Analytic Description of the Phase Transition of Inhomogeneous Multigraphs

TL;DR

This paper introduces a new analytic model for inhomogeneous multigraphs with colored vertices and weighted edges, analyzing phase transitions and critical phenomena in complex components, with applications to coloring and satisfiability problems.

Contribution

It extends existing inhomogeneous random graph models using analytic combinatorics to analyze phase transitions and applies these results to various combinatorial and computational problems.

Findings

Characterization of the birth of complex components in the new multigraph model

New proofs for known results on 2-colorability and multigraph enumeration

New insights into the phase transition of quantified 2-Xor satisfiability

Abstract

We introduce a new model of random multigraphs with colored vertices and weighted edges. It is similar to the "inhomogeneous random graph model" of S\"oderberg (2002), extended by Bollob\'as, Janson and Riordan (2007). By means of analytic combinatorics, we then analyze the birth of "complex components", which are components with at least two cycles. We apply those results to give a complete picture of the finite size scaling and the critical exponents associated to a rather broad family of decision problems. As applications, we derive new proofs of known results on the 2-colorability problem, already investigated by Pittel and Yeum (2010), and on the enumeration of properly q-colored multigraphs, analyzed by Wright (1972). We also obtain new results on the phase transition of the satisfiability of quantified 2-Xor-formulas, a problem introduced by Creignou, Daud\'e and Egly (2007).