Local Minima in Disordered Mean-Field Ferromagnets

Abstract

We consider the complexity of random ferromagnetic landscapes on the hypercube {±1} N given by Ising models on the complete graph with i.i.d. non-negative edge-weights. This includes, in particular, the case of Bernoulli disorder corresponding to the Ising model on a dense random graph G(N, p). Previous results had shown that, with high probability as N → ∞, the gradient search (energy-lowering) algorithm, initialized uniformly at random, converges to one of the homogeneous global minima (all-plus or all-minus). Here, we devise two modified algorithms tailored to explore the landscape at near-zero magnetizations (where the effect of the ferromagnetic drift is minimized).With these, we numerically verify the landscape complexity of random ferromagnets, finding a diverging number of (1-spin-flip-stable) local minima as N → ∞. We then investigate some of the properties of these local minima (e.g., typical energy and magnetization) and compare to the situation where the edge-weights are drawn from a heavy-tailed distribution.

Friendly bisections of random graphs

“…In a cut of a graph, i.e., a partition of its vertex set into two parts, we call a vertex friendly if it has more neighbours in its own part than across, and unfriendly otherwise. Questions about finding friendly and unfriendly partitions of graphs, i.e., partitions in which all (or almost all) the vertices are friendly or unfriendly, have been investigated in various contexts: in combinatorics, on account of their inherent interest [5,10,19,26,30,34,36], in computer science, as 'local' analogues of important NP-complete partitioning problems [4,13], in probability and statistical physics, owing to their connections to spin glasses [1,18,20,32], and in logic and set theory [2,31]; this list is merely a representative sample (and by no means exhaustive) since such partitions have been studied extremely broadly. On the other hand, when it comes to finding friendly or unfriendly bisections, i.e., partitions into two parts whose sizes differ by at most one, much less is known.…”

Untitled

“…In a cut of a graph, i.e., a partition of its vertex set into two parts, we call a vertex friendly if it has more neighbours in its own part than across, and unfriendly otherwise. Questions about finding friendly and unfriendly partitions of graphs, i.e., partitions in which all (or almost all) the vertices are friendly or unfriendly, have been investigated in various contexts: in combinatorics, on account of their inherent interest [5,10,19,26,30,34,36], in computer science, as 'local' analogues of important NP-complete partitioning problems [4,13], in probability and statistical physics, owing to their connections to spin glasses [1,18,20,32], and in logic and set theory [2,31]; this list is merely a representative sample (and by no means exhaustive) since such partitions have been studied extremely broadly. On the other hand, when it comes to finding friendly or unfriendly bisections, i.e., partitions into two parts whose sizes differ by at most one, much less is known.…”

Friendly bisections of random graphs

“…In a cut of a graph, i.e., a partition of its vertex set into two parts, we call a vertex friendly if it has more neighbours in its own part than across, and unfriendly otherwise. Questions about finding friendly and unfriendly partitions of graphs, i.e., partitions in which all (or almost all) the vertices are friendly or unfriendly, have been investigated in various contexts: in combinatorics, on account of their inherent interest [5,10,28,30], in computer science, as 'local' analogues of important NP-complete partitioning problems [4,13], in probability and statistical physics, owing to their connections to spin glasses [1,16,26], and in logic and set theory [2,25]. On the other hand, when it comes to finding friendly or unfriendly bisections, i.e., partitions into two parts whose sizes differ by at most one, there is mostly speculative folklore; our aim here is to prove one such old and well-known conjecture about random graphs due to Füredi [15], a problem that has gained some notoriety over the years, in part due to its inclusion in Green's list of 100 open problems [17,Problem 91].…”