(Dis)assortative partitions on random regular graphs

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