1. Journal Club Physical Review E
Merril Gersten
August 30, 2010

2. Dynamics of overlapping structures in modular networks Almendral, JA: Physical Review E 82: (2010)
Questions addressed:
How to find nodes which overlap 2 or more communities (partitions) in a complex graph.
Methods:
“Dynamical overlapping” in which nodes are treated as oscillators and the instantaneous frequency of an overlapping node (“synchronization interface”) oscillates between the frequencies of its associated modules.
Results/Discussion:
Method identifies nodes in three real-world networks that serve as bridges between two communities.
Computational (experimental) results support an analytical analysis.
The quantity kiout/ki can be used to identify overlapping nodes in a graph.
Interest to Lab:
Work here on edge betweenness partitioning has not addressed the overlap issue.

3. Previous work: Li, D. Physical Review Letters 101:168701 (2008)
G = graph of N coupled Kuramoto oscillators
fi = phases of oscillators
ki=ith oscillator
d = coupling strength
aij = elements of adjacency matrix of G
[1 (or 0) if link does (doesn’t) exist]
wi = frequency of oscillator
Ci = overlapping index

4. G = graph of N coupled Kuramoto oscillators
fi = phases of oscillators
ki=ith oscillator
d = coupling strength
aij = elements of adjacency matrix of G
[1 (or 0) if link does (doesn’t) exist]
wi = frequency of oscillator
Ci = overlapping index
Mi =module

5. * * Analytical Results Solution: or Solution: or = in-degree
= out-of-module links in
depends on bi’s, distribution unknown *