(Dis)assortative Partitions on Random Regular Graphs

TL;DR

This paper analyzes the existence and computational complexity of assortative and disassortative partitions in random regular graphs using the cavity method, revealing phase transitions and conjectured hardness results related to spin glass models.

Contribution

It introduces a cavity method analysis of partition problems on random regular graphs, establishing existence conditions, solution structure, and algorithmic hardness conjectures for different parameter regimes.

Findings

Existence of partitions characterized by parameters (d,H)

Frozen-1RSB structure for H > ⌈d/2⌉, implying hardness

Algorithmic ease for H ≤ ⌈d/2⌉

Abstract

We study the problem of assortative and disassortative partitions on random $d$-regular graphs. Nodes in the graph are partitioned into two non-empty groups. In the assortative partition every node requires at least $H$ of their neighbors to be in their own group. In the disassortative partition they require less than $H$ neighbors to be in their own group. Using the cavity method based on analysis of the Belief Propagation algorithm we establish for which combinations of parameters $(d,H)$ these partitions exist with high probability and for which they do not. For $H>\lceil \frac{d}{2} \rceil $ we establish that the structure of solutions to the assortative partition problems corresponds to the so-called frozen-1RSB. This entails a conjecture of algorithmic hardness of finding these partitions efficiently. For $H \le \lceil \frac{d}{2} \rceil $ we argue that the assortative partition problem is algorithmically easy on average for all $d$. Further we provide arguments about asymptotic equivalence between the assortative partition problem and the disassortative one, going trough a close relation to the problem of single-spin-flip-stable states in spin glasses. In the context of spin glasses, our results on algorithmic hardness imply a conjecture that gapped single spin flip stable states are hard to find which may be a universal reason behind the observation that physical dynamics in glassy systems display convergence to marginal stability.