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When Affinity Meets Resistance On the Topological Centrality of Edges in Complex Networks. Gyan Ranjan University of Minnesota, MN [Collaborators: Zhi-Li Zhang and Hesham Mekky.] . IMA International workshop on Complex Systems and Networks, 2012. . Overview. Motivation Geometry of networks
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When Affinity Meets ResistanceOn the Topological Centrality of Edges in Complex Networks Gyan Ranjan University of Minnesota, MN [Collaborators: Zhi-Li Zhang and Hesham Mekky.] IMA International workshop on Complex Systems and Networks, 2012.
Overview • Motivation • Geometry of networks • n-dimensional embedding • Bi-partitions of a graph • Connectivity within and across partitions • Random detours • Overhead • Real-world networks and applications IMA International workshop on Complex Systems and Networks, 2012.
Overview • Motivation • Geometry of networks • n-dimensional embedding • Bi-partitions of a graph • Connectivity within and across partitions • Random detours • Overhead • Real-world networks and applications IMA International workshop on Complex Systems and Networks, 2012.
Motivation • Complex networks • Study of entities and inter-connections • Applicable to several fields • Biology, structural analysis, world-wide-web • Notion of centrality • Position of entities and inter-connections • Page-rank of Google • Utility • Roles and functions of entities and inter-connections • Structure determines functionality IMA International workshop on Complex Systems and Networks, 2012.
Cart before the Horse • Centrality of nodes: Red to blue to white, decreasing order [1]. IMA International workshop on Complex Systems and Networks, 2012. Western states power grid Network sciences co-authorship
State of the Art • Node centrality measures • Degree, Joint-degree • Local influence • Shortest paths based • Random-walks based • Page Rank • Sub-graph centrality • Edge centrality • Shortest paths based [Explicit] • Combination of node centralities of end-points [Implicit] • Joint degree across the edge • Our approach • A geometric and topological view of network structure • Generic, unifies several approaches into one IMA International workshop on Complex Systems and Networks, 2012.
Overview • Motivation • Geometry of networks • n-dimensional embedding • Bi-partitions of a graph • Connectivity within and across partitions • Random detours • Overhead • Example and real-world networks IMA International workshop on Complex Systems and Networks, 2012.
Definitions • Network as a graph G(V, E) • Simple, connected and unweighted [for simplicity] • Extends to weighted networks/graphs • wij is the weight of edge eij • Topological dimensions • |V(G)| = n [Order of the graph] • |E(G)| = m [Number of edges] • Vol(G) = 2 m [Volume of the graph] • d(i) = Degree of node i IMA International workshop on Complex Systems and Networks, 2012.
The Graph and Algebra • For a graph G(V, E) • [A]nxn = Adjacency matrix of G(V, E) • aij = 1 if in E(G), 0 otherwise • [D] nxn = Degree matrix of G(V, E) • [L] nxn = D – A = Laplacian matrix of G(V, E) • Structure of L • Symmetric, centered and positive semi-definite • L • U • Lambda IMA International workshop on Complex Systems and Networks, 2012.
Geometry of Networks • The Moore-Penrose pseudo-inverse of L • Lp • where • In this n-dimensional space [2]: • x • x • x IMA International workshop on Complex Systems and Networks, 2012.
Overview • Motivation • Geometry of networks • n-dimensional embedding • Bi-partitions of a graph • Connectivity within and across partitions • Random detours • Overhead • Real-world networks and applications IMA International workshop on Complex Systems and Networks, 2012.
Bi-Partitions of a Network • Connected bi-partitions of G(V, E) • P(S, S’): a cut with two connected sub-graphs • V(S), V(S’) and E(S, S’) : nodes and edges • T(G), T(S) and T(S’) : Spanning trees • T set of spanning trees in S and S’ respectively • set of connected bi-partitions • Represents a reduced state • First point of disconnectedness • Where does a node / edge lie? IMA International workshop on Complex Systems and Networks, 2012. S S’
Bi-Partitions and L+ A measure of centrality of edge eij in E(G): • Lower the value, bigger the sub-graph in which eij lies. IMA International workshop on Complex Systems and Networks, 2012. • Lower the value, bigger the sub-graph in which i lies.
Bi-Partitions and L+ For an edge eij in E(G): [2, 3] IMA International workshop on Complex Systems and Networks, 2012. • Higher the value, more the spanning trees on which eij lies.
When the Graph is a Tree • Lower the value, closer to the tree-center i is. IMA International workshop on Complex Systems and Networks, 2012. • Lower the value, closer to the tree-center eij is.
When the Graph is a Tree IMA International workshop on Complex Systems and Networks, 2012.
Overview • Motivation • Geometry of networks • n-dimensional embedding • Bi-partitions of a graph • Connectivity within and across partitions • Random detours • Overhead • Real-world networks and applications IMA International workshop on Complex Systems and Networks, 2012.
Random Detours • Random walk from i to j • Hitting time: Hij • Commute time: Cij = Hij + Hji = Vol(G) [2, 3] • Random detour • i to j but through k • Detour overhead [1] IMA International workshop on Complex Systems and Networks, 2012.
Recurrence in Detours • Expected number of times the walker returns to source IMA International workshop on Complex Systems and Networks, 2012.
Overview • Motivation • Geometry of networks • n-dimensional embedding • Bi-partitions of a graph • Connectivity within and across partitions • Random detours • Overhead • Real-world networks and applications IMA International workshop on Complex Systems and Networks, 2012.
Wherein lies the Core IMA International workshop on Complex Systems and Networks, 2012.
The Net-Sci Network IMA International workshop on Complex Systems and Networks, 2012. Selecting edges based on centrality
The Western States Power-Grid IMA International workshop on Complex Systems and Networks, 2012. |V(G)| = 4941, |E(G)| = 6954 (a) Edges with Le+ ≤ 1/3 of mean (b) Edges with Le+ ≤ 1/2 of mean (c) Edges with Le+ ≤ mean
Extract Trees the Greedy Way IMA International workshop on Complex Systems and Networks, 2012. Spanning tree obtained through Kruskal’s algorithm on Le+ The Italian power grid network
Relaxed Balanced Bi-Partitioning IMA International workshop on Complex Systems and Networks, 2012. • Balanced connected bi-partitioning • NP-Hard problem • Relaxed version feasible • |E(S, S’)| minimization not required • Node duplication permitted
Summary of Results • Geometric approach to centrality • The eigen space of L+ • Length of position vector, angular and Euclidean distances • Notion of centrality • Based on position and connectedness • Global measure, topological connection • Applications • Core identification • Greedy tree extraction • Relaxed bi-partitioning IMA International workshop on Complex Systems and Networks, 2012.
Questions? • Thank you! IMA International workshop on Complex Systems and Networks, 2012.
Selected References • [1] G. Ranjan and Z. –L. Zhang, Geometry of Complex Networks and Topological Centrality, [arXiv 1107.0989]. • [2] F. Fouss et al., Random-walk computation of similarities between nodes of a graph with application to collaborative recommendation, IEEE Transactions on Knowledge and Data Engineering, 19, 2007. • [3] D. J. Klein and M. Randic. Resistance distance. J. Math. Chemistry, 12:81–95, 1993. IMA International workshop on Complex Systems and Networks, 2012.
Acknowledgment • The work was supported by DTRA grant HDTRA1-09-1-0050 and NSF grants CNS-0905037, CNS-1017647 and CNS-1017092. IMA International workshop on Complex Systems and Networks, 2012.