Graph Combinatorics

Abstract

In this chapter we go back to a formal level and discuss the connection between the maximum-entropy ensembles of constrained graphs considered so far and various combinatorial problems in the asymptotic limit of an infinite number of nodes. This seemingly mysterious connection is actually a natural consequence of the fact that, for any discrete combinatorial problem where we need to sample or enumerate the (microcanonical) configurations compatible with a given ‘hard’ constraint, there exists a dual (canonical) problem induced by the ‘soft’ version of the same constraint. Thus, if the microcanonical and canonical ensembles are asymptotically equivalent, one can operate in the canonical ensemble and, up to finite-size corrections, extend the results to the macrocanonical one, which is otherwise very hard to deal with. It is therefore intriguing to relate the feasibility of combinatorial problems to the property of ensemble equivalence. We show that, while graphs with a single constraint on the total number of links are ensemble-equivalent, graphs with given degree sequence are not. Unlike other examples in statistical physics, where the lack of ensemble equivalence arises from long-range interactions or non-additivity, here the novel mechansim is the extensivity of the number of constraints. We discuss important implications for graph combinatorics and for the choice of the correct ensemble in practical situations. The final result is an explicit connection between the solution of a combinatorial problem and the (non)equivalence between the two associated microcanonical and canonical ensembles.

The heavier the burden, the closer our lives come to the earth, the more real and truthful they become. Conversely, the absolute, absence of burden causes man to be lighter than air, to soar into, heights, take leave of the earth and his earthly being, and become only half real, his movements as free as they are, insignificant. What then shall we choose? Weight or lightness?

—Milan Kundera, Nesnesitelná Lehkost Bytí