On the numerical study of percolation and epidemic critical properties in networks

The static properties of the fundamental model for epidemics of diseases allowing immunity (susceptible-infected-removed model) are known to be derivable by an exact mapping to bond percolation. Yet when performing numerical simulations of these dynamics in a network a number of subtleties must be taken into account in order to correctly estimate the transition point and the associated critical properties. We expose these subtleties and identify the different quantities which play the role of criticality detector in the two dynamics.Postprint (author's final draft

Relating Topological Determinants of Complex Networks to Their Spectral Properties: Structural and Dynamical Effects

The largest eigenvalue of a network's adjacency matrix and its associated principal eigenvector are key elements for determining the topological structure and the properties of dynamical processes mediated by it. We present a physically grounded expression relating the value of the largest eigenvalue of a given network to the largest eigenvalue of two network subgraphs, considered as isolated: The hub with its immediate neighbors and the densely connected set of nodes with maximum $K$-core index. We validate this formula showing that it predicts with good accuracy the largest eigenvalue of a large set of synthetic and real-world topologies. We also present evidence of the consequences of these findings for broad classes of dynamics taking place on the networks. As a byproduct, we reveal that the spectral properties of heterogeneous networks built according to the linear preferential attachment model are qualitatively different from those of their static counterparts.Comment: 18 pages, 13 figure

Zero temperature Glauber dynamics on complex networks

We study the Glauber dynamics at zero temperature of spins placed on the vertices of an uncorrelated network with a power-law degreedistribution. Application of mean-field theory yields as main prediction that for symmetric disordered initial conditions the mean time to reach full order is finite or diverges as a logarithm of the system size N, depending on the exponent of the degree distribution. Extensive numerical simulations contradict these results and clearly show that the mean-field assumption is not appropriate to describe this problem.Comment: 20 pages, 10 figure

Relevance of backtracking paths in recurrent-state epidemic spreading on networks

The understanding of epidemics on networks has greatly benefited from the recent application of message-passing approaches, which allow us to derive exact results for irreversible spreading (i.e., diseases with permanent acquired immunity) in locally treelike topologies. This success has suggested the application of the same approach to recurrent-state epidemics, for which an individual can contract the epidemic and recover repeatedly. The underlying assumption is that backtracking paths (i.e., an individual is reinfected by a neighbor he or she previously infected) do not play a relevant role. In this paper we show that this is not the case for recurrent-state epidemics since the neglect of backtracking paths leads to a formula for the epidemic threshold that is qualitatively incorrect in the large size limit. Moreover, we define a modified recurrent-state dynamics which explicitly forbids direct backtracking events and show that this modification completely upsets the phenomenology.Postprint (published version

Eigenvector localization in real networks and its implications for epidemic spreading

The spectral properties of the adjacency matrix, in particular its largest eigenvalue and the associated principal eigenvector, dominate many structural and dynamical properties of complex networks. Here we focus on the localization properties of the principal eigenvector in real networks. We show that in most cases it is either localized on the star defined by the node with largest degree (hub) and its nearest neighbors, or on the densely connected subgraph defined by the maximum $K$-core in a $K$-core decomposition. The localization of the principal eigenvector is often strongly correlated with the value of the largest eigenvalue, which is given by the local eigenvalue of the corresponding localization subgraph, but different scenarios sometimes occur. We additionally show that simple targeted immunization strategies for epidemic spreading are extremely sensitive to the actual localization set.Comment: 13 pages, 6 figure

Cumulative Merging Percolation and the epidemic transition of the Susceptible-Infected-Susceptible model in networks

We consider cumulative merging percolation (CMP), a long-range percolation process describing the iterative merging of clusters in networks, depending on their mass and mutual distance. For a specific class of CMP processes, which represents a generalization of degree-ordered percolation, we derive a scaling solution on uncorrelated complex networks, unveiling the existence of diverse mechanisms leading to the formation of a percolating cluster. The scaling solution accurately reproduces universal properties of the transition. This finding is used to infer the critical properties of the Susceptible-Infected-Susceptible (SIS) model for epidemics in infinite and finite power-law distributed networks. Here discrepancies between analytical approaches and numerical results regarding the finite size scaling of the epidemic threshold are a crucial open issue in the literature. We find that the scaling exponent assumes a nontrivial value during a long preasymptotic regime. We calculate this value, finding good agreement with numerical evidence. We also show that the crossover to the true asymptotic regime occurs for sizes much beyond currently feasible simulations. Our findings allow us to rationalize and reconcile all previously published results (both analytical and numerical), thus ending a long-standing debate.Comment: 14 pages, 9 figures, final accepted versio

Irrelevance of information outflow in opinion dynamics models

The Sznajd model for opinion dynamics has attracted a large interest as a simple realization of the psychological principle of social validation. As its most salient feature, it has been claimed that the Sznajd model is qualitatively different from other ordering processes, because it is the only one featuring outflow of information as opposed to inflow. We show that this claim is unfounded by presenting a generalized zero-temperature Glauber-type of dynamics which yields results indistinguishable from those of the Sznajd model. In one-dimension we also derive an exact expression for the exit probability of the Sznajd model, that turns out to coincide with the result of an analytical approach based on the Kirkwood approximation. This observation raises interesting questions about the applicability and limitations of this approach.Comment: 5 pages, 4 figure

Cumulative merging percolation: a long-range percolation process in networks

Percolation on networks is a common framework to model a wide range of processes, from cascading failures to epidemic spreading. Standard percolation assumes short-range interactions, implying that nodes can merge into clusters only if they are nearest neighbors. Cumulative merging percolation (CMP) is a percolation process that assumes long-range interactions such that nodes can merge into clusters even if they are topologically distant. Hence, in CMP clusters do not coincide with the topologically connected components of the network. Previous work has shown that a specific formulation of CMP features peculiar mechanisms for the formation of the giant cluster and allows one to model different network dynamics such as recurrent epidemic processes. Here we develop a more general formulation of CMP in terms of the functional form of the cluster interaction range, showing an even richer phase transition scenario with competition of different mechanisms resulting in crossover phenomena. Our analytic predictions are confirmed by numerical simulations.Postprint (author's final draft

Competing activation mechanisms in epidemics on networks

In contrast to previous common wisdom that epidemic activity in heterogeneous networks is dominated by the hubs with the largest number of connections, recent research has pointed out the role that the innermost, dense core of the network plays in sustaining epidemic processes. Here we show that the mechanism responsible of spreading depends on the nature of the process. Epidemics with a transient state are boosted by the innermost core. Contrarily, epidemics allowing a steady state present a dual scenario, where either the hub independently sustains activity and propagates it to the rest of the system, or, alternatively, the innermost network core collectively turns into the active state, maintaining it globally. In uncorrelated networks the former mechanism dominates if the degree distribution decays with an exponent larger than 5/2, and the latter otherwise. Topological correlations, rife in real networks, may perturb this picture, mixing the role of both mechanisms.Comment: 32 pages, 10 figure