Equilibrium statistical mechanics on correlated random graphs

Abstract

Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift [−1, +1] → [0, +1] in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents (the Hebbian kernel)-crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality: both a statistical mechanics and a topological phase diagrams are obtained. Overall the picture depicted from statistical mechanics is quite intuitive: at least at equilibrium, dilution (of whatever kind) simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved. 1As we will see, small values of a give rise to highly correlated, diluted networks, while, as a gets larger the network gets more and more connected and correlation among links vanishes.Even though the theory is defined at each finite V and L, as standard in statistical mechanics, we are interested in the large V behavior (such that, under central limit theorem permissions, deviations from averaged values become negligible and the theory predictive). To this task we find meaningful to let even L diverge linearly with the system size (to bridge conceptually to high storage neural networks), such that lim V →∞ L/V = α defines α as another control parameter. Finally, since we are interested in the regime of large V and large L we will often confuse V with V − 1 and L with L − 1. 4

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