Abstract
The Contact Process has been studied on complex networks exhibiting different kinds of quenched disorder. Numerical evidence is found for Griffiths phases and other rare region effects, in Erd\H os R\'enyi networks, leading rather generically to anomalously slow (algebraic, logarithmic,...) relaxation. More surprisingly, it turns out that Griffiths phases can also emerge in the absence of quenched disorder, as a consequence of sole topological heterogeneity in networks with finite topological dimension. In case of scale-free networks, exhibiting infinite topological dimension, slow dynamics can be observed on tree-like structures and a superimposed weight pattern. In the infinite size limit the correlated subspaces of vertices seem to cause a smeared phase transition. These results have a broad spectrum of implications for propagation phenomena and other dynamical process on networks and are relevant for the analysis of both models and empirical data.
Highlights
Nonequilibrium systems have been a central research topic of statistical mechanics [1,2,3]
The Contact Process has been studied on complex networks exhibiting different kinds of quenched disorder
In case of scalefree networks, exhibiting infinite topological dimension, slow dynamics can be observed on tree-like structures and a superimposed weight pattern
Summary
Nonequilibrium systems have been a central research topic of statistical mechanics [1,2,3]. Many network models exhibit infinite topological dimension (d), simple mean-field approximations cannot capture several important features [23,24,25,26,27,28] Very recently it has been conjectured [29,30,31] that generic slow (power-law, or logarithmic) dynamics is observable only in networks with finite d. Systematic finite scaling study revealed that these power-laws saturate in the N → ∞ thermodynamic limit, suggesting smeared phase transitions known from Euclidean, regular systems if the correlated subspaces can undergo phase transitions themselves, when they are effectively above the lower critical dimension of the problem : dRR > dc− [44] In this case, the dynamics of the locally ordered RR-s completely freezes, and they develop a truly static order parameter. In infinite dimensional networks such RR-s can be embedded as a percolation analysis confirmed this [45]
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