when affinity meets resistance on the topological centrality of edges in complex networks
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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.