Dynamic vaccination in partially overlapped multiplex network.

In this work we propose and investigate a strategy of vaccination which we call "dynamic vaccination." In our model, susceptible people become aware that one or more of their contacts are infected and thereby get vaccinated with probability ω, before having physical contact with any infected patient. Then the nonvaccinated individuals will be infected with probability ß. We apply the strategy to the susceptible-infected-recovered epidemic model in a multiplex network composed by two networks, where a fraction q of the nodes acts in both networks. We map this model of dynamic vaccination into bond percolation model and use the generating functions framework to predict theoretically the behavior of the relevant magnitudes of the system at the steady state. We find a perfect agreement between the solutions of the theoretical equations and the results of stochastic simulations. In addition, we find an interesting phase diagram in the plane ß-ω, which is composed of an epidemic and a nonepidemic phase, separated by a critical threshold line ß_{c}, which depends on q. As q decreases, ß_{c} increases, i.e., as the overlap decreases, the system is more disconnected, and therefore more virulent diseases are needed to spread epidemics. Surprisingly, we find that, for all values of q, a region in the diagram where the vaccination is so efficient that, regardless of the virulence of the disease, it never becomes an epidemic. We compare our strategy with random immunization and find that, using the same amount of vaccines for both scenarios, we obtain that the spread of disease is much lower in the case of dynamic vaccination when compared to random immunization. Furthermore, we also compare our strategy with targeted immunization and we find that, depending on ω, dynamic vaccination will perform significantly better and in some cases will stop the disease before it becomes an epidemic.

Failure and recovery in dynamical networks.

Failure, damage spread and recovery crucially underlie many spatially embedded networked systems ranging from transportation structures to the human body. Here we study the interplay between spontaneous damage, induced failure and recovery in both embedded and non-embedded networks. In our model the network's components follow three realistic processes that capture these features: (i) spontaneous failure of a component independent of the neighborhood (internal failure), (ii) failure induced by failed neighboring nodes (external failure) and (iii) spontaneous recovery of a component. We identify a metastable domain in the global network phase diagram spanned by the model's control parameters where dramatic hysteresis effects and random switching between two coexisting states are observed. This dynamics depends on the characteristic link length of the embedded system. For the Euclidean lattice in particular, hysteresis and switching only occur in an extremely narrow region of the parameter space compared to random networks. We develop a unifying theory which links the dynamics of our model to contact processes. Our unifying framework may help to better understand controllability in spatially embedded and random networks where spontaneous recovery of components can mitigate spontaneous failure and damage spread in dynamical networks.

Recovery of Interdependent Networks.

Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. An important question that has not been addressed is how to repair a failing system before it suffers total breakdown. Here we introduce a recovery strategy for nodes and develop an analytic and numerical framework for studying the concurrent failure and recovery of a system of interdependent networks based on an efficient and practically reasonable strategy. Our strategy consists of repairing a fraction of failed nodes, with probability of recovery γ, that are neighbors of the largest connected component of each constituent network. We find that, for a given initial failure of a fraction 1 - p of nodes, there is a critical probability of recovery above which the cascade is halted and the system fully restores to its initial state and below which the system abruptly collapses. As a consequence we find in the plane γ - p of the phase diagram three distinct phases. A phase in which the system never collapses without being restored, another phase in which the recovery strategy avoids the breakdown, and a phase in which even the repairing process cannot prevent system collapse.

Assortativity and leadership emerge from anti-preferential attachment in heterogeneous networks.

Real-world networks have distinct topologies, with marked deviations from purely random networks. Many of them exhibit degree-assortativity, with nodes of similar degree more likely to link to one another. Though microscopic mechanisms have been suggested for the emergence of other topological features, assortativity has proven elusive. Assortativity can be artificially implanted in a network via degree-preserving link permutations, however this destroys the graph's hierarchical clustering and does not correspond to any microscopic mechanism. Here, we propose the first generative model which creates heterogeneous networks with scale-free-like properties in degree and clustering distributions and tunable realistic assortativity. Two distinct populations of nodes are incrementally added to an initial network by selecting a subgraph to connect to at random. One population (the followers) follows preferential attachment, while the other population (the potential leaders) connects via anti-preferential attachment: they link to lower degree nodes when added to the network. By selecting the lower degree nodes, the potential leader nodes maintain high visibility during the growth process, eventually growing into hubs. The evolution of links in Facebook empirically validates the connection between the initial anti-preferential attachment and long term high degree. In this way, our work sheds new light on the structure and evolution of social networks.

Comment on "Percolation transitions are not always sharpened by making networks interdependent".

A Comment on the Letter by S. W. Son, P. Grassberger, and M. Paczuski, Phys. Rev. Lett. 107, 195702 (2011). The authors of the Letter offer a Reply.

Optimal transport exponent in spatially embedded networks.

The imposition of a cost constraint for constructing the optimal navigation structure surely represents a crucial ingredient in the design and development of any realistic navigation network. Previous works have focused on optimal transport in small-world networks built from two-dimensional lattices by adding long-range connections with Manhattan length r(ij) taken from the distribution P(ij)~r(ij)(-α), where α is a variable exponent. It has been shown that, by introducing a cost constraint on the total length of the additional links, regardless of the strategy used by the traveler (independent of whether it is based on local or global knowledge of the network structure), the best transportation condition is obtained with an exponent α=d+1, where d is the dimension of the underlying lattice. Here we present further support, through a high-performance real-time algorithm, on the validity of this conjecture in three-dimensional regular as well as in two-dimensional critical percolation clusters. Our results clearly indicate that cost constraint in the navigation problem provides a proper theoretical framework to justify the evolving topologies of real complex network structures, as recently demonstrated for the networks of the US airports and the human brain activity.

Stability of climate networks with time.

The pattern of local daily fluctuations of climate fields such as temperatures and geopotential heights is not stable and hard to predict. Surprisingly, we find that the observed relations between such fluctuations in different geographical regions yields a very robust network pattern that remains highly stable during time. Using a new systematic methodology we track the origins of the network stability. It is found that about half of this network stability is due to the spatial 2D embedding of the network, and half is due to physical coupling between climate in different locations. We also find that around the equator, the contribution of the physical coupling is significantly less pronounced compared to off-equatorial regimes. Finally, we show that there is a gradual monotonic modification of the network pattern as a function of altitude difference.

Epidemics on interconnected networks.

Populations are seldom completely isolated from their environment. Individuals in a particular geographic or social region may be considered a distinct network due to strong local ties but will also interact with individuals in other networks. We study the susceptible-infected-recovered process on interconnected network systems and find two distinct regimes. In strongly coupled network systems, epidemics occur simultaneously across the entire system at a critical infection strength ß(c), below which the disease does not spread. In contrast, in weakly coupled network systems, a mixed phase exists below ß(c) of the coupled network system, where an epidemic occurs in one network but does not spread to the coupled network. We derive an expression for the network and disease parameters that allow this mixed phase and verify it numerically. Public health implications of communities comprising these two classes of network systems are also mentioned.

Robustness of a network formed by n interdependent networks with a one-to-one correspondence of dependent nodes.

Many real-world networks interact with and depend upon other networks. We develop an analytical framework for studying a network formed by n fully interdependent randomly connected networks, each composed of the same number of nodes N. The dependency links connecting nodes from different networks establish a unique one-to-one correspondence between the nodes of one network and the nodes of the other network. We study the dynamics of the cascades of failures in such a network of networks (NON) caused by a random initial attack on one of the networks, after which a fraction p of its nodes survives. We find for the fully interdependent loopless NON that the final state of the NON does not depend on the dynamics of the cascades but is determined by a uniquely defined mutual giant component of the NON, which generalizes both the giant component of regular percolation of a single network (n=1) and the recently studied case of the mutual giant component of two interdependent networks (n=2). We also find that the mutual giant component does not depend on the topology of the NON and express it in terms of generating functions of the degree distributions of the network. Our results show that, for any n≥2 there exists a critical p=p(c)>0 below which the mutual giant component abruptly collapses from a finite nonzero value for p≥p(c) to zero for p2, a RR NON is stable for any n with p(c)<1). This results arises from the critical role played by singly connected nodes which exist in an ER NON and enhance the cascading failures, but do not exist in a RR NON.

Emergence of El Niño as an autonomous component in the climate network.

We construct and analyze a climate network which represents the interdependent structure of the climate in different geographical zones and find that the network responds in a unique way to El Niño events. Analyzing the dynamics of the climate network shows that when El Niño events begin, the El Niño basin partially loses its influence on its surroundings. After typically three months, this influence is restored while the basin loses almost all dependence on its surroundings and becomes autonomous. The formation of an autonomous basin is the missing link to understand the seemingly contradicting phenomena of the afore-noticed weakening of the interdependencies in the climate network during El Niño and the known impact of the anomalies inside the El Niño basin on the global climate system.

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

Dynamics of overlapping structures in modular networks.

Modularity is a fundamental feature of real networks, being intimately bounded to their functionality, i.e., to their capability of performing parallel tasks in a coordinated way. Although the modular structure of real graphs has been intensively studied, very little is known on the interactions between functional modules of a graph. Here, we present a general method based on synchronization of networking oscillators, that is able to detect overlapping structures in multimodular environments. We furthermore report the full analytical and theoretical description on the relationship between the overlapping dynamics and the underlying network topology. The method is illustrated by means of a series of applications.

Towards design principles for optimal transport networks.

We investigate the navigation problem in lattices with long-range connections and subject to a cost constraint. Our network is built from a regular two-dimensional (d=2) square lattice to be improved by adding long-range connections (shortcuts) with probability P(ij) approximately r(ij)(-alpha), where r(ij) is the Manhattan distance between sites i and j, and alpha is a variable exponent. We introduce a cost constraint on the total length of the additional links and find optimal transport in the system for alpha=d+1 established here for d=1 and d=2. Remarkably, this condition remains optimal, regardless of the strategy used for navigation, being based on local or global knowledge of the network structure, in sharp contrast with the results obtained for unconstrained navigation using global or local information, where the optimal conditions are alpha=0 and alpha=d, respectively. The validity of our results is supported by data on the U.S. airport network.

Synchronization interfaces and overlapping communities in complex networks.

We show that a complex network of phase oscillators may display interfaces between domains (clusters) of synchronized oscillations. The emergence and dynamics of these interfaces are studied for graphs composed of either dynamical domains (influenced by different forcing processes), or structural domains (modular networks). The obtained results allow us to give a functional definition of overlapping structures in modular networks, and suggest a practical method able to give information on overlapping clusters in both artificially constructed and real world modular networks.

Climate networks around the globe are significantly affected by El Niño.

The temperatures in different zones in the world do not show significant changes due to El Niño except when measured in a restricted area in the Pacific Ocean. We find, in contrast, that the dynamics of a climate network based on the same temperature records in various geographical zones in the world is significantly influenced by El Niño. During El Niño many links of the network are broken, and the number of surviving links comprises a specific and sensitive measure for El Niño events. While during non-El Niño periods these links which represent correlations between temperatures in different sites are more stable, fast fluctuations of the correlations observed during El Niño periods cause the links to break.

Optimization of network robustness to waves of targeted and random attacks.

We study the robustness of complex networks to multiple waves of simultaneous (i) targeted attacks in which the highest degree nodes are removed and (ii) random attacks (or failures) in which fractions p(t) and p(r) , respectively, of the nodes are removed until the network collapses. We find that the network design which optimizes network robustness has a bimodal degree distribution, with a fraction r of the nodes having degree k2 = ((k)-1+r)/r and the remainder of the nodes having degree k1=1, where k is the average degree of all the nodes. We find that the optimal value of r is of the order of p(t)/p(r) for p(t)/p(r) << 1.

Localization transition on complex networks via spectral statistics.

The spectral statistics of complex networks are numerically studied. The features of the Anderson metal-insulator transition are found to be similar for a wide range of different networks. A metal-insulator transition as a function of the disorder can be observed for different classes of complex networks for which the average connectivity is small. The critical index of the transition corresponds to the mean field expectation. When the connectivity is higher, the amount of disorder needed to reach a certain degree of localization is proportional to the average connectivity, though a precise transition cannot be identified. The absence of a clear transition at high connectivity is probably due to the very compact structure of the highly connected networks, resulting in a small diameter even for a large number of sites.