these network represent binary interactions between the pair of proteinsrepresented as nodes. These undirected and unweighted networks are disconnected with several components, however,they both include a giant component. The giant component consists of 8152 nodes and 52036 edges in the humanPPI network, and of 1306 nodes and 2278 edges in fission yeast PPI network. We have computed the Forman-Riccicurvature, Ollivier-Ricci curvature and degree difference of edges in these networks, and their distributions are shownin Figure 1.In the human PPI network, while Ollivier-Ricci curvature has a unimodal distribution, the bimodal distribution ofForman-Ricci curvature in Figure 1 signals an evident heterogeneity in the space of protein-protein interactions in thegiant component; a major group of interactions are distributed around a relatively small-valued mode, and a smallgroup of interactions between proteins that are, in average, involved in a signficantly larger number of interactions.The degree difference distribution indicates that, although the majority of interactions are between proteins withrelatively similar degree, a noticeable proportion of the edges have a considerably large degree difference, which canbe as large as 497. This observation is in line with the fact that this network is moderately disassortative withassortativity value ∼ − . ∼ − .
4. DIRECTED GRAPHS
It is not only the case that the preceding constructions extend to directed graphs, but in fact, they become even morenatural in that context. Curvature concepts for directed graphs have been systematically developed and evaluated in[14] (see also see [12]). Here, we shall formulate the concepts in such a manner that they will naturally generalize tohypergraphs.Thus, let e = [ v, w ] be a directed edge with tail v and head w , that is, going from v to w . The input of e at its tail v then are all edges that have v as their head; let their number be deg in ( e ). Similarly, deg out ( e ) denotes the number F(e) F r e qu e n c y a. F(e) F r e qu e n c y b. ד (e) F r e qu e n c y c. ד (e) F r e qu e n c y d. O(e) F r e qu e n c y e. O(e) F r e qu e n c y f. Human Fission Yeast
FIG. 1. The distribution of (a,b)
Forman-Ricci curvature, (c,d) degree difference, and (e,f )
Ollivier-Ricci curvature in thegiant components of the binary protein interaction networks in human (left) and fission yeast (right), respectively. In each case,protein-protein interactions are represented via an undirected and unweighted graph. The nodes and edges represent proteinsand binary interactions between them, respectively. The giant component of the human network has 8152 nodes and 52036edges, while of the fission yeast network has 1306 nodes and 2278 edges. of output edges of e , that is, those that have w as their tail. We may then put [39] F → ( e ) := 2 − deg in ( e ) − deg out ( e ) . (9)We could also form alternative expressions by considering the numbers of edges that have v as their tail and/or ofthose that have w as their head. Similarly for the next expression, the directed degree difference [6] (cid:107) → ( e ) := deg out ( e ) − deg in ( e ) . (10) F → ( e ) now is very negative, or equivalently, deg in ( e )+deg out ( e ) is very large for those edges that receive a lot of inputand produce a lot of output. (cid:107) → ( e ) is positive for those edges that are productive in the sense that they producemore output than they receive input, or that lead to more diversification. It is negative for edges that are receptive,that is collect more input than emit as output.Likewise, we can define the Ollivier-Ricci curvature O → ( e ) of a directed edge [35] by computing the optimaltransportation distance between its input and its output. When there are no shorter connections between inputsand outputs than those going through e itself, then O → ( e ) assumes its smallest possible value −
2. In contrast, wheninputs coincide with outputs, that is, if there is a directed triangle from a vertex u to itself, where u produces both aninput of e and receives an output of e , then this yields a positive contribution. In fact, formula (8) perfectly extendsto the directed case.Let us recall the procedure in detail. For the directed edge e [ v, w ], we define two measures, (cid:40) if e has no incoming edges: µ in ( e ) = 1if e has n incoming edges: µ in ( e ) = n for each incoming edge (11) (cid:40) if e has no outgoing edges: µ out ( e ) = 1if e has n outgoing edges: µ out ( e ) = n for each outgoing edge (12)and µ in ( e (cid:48) ) = µ out ( e (cid:48) ) = 0 for all edges e (cid:48) not occurring in those formulae . The first cases, that is, where there are no incoming edges at the tail or no outgoing edges at the head, that is, wherethe tail is a source or the head is a sink, represent complications that would not arise in the undirected case. Asthey are easily handled, we shall mostly ignore them. In any case, both measures are normalized to have total mass1 and thus are probability measures. We then define the distance between an edge e occurring in (11) and an edge e occuring in (12) as d e ( e , e ) =minimal number of edges needed to get from the tail of e to the head of e . (13)We then put again O → ( e ) → := 1 − W ( µ in , µ out ) (14)where W now is the 1-Wasserstein distance between µ in and µ out , W ( µ in , µ out ) := inf p ∈ Π( µ in ,µ out ) (cid:88) ( e ,e ) ∈ E × E d e ( e , e ) p ( e , e ) (15)and Π( µ in , µ out ) is the set of measures on E × E that project to µ in and µ out , respectively. E here is the set ofdirected edges of the directed graph under consideration. We again have the important formula [35] O → ( e ) = m − m − m , (16)where m i is the number of edges that have to be transported by the distance i in an optimal transport plan. Again,for two given edges in the in- and output of e , that distance might be larger than the distance (13).Thus, the directed curvature notions evaluate flows through edges. As in [14], we may also evaluate flows throughvertices by taking the difference between the sum of the Ricci curvatures of the incoming edges and that for theoutgoing edges. Moreover, in [14], also notions of augmented Forman curvature were developed for directed networks.Augmentation means that one inserts two-dimensional faces into triangles of edges. Such triangles then increase thecurvature, and thereby decrease the difference between Forman and Ollivier type curvatures. Here, however, we donot explore that direction. Transcriptional regulatory networks
To illustrate an application to directed networks, we have studied the transcriptional regulatory network (TRN) ofthe important human pathogen
Mycobacterium tuberculosis [40], with 2547 nodes and 6581 edges. The
M. tuberculosis
TRN was constructed based on ChIP-seq data for 143 transcription factors (TFs) [40]. In this directed and unweightednetwork, each directed edge signifies the regulatory control by a TF of a target gene. In other words, the source nodesin this directed network are TFs while target nodes are target genes. In Figure 2, we show the distribution ofthe Forman-Ricci curvature, Ollivier-Ricci curvature and degree difference of directed edges in the
M. tuberculosis
TRN. In Figure 2a, it is seen that the edges are densely concentrated around Forman-Ricci curvature value 0, andthis indicates that the majority of the edges have a tail vertex with small in-degree and a head vertex with smallout-degree. Likewise, most edges have zero or small value of directed degree difference, and this indicates that thein-degree of the tail vertex and the out-degree of the head vertex for most edges are rather similar. There are also 24vertices with out-degree greater than 100, which can explain the long tail in both Forman-Ricci curvature and degreedifference distributions in Figure 2. On the other hand, the Ollivier-Ricci curvature of the edges in this TRN has amultimodal distribution, with major peaks corresponding to curvature values 0, − − .
5, and − . −400 −300 −200 −100 F → (e) F r e qu e n c y ד → (e) F r e qu e n c y −2 −1 O → (e) F r e qu e n c y a. b.c. Mycobacterium tuberculosis regulatory network
FIG. 2. The distribution of (a)
Forman-Ricci curvature, (b) degree difference, and (c)
Ollivier-Ricci curvature in the tran-scriptional regulatory network of
Mycobacterium tuberculosis . There are 6581 unweighted directed edges and 2547 unweightednodes. The source in each directed edge is a transcription factor (TF) and the target is a target gene controled by the TF.
5. WEIGHTED GRAPHS
The extension of all discussed concepts to weighted graphs is straightforward. One simply counts each edge withits weight. It is therefore not necessary to develop the details here.
6. HYPERGRAPHS
The preceding concepts are set up in such a manner that they naturally extend to hypergraphs. We directly considerdirected hypergraphs. A directed hypergraph may have several nodes through which it receives inputs and severalnodes through which it produces outputs. An important example is a chemical reaction whose input nodes are callededucts and whose output nodes products. In a chemical reaction, there may exist catalyzers, that is, substances thatincrease the rate of a reaction without being modified by it. Formally, they should be counted as both input andoutput nodes. That is, the two subsets of the nodes of a directed hypergraph, its input and output nodes, need notbe disjoint. That will not constitute a problem for the formal concepts to be developed (see, for instance, [17] for adiscussion of directed hyperloops and their curvature).A directed hypergraph H = ( V, E ) consists of a set V of nodes or vertices and a set E of ordered pairs of subsets of V , not both of them being empty, called hyperedges. For a hyperedge e = ( e , e ) ∈ E , e ⊂ V is the head of e , and e ⊂ V is its tail. We let | f | be the number of vertices in f ⊂ V . We let deg in ( e ) of a hyperedge be the number ofhyperedges that have an input node of e as their head, and deg out ( e ) the number of hyperedges that have an outputnode of e as their tail. Since an input edge might connect to more than one input node of e , input hyperedges arecounted with the number of input nodes of e that they connect to, and analogously for output edges. As in (9), wethen define the Forman-Ricci curvature of a hyperedge e = ( e , e ) as [39] F → ( e ) = | e | + | e | − deg in ( e ) − deg out ( e ) . (17)(For a different definition of the Forman-Ricci curvature of a directed hypergraph, see [41].) Thus, here we count thenumber of inputs received through input nodes and the number of outputs produced at output nodes. As in [39], onecan also define different types of Forman-Ricci curvature of a directed hyperedge by arranging inputs and outputsdifferently.Similarly, as in (10), we can put (cid:107) → ( e ) := deg out ( e ) − deg in ( e ) . (18)Following [35], we can also define the Ollivier-Ricci curvature of a directed hyperedge via the Wasserstein distancebetween two probability measures associated to a directed hyperedge. As in (11) and (12), we need to define thecorresponding measures µ in , µ out . In (11) and (12), the principle that was that the total measure 1 is evenly splitamong the inputs or the outputs, resp., unless we had a source or a sink. Now, there are more demands for splitting.A directed hyperedge may in general have more than one tail or head node, and at each of them, several incomingresp. outgoing hyperedges might be found, and each them may again have more than one tail or head. The principlethen is to split the available measure at each step evenly among all the possible recipients. We shall explain theresulting splitting procedure for µ in as the one for µ out is analogous. Let the tail e of the hyperedge e = ( e , e )have η in elements. A source, that is, an element of e without incoming hyperedges, gets the weight µ in ( v ) = η in . Tohandle the others, we define the set M of masses of e = ( e , e ) as the union of the tails of hyperedges that come inat an element w ∈ e , that is, have w in their head set. We then first divide the measure η in that we can distribute atsuch a w evenly among all those incoming hyperedges, and for each such hyperedge, we divide the measure associatedwith it in that manner evenly in its tail. In that manner, we assign a measure to every element in M . Thus, wehave distributed the total measure 1 among the sources and the masses of our hyperedge. This yields µ in , and asmentioned, µ out is constructed analogously by assigning measures to the sinks, that is, those members of e and theholes, that is, the heads of hyperedges that have an element of e in their tail set. The (directed) distance d ( u, v )between a mass u and a hole v of a hyperedge e = ( e , e ) is defined as the minimal number of directed hyperedgesconnecting them. Again, it is at most 3, and this value is attained if u → e , e → v and there is no shorter way tomove from u to v than to go through e . It is 0 when u = v is at the same time a mass and a hole of e , and it is 1 if u is an input of a hyperedge and v is an output. Again, formally, we want to solve an optimal transport problem formoving the first probability measure to the second one. We thus minimize (cid:88) u → e i (cid:88) e j → v d ( u, v ) E ( u, v ) (19)over the set of all matrices E (transport plans) whose entries E ( u, v ) represent the amount of mass from µ M ( u ) movedfrom vertex u to vertex v .If m δ is the amount of mass that is moved at distance δ in an optimal transport plan, the directed Ollivier-Riccicurvature of e is defined as in (12) and becomes again as in (13) O → ( e ) = m − m − m . (20)It is bounded above by 1. This is reached when m = 1, i.e., when each mass coincides with a hole of the same size.It is bounded below by −
2, reached when m = 1, i.e., when there are no shortcuts available and each mass has to bemoved through e to reach a hole. Again, (20) can be taken as the definition of O → ( e ). While it depends on identifyingan optimal transport plan, the formula as such is obviously very simple. For applications, see [42].For instance, we can consider the red hyperedge in Figure 3. Bullets represent vertices. The green bullet in the leftis a source since it has no incoming hyperedge while the blue bullet in the right is a sink since it has not outgoinghyperedges. For representing masses and holes we use triangles and squares respectively. As the red hyperedge hastwo vertices in its tail set and each of them has at most one vertex as an incoming neighbour, the size of the massesis 1/2. In contrast, the sizes of the four holes are different. The biggest one is the sink, with mass 1/2. Another holewith the size 1/4 is the top middle vertex which already got 1/2 of the total mass. The size of the remaining hole is1/4, divided equally among the two vertices in the top right of the figure. Thus, both the triangles and the squareshave total size 1, and the task now is to move the triangles to the squares with least total cost. There are two optimalplans, leading to the negative curvature value -1/4; In one plan, m = 1 / m = 1 / m = 0 and m = 1 /
4, whilein the other m = 1 / m = 1 / m = 1 / m = 0. Note that there are also transfer plans with m = 1, andall other values 0, but they are not optimal.These two different curvature notions represent complementary tools for detecting local geometry and connectivitypatterns in directed hypergraphs. Forman curvature monotonically decreases with the number of incoming andoutgoing neighbours of input and output nodes, resp., and it therefore detects hyperedges joining highly connectednodes. Ollivier curvature, on the other hand, is controlled by the overlap of the set of masses and holes (e.g. directedtriangles) and by shortcuts between them (e.g. directed quadrangles and pentagons). We illustrate these principlesin Figure 4. We want to evaluate the curvatures of the black hyperedge in the left and the right figure in variousconstellations. Without any of the colored hyperedges, F → ( e ) = | e i | + | e j | , while O → ( e ) = 0. When the red edges arepresent, we get F → ( e ) = 0 in both the left and the right figure, whereas O → ( e ) is negative in the left case, because (cid:3) •• • (cid:3) • •4• • (cid:3) • (cid:3) FIG. 3. The red hyperedge is negatively curved as in an optimal transference plan, the size of coincident masses (triangles) andholes (quadrangles), located on the top middle vertex, is less than the size of the masses which need to be moved with distance2. Also the two colored vertices in the left and the right of the figure are a source and a sink since they have no incoming resp.outgoing hyperedges. there are no shortcuts, but positive in the right case, when the inputs of the tail coincide with the outputs of thehead. The presence of the blue edges on the right, however, makes a difference for F → ( e ), but not for O → ( e ), thatis, the former, but not the latter distinguishes between those cases. In contrast, while F → ( e ) does not distinguishbetween the presence of the blue and the green edges in the right figure, O → ( e ) sees the effect, as it is more negativein the presence of the green than in that of blue edges (the blue edges contribute to m , but the green ones to m ). FIG. 4. Illustration of the different connectivity patterns that affect F ( e ) and O ( e ). Well-connected hyperedges often play a keyrole in a network. Said hyperedges are identified by F → ( e ) = | e | + | e | − deg in ( e ) − deg out ( e ), since it decreases monotonicallywith the number of incoming neighbors to the tail and outgoing neighbors from the head of e . Nevertheless, F → ( e ) is notaffected by the presence of arcs from the former to later. O → ( e ) = m − m − m captures this complementary information.On the right, we find directed triangles, which contribute to m (black and red hyperedges), directed quadrangles to m (blackand blue hyperedges), and directed pentagons to m (black and green hyperedges). In the figure on the left, the shortest pathbetween any incoming and any outgoing neighbor, is 3. Such a connectivity pattern contributes to m . Metabolic networks
Metabolic networks are evocative examples of directed hypergraphs, where metabolites react with others to produceproducts. Both, reactants, e , and products, e , typically contain more than one substance ( | e | ≥ | e | ≥ e → e ). Since metabolic networks have been extensively studied, they present an ideal settingto illustrate how to use the hypergraph tools described here. For this, let us consider the metabolic network of Mycobacterium tuberculosis
H37Rv (version iNJ661) [43] modelled as a directed hypergraph. This network contains0 a . − − − −
300 0050100150200 F → ( e ) F r e q u e n c y b. − −
300 0 300 600050100150200 k → ( e ) F r e q u e n c y c . − − . − − . . O → ( e ) F r e q u e n c y FIG. 5. The distribution of (a) Forman-Ricci curvature, (b) degree difference, and (c) Ollivier-Ricci curvature in the metabolicnetwork of
Mycobacterium tuberculosis
H37Rv, which is represented as a directed hypergraph with nodes as
M. tuberculosis metabolites and directed hyperedges as chemical reactions. The network has 743 nodes and 1195 hyperedge edges.
939 reactions and 743 metabolites, of which 256 are reversible. Each reversible reaction ( e (cid:28) e ) was divided intotwo, a forward reaction ( e → e ), and its reverse reaction ( e ← e ). As a result, the network contains 1195 directedhyperedges. Most substrates of this metabolic network are consumed or produced by one reaction only. Also, a feware involved in more than half of the reactions ( ∼
50% require h , h2o , atp or nadhp , and ∼
57% produce h , pi , h2o , adp , or co2 ), the distributions of indegree and outdegree in Figure 6 a) summarize this behavior.Suppose that we want to investigate whether starting materials that are produced in several different ways (largeindegree in e ) produce substances that also serve as starting materials for many reactions (large outdegree in e ),that is, whether targets are transformed into key precursors. There are two aspects relevant for this question. First,we must find out if the network is assortative. Since it is a hypergraph, we use the degree difference and its distributionshows that this is mostly the case (see Figure 5 b)). Notoriously, the degree difference is ∼ ∼ F → ( e ).Figure 5 a) shows that the dominant mode is represented by curvature around zero. There are also secondary humpsassociated with more negative curvature values. Perhaps the most important reactions, however, are those thathave very low (negative) curvature values, but a degree difference near zero. In fact, the first reaction on the list isthe fundamental reaction that creates the energy storage molecule adenosine triphosphate (ATP), e : adp+h+pi → atp+h+h2o , with F → ( e ) = − (cid:107) → ( e ) = 1. Furthermore, the associated mass set M shows that there are 400precursors for the substrates of this reaction, and, based on the set of holes H , there are 464 derived metabolites. Thispair of values correspond to the upper right blue mark in Figure 6 c). Notice that this information is not given bynode degree. With few exceptions, precursors and derivatives are at distances shorter than three, and mostly aroundzero, as shown by O → ( e ) (see Figure 5 c)). For the reaction discussed here, O → ( e ) = 0 .
35 and it corresponds to thethe left most blue mark of Figure 6 d). The preceding already illustrates how a combination of the three measures thatwe have developed, F → ( e ) , O → ( e ) and (cid:107) → ( e ), can reveal the fundamental structural properties of specific reactionsinside the metabolic network. Both evaluating the statistical distributions of these three quantities and comparingthem for different networks, and analyzing those reactions that produce particularly prominent values for them inmore detail should yield deeper insight into the structure of metabolic networks.1 a . 10 Node degree F r e q u e n c y IndegreeOutdegree b . 1 2 3 4 5 6 7 812345678 | e | | e | c . 100 200 300 400100200300400 |M| | H | d . − − − −
300 0 − − , − − , , F → ( e ) O → ( e ) FIG. 6. The distribution of (a) node degree, (b) sizes of tails and heads, ( | e | , | e | ), (c) number of masses and holes, ( M , H ),and (d) (F( e ) , O( e )) values in the metabolic network of Mycobacterium tuberculosis
H37Rv, which is represented as a directedhypergraph with nodes as
M. tuberculosis metabolites and directed hyperedges as chemical reactions. The network has 743nodes and 1195 hyperedge edges.
ACKNOWLEDGMENT
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