(Dis)assortative Partitions on Random Regular Graphs
Freya Behrens, Gabriel Arpino, Yaroslav Kivva, Lenka Zdeborov\'a

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 -regular graphs. Nodes in the graph are partitioned into two non-empty groups. In the assortative partition every node requires at least of their neighbors to be in their own group. In the disassortative partition they require less than 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 these partitions exist with high probability and for which they do not. For 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 we argue that the assortative partition…
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