Correlations in interacting systems with a network topology

Abstract

We study pair correlations in cooperative systems placed on complex networks. We show that usually in these systems, the correlations between two interacting objects (e.g., spins), separated by a distance $\ell$, decay, on average, faster than $1/(\ell z_{\ell })$. Here $z_{\ell }$ is the mean number of the $\ell$-th nearest neighbors of a vertex in a network. This behavior, in particular, leads to a dramatic weakening of correlations between second and more distant neighbors on networks with fat-tailed degree distributions, which have a divergent number $z_2$ in the infinite network limit. In this case, only the pair correlations between the nearest neighbors are observable. We obtain the pair correlation function of the Ising model on a complex network and also derive our results in the framework of a phenomenological approach.

5 pages

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Cited by in corpus (7)

• Critical phenomena in complex networks
• Griffiths singularities and algebraic order in the exact solution of an Ising model on a fractal modular network
• Equations of motion approach to the spin-1/2 Ising model on the Bethe lattice
• Effective field theory for models defined over small-world networks. First and second order phase transitions
• Critical behavior and correlations on scale-free small-world networks. Application to network design
• Ising spin glass models versus Ising models: an effective mapping at high temperature II. Applications to graphs and networks
• Diversity of Cortical States at Non-Equilibrium Simulated by the Ferromagnetic Ising Model Under Metropolis Dynamics