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 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.

3. 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.

4. 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.

5. 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

6. 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.

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) 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.

9. 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.

10. Geometry of Networks The Moore-Penrose pseudo-inverse of LLp
where
In this n-dimensional space [2]: x
IMA International workshop on Complex Systems and Networks, 2012.

11. 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.

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
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’

13. 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.

14. 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.

15. 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.

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

17. 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.

18. 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.

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

20. 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.

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

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

23. 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

24. 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

25. 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

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 ].
[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 HDTRA and NSF grants CNS , CNS and CNS
IMA International workshop on Complex Systems and Networks, 2012.