Random graphs and the physical world

Abstract

Evolving random graph R∞ as introduced and studied by Erdös and Rényi, represent the limit of random f-graphs Rf when f→∞. The latter have been studied mainly by chemists. Such systems show an abrupt transition with the appearance of a giant component (1-connected subgraph) which models transitions in physical systems. It is known that further abrupt transitions in Rf (and R∞) occur with the appearance of giant k-connected subgraphs and these transitions also appear to have their counterparts in physical systems. Cycle length distributions in Rf and R∞ (following the k=1 transition) appear to be inconsistent with the use of these random graphs as physical models and have led to the use of random lattice-graphs RL(f). Results from percolation theory in physics relate to an abrupt transition for 1-connected subgraphs and lead to some interesting conclusions about the use of random lattice-graphs when these systems are compared to the transition in Rf. Further progress in applying random graph theory to model physical systems requires that similar results on abrupt transitions for k-connected subgraphs and on cycle distributions in RL(f) be obtained. We report here on progress and problems of this type in the setting of the applicability of random graphs to model highly interesting physical, chemical and biological systems.