when affinity meets resistance on the topological centrality of edges in complex networks

When Affinity Meets ResistanceOn the Topological Centrality of Edges in Complex

Networks

Gyan RanjanUniversity of Minnesota, MN

[Collaborators: Zhi-Li Zhang and Hesham Mekky.]

IMA International workshop on Com

plex 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 Com

plex 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 Com

plex 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 Com

plex Systems and

Networks, 2012.

Cart before the Horse

IMA International workshop on Com

plex Systems and

Networks, 2012.

Centrality of nodes: Red to blue to white, decreasing order [1].

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 Com

plex 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 Com

plex 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 Com

plex 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 Com

plex 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 Com

plex 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 Com

plex 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 Com

plex Systems and

Networks, 2012.

S S’

Bi-Partitions and L+

IMA International workshop on Com

plex Systems and

Networks, 2012.

Lower the value, bigger the sub-graph in which eij lies.

Lower the value, bigger the sub-graph in which i lies.

A measure of centrality of edge eij in E(G):

Bi-Partitions and L+

IMA International workshop on Com

plex Systems and

Networks, 2012.

Higher the value, more the spanning trees on which eij lies.

[2, 3]

For an edge eij in E(G):

When the Graph is a Tree

IMA International workshop on Com

plex Systems and

Networks, 2012.

Lower the value, closer to the tree-center i is.

Lower the value, closer to the tree-center eij is.

When the Graph is a Tree

IMA International workshop on Com

plex 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 Com

plex 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 Com

plex Systems and

Networks, 2012.

Recurrence in Detours

IMA International workshop on Com

plex Systems and

Networks, 2012.

Expected number of times the walker returns to source

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 Com

plex Systems and

Networks, 2012.

Wherein lies the Core

IMA International workshop on Com

plex Systems and

Networks, 2012.

The Net-Sci Network

IMA International workshop on Com

plex Systems and

Networks, 2012. Selecting edges based on centrality

The Western States Power-Grid

|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

IMA International workshop on Com

plex Systems and

Networks, 2012.

Extract Trees the Greedy Way

IMA International workshop on Com

plex Systems and

Networks, 2012. The Italian power grid network

Spanning tree obtained through Kruskal’s algorithm on Le

+

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 Com

plex Systems and

Networks, 2012.

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 Com

plex Systems and

Networks, 2012.

Questions? Thank you!

IMA International workshop on Com

plex 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 betweennodes 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 Com

plex 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 Com

plex Systems and

Networks, 2012.