Phase transitions in random Potts systems and the community detection problem: spin-glass type and dynamic perspectives

TL;DR

This paper investigates phase transitions in Potts spin glass systems related to community detection, revealing solvable, hard, and unsolvable regimes, with implications for understanding complexity and chaos in physical and computational systems.

Contribution

It introduces a broad analysis of phase transitions in Potts spin glass models applied to community detection, highlighting spin glass and dynamic transition phenomena.

Findings

Identification of solvable, hard, and unsolvable phases in community detection.

Observation of spin glass transitions at low and high temperatures.

Relation of chaotic behavior in thermodynamic systems to computational hardness.

Abstract

Phase transitions in spin glass type systems and, more recently, in related computational problems have gained broad interest in disparate arenas. In the current work, we focus on the "community detection" problem when cast in terms of a general Potts spin glass type problem. As such, our results apply to rather broad Potts spin glass type systems. Community detection describes the general problem of partitioning a complex system involving many elements into optimally decoupled "communities" of such elements. We report on phase transitions between solvable and unsolvable regimes. Solvable region may further split into "easy" and "hard" phases. Spin glass type phase transitions appear at both low and high temperatures (or noise). Low temperature transitions correspond to an "order by disorder" type effect wherein fluctuations render the system ordered or solvable. Separate transitions appear at higher temperatures into a disordered (or an unsolvable) phase. Different sorts of randomness lead to disparate behaviors. We illustrate the spin glass character of both transitions and report on memory effects. We further relate Potts type spin systems to mechanical analogs and suggest how chaotic-type behavior in general thermodynamic systems can indeed naturally arise in hard-computational problems and spin-glasses. The correspondence between the two types of transitions (spin glass and dynamic) is likely to extend across a larger spectrum of spin glass type systems and hard computational problems. We briefly discuss potential implications of these transitions in complex many body physical systems.