Rare-region effects in the contact process on networks

Abstract

Networks and dynamical processes occurring on them have become a paradigmatic representation of complex systems. Studying the role of quenched disorder, both intrinsic to nodes and topological, is a key challenge. With this in mind, here we analyze the contact process (i.e., the simplest model for propagation phenomena) with node-dependent infection rates (i.e., intrinsic quenched disorder) on complex networks. We find Griffiths phases and other rare-region effects, leading rather generically to anomalously slow (algebraic, logarithmic, etc.) relaxation, on Erdős-Rényi networks. We predict similar effects to exist for other topologies as long as a nonvanishing percolation threshold exists. More strikingly, we find that Griffiths phases can also emerge--even with constant epidemic rates--as a consequence of mere topological heterogeneity. In particular, we find Griffiths phases in finite-dimensional networks as, for instance, a family of generalized small-world networks. These results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks, and are relevant for the analysis of both models and empirical data.

Rare regions of the susceptible-infected-susceptible model on Barabási-Albert networks

“…References [16,18] concluded that finite topological dimension is a necessary condition for observing Griffiths phases and activated scaling in the case of the basic model of nonequilibrium system, the contact process. In the case of CP on certain weighed networks, numerical evidence was shown for generic power laws, but in the thermodynamic limit this phase seems to disappear and a smeared phase transition exists, due to the infinite-dimensional correlated rare regions [30].…”

“…Studies of the CP, as well as other processes [8,9], have shown that quenched disorder in networks is relevant in the dynamical systems defined on top of them. Very recently it has been shown [16][17][18] that generic slow (power-law or logarithmic) dynamics is observable by simulating CP on networks with finite d. This observation is relevant for recent developments in dynamical processes on complex networks such as the simple model of "working memory" [19], brain dynamics [20], social networks with heterogeneous communities [21], or slow relaxation in glassy systems [22].…”

“…At and below the λ 0 c rare clusters may still form, with an over-average mean degree, which are locally supercritical at a given λ and induce a slower-than-exponential but fasterthan-power-law decay of the global density. In the case of these "weak rare-region effects," numerical simulations [18] showed stretched-exponential decay, ρ(t) ∼ e −const×t y with exponents y varying from values close to 1, for small λ, to very small values (converging to 0) for λ approaching the GP. The rare-region induced dynamical behavior can be described by nonperturbative methods [26][27][28][29].…”

Slow dynamics and rare-region effects in the contact process on weighted tree networks

“…In the case of the CP on complex networks, it has recently been shown that an intrinsic quenched disorder 3 defined by a varying control parameter λ i , can induce GPs and other RR effects on random Erdős-Rényi networks [30] below (and at) the percolation threshold [17][18][19]. The authors in Refs.…”

“…The effects of network disorder have recently been extended one step further, in a series of papers [17][18][19] dealing with the possibility of observing Griffiths phase (GP) and rare-region (RR) phenomena [20,21] at the phase transition of the CP on complex networks. 1 In case of regular lattices it is well known that quenched disorder 2 can strongly alter the behavior of a phase transition, imposing an anomalously slow relaxation [23].…”