When Affinity Meets Resistance

On the Topological Centrality of Edges in Complex Networks

Gyan RanjanUniversity of Minnesota, MN

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

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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

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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

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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

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Cart before the Horse

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

Western states power grid Network sciences co-authorship

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

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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

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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

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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

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Geometry of Networks The Moore-Penrose pseudo-inverse of L

Lp

where

In this n-dimensional space [2]:

x

x

x

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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

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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?

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S S’

Bi-Partitions and L+

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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+

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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

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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

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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

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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]

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Recurrence in Detours

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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

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Wherein lies the Core

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The Net-Sci Network

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2. 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

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Extract Trees the Greedy Way

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2. 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

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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

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Questions?

Thank you!

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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.

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Acknowledgment The work was supported by DTRA grant HDTRA1-09-1-0050 and

NSF grants CNS-0905037, CNS-1017647 and CNS-1017092.

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