Comparing Three Notions of Discrete Ricci Curvature on Biological Networks
CComparing Three Notions of Discrete Ricci Curvature onBiological Networks
Maryam Pouryahya, James Mathews, Allen Tannenbaum ∗†‡
Abstract
In the present work, we study the properties of biological networks by applying analogous notions of fun-damental concepts in Riemannian geometry and optimal mass transport to discrete networks described byweighted graphs. Specifically, we employ possible generalizations of the notion of Ricci curvature on Rie-mannian manifold to discrete spaces in order to infer certain robustness properties of the networks of interest.We compare three possible discrete notions of Ricci curvature (Olivier Ricci curvature, Bakry- ´Emery Riccicurvature, and Forman Ricci curvature) on some model and biological networks. While the exact relation-ship of each of the three definitions of curvature to one another is still not known, they do yield similarresults on our biological networks of interest. These notions are initially defined on positively weightedgraphs; however, Forman-Ricci curvature can also be defined on directed positively weighted networks. Wewill generalize this notion of directed Forman Ricci curvature on the network to a form that also considersthe signs of the directions (e. g., activator and repressor in transcription networks). We call this notionthe signed-control Ricci curvature . Given real biological networks are almost always directed and possesspositive and negative controls, the notion of signed-control curvature can elucidate the network-based studyof these complex networks. Finally, we compare the results of these notions of Ricci curvature on networksto some known network measures, namely, betweenness centrality and clustering coefficients on graphs.
In recent years, there have been tremendous efforts to elucidate the complex mechanisms of biologicalnetworks by investigating the interactions of different genetic and epigenetic factors. Mathematical toolscan help significantly with overcoming many of the challenges and can facilitate better understanding of thecomplexities of such networks. This has led to the emergence of the field of network and systems biology.The mathematical methods and tools employed in networks are quite diverse and heterogeneous rangingfrom graph theory as abstract representations of pairwise interactions to complicated systems of partialdifferential equations that try to capture all details of biological interactions. ∗ M. Pouryahya is with the Department of Applied Mathematics & Statistics, Stony Brook University, NY; email:[email protected] † J. Matthews is with the Department of Medical Physics, Memorial Sloan Kettering Cancer Center, NY; email: [email protected] ‡ A. Tannenbaum is with the Departments of Computer Science and Applied Mathematics & Statistics, Stony Brook University,NY; email: [email protected] a r X i v : . [ q - b i o . M N ] D ec number of notions have been employed to investigate the structure of networks such as centrality,clustering coefficient, and network connectivity. Our study is based on several extensions of the notionof Ricci curvature on Riemannian manifolds to discrete spaces [20, 31, 38]. The Ricci curvature tensorof a manifold encodes the degree to which the geometry determined by a given Riemannian metric mightdiffer from that of ordinary Euclidean space. In Riemannian geometry, sectional curvature is defined on two-dimensional tangent planes and expresses the convexity property of the distance function between geodesics.The essential notion of Ricci curvature is the average of sectional curvatures of all tangent planes with somegiven direction [14]. One important aspect of Ricci curvature is that it can control the eigenvalues of theLaplace-Beltrami operator [26].Our use of curvature is very much involved with the theory of optimal mass transport [43, 54]. As notedby [25, 40], the space of probability densities on a metric measure space can be endowed with a naturalRiemannian metric via the Wasserstein 2-distance This is based on a seminal dynamical characterization ofthe Wasserstein metric given by Benamou and Brenier [6] on the space of probability densities. Since onehas a Riemannian-type metric, one can subsequently define a related notion of geodesics on the space ofprobability densities. As noted by Lott-Sturm-Villani [35, 49], if the underlying space is itself a Riemannianmanifold, then via the Bochner formula, one can show that there is a fundamental relationship betweenchanges in entropy, Ricci curvature, and the Wasserstein distance. In conjunction with the Fluctuation The-orem [13], we have considered increases of the Ricci curvature as being positive correlated with increases insystem functional robustness, herein expressed as ∆ Ric × ∆ R ≥ . An appealing feature of this correlationis that we can distinguish the cancer networks and the important nodes (targets) within such networks by afairly straightforward computation of Ricci curvature on the networks as previously done in [42, 46, 52].There have been several efforts to define the notion of Ricci curvature that is applicable to more generalmetric spaces [35, 49] [15, 38]. In fact, graphs are ideally suited for explaining the meaning of curvature andthe appeal of this discrete feature is the fairly straightforward computation in many cases. In the presentnote, we employ three different approaches to define the Ricci curvature on discrete spaces, each of whichis very relevant to biological networks. Our first approach is based on a definition proposed by Olivier [38]in a general framework of finite Markov processes or equivalently weighted graphs. Closely related workhas been done in [34]. The second approach follows the Γ -calculus of Bakry- ´Emery. In [4], Bakry and´Emery suggested a notion analogous to curvature in the very general framework of a Markov semigroup.Following [31], these attempts lead us to define the Bakry- ´Emery Ricci curvature on graphs. The thirdgeneralization of Ricci Curvature is defined by Forman [20] for a large class of topological spaces called cellcomplexes . Forman Ricci curvature is based on a proper interpretation of the so called Bochner-Weitzenb¨ockdecomposition of Laplace operator [14]. For graphs, as 1-dimensional cell complexes, we may apply thisdefinition to study the curvature of the networks of interest [56].In this work, we compare the results of all three of these notions of Ricci curvature on networks modeledas weighted graphs. Ollivier Ricci curvature and Forman Ricci curvature are originally edge-based notions,but Bakry- ´Emery Ricci curvature is initially defined for an edge within the network. However, the scalarOllivier/ Forman Ricci curvature can be defined for a given node in the network. Also, the advantage ofForman Ricci curvature is that it can also be calculated for a “directed” edge in the network [48]. We furthergeneralize this definition to apply the Forman Ricci curvature to the transcription network of E. Coli . In thisnetwork, in addition to the direction, there is a sign associated to each edge. In other words, the direction canhave positive (activator in
E. Coli network) or negative (inhibitor in the
E. Coli network) control. We willdefine the signed-control curvature that considers both the directionality and the signs associated to themand will calculate this notion in the
E. Coli transcription network. We regard the latter notion of curvature asone of the key contributions in the present work, that will be exploited in the future to study the properties2f various networks.In the subsequent sections, first, we review the Wasserstein distance and the Riemannian structure itinduces on the space of probability densitie. Via the Fluctuation Theorem [13], we conclude that increas-ing the Ricci curvature is positively correlated with increasing the robustness. This correlation helps us toquantify the difference between robust and normal networks by computation of their Ricci curvature. Wethen discuss more details of each of the three notions of the Ricci curvature on graphs. We apply thesenotions on some real networks such as the cancer network generated by TCGA data and the
E. Coli tran-scription network. We rank the top genes in breast carcinoma with respect to three notions Ricci curvatureand compare the significant genes with these three notions. Finally, we explore the correlation between thediscrete Ricci curvature on graphs and some known network measures, namely, betweenness centrality andclustering coefficient. Interestingly, the results are similar across all three notions of Ricci curvature.
In this section, we review some fundamental concepts with regards to the theory of optimal mass transportand Riemannian geometry. We focus on an alternative formulation of Wasserstein distance by Benamou andBrenier. This distance enables us to define a Riemannian structure on the space of probability densities dueto Jordan et al. [25] and Otto [40]. Using this structure, following the work of Lott-Sturm-Villani [35, 49]we elucidate the relationship between the Ricci curvature and the entropy.
The problem of optimal transport was first proposed by the civil engineer Monge in the 1780’s asks for theminimal transportation cost to move a pile of soil (“d´eblais”) to an excavation (“remblais”) [18, 43, 53, 54].Considering measure spaces ( X , µ ) and ( Y , ν ) and the cost c : X × Y → R + , Monge’s problem is to find a(mass preserving) transport map T : X → Y that minimizes (cid:90) X c ( x, T ( x )) dµ ( x ) . Because of the difficult constraints on the mapping T, Monge’s problem can be non-trivial to solveand even ill-posed. In 1940’s, Kantorovich proposed a relaxation to the Monge problem that allows one toemploy linear programming [18]. Let Π( µ, ν ) denote the set of all couplings between µ and ν , that is alljoint probability measures π on X × Y whose marginals are µ and ν . Accordingly, the Kantorovich problemwas the following linear programming problem: inf π ∈ Π( µ,ν ) (cid:90) X ×Y c ( x, y ) dπ ( x, y ) . The cost function was originally defined in a distance form on a Polish metric space ( X , d ) . This leadsus to the following distance known as Wasserstein distance. For any two probability measures µ , ν on X ,and p ∈ [1 , ∞ ) , the p -Wasserstein distance between µ and ν is defined as W p ( µ, ν ) = (cid:16) inf π ∈ Π( µ,ν ) (cid:90) X ×X d ( x, y ) p dπ ( x, y ) (cid:17) /p = inf E (cid:2) d ( X, Y ) p (cid:3) /p . ( X, Y ) of possible couplings of ( µ, ν ) .Satisfying the three axioms of the distance function, W p defines a distance on X [54].The -Wasserstein distance is also known as the Kantorovich-Rubinstein distance, or
Earth Mover’s distance among computer scientists. In [39], Ollivier used the -Wasserstein distance to define the (Ollivier-)Ricci curvature, which we will discuss in 3.1. The -Wasserstein distance has some very remarkableproperties and can be related to fluid mechanics. In a seminal paper, Benamou and Brenier suggestedan alternative numerical method to calculate the -Wasserstein distance [6]. They proposed a “dynamical”time-dependent formulation to Monge-Kantorovich Problem where one seeks for the geodesic that evolvesin time between the two densities. More precisely, we assume Here ρ ( t, x ) is the density of particles whichinterpolates between ρ and ρ and v ( t, x ) is the velocity field at time t and position x . Benamou andBrenier [6] formulated a dynamic computational fluid version of mass transport given by minimizing thekinetic energy of a system of particles interpolating two densities subject to a continuity constraint. Thisformulation is equivalent to the Monge-Kantorovich theory for the Wasserstein 2-metric. More precisely, W ( ρ , ρ ) = (cid:16) inf ρ,v (cid:110) (cid:90) R n (cid:90) ρ ( t, x ) | v ( t, x ) | dtdx : ρ (0 , . ) = ρ , ρ (1 , . ) = ρ (cid:111)(cid:17) / subject to the continuity equation: ∂ t ρ + ∇ · ( ρv ) = 0 (1)where ∇· stands for the divergence operator. It turns out that the optimal conditions satisfy: v ( t, x ) = ∇ ψ ( t, x ) , (2)where ψ is the Lagrange multiplier of the constrained optimization problem (1).We note that more generally, the Wasserstein distance defined above may be extended to a (completeconnected) Riemannian manifold for the space of probability densities (see our discussion below). Theabove distance endows the space of probability measures with a Riemannian structure as well via the seminalwork of [25, 40] to which we refer the interested reader for all the details. Accordingly, one can define anotion of geodesic on the probability space. Assume that M is a complete connected Riemannian manifold equipped with metric g . The Ricci curvaturetensor provides a way of measuring the degree to which the geometry determined by a given Riemannianmetric might differ from that of ordinary Euclidean space. We refer the reader to [14] for all the technicaldetails. We just remark here that curvature controls the local behavior of the geodesics. Generally, in theneighborhoods with positive curvature the geodesics converge, whereas when the curvature is negative theydiverge. A lower bounds on the Ricci curvature prevents geodesics from diverging too fast and geodesicballs from growing too fast in volume. In fact, the lower Ricci curvature bounds estimate the tendency ofgeodesics to converge. Interestingly, optimal transport offers a formulation of lower Ricci curvature boundsin terms of entropy that we review here.For the measure µ , the Boltzmann entropy is defined as follows:Ent ( µ ) = − (cid:90) M ρ log ρ dvol , ρ = dµ/ dvol. Lott, Sturm and Villani [35, 49] discovered a beautiful connection between Riccicurvature and the Boltzmann entropy: Ric ≥ k if and only if the entropy functional is displacement k -concave along the -Wasserstein geodesics, i.e. for all µ , µ ∈ P ( M ) and t ∈ [0 , we have:Ent ( µ t ) ≥ (1 − t ) Ent ( µ ) + t Ent ( µ ) + k t (1 − t )2 W ( µ , µ ) , where ( µ t ) ≤ t ≤ is the -Wasserstein geodesics between µ and µ . In fact, this inequality indicates thepositive correlation between entropy and curvature which we express as: ∆ Ent × ∆ Ric ≥ . In this section, we review the relation between robustness and the network entropy. We define robustness asthe ability of a system to functionally adapt to perturbations in its environment. As discussed by Alon [2],biological circuits have robust designs for which their essential function is nearly independent of biochemi-cal parameters that tend to vary from cell to cell. Robustness is a relative measure. The work done in [13,57]shows that in many cases the normal protein interaction networks are less robust than their cancerous ana-logues.In [13], the authors proposed that robustness can be characterized by the fluctuation decay rate afterrandom perturbations. Their approach is as follows: Consider some perturbation in a given observable (e.g.,for a graph in a node or an edge). Such changes will deviate the stationary state from its unperturbed valueat time . Now let p (cid:15) ( t ) define the probability that the sample mean’s deviation is more than (cid:15) from theoriginal value at time t . As t increase, p (cid:15) ( t ) converges to zero. So one defines the fluctuation decay rate,noted by R as follows: R := lim t →∞ ,(cid:15) → ( − t log p (cid:15) ( t )) . Thus, a larger value of R corresponds to a smaller deviation from the steady-state condition, but this is alsocorrelated with the larger values of entropy via the fluctuation theorem. In fact, the fluctuation theoremindicates that the rate of decay of fluctuations to the steady state is positively correlated with entropy at thesteady sate. In other words, the fluctuation theorem implies that an increase in entropy entails a greaterrobustness to the perturbation of the network. This indicates that the network entropy and robustness arepositively correlated: ∆ Ent × ∆ R ≥ , As we previously remarked, the work of Lott-Sturm-Villani [35, 49] indicates the positively correlationof increases in entropy and Ricci curvature. Given the Fluctuation Theorem, one can conclude that Riccicurvature and robustness of a network are positively correlated: ∆ R × ∆ Ric ≥ (3)In conclusion, this correlation enables us to identify the network robustness and to quantify the differencebetween robust and normal networks by a fairly straightforward computation of Ricci curvature on thenetworks as was done in [46, 52]. In the following section, we study three formulations of the discrete Riccicurvature on weighted undirected graphs. 5 Discrete versions of Ricci curvature
In this section, we define the Ricci curvature on discrete metric measure spaces including weighted graphs.Specifically, we assume that our network is presented by an undirected and positively weighted graph, G = ( V, E ) , where V is the set of n vertices (nodes) in the network and E is the set of all edges (links)connecting them. We set d x := (cid:88) z w xz µ x ( y ) := w xy d x , (4)The sum is taken over all neighbors z of x , and w xy denotes the weight of an edge connecting x and y (it is taken as zero if there is no connecting edge between x and y ). Note that since the graph is undirected,we have that w xy = w yx . Let W G = ( w xy ) ≤ x,y ≤ n be the matrix of weights, and D G be an n × n diagonalmatrix whose diagonal is ( d x ) for all x ∈ E . Now, the Laplacian matrix is defined as L G := D G − W G . We define the combinatorial Laplacian matrix , L G , to be the negative of the Laplacian matrix. L G is alwaysnegative semi-definite. One of the key definitions of Ricci curvature on a discrete metric measure space, is via the
Ollivier-Riccicurvature [39]. As discussed by Ollivier, the Ricci curvature of a metric measure space can be definedby comparing the distance between small spheres and the distance between their centers. He then extendsthis idea from the geodesic sphere to an associated probability measure at the point on a metric space X .Consider the graph metric d : V × V → R + on the set of vertices V where d ( x, y ) is the number of edges inthe shortest path connecting x and y . For any two distinct points x, y ∈ V , the Ollivier-Ricci (OR) curvature is defined as follows: k ( x, y ) := 1 − W ( µ x , µ y ) d ( x, y ) (5)where µ x , µ y are defined in (4). Here, we used the Hungarian algorithm [45] to compute the EarthMover Distance on our reference networks.Using this edge based notion of curvature, we can also define the scalar curvature of a given node in thegraph as follows: S OR ( x ) := (cid:88) y k ( x, y ) , where the sum is taken over all neighbors of x . In Riemannian manifold many consequences of Ricci curvature lower bound comes from the well-knownBochner formula. For a differentiable function f on the complete Riemannian manifold ( M, g ) we have:
12 ∆ |∇ f | − (cid:104)∇ f, ∇ (∆ f ) (cid:105) = (cid:107) Hess f (cid:107) HS + Ric ( ∇ f, ∇ f ) , (6)6here HS stands for Hilbert-Schmidt norm, i.e. for matrix A with its adjoint A ∗ , (cid:107) A (cid:107) HS = (cid:112) tr ( A ∗ A ) .The formula has a more general version for 1-forms which is the Bochner-Weitzenb¨ock formula and it isused by Forman to define a combinatorial Ricci curvature with we will review in the following Section 3.3.Applying the Cauchy-Schwarz’s inequality, (cid:107) Hess f (cid:107) HS ≥ (∆ f ) n , to (6) we obtain:
12 ∆ |∇ f | − (cid:104)∇ f, ∇ (∆ f ) (cid:105) ≥ (∆ f ) N + Ric ( ∇ f, ∇ f ) , for N ≥ n . Therefore; by taking N → ∞ we can use:
12 ∆ |∇ f | − (cid:104)∇ f, ∇ (∆ f ) (cid:105) ≥ k |∇ f | , (7)to characterize Ric ≥ k .Inspired by the Bochner formula, the Γ calculus was formulated by Bakry- ´Emery [4] in a way that the Γ( f, f ) is an analogue of |∇ f | , and Γ ( f, f ) is an analogue of ∆ |∇ f | − (cid:104)∇ (∆ f ) , ∇(cid:105) in (7). Moreover,the authors of [31] used this Γ calculus to give a notion of curvature on graphs. Assume we have the graph G = ( V, E ) and the function f : V → R on the set of vertices, the discrete Laplacian ∆ is defined by: ∆ f ( x ) = (cid:88) ( x,y ) ∈ E w xy ( f ( y ) − f ( x )) , (8)For a graph of finite number of edges and vertices, this definition coincide with the combinatorial Lapla-cian matrix, L G that we defined before in Section 3. That is, writing f as a column vector, we have: ∆ f ( x ) = L G f ( x ) where L G f ( x ) denotes the x th entry of the product vector L G f . Now we define the two following bilinearoperators for the functions f, g : V → R and x ∈ V : Γ( f, g )( x ) := 1 / L G ( f · g )( x ) − f ( x ) L G g ( x ) − g ( x ) L G f ( x )] , Γ ( f, g )( x ) := 1 / L G (Γ( f, g )( x )) − Γ( f, L G g )( x ) − Γ( g, L G f )( x )] . (9)Following the Bochner formula and the inequality ( ), we define the Bakry- ´Emery-Ricci (BER) curvature of a given node x ∈ V as the maximum value of k ( x ) such that: Γ ( f )( x ) ≥ k ( x )Γ( f )( x ) , ∀ f. (10)where Γ( f ) := Γ( f, f ) and Γ ( f ) := Γ ( f, f ) . The later definition only makes sense if Γ is positivesemi-definite at x . Also, applying Γ( f ) : V → R , to the operator (8) we will have: Γ( f )( x ) = (cid:88) y ∼ x w xy ( f ( x ) − f ( y )) . Therefore, as long as our graph has positive weights we can use (10) to find the BER Ricci curvature ofa given node. Here, we used the convex optimization package CVX [7] to find the BER curvature on ourreference networks. 7 .3 Forman-Ricci Curvature
Forman’s discretization of Ricci curvature is based on the Bochner-Weitzenb¨ock decomposition of theLaplace operator which is a generalization of Bochner formula to 1-forms [20] . It is defined in generalfor CW-complexes. This class of spaces is broader than simplicial complexes and most importantly it in-cludes graphs (1-dimensional CW complexes). A p -cell in a CW-complex is a space that is homeomorphicto an open disk of dimension p . We say that two p -cells α and α are parallel neighbors if they only satisfyone of the following conditions:1. There is a ( p + 1) -cell β such that α < β and α < β .2. There is a ( p − -cell γ such that γ < α and γ < α .where by α < β we mean that α is contained in the boundary of β . Forman considers the following cellularchain complex on M : → C n ( M, R ) ∂ −→ C n − ( M, R ) ∂ −→ . . . ∂ −→ C ( M, R ) → Replacing the derivatives by the boundary operator and the p -forms by p -cells, Forman defines an analogousnotion of Laplace-Beltrami operator on C p ( M ) as follows: (cid:3) p = ∂∂ ∗ + ∂ ∗ ∂ : C p ( M ) → C p ( M ) . where ∂ ∗ is the formal adjoint operator of ∂ .The Bochner-Weitzenb¨ock theorem gives the following decomposition of Laplace-Beltrami operator on p -forms [26]: ∆ p = ∇ ∗ p ∇ p + F p where ∇ p is the covariant derivative operator and F p is a linear operator involving only curvature. Specially,for a -form, ω ∈ T ∗ x ( M ) , we have: F ( ω ) = (cid:104) F ( x ) ω, ω (cid:105) = Ric ( ω ) . (11)In combinatorial settings, Forman [20] shows that there is a canonical decomposition: (cid:3) p = B p + F p , where B p is a nonnegative operator. In analogy with Bochner-Wetzenb¨rock formula, for any p -cell α , hedefines: F p ( α ) = (cid:104) F p ( α ) , α (cid:105) . He finally reduces this definition to an explicit calculation of F p as follows: F p ( α ) := { ( p + 1) -cells β > α } + { ( p − -cells γ < α }− { parallel neighbors of α } . Similar to formula (11), for the special case of p = 1 , Forman defines the Ricci curvature of each edge e (1-cell) by Ric ( e ) = F ( e ) ( e ) = { -cells f > e } + 2 − { parallel neighbors of e } . Forman also extends this definition to the weighted case where each cell has been assigned a positive weight.We can use this formula to define the curvature of a network [48, 56], . Assume that as before our networkis presented by a weighted graph G = ( V, E ) . The Forman-Ricci (FR) Curvature of the graph G is definedas follows: Ric ( x, y ) = w xy (cid:32) w x w xy − (cid:88) z (cid:54) = y w x √ w xy w xz (cid:33) + w xy (cid:32) w y w xy − (cid:88) s (cid:54) = y w y √ w xy w sy (cid:33) , (12)where the sum is taken over all neighbors z of x and s of y except y and x , respectively. w x denotes theweights associated with the node x , which is calculated as the sum of the weights of the incident edges ( d x in (4)) divided by the degree of the node. w y is defined similarly.Furthermore, similar to OR curvature one can define the scalar FR curvature for a given node x in thenetwork as follows: S F R ( x ) := (cid:88) y Ric ( x, y ) , (13)where, again, the sum is taken over all neighbors of x . In this section, we apply the Ricci curvature analysis to some model and real networks. We investigateseven networks composed of cancer specific genes provided by Memorial Sloan Kettering Cancer Cen-ter [44] which have been a subject of investigation in previous work [41, 46, 52]. The data initially consistsof genes and RNA-Seq data from 3000 samples of primary tumor and adjacent normal sample fromseven distinct tissues. We chose cancer-related genes of this data from COSMIC cancer gene mutationdata [19]. The TCGA gene expression (The cancer Genome Atlas, https://cancergenome.nih.gov/) is derivedfor seven corresponding tumor types: breast invasive carcinoma [BRCA], head and neck squamous cell car-cinoma [HNSC], kidney renal papillary cell carcinoma [KIRP], liver hepatocellular carcinoma [LIHC], lungadenocarcinoma [LUAD], prostate adenocarcinoma [PRAD], and thyroid carcinoma [THCA]. The networkis constructed by (Spearman) correlation values of gene-to-gene expressions in cancerous and normal tissuesacross all samples within a given genotype. We further used the transformation of (1+ corr ( i,j ))2 for the genes i and j E. coli derivedfrom the Uri Alon laboratory website at the Weizmann Institute [3]. In this network the ‘nodes’ are operons9hich contain one or more structural genes transcribed on the same mRNA. Each ‘edge’ is directed from anoperon that encodes a transcription factor to an operon that it directly regulates [47]. This network consistsof 423 nodes (operons) and 578 edges. In addition to the direction of the regulation, the regulator couldbe an activator (promotor) or repressor (inhibitor). We generalize the FR curvature to the one that alsoconsiders the signs of these directions. As discussed by Uri Alon [2] the transcription network of E. Coli asa biological network exhibits a great degree of robustness.Finally, we compare the notion of Ricci curvature on networks to two common measurement on net-works, betweenness centrality and clustering coefficient. To this end, we consider the scale-free network ofBarab´asi-Albert (BA) model [1] and the network of breast cancer mentioned above. The BA model is analgorithm for generating a random scale-free network, in which the degree distribution (for large values of k ) is a power law of the form: P ( k ) ∼ k − where P ( k ) is the fraction of nodes in the network having k connections to other nodes. The network beginswith an initial connected network. New nodes are connected to existing nodes at a time with a probabilitythat is proportional to the number of links that the existing nodes already have. Consequently, the new nodesprefer to attach themselves to the already heavily linked nodes known as hubs. We discuss the results ofthese investigations in the following sections. We investigate the three notions of Ricci curvature on the seven different genotype networks we discussedearlier. Table 1 provides the results of the average deference between cancer and normal network (cancer-normal) in terms of the Ricci curvature. It has been posited in several studies [13, 57] that cancer networksexhibit a higher degree of robustness than their analogous normal tissue network. Here, we investigate thishypothesis by comparing the three discrete curvatures on cancer and normal networks of the TCGA data.As it is shown in Table 1, all three Ricci curvatures (OR, BER and FR curvature) have higher values in theseven cancer networks compared to the normal ones.
Top ranked genes in breast carcinoma
We also ranked the top genes in breast carcinoma with respect to three notions of Ricci curvature. Thesegenes has been ranked based on their curvature’s difference in cancer and normal networks (cancer- normal).In fact, these are the nodes expected to have the most contribution in the robustness of the cancer networkcompared to the normal network. Since the OR and FR curvature are initially edge based, we provide thetop pair in Tables 3 and 5. Table 4 provides the top node-based BER curvature. There are some similaritiesbetween the top ranked genes with respect to FR curvature and BER curvature, namely, ALDH2, NDRG,CLTCL1, KIF5B, PPARG, PTPN11, JAK1, PIK3CA, SDHB, EPS15, ERG, and HIP. There are some simi-larities between the top ranked genes with respect to FR curvature and OR curvature, namely IDH1, RUNX1,HMGA1, SDHB, EPS15, and ERG. There are three genes, SDHB, EPS15, and ERG found among the topranked genes with respect to all three FR, BER and OR curvatures.A number of these genes have known clinical implications with regards to breast cancer. For example,the EPS15 gene encodes for the Epidermal growth factor receptor subset 15 (EPS15 protein) [12]. EPS15plays a crucial role in the degradation of growth factor receptors. Furthermore, the development of a widevariety of malignancies, including breast cancer, is known to be associated with the over-expression of re-ceptor tyrosine kinases (RTKs), which are largely comprised of growth factor receptors. Furthermore, theprognostic value of EPS15 has been reported given the notion that over-expression of EPS15 is significantly10 ancer Type ∆ Av-erageORCur-va-ture ∆ Av-erageBERCur-va-ture ∆ Av-erageFRCurvature
Breast Carcinoma 0.012 0.182 13.022Head/Neck Carci-noma 0.004 0.116 9.100Kidney Carcinoma 0.010 0.217 7.711Liver Carcinoma 0.008 0.227 3.136Lung Adenocarci-noma 0.013 0.320 7.898Prostate Adenocar-cinoma 0.009 0.179 7.368Thyroid Carcinoma 0.006 0.133 2.969Table 1: All seven cancer networks have a higher average Ricci Curvature than the complementary normalnetworks.associated with a favorable clinical outcome with regards to breast cancer. Also, mutations of the Suc-cinate Dehydrogenase B (SDHB)gene lead to a reduction in the amount of SDHB protein in the cell andloss of SDH enzyme activity. Within mitochondria, the SDH enzyme plays an important role in energymetabolism, linking the the Krebs cycle with oxidative phosphorylation. Loss of function of this gene andsubsequent decreased SDH enzyme activity results in abnormal hypoxia signaling and increased formationof tumors. Furthermore, the SDHB gene is a noted tumor suppressor gene and has been identified as a cul-prit in various cancers including Gastrointestinal Stromal Tumors (GIST), malignant pheochromocytomaand paraganglionomas, as well as renal cell carcinoma. There are case reports in the literature of breastcancer involving SDHB mutations, although further studies in this regard are warranted [30, 32]. Of note,one could provide this comparison by the node-based scalar OR and FR curvature as we discussed before.However, doing so seems to miss some of the original information; therefore, we prefer to find the top geneswith regards to the original results. Also, there are some important cancer-related gene mutations knownto play a significant role in breast carcinoma such as BRCA1 and BRCA2 which are not ranked among thetop ranked genes with regards to curvature. This potentially indicates the limitation of node-wise study ofthe curvature on the networks compared to the global one. This could be due to the fact that the node-wisemeasurement in all three methods only considers the adjacent vertices and some non-adjacent pathways thatcould possibly have a significant contribution in the robustness of the network, might be neglected. Here,the subject of interest is more the similarities of the results with respect to the three notions of curvaturesthat we discussed above.
E. Coli
Network
In the previous sections, we compared three notions of network curvature that apply in the unstructured case,where the network is modeled by an undirected graph. All three notions admit a generalization in whichpositive numerical edge weights and node weights were taken into account. In this section, we will discuss a11urther generalization of the Forman-Ricci (FR) curvature in which edge directions and edge “control types”are also taken into account.Real biological networks are almost always directed, in the sense that the edges represent a relationshipof action of the agent or quantity labeled by one node upon the agent or quantity labeled by the other. Inpractice the nature of this action or influence is either positive (promotion, activation) or negative (repression,inhibition). We will use the terms positive control and negative control for these properties. A networkaugmented with such edge properties will be called a signed-control network . In this section we focus ondirected, weighted, signed-control networks.As already indicated, the authors [56] and [48] imported FR curvature into the domain of network anal-ysis, and demonstrated a certain amount of utility of this tool (recall formula 12). A proposed generalizationto the directed case appears in [48], consisting of a simple restriction on the terms appearing in 12. Namely,for a given edge e , one calculates the ordinary FR curvature of the “source-free and drain-free” subnetworkdefined near e . By definition this subnetwork contains only those directed edges terminating at the sourceof e or emanating from the target of e . The authors call these edges the edges “consistent with e ”.We reiterate that we assume that the networks which we consider are positively weighted. Each edgeweight is interpreted as a quantification of the strength or amplitude of the action represented by the edge.Though mapping directed, positively-weighted, signed-control networks onto directed, signed-weighted net-works by signing each edge weight with the control type may have some utility, we think a different approachis more natural.Figure 1: The green edge is the edge e under consideration. The dashed edges do not contribute to thedirected FR curvature at e .Figure 1 illustrates the classification of the edges in the first-neighbor neighborhood of a given edge e which leads to the modification of formula 12 for directed networks. The dashed arrows indicate edgeswhose corresponding terms are omitted.Figure 2 shows the analogous classification in the case of directed signed-control networks. Positivecontrol is indicated with an ordinary arrow, while negative control is indicated by a flat arrowhead (thissymbol is consistent with usage in [2]).We seek a further modification which takes account of the control types for each edge. We choose tointroduce signs to the terms according to the following heuristic, closely resembling the heuristic of Sreejith et al [48] in establishing the directed generalization. As before, each term corresponds to a certain edgeadjacent to e , and we only consider the edges consistent with e in the directed sense. In case e and theadjacent edge are both of positive control, the total effect of the pair is considered to be of positive control,and we proceed as in the case or ordinary directed networks, making no alteration to the corresponding term.In case exactly one of e and the adjacent edge is of negative control, the total effect of the pair is consideredto be of negative control, and in this case we multiply the corresponding term by the sign − . In case both e and the adjacent edge are of negative control, the total effect of the pair is considered to be of positivecontrol, and in this case no alteration is made to the corresponding term.We call the resulting curvature notion signed-control curvature as an abbreviation for directed, signed-control Forman-Ricci curvature , in which the presence of directions and the relation to Forman-Ricci cur-vature are understood. Also, the scalar signed-control curvature of a given node in the network is defined as12he sum of the curvatures of the adjacent edges to that node as before 13.Figure 2: As in figure 1, the dashed edges do not contribute terms to the curvature at green edge e . In therespective cases, e of positive and negative control, the contribution of each neighbor is signed as indicated.Notice that the resulting formula is not identical with the formula one would obtain by mapping theoriginal directed, weighted, signed-control network onto a directed, signed-weighted network and calculat-ing the FR curvature. In that case the edges of negative control would always appear with a negative sign,whereas in our case this sign depends also on the control type of the edge e under consideration.The readers familiar with category theory may wish to conceptualize our construction more succinctlyby observing that our signed-control networks can be regarded as generating freely a mathematical categorytogether with a functor to the groupoid {± } (and another functor recording the edge weights). Then thecurvature of an edge (morphism) is a certain sum over composable neighboring morphisms, signed by thevalues of the functor on the composition. This point of view is convenient but not essential.We apply the signed-control curvature to the directed signed-control network of E. Coli gene productsand their transcription regulation relations, obtained from the Alon website [3]. Table 2 provides averageFR curvature on undirected, directed, and directed signed control forms of
E. Coli network. Moreover, thesenotions are provided for five operons of the network with highest degree (sum of In-degree and out-degree).A visualization of the largest connected component of the network is presented in Figure 3. As it is shownin the table considering the directions and the signs associated to these directions can have a great impacton the curvature values.
We are also interested in the relationship between Ricci curvature and common measures of a network. Westudy the correlation between Ricci curvature and two important network measures, namely, betweennesscentrality and clustering coefficient.Betweenness centrality (of a node) quantifies the number of times a node acts as a bridge along thegeodesic path between two other nodes. More precisely, betweenness centrality of a vertex v is defined13igure 3: The largest connected component of the E. Coli transcription network (via Gephi). The networkconsists of 423 nodes and 578 edges. The edges are: 58% activator, 37% repressor and 5% dual (bothactivator and repressor). The dual edges are considered as activators in the calculation of signed-controlcurvature. Here, the size of the nodes are proportional to the node-degree and the colors of the nodescorrespond to the scalar signed-control curvature of that node. Also, the color of the edges represents thesign of the regulation. 14able 2: Forman-Ricci curvature has been calculated on E. Coli transcription network in three settings: FRcurvature on the undirected network, directed FR curvature that consider the directions of the edges, andsigned-control FR curvature on the directed and signed-control network. In addition to the average curvatureof the E. Coli network, we provide the curvature of five hubs in the network. The degree (sum of in-degreeand out-degree) of these nodes has been included in parentheses following the name of the operons.
Operon (de-gree) UndirectedFRcurva-ture DirectedFRcur-va-ture Signed-control(FR)curvature
Average curva-ture of network -11.713 -1.008 0.967crp (74) -366 -214 -110yhdG-fis (28) -499 -58 154rpoE-rseABC(26) -713 -62 -62fnr (24) -26 -72 88himA (23) -3 -56 68as [8]: BC v = (cid:88) s (cid:54) = v (cid:54) = t ∈ V δ st ( v ) δ st where δ st is the total number of geodesic paths from node s to node t and δ st ( v ) is the number of those pathsthat pass through v .Figure 4 shows a correlation between three notions of curvature and betweenness centrality in a scalefree network model and the cancer network of breast carcinoma. The correlation is slightly stronger inthe scale free network but overall there is weak correlation between these two measured on the networkwhich suggests that they are measuring different assets in the network. It is also interesting to note that theoverall shape of the correlation plot and correlation coefficient value is very similar with different discretenotions of Ricci curvature for the two networks. This can also help us to understand that these curvaturesare presenting the same measure on the complex networks.On the other hand, a clustering coefficient is a measure of the relative local frequency of triangles withina graph. The local clustering coefficient of a vertex v in graph is introduced by Watts-Strogatz [55] asfollows: C v = number of triangles connected to node v number of triples centered around node v where a triple centered around node v is a set of two edges connected to node v (If the degree of node v is 0or 1, which gives us C v = 0 / , we set C v = 0 ). By definition, ≤ C v ≤ .As discussed by Jost and Liu [26], local clustering coefficients can define a lower boundary for theOllivier-Ricci curvature. In fact, they discuss that a lower Ricci curvature bound prevents geodesics fromdiverging too fast and balls from growing too fast in volume. An analogue of geodesics starting in differentdirections, but eventually approaching each other, could be a triangle on a graph. Therefore, they expectthat the Ricci curvature on a graph be related to the relative abundance of triangles. However, in practice15 a)(b) Figure 4: (a) Scale-free Network (b) Breast Cancer Network, (Pearson) correlation between three notionsof curvature and betweenness centrality for each node of the network. The shapes of the correlation andthe values of the correlation coefficient (R) are similar with these three different notions of discrete Riccicurvature.the correlation between clustering coefficient and Ricci curvature on a graph is not very strong. This can beseen in Fig. 5. Again, this suggests that these two are in fact measuring the network differently. Also, theplots for each network with three different Ricci curvature look pretty similar.
We believe that Ricci curvature can help us to study the robustness of complex networks. Consequently, wecan quantify the differences between normal and robust networks by computing their Ricci curvature. Wereviewed three techniques that can be used for measuring the Ricci curvature on networks. Our methods arevery much intertwined with the theory of optimal mass transport. In our findings, all seven studied cancernetworks have shown a higher average Ricci curvature than the normal complementary networks for allmethods of computation. There is not a very strong correlation between Ricci curvature and betweennesscentrality nor the clustering coefficient. However, all of the results are consistent with three different notionsof discrete Ricci curvature on networks. Therefore, understanding the geometry of the networks through theRicci curvature yields a novel measurement which is connected to the entropy and robustness of networks.The first point of interest for future work is to repeat this study on larger networks, using data consistingof both cancer and non-cancer-related genes. Ranking the nodes by their Ricci curvature, we would expect16 a)(b)
Figure 5: (a) Scale-free Network (b) Breast Cancer Network, (Pearson) correlation between three notionsof curvature and clustering coefficient for each node of the network. The shapes of the correlation andthe values of the correlation coefficient (R) are similar with these three different notions of discrete Riccicurvature.to find the cancer-related genes clustered at the top of the list and the non-cancer-related genes clustered atthe bottom. This would provide additional support for our definition of Ricci curvature and any unexpectedrankings could possibly provide some new insight. Furthermore, we are interested in generalizing thiswork from scalar-valued densities to the vector-valued ones. We are interested in applying some recentresults [9] on “vector-valued” mass transport theory to “multiomics” cancer networks in which both proteinand genomic data are combined rather than treated as independent entities. We also want to derive morecomputationally efficient methods for computing the 1-Wasserstein metric as a clustering technique that hasfound some success in a recent work on sarcoma [10].In summary, our study shows the consistency of the results using three very different discrete analoguesof Ricci curvature on biological cancer networks. Perhaps more importantly, we introduced the notion of signed-control curvature on (directed) signed-control networks which can be applied to many real biologi-cal networks equipped with positive/ negative controls and directions.
Acknowledgements
This project was supported by AFOSR grants (FA9550-15-1-0045 and FA9550-17-1-0435), ARO grant(W911NF-17-1-049), grants from the National Center for Research Resources (P41-RR-013218) and the17ational Institute of Biomedical Imaging and Bioengineering (P41-EB-015902), NCI grant (1U24CA18092401A1),NIA grant (R01 AG053991), MSK Cancer Center Support Grant/Core Grant (P30 CA008748), and a grantfrom the Breast Cancer Research Foundation.
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Scientific Reports (2012).20 eneRank-ing Gene A Gene B ∆∆
Scientific Reports (2012).20 eneRank-ing Gene A Gene B ∆∆ ORCurvature(Cancer-Normal)1 RNF43 RSPO3 0.35042 RNF43 RSPO2 0.34443 ERG ETV1 0.30124 GPC3 PTCH1 0.30015 SDC4 GPC3 0.29016 POT1 SBDS 0.27967 FGFR2 KDR 0.25388 ERG FOXA1 0.24609 SDC4 EXT1 0.241010 MYC SDHD 0.240811 TAL1 RUNX1 0.216712 SDC4 EXT2 0.216513 NUP214 ELN 0.213214 TAL1 TCF3 0.212315 PDGFB COL2A1 0.203616 IDH1 IDH2 0.201217 SDHB HMGA1 0.200718 TAL1 TRIM27 0.192919 EPS15 MLLT4 0.190920 FUBP1 PICALM 0.1899Table 3: The top 20 pairs of genes with respect to Olivier-Ricci curvature.21 eneRank-ing
Gene ∆ BERCurvature(Cancer-Normal) GeneRank-ing Gene ∆ BERCurvature(Cancer-Normal)1 PICALM 7.4910 21 PDGFRB 2.87962 CLTCL1 4.9102 22 JAK2 2.76673 EPS15 4.3210 23 RPN1 2.66854 KIF5B 4.1284 24 DCTN 2.43045 CLTC 4.0657 25 TBL1XR1 2.39596 PTPN11 4.0465 26 ABL1 2.39437 YWHAE 3.8416 27 PIM1 2.38798 EGFR 3.8357 28 PBRM1 2.34549 JAK1 3.7590 29 TFRC 2.316210 MSN 3.6079 30 NDRG1 2.225111 CDC73 3.5274 31 LCK 2.178812 PIK3CA 3.4499 32 KIT 2.160213 XPO1 3.4274 33 FGFR1 2.136714 ALDH2 3.3854 34 STAT5B 2.050615 SDHB 3.2626 35 ERG 2.044116 GNAS 3.1372 36 KDR 2.038717 AKT1 3.1279 37 PPARG 2.003418 MAP2K1 3.0754 38 SYK 1.991919 CBL 3.0287 39 HIP1 1.989720 PML 3.0043 40 CUX1 1.9819Table 4: The top 40 genes with respect to local BER curvature.22 eneRank-ing
Gene A Gene B ∆∆