Multiresolution community detection for megascale networks by information-based replica correlations

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 80, Issue 1, 2009, Pages

  Ronhovde, Peter ^a   Nussinov, Zohar ^a  

^a WASHINGTON UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMMUNITY DETECTION; COMMUNITY DETECTION ALGORITHMS; MULTI-RESOLUTION ALGORITHMS; MULTI-RESOLUTIONS; MULTIPLE RESOLUTIONS; MULTIRESOLUTION STRUCTURES; NORMALIZED MUTUAL INFORMATION; OPTIMIZATION PROBLEMS; POTTS MODELS; QUANTITATIVE ESTIMATES; RELATIVE STRENGTH; RESOLUTION LIMITS; SINGLE PROCESSORS; SUPERLINEAR;

ALGORITHMS; MULTIRESOLUTION ANALYSIS; SIGNAL DETECTION;

GRAPH THEORY;

EID: 68949120793     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.80.016109     Document Type: Article

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51. Data for the constructed systems examined in this paper (Figs. 1 2 3 4) can be found at http://physics.wustl.edu/zohar/communitydetection/.
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64. As a specific realization, consider a symmetric adjacency matrix A for a four node graph given by A12 = A13 = A14 = A23 = A24 =1. The permutation P34 that exchanges 3 with 4 leaves A invariant, so the graph itself is invariant. If we have two solutions B and C that place i and j in different communities then IN (B,C) 1 despite the invariance of A under the permutation. Unless one can distinguish between the nodes via external means, community groupings of the four nodes such as (123)(4) and (123)(3) correspond to the same breaking of the lattice (the graph is symmetric under the permutation), but they have a relative mutual information that differs from the self mutual information [(8-3log3) /4 vs (10-6log3) /4 for the two groupings, respectively].