Convergence of Maximum Bisection Ratio of Sparse Random Graphs

TL;DR

This paper proves that the maximum bisection ratio of large sparse random graphs converges to a deterministic limit and extends the result to certain spin glass models, generalizing the graph interpolation method.

Contribution

It establishes convergence of the maximum bisection ratio for a broad class of sparse random graphs and extends the method to non-additive graph parameters.

Findings

Maximum bisection ratio converges almost surely.

Results apply to random regular and Erdős-Rényi graphs.

Extension to 2-spin spin glasses in specific regimes.

Abstract

We consider sequences of large sparse random graphs whose degree distribution approaches a limit with finite mean. This model includes both the random regular graphs and the Erd\"os-Renyi graphs of constant average degree. We prove that the maximum bisection ratio of such a graph sequence converges almost surely to a deterministic limit. We extend this result to so-called 2-spin spin glasses in the paramagnetic to ferromagnetic regime. Our work generalizes the graph interpolation method to some non-additive graph parameters.