Lidia A. Braunstein

Optimal Paths in Disordered Complex Networks

We study the optimal distance in networks, l(opt), defined as the length of the path minimizing the total weight, in the presence of disorder. Disorder is introduced by assigning random weights to the links or nodes. For strong disorder, where the maximal weight along the path dominates the sum, we find that l(opt) approximately N(1/3) in both Erdos-Rényi (ER) and Watts-Strogatz (WS) networks. For scale-free (SF) networks, with degree distribution P(k) approximately k(-lambda), we find that l(opt) scales as N((lambda-3)/(lambda-1)) for 3<lambda<4 and as N(1/3) for lambda> or =4. Thus, for these networks, the small-world nature is destroyed. For 2<lambda<3, our numerical results suggest that l(opt) scales as ln(lambda-1N. We also find numerically that for weak disorder l(opt) approximately ln(N for both the ER and WS models as well as for SF networks.

Structure of shells in complex networks

We define shell l in a network as the set of nodes at distance l with respect to a given node and define rl as the fraction of nodes outside shell l . In a transport process, information or disease usually diffuses from a random node and reach nodes shell after shell. Thus, understanding the shell structure is crucial for the study of the transport property of networks. We study the statistical properties of the shells of a randomly chosen node. For a randomly connected network with given degree distribution, we derive analytically the degree distribution and average degree of the nodes residing outside shell l as a function of rl. Further, we find that rl follows an iterative functional form rl=phi(rl-1) , where phi is expressed in terms of the generating function of the original degree distribution of the network. Our results can explain the power-law distribution of the number of nodes Bl found in shells with l larger than the network diameter d , which is the average distance between all pairs of nodes. For real-world networks the theoretical prediction of rl deviates from the empirical rl. We introduce a network correlation function c(rl) identical with rl/phi(rl-1) to characterize the correlations in the network, where rl is the empirical value and phi(rl-1) is the theoretical prediction. c(rl)=1 indicates perfect agreement between empirical results and theory. We apply c(rl) to several model and real-world networks. We find that the networks fall into two distinct classes: (i) a class of poorly connected networks with c(rl)>1 , where a larger (smaller) fraction of nodes resides outside (inside) distance l from a given node than in randomly connected networks with the same degree distributions. Examples include the Watts-Strogatz model and networks characterizing human collaborations such as citation networks and the actor collaboration network; (ii) a class of well-connected networks with c(rl)<1 . Examples include the Barabási-Albert model and the autonomous system Internet network.

Epidemics in partially overlapped multiplex networks.

Many real networks exhibit a layered structure in which links in each layer reflect the function of nodes on different environments. These multiple types of links are usually represented by a multiplex network in which each layer has a different topology. In real-world networks, however, not all nodes are present on every layer. To generate a more realistic scenario, we use a generalized multiplex network and assume that only a fraction of the nodes are shared by the layers. We develop a theoretical framework for a branching process to describe the spread of an epidemic on these partially overlapped multiplex networks. This allows us to obtain the fraction of infected individuals as a function of the effective probability that the disease will be transmitted . We also theoretically determine the dependence of the epidemic threshold on the fraction of shared nodes in a system composed of two layers. We find that in the limit of the threshold is dominated by the layer with the smaller isolated threshold. Although a system of two completely isolated networks is nearly indistinguishable from a system of two networks that share just a few nodes, we find that the presence of these few shared nodes causes the epidemic threshold of the isolated network with the lower propagating capacity to change discontinuously and to acquire the threshold of the other network.

Unification of theoretical approaches for epidemic spreading on complex networks

Models of epidemic spreading on complex networks have attracted great attention among researchers in physics, mathematics, and epidemiology due to their success in predicting and controlling scenarios of epidemic spreading in real-world scenarios. To understand the interplay between epidemic spreading and the topology of a contact network, several outstanding theoretical approaches have been developed. An accurate theoretical approach describing the spreading dynamics must take both the network topology and dynamical correlations into consideration at the expense of increasing the complexity of the equations. In this short survey we unify the most widely used theoretical approaches for epidemic spreading on complex networks in terms of increasing complexity, including the mean-field, the heterogeneous mean-field, the quench mean-field, dynamical message-passing, link percolation, and pairwise approximation. We build connections among these approaches to provide new insights into developing an accurate theoretical approach to spreading dynamics on complex networks.

Optimal paths in complex networks with correlated weights: The worldwide airport network

We study complex networks with weights w(ij) associated with each link connecting node i and j. The weights are chosen to be correlated with the network topology in the form found in two real world examples: (a) the worldwide airport network and (b) the E. Coli metabolic network. Here w(ij) approximately equals x(ij)(k(i)k(j))alpha, where k(i) and k(j) are the degrees of nodes i and j , x(ij) is a random number, and alpha represents the strength of the correlations. The case alpha >0 represents correlation between weights and degree, while alpha< 0 represents anticorrelation and the case alpha=0 reduces to the case of no correlations. We study the scaling of the lengths of the optimal paths, l(opt), with the system size N in strong disorder for scale-free networks for different alpha. We find two different universality classes for l(opt), in strong disorder depending on alpha: (i) if alpha >0 , then for lambda >2 the scaling law l(opt) approximately equals N(1/3), where lambda is the power-law exponent of the degree distribution of scale-free networks, and (ii) if alpha< or =0 , then l(opt) approximately equals N((nu)(opt)) with nu(opt) identical to its value for the uncorrelated case alpha=0. We calculate the robustness of correlated scale-free networks with different alpha and find the networks with alpha< 0 to be the most robust networks when compared to the other values of alpha. We propose an analytical method to study percolation phenomena on networks with this kind of correlation, and our numerical results suggest that for scale-free networks with alpha< 0 , the percolation threshold p(c) is finite for lambda >3, which belongs to the same universality class as alpha=0 . We compare our simulation results with the real worldwide airport network, and we find good agreement.

Suppressing disease spreading by using information diffusion on multiplex networks

Although there is always an interplay between the dynamics of information diffusion and disease spreading, the empirical research on the systemic coevolution mechanisms connecting these two spreading dynamics is still lacking. Here we investigate the coevolution mechanisms and dynamics between information and disease spreading by utilizing real data and a proposed spreading model on multiplex network. Our empirical analysis finds asymmetrical interactions between the information and disease spreading dynamics. Our results obtained from both the theoretical framework and extensive stochastic numerical simulations suggest that an information outbreak can be triggered in a communication network by its own spreading dynamics or by a disease outbreak on a contact network, but that the disease threshold is not affected by information spreading. Our key finding is that there is an optimal information transmission rate that markedly suppresses the disease spreading. We find that the time evolution of the dynamics in the proposed model qualitatively agrees with the real-world spreading processes at the optimal information transmission rate.

Non-consensus Opinion Models on Complex Networks

Social dynamic opinion models have been widely studied to understand how interactions among individuals cause opinions to evolve. Most opinion models that utilize spin interaction models usually produce a consensus steady state in which only one opinion exists. Because in reality different opinions usually coexist, we focus on non-consensus opinion models in which above a certain threshold two opinions coexist in a stable relationship. We revisit and extend the non-consensus opinion (NCO) model introduced by Shao et al. (Phys. Rev. Lett. 103:01870, 2009). The NCO model in random networks displays a second order phase transition that belongs to regular mean field percolation and is characterized by the appearance (above a certain threshold) of a large spanning cluster of the minority opinion. We generalize the NCO model by adding a weight factor W to each individual’s original opinion when determining their future opinion (NCOW model). We find that as W increases the minority opinion holders tend to form stable clusters with a smaller initial minority fraction than in the NCO model. We also revisit another non-consensus opinion model based on the NCO model, the inflexible contrarian opinion (ICO) model (Li et al. in Phys. Rev. E 84:066101, 2011), which introduces inflexible contrarians to model the competition between two opinions in a steady state. Inflexible contrarians are individuals that never change their original opinion but may influence the opinions of others. To place the inflexible contrarians in the ICO model we use two different strategies, random placement and one in which high-degree nodes are targeted. The inflexible contrarians effectively decrease the size of the largest rival-opinion cluster in both strategies, but the effect is more pronounced under the targeted method. All of the above models have previously been explored in terms of a single network, but human communities are usually interconnected, not isolated. Because opinions propagate not only within single networks but also between networks, and because the rules of opinion formation within a network may differ from those between networks, we study here the opinion dynamics in coupled networks. Each network represents a social group or community and the interdependent links joining individuals from different networks may be social ties that are unusually strong, e.g., married couples. We apply the non-consensus opinion (NCO) rule on each individual network and the global majority rule on interdependent pairs such that two interdependent agents with different opinions will, due to the influence of mass media, follow the majority opinion of the entire population. The opinion interactions within each network and the interdependent links across networks interlace periodically until a steady state is reached. We find that the interdependent links effectively force the system from a second order phase transition, which is characteristic of the NCO model on a single network, to a hybrid phase transition, i.e., a mix of second-order and abrupt jump-like transitions that ultimately becomes, as we increase the percentage of interdependent agents, a pure abrupt transition. We conclude that for the NCO model on coupled networks, interactions through interdependent links could push the non-consensus opinion model to a consensus opinion model, which mimics the reality that increased mass communication causes people to hold opinions that are increasingly similar. We also find that the effect of interdependent links is more pronounced in interdependent scale free networks than in interdependent Erdős Rényi networks.

Immunization strategy for epidemic spreading on multilayer networks

In many real-world complex systems, individuals have many kind of interactions among them, suggesting that it is necessary to consider a layered structure framework to model systems such as social interactions. This structure can be captured by multilayer networks and can have major effects on the spreading of process that occurs over them, such as epidemics. In this Letter we study a targeted immunization strategy for epidemic spreading over a multilayer network. We apply the strategy in one of the layers and study its effect in all layers of the network disregarding degree-degree correlation among layers. We found that the targeted strategy is not as efficient as in isolated networks, due to the fact that in order to stop the spreading of the disease it is necessary to immunize more than the 80 % of the individuals. However, the size of the epidemic is drastically reduced in the layer where the immunization strategy is applied compared to the case with no mitigation strategy. Thus, the immunization strategy has a major effect on the layer were it is applied, but does not efficiently protect the individuals of other layers.

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<w(c)) where the disease reaches a large fraction of the population from a phase (w≥w(c)) where the disease does not spread out. We find that in our model the topology of the network strongly affects the size of the propagation and that w(c) increases with the mean degree and heterogeneity of the network. We also find that w(c) is reduced if we perform a preferential rewiring, in which the rewiring probability is proportional to the degree of infected nodes.

Epidemic Model with Isolation in Multilayer Networks

The Susceptible-Infected-Recovered (SIR) model has successfully mimicked the propagation of such airborne diseases as influenza A (H1N1). Although the SIR model has recently been studied in a multilayer networks configuration, in almost all the research the isolation of infected individuals is disregarded. Hence we focus our study in an epidemic model in a two-layer network, and we use an isolation parameter w to measure the effect of quarantining infected individuals from both layers during an isolation period tw. We call this process the Susceptible-Infected-Isolated-Recovered (SIIR) model. Using the framework of link percolation we find that isolation increases the critical epidemic threshold of the disease because the time in which infection can spread is reduced. In this scenario we find that this threshold increases with w and tw. When the isolation period is maximum there is a critical threshold for w above which the disease never becomes an epidemic. We simulate the process and find an excellent agreement with the theoretical results.