Equilibrium statistical mechanics on correlated random graphs

Abstract

Biological and social networks have recently attracted great attention from physicists.Among several aspects, two main ones may be stressed: a non-trivial topology of the graphdescribing the mutual interactions between agents and, typically, imitative, weighted,interactions. Despite such aspects being widely accepted and empirically confirmed, theschemes currently exploited in order to generate the expected topology are based on apriori assumptions and, in most cases, implement constant intensities for links.Here we propose a simple shift in the definition of patterns in a Hopfield model: a straightforward effect is the conversionof frustration into dilution. In fact, we show that by varying the bias of patterndistribution, the network topology (generated by the reciprocal affinities among agents,i.e. the Hebbian rule) crosses various well-known regimes, ranging from fully connected, toan extreme dilution scenario, then to completely disconnected. These features, as well assmall-world properties, are, in this context, emergent and no longer imposed a priori.The model is throughout investigated also from a thermodynamics perspective: the Isingmodel defined on the resulting graph is analytically solved (at a replica symmetric level) byextending the double stochastic stability technique, and presented together with itsfluctuation theory for a picture of criticality. Overall, our findings show that, at least atequilibrium, dilution (of whatever kind) simply decreases the strength of the coupling feltby the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The maindifference with respect to previous investigations is that, within our approach, replicasdo not appear: instead of (multi)-overlaps as order parameters, we introduce aclass of magnetizations on all the possible subgraphs belonging to the main oneinvestigated: as a consequence, for these objects a closure for a self-consistent relation isachieved.