Spectral analysis and slow spreading dynamics on complex networks

Abstract

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 meanfield (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.

Heterogeneous pair-approximation for the contact process on complex networks

“…It is well known that cluster approximations of higher orders improve the critical point estimates but do not change the critical exponents in lattice systems [30]. As expected, the pair HMF theory for the CP yields the same scaling exponents as the one-vertex approximation [12,14,15], (4) changing only the amplitudes and the finite-size corrections to the scaling as we will show in this section.…”

“…The accurate theoretical understanding of dynamical systems in the form of reaction-diffusion processes running on the top of complex networks rates among the hottest issues in complex network theory [1][2][3][4][5][6][7][8][9][10][11][12][13]. Much effort has been devoted to the criticality of the ensuing absorbing state phase transition observed in the contact process (CP) [11][12][13][14][15] and in the susceptibleinfected-susceptible (SIS) [1,[3][4][5][6][7]10] models, mainly based on perturbative approaches around the transition point [1-4, 7, 9, 12], even though non-perturbative analyses have recently been performed [10].…”

Critical dynamics on a large human Open Connectome network

“…It has been conjectured that network heterogeneity * odor@mfa.kfki.hu can cause GPs if the topological (graph) dimension D, defined by N r ∼ r D , where N r is the number of (j ) nodes within topological distance r = d(i,j ) from an arbitrary origin (i), is finite [16]. This hypothesis was pronounced for the contact process (CP) [17], but subsequent studies found numerical evidence for its validity in the case of more general spreading models [18][19][20]. Recently, a GP has been reported in synthetic brain networks [21][22][23] with finite D. At first sight this seems to exclude relevant disorder effects in the so-called small-world network models.…”

Wavelet correlations to reveal multiscale coupling in geophysical systems

“…Multiscale interactions have recently received extensive attention in the literature and have been proposed as a mechanism for the triggering of extreme events [Miralles et al, 2014;Peters et al, 2004;Raffa et al, 2008], abrupt regime transitions [Okin et al, 2009;Peters et al, 2007], and patterns formation [Scanlon et al, 2007;Guttal and Jayaprakash, 2009]. Examples of this increasing interest for multiscale and cross-scale interactions can be found in ecology [Allen and Holling, 2013;Moritz et al, 2005;Cash et al, 2006;Peters et al, 2007;Raffa et al, 2008;Scanlon et al, 2007;Thrush et al, 2013;Werner et al, 2014] and climate dynamics [Holbrook et al, 2014;Debra et al, 2007;Molini et al, 2010a;Okin et al, 2009;Rial et al, 2004] and also in fields other than geosciences such as network morphology [Ódor, 2013;Pastor-Satorras and Vespignani, 2001] and econometrics [Nikkinen et al, 2011]. Most of these studies are based on minimalist models of interaction across multiple temporal and spatial scales [Allen and Holling, 2002;Peters et al, 2004Peters et al, , 2007 or-when some kind of data-driven approach is attempted-on classical scaling statistics more able to resolve the scale-dependent structure 10.1002/2015JD023265 of the considered processes rather than the nature of the coupling across the diverse scales.…”