Topological implications of negative curvature for biological and social networks

Transcription

1 Topological implications of negative curvature for biological and social networks Bhaskar DasGupta Department of Computer Science University of Illinois at Chicago Chicago, IL November 29, 2014 Joint work with Réka Albert (Penn State) Nasim Mobasheri (UIC) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

2 Outline of talk 1 Introduction 2 Basic definitions and notations 3 Computing hyperbolicity for real networks 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

3 Introduction Various network measures Graph-theoretical analysis leads to useful insights for many complex systems, such as World-Wide Web social network of jazz musicians metabolic networks protein-protein interaction networks Examples of useful network measures for such analyses degree based, e.g. maximum/minimum/average degree, degree distribution, connectivity based, e.g. clustering coefficient, largest cliques or densest sub-graphs, geodesic based, e.g. diameter, betweenness centrality, other more complex measures Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

4 Introduction network curvature as a network measure network measure for this talk network curvature via (Gromov) hyperbolicity measure originally proposed by Gromov in 1987 in the context of group theory observed that many results concerning the fundamental group of a Riemann surface hold true in a more general context defined for infinite continuous metric space with bounded local geometry via properties of geodesics can be related to standard scalar curvature of Hyperbolic manifold adopted to finite graphs using a so-called 4-node condition Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

5 Outline of talk 1 Introduction 2 Basic definitions and notations 3 Computing hyperbolicity for real networks 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

6 Basic definitions and notations Graphs, geodesics and related notations Graphs, geodesics and related notations G= G = (V,E) connected undirected graph of n 4 nodes ) between nodes u and v u P v path P ( u 0,u 1,...,u k 1,u k =u = v l(p ) length (number of edges) of the path u P v P u i u j sub-path ( ) u ui i,u i+1,...,u j of P between nodes ui u i and u j u s v a shortest path between nodes u and v length of a shortest path between nodes u and v d u,v u 2 u 4 P u 2 u6 P u 2 u6 is the path P ( ) u 2,u 4,u 5,u 6 u 1 u 3 u 5 u 6 l(p )= = 3 d u2,u 6 d u2,u 6 = 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

7 Basic definitions and notations 4 node condition (Gromov, 1987) d u2,u 4 Consider four nodes u 1,u 2,u 3,u 4 and the six shortest paths among pairs of these nodes d u2,u 3 u 2 u 1 u 4 u 3 d u1,u 4 d u1,u 2 d u3,u 4 d u3,u 4 d u1,u 3 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

8 Basic definitions and notations 4 node condition (Gromov, 1987) d u2,u 4 Consider four nodes u 1,u 2,u 3,u 4 and the six shortest paths among pairs of these nodes Assume, without loss of generality, that d u1,u 4 + d u2,u 3 d u1,u }{{} 3 + d u2,u4 d u1,u }{{} 2 + d u3,u }{{} 4 =L =M =S d u2,u 3 u 2 u 1 u 4 u 3 d u1,u 2 d u3,u 4 d u1,u 3 d u1,u 4 d u3,u Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

9 Basic definitions and notations 4 node condition (Gromov, 1987) d u2,u 4 Consider four nodes u 1,u 2,u 3,u 4 and the six shortest paths among pairs of these nodes Assume, without loss of generality, that d u1,u 4 + d u2,u 3 d u1,u }{{} 3 + d u2,u4 d u1,u }{{} 2 + d u3,u }{{} 4 =L =M =S d u2,u 3 u 2 u 1 u 4 u 3 d u1,u 2 d u3,u 4 d u1,u 3 d u1,u 4 d u3,u Let δ u1,u 2,u 3,u 4 = L M 2 + ( + ) 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

10 Basic definitions and notations 4 node condition (Gromov, 1987) d u2,u 4 Consider four nodes u 1,u 2,u 3,u 4 and the six shortest paths among pairs of these nodes Assume, without loss of generality, that d u1,u 4 + d u2,u 3 d u1,u }{{} 3 + d u2,u4 d u1,u }{{} 2 + d u3,u }{{} 4 =L =M =S d u2,u 3 u 2 u 1 u 4 u 3 d u1,u 2 d u3,u 4 d u1,u 3 d u1,u 4 d u3,u Let δ u1,u 2,u 3,u 4 = L M 2 + ( + ) 2 Definition (hyperbolicity of G) { } δ(g) = max δu1,u u 1,u 2,u 3,u 2,u 3,u 4 4 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

11 Basic definitions and notations Hyperbolic graphs (graphs of negative curvature) Definition ( -hyperbolic graphs) G is -hyperbolic provided δ(g) Definition (Hyperbolic graphs) If is a constant independent of graph parameters, then a -hyperbolic graph is simply called a hyperbolic graph Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

12 Basic definitions and notations Hyperbolic graphs (graphs of negative curvature) Definition ( -hyperbolic graphs) G is -hyperbolic provided δ(g) Definition (Hyperbolic graphs) If is a constant independent of graph parameters, then a -hyperbolic graph is simply called a hyperbolic graph Example (Hyperbolic and non-hyperbolic graphs) Tree: (G)=00 hyperbolic graph Chordal (triangulated) graph: (G)=1/2 hyperbolic graph Simple cycle: (G)= n/4 = non-hyperbolic graph n= 10 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

13 Basic definitions and notations Hyperbolicity of real-world networks Are there real-world networks that are hyperbolic? Yes, for example: Preferential attachment networks were shown to be scaled hyperbolic [Jonckheere and Lohsoonthorn, 2004; Jonckheere, Lohsoonthorn and Bonahon, 2007] Networks of high power transceivers in a wireless sensor network were empirically observed to have a tendency to be hyperbolic [Ariaei, Lou, Jonckeere, Krishnamachari and Zuniga, 2008] Communication networks at the IP layer and at other levels were empirically observed to be hyperbolic [Narayan and Saniee, 2011] Extreme congestion at a very limited number of nodes in a very large traffic network was shown to be caused due to hyperbolicity of the network together with minimum length routing [Jonckheerea, Loua, Bonahona and Baryshnikova, 2011] Topology of Internet can be effectively mapped to a hyperbolic space [Bogun, Papadopoulos and Krioukov, 2010] Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

14 Basic definitions and notations Average hyperbolicity measure, computational issues Definition (average hyperbolicity) δ ave (G)= ( 1 n 4) expected value of δ u1,u δ 2,u 3,u 4 if u1,u 2,u 3,u 4 u 1,u 2,u 3,u 4 are picked uniformly at random δ u1,u 2,u 3,u 4 u 1,u 2,u 3,u 4 δ u1,u 2,u 3,u 4 Computation of δ(g) and δ ave (G) Trivially in O ( n 4) time Floyd Warshall algorithm Compute all-pairs shortest paths O ( n 3) time For each combination u 1,u 2,u 3,u 4, compute δ u1,u 2,u 3,u 4 O ( n 4) time Open problem: can we compute in O ( n 4 ε) time? Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

15 Outline of talk 1 Introduction 2 Basic definitions and notations 3 Computing hyperbolicity for real networks 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

16 Computing hyperbolicity for real networks Direct calculation Real networks used for empirical validation 20 well-known biological and social networks 11 biological networks that include 3 transcriptional regulatory, 5 signalling, 1 metabolic, 1 immune response and 1 oriented protein-protein interaction networks 9 social networks range from interactions in dolphin communities to the social network of jazz musicians hyperbolicity of the biological and directed social networks was computed by ignoring the direction of edges hyperbolicity values were calculated by writing codes in C using standard algorithmic procedures Next slide: List of 20 networks Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

17 Computing hyperbolicity for real networks Direct calculation 11 biological networks # nodes # edges 1. E. coli transcriptional Mammalian signaling E. coli transcriptional T-LGL signaling S. cerevisiae transcriptional C. elegans metabolic Drosophila segment polarity (6 cells) 8. ABA signaling Immune response network T cell receptor signaling Oriented yeast PPI social networks # nodes # edges 1. Dolphin social network American College Football Zachary Karate Club Books about US politics Sawmill communication network 6. Jazz Musician network Visiting ties in San Juan World Soccer Data, Paris Les Miserables characters Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

18 Computing hyperbolicity for real networks Direct calculation Biological networks social networks Average degree δ ave δ 1. E. coli transcriptional Mammalian Signaling E. Coli transcriptional T LGL signaling S. cerevisiae transcriptional C. elegans Metabolic Drosophila segment polarity ABA signaling Immune Response Network T Cell Receptor Signalling Oriented yeast PPI Average degree δ ave δ 1. Dolphins social network American College Football Zachary Karate Club Books about US Politics Sawmill communication Jazz musician Visiting ties in San Juan World Soccer data, Les Miserable Hyperbolicity values of almost all networks are small For all networks δ ave is one or two orders of magnitude smaller than δ Intuitively, this suggests that value of δ may be a rare deviation from typical values of δ u1,u 2,u 3,u 4 for most combinations of nodes {u 1,u 2,u 3,u 4 } No systematic dependence of δ on number of nodes/edges or average degree Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

19 Computing hyperbolicity for real networks Direct calculation Definition (Diameter of a graph) { } D= = max u,v du,v longest shortest path Fact δ D/2 δ small diameter implies small hyperbolicity We found no systematic dependence of δ on D Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

20 Computing hyperbolicity for real networks Direct calculation Definition (Diameter of a graph) { } D= = max u,v du,v longest shortest path Fact δ D/2 δ small diameter implies small hyperbolicity We found no systematic dependence of δ on D For more rigorous checks of hyperbolicity of finite graphs and for evaluation of statistical significance of the hyperbolicity measure see our paper R. Albert, B. DasGupta and N. Mobasheri, Topological implications of negative curvature for biological and social networks. Physical Review E 89(3), (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

21 Outline of talk 1 Introduction 2 Basic definitions and notations 3 Computing hyperbolicity for real networks 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

22 We discuss topological implications of hyperbolicity somewhat informally Precise Theorems and their proofs are available in our paper R. Albert, B. DasGupta and N. Mobasheri, Topological implications of negative curvature for biological and social networks. Physical Review E 89(3), (2014) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

23 Outline of talk 1 Introduction 2 Basic definitions and notations 3 Computing hyperbolicity for real networks 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

24 Hyperbolicity and crosstalk in regulatory networks Definition (Path chord and chord) u 5 u 0 u 5 u 0 u 4 p a t h - c h o r d v u 1 u 4 c h o r d u 1 u 3 u 2 u 3 u 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

25 Hyperbolicity and crosstalk in regulatory networks Definition (Path chord and chord) u 5 u 0 u 5 u 0 u 4 p a t h - c h o r d v u 1 u 4 c h o r d u 1 u 3 u 2 u 3 u 2 Theorem (large cycle without path-chord imply large hyperbolicity) G has a cycle of k nodes which has no path-chord Corollary δ δ k/4 k/4 Any cycle containing more than 4δ nodes must have a path-chord Example δ<1 G is chordal graph Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

26 Hyperbolicity and crosstalk in regulatory networks An example of a regulatory network Network associated to the Drosophila segment polarity G. von Dassow, E. Meir, E. M. Munro and G. M. Odell, Nature 406, (2000) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

27 Hyperbolicity and crosstalk in regulatory networks Hyperbolicity and crosstalk in regulatory networks short-cuts in long feedback loops node regulates itself through a long feedback loop this loop must have a path-chord a shorter feedback cycle through the same node interpreting chord or short path-chord as crosstalk source regulates target through two long paths must exist a crosstalk path between these two paths c r o s s t a l k a node source target number of crosstalk paths increases at least linearly with total length of two paths Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

28 Outline of talk 1 Introduction 2 Basic definitions and notations 3 Computing hyperbolicity for real networks 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

29 Geodesic triangles and crosstalk paths Geodesic triangles and crosstalk paths u 1 u 0 u 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

30 Geodesic triangles and crosstalk paths Geodesic triangles and crosstalk paths u 1 shortest path u 0 u 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

31 Geodesic triangles and crosstalk paths Geodesic triangles and crosstalk paths du0,u d u0,u0,1 = 1 + d u0,u2 d u1,u2 2 du1,u d u1,u0,1 = 2 + d u1,u0 d u2,u0 2 u 1 shortest path d u1,u 0,1 =d u1,u 1,2 d u0,u 0,1 =d u0,u 0,2 u 0,1 d u2,u 0,2 =d u2,u 1,2 u 1,2 u 0 u 0,2 u 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

32 Geodesic triangles and crosstalk paths Geodesic triangles and crosstalk paths { } v in one path v in the other path such that d v,v max 6δ, 2 du0,u d u0,u0,1 = 1 + d u0,u2 d u1,u2 2 u 1 shortest path du1,u d u1,u0,1 = 2 + d u1,u0 d u2,u0 2 d u1,u 0,1 =d u1,u 1,2 d u0,u 0,1 =d u0,u 0,2 u 0,1 d v,v d u2,u 0,2 =d u2,u 1,2 u 1,2 d v,v d v,v u 0 u 0,2 u 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

33 Implications of geodesic triangles and crosstalk paths for regulatory networks Implications of geodesic triangles for regulatory networks Consider feedback or feed-forward loop formed by the shortest paths among three nodes We can expect short cross-talk paths between these shortest paths Feedback/feed-forward loop is nested with additional feedback/feed-forward loops Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

34 Implications of geodesic triangles and crosstalk paths for regulatory networks Implications of geodesic triangles for regulatory networks Consider feedback or feed-forward loop formed by the shortest paths among three nodes We can expect short cross-talk paths between these shortest paths Feedback/feed-forward loop is nested with additional feedback/feed-forward loops Empirical evidence [R. Albert, Journal of Cell Science 118, (2005)] Network motifs a are often nested Two generations of nested assembly for a common E. coli motif [DeDeo and Krakauer, 2012] a e.g., feed-forward or feedback loops of small number of nodes Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

35 Hausdorff distance between shortest paths Definition (Hausdorff distance between two paths P 1 P 1 and P 2 ) { d H (P 1,P 2 ) def { { } = max max min d v1,v 2 }, } max min d v1,v 2 v 1 P 1 v 2 P 2 v 2 P 2 v 1 P 1 small Hausdorff distance implies every node of either path is close to some node of the other path Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

36 Hausdorff distance between shortest paths Definition (Hausdorff distance between two paths P 1 P 1 and P 2 ) { d H (P 1,P 2 ) def { { } = max max min d v1,v 2 }, } max min d v1,v 2 v 1 P 1 v 2 P 2 v 2 P 2 v 1 P 1 small Hausdorff distance implies every node of either path is close to some node of the other path d v,v d v,v u 0 u 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

37 Hausdorff distance between shortest paths Definition (Hausdorff distance between two paths P 1 P 1 and P 2 ) { d H (P 1,P 2 ) def { { } = max max min d v1,v 2 }, } max min d v1,v 2 v 1 P 1 v 2 P 2 v 2 P 2 v 1 P 1 small Hausdorff distance implies every node of either path is close to some node of the other path P 2 d v,v d v,v u 0 u 2 P 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

38 Hausdorff distance between shortest paths Definition (Hausdorff distance between two paths P 1 P 1 and P 2 ) { d H (P 1,P 2 ) def { { } = max max min d v1,v 2 }, } max min d v1,v 2 v 1 P 1 v 2 P 2 v 2 P 2 v 1 P 1 small Hausdorff distance implies every node of either path is close to some node of the other path P 2 d H (P 1,P 2 ) max { 6δ, 2 } d v,v d v,v u 0 u 2 P 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

39 Hausdorff distance between shortest paths this result versus our previous path-chord result path-chord result P 2 this result d H (P 1,P 2 ) max{ 6δ, 2 } P 2 u 0 u 2 P 1 long cycle there is a path chord d v,v u 0 u 2 P 1 d v,v Which result is more general in nature? Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

40 Hausdorff distance between shortest paths this result versus our previous path-chord result path-chord result P 2 u 0 u 2 P 1 long cycle there is a path chord u 0 this result d H (P 1,P 2 ) max{ 6δ, 2 } d v,v P 2 u 0 u 2 P 1 d v,v u 2 Which result is more general in nature? Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

41 A notational simplification A notational simplification unless G is a tree or a complete graph (K K n ), δ>0 δ>0 δ 1/2 δ 1/2 max{ { 6δ, 2 }=6δ Hence, we will simply write 6δ instead of max{ { 6δ, 2 } Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

42 Distance between geodesic and arbitrary path s P Distance from a shortest path u 0 u 1 to another arbitrary path u 0 u1 n is the number of nodes in the graph l(p ) is length of path P P u 0 P u 0 u1 arbitrary node u 0 v u 1 s shortest path u 0 u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

43 Distance between geodesic and arbitrary path s P Distance from a shortest path u 0 u 1 to another arbitrary path u 0 u1 n is the number of nodes in the graph l(p ) is length of path P P u 0 P u 0 u1 v d v,v 6δ log 2 l(p ) }{{} < 6δ log 2 n O(logn) if δ is constant v d v,v d v,v arbitrary node u 0 v u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

44 Distance between geodesic and arbitrary path An interesting implication of this bound v d v,v 6δ log 2 l(p ) P v d v,v d v,v u 0 v u 1 shortest path Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

45 = γ Implications of hyperbolicity Distance between geodesic and arbitrary path An interesting implication of this bound assume v P d v,v γ P u 0 v u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

46 = γ Implications of hyperbolicity Distance between geodesic and arbitrary path An interesting implication of this bound P assume v P d v,v γ γ ( l(p ) 2 6δ = Ω (2 2 Ω(γ)) if δ is constant u 0 v u 1 Next: better bounds for approximately short paths Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

47 Approximately short path Why consider approximately short paths? Regulatory networks Up/down-regulation of a target node is mediated by two or more close to shortest paths starting from the same regulator node Additional very long paths between the same regulator and target node do not contribute significantly to the target node s regulation source target Definition ε-additive-approximate short path P l(p ) length of P length of shortest path +ε Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

48 Approximately short path Why consider approximately short paths? Algorithmic efficiency reasons Approximate short path may be faster to compute as opposed to exact shortest path Routing and navigation problems (traffic networks) Routing via approximate short path P Definition µ-approximate short path u 0 uk = ( ) u 0,u 1,...,u k l ( P ) u i u j µ length of sub-path from u i to u j d ui,u j distance between u i and u j for all 0 i < j k v1 v3 2-approximate path u 0 u P 7 = ( ) u 0,u 1,...,u 7 s shortest path u 0 u 7 u 0 u0=v0 u1 u2 u4 u5 u6 u7=v4 u3=v2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

49 Distance between geodesic and approximately short path Distance from shortest path to an approximately short }{{} ε-additive approximate or, µ-approximate P P u 0 u1 P path u 0 u1 v d v,v arbitrary node u 0 v u 1 s shortest path u 0 u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

50 Distance between geodesic and approximately short path Distance from shortest path to an approximately short }{{} ε-additive approximate or, µ-approximate P P u 0 u1 P path u 0 u1 v d v,v arbitrary node u 0 v u 1 s shortest path u 0 u 1 P u 0 u1 is ε-additive approximate ( ) ( v v d v,v 6δ+2 log 2 8 ( 6δ+2 ) [ ] ) log 2 (6δ+2) (4+2ε) + 1+ ε 2 O ( δlog ( ε+δ logε )) depends only on δ and ε short crosstalk path for small ε and δ Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

51 Distance between geodesic and approximately short path Distance from shortest path to an approximately short }{{} ε-additive approximate or, µ-approximate P P u 0 u1 P path u 0 u1 v d v,v arbitrary node u 0 v u 1 s shortest path u 0 u 1 P u 0 u1 is µ-approximate ( ) ( (6µ+2 v v )( ) [ ( ) ] ) d v,v 6δ+2 log 2 6δ+2 log2 (6δ+2) 3µ+1 µ +µ O ( δ log ( µδ )) depends only on δ and µ short crosstalk path for small µ and δ Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

52 Distance between geodesic and approximately short path Contrast the new bounds with the old bound of d v,v = O ( δ logl(p ) ) d u0,u 1 is the length of a shortest path between u 0 and u 1 u 0 u 1 P u 0 u1 P u 0 u1 is ε-additive approximate l(p ) d u0,u 1 + ε P u 0 u1 P u 0 u1 is µ-approximate l(p ) µd u0,u 1 Old bound O ( δ log ( )) ε+d u0,u 1 Old bound O ( δ ( log ( ))) µd u0,u 1 New bound O ( δlog ( ε+δ logε )) no dependency on d u0,u 1 New bound O ( δ log ( µδ )) no dependency on d u0,u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

53 Distance between geodesic and approximately short path P Distance from an approximately short path u }{{} 0 u1 to a shortest path ε-additive approximate or, µ-approximate for simplified exposition, we show bounds only in asymptotic O( ) notation please refer to our paper for more precise bounds P P u 0 u1 v arbitrary node d v,v u 0 v u 1 s shortest path u 0 u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

54 Distance between geodesic and approximately short path P Distance from an approximately short path u }{{} 0 u1 to a shortest path ε-additive approximate or, µ-approximate for simplified exposition, we show bounds only in asymptotic O( ) notation please refer to our paper for more precise bounds P P u 0 u1 v arbitrary node d v,v u 0 v u 1 s shortest path u 0 u 1 P u 0 u1 is ε-additive approximate v v v d v,v O( O ( ε+δlog ( ε+δ logε )) depends only on δ and ε Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

55 Distance between geodesic and approximately short path P Distance from an approximately short path u }{{} 0 u1 to a shortest path ε-additive approximate or, µ-approximate for simplified exposition, we show bounds only in asymptotic O( ) notation please refer to our paper for more precise bounds P P u 0 u1 v arbitrary node d v,v u 0 v u 1 s shortest path u 0 u 1 P u 0 u1 is µ-approximate v v v d v,v O( O ( µδ log ( µδ )) depends only on δ and µ Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

56 Distance between geodesic and approximately short path P 1 Distance from approximate short path P }{{ 1 } arbitrary node v P 2 to approximate short path P }{{ 2 } nearest node v P 2 u 0 u 1 arbitrary node v P 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

57 Distance between geodesic and approximately short path P 1 Distance from approximate short path P }{{ 1 } arbitrary node v P 2 to approximate short path P }{{ 2 } nearest node v P 2 go to any shortest path v u 0 u 1 shortest path arbitrary node v P 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

58 Distance between geodesic and approximately short path P 1 Distance from approximate short path P }{{ 1 } arbitrary node v P 2 to approximate short path P }{{ 2 } nearest node v P 2 v continue to the other path v u 0 u 1 shortest path arbitrary node v P 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

59 Distance between geodesic and approximately short path P 1 P 2 Distance from approximate short path P 1 to approximate short path P 2 }{{}}{{} arbitrary node v nearest node v we sometimes overestimate quantities to simplify expression P 1 is ε 1 -additive approximate P 2 is ε 2 -additive approximate O ( ε (ε δlog(ε ε 2 )+δlogδ ) P 1 is ε-additive approximate P 2 is µ-approximate O ( ε+δlog ( εµ ) + δ 2 loglogε logε ) P 1 is µ-approximate P 2 is ε-additive approximate O ( µδlog ( µδ ) + ε+δlogε ε ) P 1 is µ 1 -approximate P 2 is µ 2 -approximate O ( µ 1 δlog ( µ 1 δ ) ) + δlogµ 2 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

60 Distance between geodesic and approximately short path Interesting implications of these improved bounds approximately short path P v d v,v d v,v u 0 v u 1 shortest path Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

61 Distance between geodesic and approximately short path assume v P d v,v γ Interesting implications of these improved bounds approximately short path P = γ u 0 v u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

62 Distance between geodesic and approximately short path Interesting implications of these improved bounds assume v P d v,v γ if P is ε-additive-approximate short then ( ) 2 ε=ω Ω γ /δ logδ δ approximately short path P = γ u 0 v u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

63 Distance between geodesic and approximately short path Interesting implications of these improved bounds assume v P d v,v γ if P is µ-approximate short then ( ) 2 µ=ω γ /δ γ approximately short path P = γ u 0 v u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

64 Distance between geodesic and approximately short path To wrap it up, approximate shortest paths look like the following cartoon Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

65 Distance between geodesic and approximately short path To wrap it up, approximate shortest paths look like the following cartoon Interpretation for regulatory networks It is reasonable to assume that, when up- or down-regulation of a target node is mediated by two or more approximate short paths starting from the same regulator node, additional very long paths between the same regulator and target node do not contribute significantly to the target node s regulation We refer to the short paths as relevant, and to the long paths as irrelevant Then, our finding can be summarized by saying that almost all relevant paths between two nodes have crosstalk paths between each other Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

66 Outline of talk 1 Introduction 2 Basic definitions and notations 3 Computing hyperbolicity for real networks 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

67 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ 4 α>0 r > 3(κ 2) δ Example: 5 1 9δ+1 u 0 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

68 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ 4 α>0 r > 3(κ 2) δ Example: 5 1 9δ+1 u 2 r u 0 3κδ u 1 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

69 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ 4 α>0 r > 3(κ 2) δ Example: 5 1 9δ+1 r + α u 2 α u 3 r u 0 3κδ u 1 α u 4 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

70 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ 4 α>0 r > 3(κ 2) δ Example: 5 1 9δ+1 consider any shortest path P between u 3 and u 4 P must look like this u 4 α u 3 u 2 r u 0 3κδ u 1 α u 4 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

71 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ 4 α>0 r > 3(κ 2) δ Example: 5 1 9δ+1 consider any shortest path P between u 3 and u 4 P must look like this u 4 α u 3 γ=d u0,v r ( 3 2 κ 1) δ r Θ(κδ) r u 0 v u 2 P γ u 1 α u 4 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

72 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes integer parameters used in this result κ 4 α>0 r > 3(κ 2) δ Example: 5 1 9δ+1 consider any shortest path P between u 3 and u 4 P must look like this u 4 α u 3 γ=d u0,v r ( 3 2 κ 1) δ r Θ(κδ) l(p ) (3κ 2)δ+2α Ω(κδ+α) r u 0 γ v u 2 u 1 P α u 4 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

73 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes Corollary (of previous results) consider any path P between u 3 and u 4 suppose that P does not intersect the shaded region α u 3 u 2 r u 0 3κδ P u 1 α u 4 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

74 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes Corollary (of previous results) α l(p ) 2 6δ + κ 4 2 Ω( α δ + κ) consider any path P between u 3 and u 4 suppose that P does not intersect the shaded region u 2 α u 3 r u 0 3κδ P u 1 very long path α u 4 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

75 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes Corollary (of previous results) α l(p ) 2 6δ + κ 4 2 Ω( α δ + κ) P ε-additive-approximate ε> 2 α 6δ + κ 4 log 2 (48δ) 48δ Ω ( 2 Θ(α+κ)) if δ is constant r u 0 consider any path P between u 3 and u 4 suppose that P does not intersect the shaded region u 2 u 1 α 3κδ u 3 P very long path α u 4 Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

76 Identifying essential edges and nodes in regulatory networks Influence of a node on the geodesics between other pair of nodes Corollary (of previous results) α l(p ) 2 6δ + κ 4 2 Ω( α δ + κ) P ε-additive-approximate ε> 2 α 6δ + κ 4 log 2 (48δ) 48δ Ω ( 2 Θ(α+κ)) if δ is constant P µ-approximate µ α 2 6δ + κ 4 ( 12α+6δ 3κ 26 ) ( ) 2 Ω Θ ( α δ +κ ) α+κδ r u 0 consider any path P between u 3 and u 4 suppose that P does not intersect the shaded region u 2 u 1 α 3κδ α u 3 u 4 P very long path Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

77 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks u source shortest path u target Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

78 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks u source u middle u target Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

79 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks u source u middle ξ=o(δ) u target Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

80 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks All shortest paths between u source and u target must intersect the ξ-neighborhood Therefore, knocking out nodes in ξ-neighborhood cuts off all shortest regulatory paths between u source and u target u source sho rtest path u middle shortest path u target sh o rtest pa th ξ=o(δ) shortest pat h Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

81 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks But, it gets even more interesting! u source sho rtest path u middle shortest path u target sh o rtest pa th ξ=o(δ) shortest pat h Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

82 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks But, it gets even more interesting! shifting the ξ-neighborhood does not change claim shortest path shortest path u source u target shortest path shortest path Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

83 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks how about enlarging the ξ-neighborhood? u source sho rtest path u middle shortest path u target sh o rtest pa th ξ=o(δ) sho rtest pa t h Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

84 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks how about enlarging the ξ-neighborhood? approximately short paths start intersecting the neighborhood u source approxima te shortest path sho ly short r te st path u middle 2 ξ approxi m ately shortest shortest pat h p a t h short u target Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

85 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks Consider a ball (neighborhood) of radius ξlogn (n is the number of nodes) u source u middle ξ logn u target Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

86 Identifying essential edges and nodes in regulatory networks Interesting implications of these bounds for regulatory networks Consider a ball (neighborhood) of radius ξlogn (n is the number of nodes) All paths intersect the neighborhood u source u middle ξ logn u target Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

87 Identifying essential edges and nodes in regulatory networks Empirical estimation of neighborhoods and number of essential nodes We empirically investigated these claims on relevant paths passing through a neighborhood of a central node for the following two biological networks: E. coli transcriptional T-LGL signaling by selecting a few biologically relevant source-target pairs Our results show much better bounds for real networks compared to the worst-case pessimistic bounds in the mathematical theorems see our paper for further details Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

88 i m a p p i c r Implications of hyperbolicity Identifying essential edges and nodes in regulatory networks The following cartoon informally depicts some of the preceding discussions a x o t e g e o d e s Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

89 Identifying essential edges and nodes in regulatory networks The following cartoon informally depicts some of the preceding discussions t e r o x i m a p p a g e o d e s i c Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

90 Identifying essential edges and nodes in regulatory networks The following cartoon informally depicts some of the preceding discussions t e a p p r o x i m a g e o d e s i c the further we move from the central node the more a shortest path bends inward towards the central node Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

91 Identifying essential edges and nodes in regulatory networks eavesdropper Å «eavesdropper may succeed with limited sensor range eavesdropper need not be a hub Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

92 Identifying essential edges and nodes in regulatory networks Đ Traffic network Đ need not be a hub Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

93 Outline of talk 1 Introduction 2 Basic definitions and notations 3 Computing hyperbolicity for real networks 4 Implications of hyperbolicity of networks Hyperbolicity and crosstalk in regulatory networks Geodesic triangles and crosstalk paths Identifying essential edges and nodes in regulatory networks A social network application Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

94 Effect of hyperbolicity on structural holes in social networks Visual illustration of a well-known social network Zachary s Karate Club ( Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

95 Effect of hyperbolicity on structural holes in social networks Structural hole in a social network [Burt, 1995; Borgatti, 1997] Definition (Adjacency matrix of an undirected unweighted graph) u v au,v { 1, if {u, v} is an edge a u,v = 0, otherwise Definition (measure of structural hole at node u [Burt, 1995; Borgatti, 1997]) v V M u def == max x u a u,v + a v,u { } a u,x + a x,u [ (assume u has degree at least 2) a u,y + a y,u a ( ) v,y + a y,v { } au,x + a x,u av,z + a too complicated z,v 1 y V y u,v x u max z y Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

96 Effect of hyperbolicity on structural holes in social networks Structural hole in a social network [Burt, 1995; Borgatti, 1997] Definition (Adjacency matrix of an undirected unweighted graph) u v au,v { 1, if {u, v} is an edge a u,v = 0, otherwise Definition (measure of structural hole at node u [Burt, 1995; Borgatti, 1997]) (assume u has degree at least 2) Let Nbr(u) be set of nodes adjacent to u M u = Nbr(u) a v,y v,y Nbr(u) Nbr(u) Next: An intuitive interpretation of Mu Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

97 Effect of hyperbolicity on structural holes in social networks An intuitive interpretation ofm M u Definition (weak dominance ρ,λ weak ) Nodes v, y are weakly (ρ,λ)-dominated by node u provided ρ< d u,v,d u,y ρ+λ, and for at least one shortest path P between v and y, P contains a node z such that d u,z ρ Definition (strong dominance ρ,λ strong ) Nodes v, y are strongly (ρ,λ)-dominated by node u provided ρ< d u,v,d u,y ρ+λ, and for every shortest path P between v and y, P contains a node z such that d u,z ρ y λ=2 y λ=2 v ρ=1 u v ρ=1 u Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

98 Effect of hyperbolicity on structural holes in social networks An intuitive interpretation ofm M u Notation (boundary of the ξ-neighborhood of node u) B ξ (u)= { v d u,v = ξ } the set of all nodes at a distance of precisely ξ from u M u Observation number of pairs of nodes = E v, y such that v is selected uniformly ran- v, y is weakly (0, 1 )-dominated by u domly from B j (u) ρ λ 0 < j 1 ρ λ E number of pairs of nodes v, y such that v is selected uniformly ran- v, y is strongly (0,1)-dominated by u domly from B j (u) 0< j 1 always true equality does not hold in general u Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52 y v

99 Effect of hyperbolicity on structural holes in social networks GeneralizeM M u tom M u,ρ,λ for larger ball of influence of a node replace (0,1) by (ρ,λ) M u = E number of pairs of nodes v, y such that v, y is weakly (0, 1 )-dominated by u ρ λ v is selected uniformly randomly from B j (u) 0 < j 1 ρ λ M u,ρ,λ = E number of pairs of nodes v, y such that v is selected uniformly ran- v, y is weakly (ρ,λ)-dominated by u domly from B j (u) ρ< j λ Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

100 Effect of hyperbolicity on structural holes in social networks GeneralizeM M u tom M u,ρ,λ for larger ball of influence of a node replace (0,1) by (ρ,λ) Lemma (equivalence of strong and weak domination) If λ 6δlog 2 n then M u,ρ,λ == def E number of pairs of nodes v, y such that v is selected uniformly ran- v, y is weakly (ρ,λ)-dominated by u domly from B j (u) ρ< j λ = E number of pairs of nodes v, y such that v is selected uniformly ran- v, y is strongly (ρ,λ)-dominated by u domly from B j (u) ρ< j λ equality holds now Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

101 Effect of hyperbolicity on structural holes in social networks Lemma (equivalence of strong and weak domination) If λ 6δ log 2 n then M u,ρ,λ == def E number of pairs of nodes v, y such that v is selected uniformly ran- v, y is weakly (ρ,λ)-dominated by u domly from B j (u) ρ< j λ = E number of pairs of nodes v, y such that v is selected uniformly ran- v, y is strongly (ρ,λ)-dominated by u domly from B j (u) ρ< j λ What does this lemma mean intuitively? Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

102 Effect of hyperbolicity on structural holes in social networks What does this lemma mean intuitively? B ρ (u) ρ u λ 6δ log 2 n B ρ+λ (u) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

103 Effect of hyperbolicity on structural holes in social networks What does this lemma mean intuitively? B ρ (u) ρ u λ 6δ log 2 n v Plato either all the shortest paths are completely inside B ρ+λ (u) y Socrates B ρ+λ (u) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

104 Effect of hyperbolicity on structural holes in social networks What does this lemma mean intuitively? or all the shortest paths are completely outside of B ρ+λ (u) B ρ (u) ρ u λ 6δ log 2 n v Plato y Socrates B ρ+λ (u) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

105 Effect of hyperbolicity on structural holes in social networks What does this lemma mean intuitively? but not both! B ρ (u) ρ u λ 6δ log 2 n v Plato y Socrates B ρ+λ (u) Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52

106 Final slide Thank you for your attention Questions?? Bhaskar DasGupta (UIC) Negative curvature for networks November 29, / 52