Griffiths Phases on Complex Networks

Abstract

Quenched disorder is known to play a relevant role in dynamical processes and phase transitions. Its effects on the dynamics of complex networks have hardly been studied. Aimed at filling this gap, we analyze the contact process, i.e., the simplest propagation model, with quenched disorder on complex networks. We find Griffiths phases and other rare-region effects, leading rather generically to anomalously slow (algebraic, logarithmic, …) relaxation, on Erdos-Rényi networks. Similar effects are predicted to exist for other topologies with a finite percolation threshold. More surprisingly, we find that Griffiths phases can also emerge in the absence of quenched disorder, as a consequence of topological heterogeneity in networks with finite topological dimension. These results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks.

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].…”

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

“…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].…”

“…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 authors in Refs. [17,19] also considered the issue of the effects of pure topological disorder for a CP with constant λ. By means of theoretical arguments and numerical simulations, they conjectured that RR effects should be relevant only on networks with a finite topological dimension D, defined in terms of the number of vertices at a topological distance from a given source, N ( ) ∼ D .…”