1. 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.

2. 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.

3. 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.

4. 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.

5. Cart before the Horse IMA International workshop on Complex Systems and Networks, 2012. Centrality of nodes: Red to blue to white, decreasing order [1]. Western states power gridNetwork sciences co-authorship

6. State of 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.

7. 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.

8. Definitions Network as a graph G(V, E) Simple, connected and unweighted [for simplicity] Extends to weighted networks/graphs w ij is the weight of edge e ij 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.

9. The Graph and Algebra For a graph G(V, E) [A] nxn = Adjacency matrix of G(V, E) a ij = 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.

10. 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.

11. 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.

12. 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 Tset 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’

13. Bi-Partitions and L + IMA International workshop on Complex Systems and Networks, 2012. Lower the value, bigger the sub-graph in which e ij lies. Lower the value, bigger the sub-graph in which i lies. A measure of centrality of edge e ij in E(G):

14. Bi-Partitions and L + IMA International workshop on Complex Systems and Networks, 2012. Higher the value, more the spanning trees on which e ij lies. [2, 3] For an edge e ij in E(G):

15. When the Graph is a Tree IMA International workshop on Complex Systems and Networks, 2012. Lower the value, closer to the tree-center i is. Lower the value, closer to the tree-center e ij is.

16. When the Graph is a Tree IMA International workshop on Complex Systems and Networks, 2012.

17. 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.

18. Random Detours Random walk from i to j Hitting time: H ij Commute time: C ij = H ij + H ji = Vol(G) [2, 3] Random detour i to j but through k Detour overhead [1] IMA International workshop on Complex Systems and Networks, 2012.

19. Recurrence in Detours IMA International workshop on Complex Systems and Networks, 2012. Expected number of times the walker returns to source

20. 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.

21. Wherein lies the Core IMA International workshop on Complex Systems and Networks, 2012.

22. The Net-Sci Network IMA International workshop on Complex Systems and Networks, 2012. Selecting edges based on centrality

23. The Western States Power-Grid |V(G)| = 4941, |E(G)| = 6954 (a) Edges with L e + ≤ 1/3 of mean (b) Edges with L e + ≤ 1/2 of mean (c) Edges with L e + ≤ mean IMA International workshop on Complex Systems and Networks, 2012.

24. Extract Trees the Greedy Way IMA International workshop on Complex Systems and Networks, 2012. The Italian power grid network Spanning tree obtained through Kruskal’s algorithm on L e +

25. Relaxed Balanced Bi-Partitioning Balanced connected bi-partitioning NP-Hard problem Relaxed version feasible |E(S, S’)| minimization not required Node duplication permitted IMA International workshop on Complex Systems and Networks, 2012.

26. 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.

27. Questions? Thank you! IMA International workshop on Complex Systems and Networks, 2012.

28. 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.

29. 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.