Triple point in correlated interdependent networks.

Many real-world networks depend on other networks, often in nontrivial ways, to maintain their functionality. These interdependent "networks of networks" are often extremely fragile. When a fraction 1-p of nodes in one network randomly fails, the damage propagates to nodes in networks that are interdependent and a dynamic failure cascade occurs that affects the entire system. We present dynamic equations for two interdependent networks that allow us to reproduce the failure cascade for an arbitrary pattern of interdependency. We study the "rich club" effect found in many real interdependent network systems in which the high-degree nodes are extremely interdependent, correlating a fraction α of the higher-degree nodes on each network. We find a rich phase diagram in the plane p-α, with a triple point reminiscent of the triple point of liquids that separates a nonfunctional phase from two functional phases.

Slow epidemic extinction in populations with heterogeneous infection rates.

We explore how heterogeneity in the intensity of interactions between people affects epidemic spreading. For that, we study the susceptible-infected-susceptible model on a complex network, where a link connecting individuals i and j is endowed with an infection rate ß(ij)=λw(ij) proportional to the intensity of their contact w(ij), with a distribution P(w(ij)) taken from face-to-face experiments analyzed in Cattuto et al. [PLoS ONE 5, e11596 (2010)]. We find an extremely slow decay of the fraction of infected individuals, for a wide range of the control parameter λ. Using a distribution of width a we identify two large regions in the a-λ space with anomalous behaviors, which are reminiscent of rare region effects (Griffiths phases) found in models with quenched disorder. We show that the slow approach to extinction is caused by isolated small groups of highly interacting individuals, which keep epidemics alive for very long times. A mean-field approximation and a percolation approach capture with very good accuracy the absorbing-active transition line for weak (small a) and strong (large a) disorder, respectively.

Intermittent social distancing strategy for epidemic control.

We study the critical effect of an intermittent social distancing strategy on the propagation of epidemics in adaptive complex networks. We characterize the effect of our strategy in the framework of the susceptible-infected-recovered model. In our model, based on local information, a susceptible individual interrupts the contact with an infected individual with a probability σ and restores it after a fixed time t(b). We find that, depending on the network topology, in our social distancing strategy there exists a cutoff threshold σ(c) beyond which the epidemic phase disappears. Our results are supported by a theoretical framework and extensive simulations of the model. Furthermore we show that this strategy is very efficient because it leads to a "susceptible herd behavior" that protects a large fraction of susceptible individuals. We explain our results using percolation arguments.

Quarantine-generated phase transition in epidemic spreading.

We study the critical effect of quarantine on the propagation of epidemics on an adaptive network of social contacts. For this purpose, we analyze the susceptible-infected-recovered model in the presence of quarantine, where susceptible individuals protect themselves by disconnecting their links to infected neighbors with probability w and reconnecting them to other susceptible individuals chosen at random. Starting from a single infected individual, we show by an analytical approach and simulations that there is a phase transition at a critical rewiring (quarantine) threshold w(c) separating a phase (w

Conservative model for synchronization problems in complex networks.

In this paper we study the scaling behavior of the interface fluctuations (roughness) for a discrete model with conservative noise on complex networks. Conservative noise is a noise which has no external flux of deposition on the surface and the whole process is due to the diffusion. It was found that in Euclidean lattices the roughness of the steady state W(s) does not depend on the system size. Here, we find that for scale-free networks of N nodes, characterized by a degree distribution P(k) approximately k(-lambda), W(s) is independent of N for any lambda. This behavior is very different than the one found by Pastore y Piontti [Phys. Rev. E 76, 046117 (2007)] for a discrete model with nonconservative noise, which implies an external flux, where W(s) approximately ln N for lambda<3 , and was explained by nonlinear terms in the analytical evolution equation for the interface [La Rocca, Phys. Rev. E 77, 046120 (2008)]. In this work we show that in these processes with conservative noise the nonlinear terms are not relevant to describe the scaling behavior of W(s).

Evolution equation for a model of surface relaxation in complex networks.

In this paper we derive analytically the evolution equation of the interface for a model of surface growth with relaxation to the minimum (SRM) in complex networks. We were inspired by the disagreement between the scaling results of the steady state of the fluctuations between the discrete SRM model and the Edward-Wilkinson process found in scale-free networks with degree distribution P(k) approximately k(-lambda) for lambda<3 [Pastore y Piontti, Phys. Rev. E 76, 046117 (2007)]. Even though for Euclidean lattices the evolution equation is linear, we find that in complex heterogeneous networks nonlinear terms appear due to the heterogeneity and the lack of symmetry of the network; they produce a logarithmic divergency of the saturation roughness with the system size as found by Pastore y Piontti for lambda<3.

Discrete surface growth process as a synchronization mechanism for scale-free complex networks.

We consider the discrete surface growth process with relaxation to the minimum [F. Family, J. Phys. A 19, L441 (1986)] as a possible synchronization mechanism on scale-free networks, characterized by a degree distribution P(k) approximately k;{-lambda} , where k is the degree of a node and lambda its broadness, and compare it with the usually applied Edward-Wilkinson process (EW) [S. F. Edwards and D. R. Wilkinson, Proc. R. Soc. London, Ser. A 381, 17 (1982)]. In spite of both processes belonging to the same universality class for Euclidean lattices, in this work we demonstrate that for scale-free networks with exponents lambda<3 the scaling behavior of the roughness in the saturation cannot be explained by the EW process. Moreover, we show that for these ubiquitous cases the Edward-Wilkinson process enhances spontaneously the synchronization when the system size is increased. This nonphysical result is mainly due to finite size effects due to the underlying network. Contrarily, the discrete surface growth process does not present this flaw and is applicable for every lambda .