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 Bi-partitions of a graph 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 Bi-partitions of a graph 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 Notion of centrality Utility 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 Edge centrality Our approach 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) Topological dimensions 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) Structure of L [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 IMA International workshop on Complex Systems and Networks, 2012.

Overview Motivation Geometry of networks Bi-partitions of a graph 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 Bi-partitions of a graph 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 Random detour 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 Bi-partitions of a graph 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 Selecting edges based on centrality 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 (c) Edges with Le+ ≤ mean (a) Edges with Le+ ≤ 1/3 of mean (b) Edges with Le+ ≤ 1/2 of 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.