The hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for ...synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 836733 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law-tailed synchronization durations, with τ
≃ 1.2(1), away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: τ
≃ 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τ
≤ 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.
I provide numerical evidence for the robustness of the Griffiths phase (GP) reported previously in dynamical threshold model simulations on a large human brain network with N=836733 connected nodes. ...The model, with equalized network sensitivity, is extended in two ways: introduction of refractory states or by randomized time-dependent thresholds. The nonuniversal power-law dynamics in an extended control parameter region survives these modifications for a short refractory state and weak disorder. In case of temporal disorder the GP shrinks and for stronger heterogeneity disappears, leaving behind a mean-field type of critical transition. Activity avalanche size distributions below the critical point decay faster than in the original model, but the addition of inhibitory interactions sets it back to the range of experimental values.
The structural human connectome (i.e. the network of fiber connections in the brain) can be analyzed at ever finer spatial resolution thanks to advances in neuroimaging. Here we analyze several large ...data sets for the human brain network made available by the Open Connectome Project. We apply statistical model selection to characterize the degree distributions of graphs containing up to nodes and edges. A three-parameter generalized Weibull (also known as a stretched exponential) distribution is a good fit to most of the observed degree distributions. For almost all networks, simple power laws cannot fit the data, but in some cases there is statistical support for power laws with an exponential cutoff. We also calculate the topological (graph) dimension D and the small-world coefficient σ of these networks. While σ suggests a small-world topology, we found that D < 4 showing that long-distance connections provide only a small correction to the topology of the embedding three-dimensional space.
The critical behavior of the nondiffusive susceptible-infected-recovered model on lattices had been well established in virtue of its duality symmetry. By performing simulations and scaling analyses ...for the diffusive variant on the two-dimensional lattice, we show that diffusion for all agents, while rendering this symmetry destroyed, constitutes a singular perturbation that induces asymptotically distinct dynamical and stationary critical behavior from the nondiffusive model. In particular, the manifested crossover behavior in the effective mean-square radius exponents reveals that slow crossover behavior in general diffusive multispecies reaction systems may be ascribed to the interference of multiple length scales and timescales at early times.
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 Erdos ...Renyi 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 scalefree 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.
Bursty dynamics of agents is shown to appear at criticality or in extended Griffiths phases, even in case of Poisson processes. I provide numerical evidence for a power-law type of intercommunication ...time distributions by simulating the contact process and the susceptible-infected-susceptible model. This observation suggests that in the case of nonstationary bursty systems, the observed non-Poissonian behavior can emerge as a consequence of an underlying hidden Poissonian network process, which is either critical or exhibits strong rare-region effects. On the contrary, in time-varying networks, rare-region effects do not cause deviation from the mean-field behavior, and heterogeneity-induced burstyness is absent.
Universal scaling behavior is an attractive feature in statistical physics because a wide range of models can be classified purely in terms of their collective behavior due to a diverging correlation ...length. This book provides a comprehensive overview of dynamical universality classes occurring in nonequilibrium systems defined on regular lattices. The factors determining these diverse universality classes have yet to be fully understood, but the book attempts to summarize our present knowledge, taking them into account systematically.
The susceptible-infected-susceptible (SIS) model is one of the simplest memoryless systems for describing information or epidemic spreading phenomena with competing creation and spontaneous ...annihilation reactions. The effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean-field (QMF) approximations in scale-free networks. QMF can take into account topological heterogeneity and clustering effects of the activity in the steady state by spectral decomposition analysis of the adjacency matrix. Therefore, it can provide predictions on possible rare-region effects, thus on the occurrence of slow dynamics. I compare QMF results of SIS with simulations on various large dimensional graphs. In particular, I show that for Erdős-Rényi graphs this method predicts correctly the occurrence of rare-region effects. It also provides a good estimate for the epidemic threshold in case of percolating graphs. Griffiths Phases emerge if the graph is fragmented or if we apply a strong, exponentially suppressing weighting scheme on the edges. The latter model describes the connection time distributions in the face-to-face experiments. In case of a generalized Barabási-Albert type of network with aging connections, strong rare-region effects and numerical evidence for Griffiths Phase dynamics are shown. The dynamical simulation results agree well with the predictions of the spectral analysis applied for the weighted adjacency matrices.
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