(Dis)assortative partitions on random regular graphs. (30th September 2022) - Behrens, Freya; Arpino, Gabriel; Kivva, Yaroslav; Zdeborová, Lenka
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 > ⌈ d 2 ⌉ we establish that the structure of solutions to the assortative partition problems corresponds to the so-called frozen-one-step replica symmetry breaking. This entails a conjecture of algorithmic hardness of finding these partitions efficiently. For H ⩽ ⌈ d 2 ⌉ 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 through 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.