Global disorder transition in the community structure of large-q Potts systems
arXiv:1204.3649 · doi:10.1209/0295-5075/99/38006
Abstract
We examine a global disorder transition when identifying community structure in an arbitrary complex network. Earlier, we illustrated [Phil. Mag. 92, 406 (2012)] that "community detection" (CD) generally exhibits disordered (or unsolvable) and ordered (solvable) phases of both high and low computational complexity along with corresponding transitions from regular to chaotic dynamics in derived systems. Using an exact generalized dimensional reduction inequality, multivariate Tutte polynomials, and other considerations, we illustrate how increasing the number of communities q emulates increasing the heat bath temperature T for a general weighted Potts model, leading to global disorder in the community structure of arbitrary large graphs. Dimensional reduction bounds lead to results similar to those suggested by mean-field type approaches. Large systems tend toward global insolvability in the limit of large q above a crossover temperature where |J_e| is a typical interaction strength, L is the number of edges, and N is the number of nodes. For practical system sizes, a solvable phase is generally accessible at low T. The global nature of the disorder transition does not preclude solutions by local CD algorithms (even those that employ global cost function parameters) as long as community evaluations are locally determined.
6 pages, 2 figures; accepted version with minor editing, corrected indices in one equation, results unchanged, revised references
References in corpus (23)
- Fast unfolding of communities in large networks
- Maps of random walks on complex networks reveal community structure
- Near linear time algorithm to detect community structures in large-scale networks
- Resolution limit in community detection
- Critical phenomena in complex networks
- Statistical Mechanics of Community Detection
- Detecting the overlapping and hierarchical community structure of complex networks
- Network Physiology reveals relations between network topology and physiological function
- Community detection in networks with positive and negative links
- Analysis of the structure of complex networks at different resolution levels
- Multilevel compression of random walks on networks reveals hierarchical organization in large integrated systems
- Phase transition in the detection of modules in sparse networks
- Narrow scope for resolution-limit-free community detection
- Detecting network communities by propagating labels under constraints
- Graph spectra and the detectability of community structure in networks
- A sequential algorithm for fast clique percolation
- Community Detection as an Inference Problem
- Limited resolution in complex network community detection with Potts model approach
- Spectral methods for the detection of network community structure: a comparative analysis
- (Un)detectable cluster structure in sparse networks
- Mean-field driven first-order phase transitions in systems with long-range interactions
- Community Detection in Complex Networks by Dynamical Simplex Evolution
- Disorder driven phase transitions of the large q-state Potts model in 3d