Criticality on networks with topology-dependent interactions

Abstract

Weighted scale-free networks with topology-dependent interactions are studied. It is shown that the possible universality classes of critical behaviour, which are known to depend on topology, can also be explored by tuning the form of the interactions at fixed topology. For a model of opinion formation, simple mean field and scaling arguments show that a mapping ${\gamma }^{\prime }=(\gamma -\mu )/(1-\mu )$ describes how a shift of the standard exponent $\gamma$ of the degree distribution can absorb the effect of degree-dependent pair interactions $J_{ij}\propto (k_i{k}_j{)}^{-\mu }$, where $k_i$ stands for the degree of vertex $i$. This prediction is verified by extensive numerical investigations using the cavity method and Monte Carlo simulations. The critical temperature of the model is obtained through the Bethe-Peierls approximation and with the replica technique. The mapping can be extended to nonequilibrium models such as those describing the spreading of a disease on a network.

20 pages, 14 figures

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• Localization properties of a tight-binding electronic model on the Apollonian network
• Spin models on random graphs with controlled topologies beyond degree constraints
• Percolation on bipartite scale-free networks
• Scaling behavior of the contact process in networks with long-range connections
• A 3-states magnetic model of binary decisions in sociophysics
• Rounding of first-order phase transitions and optimal cooperation in scale-free networks
• Distinct dynamical behavior in Erdős-Rényi networks, regular random networks, ring lattices, and all-to-all neuronal networks
• Phase transitions in Ising model induced by weight redistribution on weighted regular networks
• Ising model for distribution networks
• Heterogeneous nucleation on complex networks with mobile impurities
• Stopping explosion by penalising transmission to hubs in scale-free spatial random graphs
• Degree-penalized contact processes