Epidemic spreading on networks with overlapping community structure

Many real networks are characterized by overlapping community structures in which vertices may belong to more than one community. In this paper, we propose a network model with overlapping community structure. The analytical and numerical results show that the connectivity distribution of this network follows a power law. We employ this network to investigate the impact of overlapping community structure on susceptible-infected-susceptible (SIS) epidemic spreading process. The simulation results indicate that significant overlapping community structure results in a major infection prevalence and leads to a peak of the spread velocity in the early stages of the emerging infection.     

Epidemic spreading on weighted complex networks

Nowadays, the emergence of online services provides various multi-relation information to support the comprehensive understanding of the epidemic spreading process. In this Letter, we consider the edge weights to represent such multi-role relations. In addition, we perform detailed analysis of two representative metrics, outbreak threshold and epidemic prevalence, on SIS and SIR models. Both theoretical and simulation results find good agreements with each other. Furthermore, experiments show that, on fully mixed networks, the weight distribution on edges would not affect the epidemic results once the average weight of whole network is fixed. This work may shed some light on the in-depth understanding of epidemic spreading on multi-relation and weighted networks.     

A modified SIS model with an infective medium on complex networks and its global stability

In this paper, we propose a modified susceptible-infected-susceptible model with an infective medium, which describes epidemics transmitted through an infective medium on complex networks. We examine epidemic thresholds for disease spreading by using this new model and compare it with the standard SIS model and another SIS model having an infective medium. We also study and compare the effects of the uniform immunization scheme on different models. We finally give some necessary and sufficient conditions for the global stability of the new model.     

Dynamical diversity induced by individual responsive immunization

A voluntary vaccination allows for a healthy individual to choose vaccination according to the individual’s local information. Hence, vaccination has the potential to provide a complex negative feedback (non-infection decreases propensity for vaccination, hence increasing infection and vice versa). In this paper, we investigate a kind of SIS epidemic model with a deterministic and voluntary vaccination scheme in scale-free networks. We first study a threshold model with no historical information. By using the comparative method we confirm that under some conditions there exist two critical values of infection rates to determine three kinds of epidemic dynamical behaviors: the epidemic spread, the asymptotical decay and the exponential decay. Furthermore, a mean-field approximation model can predict the maximal infection level but cannot predict the existence of two critical infection rates. In numerical simulations, we observe a maximum in epidemic duration as a function of the model parameter. A similar phenomenon has been found in the model with historical information. Finally, we study a degree-weighted model with a nonnegative exponent α$\alpha$ where α=0$\alpha =0$ corresponds to the threshold model. We find that at the steady state the infection density increases with α$\alpha$, while the variation of the vaccination fraction is less straightforward.     

Roles of different update strategies in the vaccination behavior on two-layered networks

Epidemic spreading has been widely investigated over the past decades. And voluntary vaccination has been often utilized to explore dynamical process in epidemics where vaccines are available. In this letter, we establish a framework considering conformity motivated update as well as myopic best response motivated update on a family network which is demonstrated by a two-layered network. Extensive numerical simulations are conducted to study the dynamics of epidemic spreading under the aforementioned update rules, from which we discover the oscillation phenomenon under the pure myopic best response condition and the amplitude diversification phenomenon under the mixing of conformity and myopic best response motivated conditions. Moreover, we find that smaller overlapping fraction of links on two-layered network shall promote the epidemic propagation. The current findings can shed some lights on the evolution of epidemic spreading process in the real-world scenarios.     

Epidemic spreading in a scale-free network of regular lattices

The susceptible-infected-susceptible (SIS) epidemics in a scale-free network in which each node is a square lattice itself is investigated through large-scale computer simulations. The model combines a local contact process among individuals in a node (or city) with stochastic long-range infections due to people traveling between cities interconnected by the national transportation scale-free network. A nonzero epidemic threshold is found and it is approached with a power-law behavior by the density of infected individuals, as observed in the small-world network of Watts and Strogatz. Also, the epidemic propagation follows a 1/f$1/f$, hierarchical dynamics from the highly connected square lattices to the smaller degree nodes in outbreaks with sizes distributed accordingly a Gaussian function.     

Epidemic spreading in lattice-embedded scale-free networks

We study geographical effects on the spread of diseases in lattice-embedded scale-free networks. The geographical structure is represented by the connecting probability of two nodes that is related to the Euclidean distance between them in the lattice. By studying the standard susceptible-infected model, we found that the geographical structure has great influences on the temporal behavior of epidemic outbreaks and the propagation in the underlying network: the more geographically constrained the network is, the more smoothly the epidemic spreads, which is different from the clearly hierarchical dynamics that the infection pervades the networks in a progressive cascade across smaller-degree classes in Barabási–Albert scale-free networks.     

Modelling the spread of sexually transmitted diseases on scale-free networks

In this paper a new model for the spread of sexually transmitted diseases (STDs) is presented. The dynamic behaviors of the model on a heterogenous scale-free (SF) network are considered, where the absence of a threshold on the SF network is demonstrated, and the stability of the disease-free equilibrium is obtained. Three immunization strategies, uniform immunization, proportional immunization and targeted immunization, are applied in this model. Analytical and simulated results are given to show that the proportional immunization strategy in the model is effective on SF networks.     

Epidemic spreading with nonlinear infectivity in weighted scale-free networks

In this paper, we investigate the epidemic spreading for the SIR model in weighted scale-free networks with the nonlinear infectivity and weighted transmission rate. Concretely, we introduce the infectivity exponent α and the weight exponent β into the epidemic system, then examine the impact of α and β on the epidemic spreading. We show that one can adjust the values of α and β to rebuild a nonzero finite epidemic threshold. Furthermore, we also find the infectivity exponent α has a stronger effect not only on the epidemic threshold, but also on the epidemic prevalence. In addition, it is also interesting to see that the absence of the epidemic threshold appears not very dejected, since the prevalence grows much more slowly as the transmission rate λ increases.     

Epidemic spreading on uncorrelated heterogenous networks with non-uniform transmission

We investigate the combinational effects of non-uniform transmission of edges and the network structure in susceptible-infected-removed models of the epidemic spreading. Here the degree-degree correlation is neglected and the transmission of individual edge depends on the degree of infected node. With the additional assumption that we can compartment a independent factor λ₀ from the transmission, we analyzed the effects of degree distribution P(k) and transmission distribution λ(k) on the threshold of λ₀.     

Competitive cluster growth on networks: Complex dynamics and survival strategies

We extend the study of a model of competitive cluster growth in an active medium from a regular topology to a complex network topology; this is done by adding nonlocal connections with probability p to sites on a regular lattice, thus enabling one to interpolate between regularity and full randomness. The model on networks demonstrates high sensitivity to small changes in initial configurations, which we characterize using damage spreading. The main focus of this paper is, however, the devising of survival strategies through selective networking, to alter the fate of an arbitrarily chosen cluster: whether this be to revive a dying cluster to life, or to make a weak survivor into a stronger one. Although such goals are typically achieved by networking with relatively small clusters, our results suggest that it ought to be possible also to network successfully with peers and larger clusters. The main indication of this comes from the probability distributions of mass differences between survivors and their immediate neighbours, which show an interesting universality; they suggest strategies for winning against the odds.     

Epidemic threshold and immunization on generalized networks

The susceptible–infected–susceptible (SIS) model is widely adopted in the studies of epidemic dynamics. When it is applied on contact networks, these networks mostly consist of nodes connected by undirected and unweighted edges following certain statistical properties, whereas in this article we consider the threshold and immunization problem for the SIS model on generalized networks that may contain different kinds of nodes and edges which are very possible in the real situation. We proved that an epidemic will become extinct if and only if the spectral radius of the corresponding parameterized adjacent matrix (PAM) is smaller than 1. Based on this result, we can evaluate the efficiency of immune strategies and take several prevailing ones as examples. In addition, we also develop methods that can precisely find the optimal immune strategies for networks with the given PAM.     

Growing community networks with local events

The study of community networks has attracted considerable attention recently. In this paper, we propose an evolving community network model based on local processes, the addition of new nodes intra-community and new links intra- or inter-community. Employing growth and preferential attachment mechanisms, we generate networks with a generalized power-law distribution of nodes’ degrees.     

Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks

We investigate the effects of delaying the time to recovery (delayed recovery) and of nonuniform transmission on the propagation of diseases on structured populations. Through a mean-field approximation and large-scale numerical simulations, we find that postponing the transition from the infectious to the recovered states can largely reduce the epidemic threshold, therefore promoting the outbreak of epidemics. On the other hand, if we consider nonuniform transmission among individuals, the epidemic threshold increases, thus inhibiting the spreading process. When both mechanisms are at work, the latter might prevail, hence resulting in an increase of the epidemic threshold with respect to the standard case, in which both ingredients are absent. Our findings are of interest for a better understanding of how diseases propagate on structured populations and to a further design of efficient immunization strategies.     

Phase diagram of epidemic spreading — unimodal vs. bimodal probability distributions

Disease spreading on complex networks is studied in SIR model. Simulations on empirical complex networks reveal two specific regimes of disease spreading: local containment and epidemic outbreak. The variables measuring the extent of disease spreading are in general characterized by a bimodal probability distribution. Phase diagrams of disease spreading for empirical complex networks are introduced. A theoretical model of disease spreading on m-ary tree is investigated both analytically and in simulations. It is shown that the model reproduces qualitative features of phase diagrams of disease spreading observed in empirical complex networks. The role of tree-like structure of complex networks in disease spreading is discussed.     

Modelling epidemics on networks

Infectious disease remains, despite centuries of work to control and mitigate its effects, a major problem facing humanity. This paper reviews the mathematical modelling of infectious disease epidemics on networks, starting from the simplest Erdös–Rényi random graphs, and building up structure in the form of correlations, heterogeneity and preference, paying particular attention to the links between random graph theory, percolation and dynamical systems representing transmission. Finally, the problems posed by networks with a large number of short closed loops are discussed.     

The spreading of opposite opinions on online social networks with authoritative nodes

The study of opinion dynamics, such as spreading and controlling of rumors, has become an important issue on social networks. Numerous models have been devised to describe this process, including epidemic models and spin models, which mainly focus on how opinions spread and interact with each other, respectively. In this paper, we propose a model that combines the spreading stage and the interaction stage for opinions to illustrate the process of dispelling a rumor. Moreover, we set up authoritative nodes, which disseminate positive opinion to counterbalance the negative opinion prevailing on online social networking sites. With analysis of the relationship among positive opinion proportion, opinion strength and the density of authoritative nodes in networks with different topologies, we demonstrate that the positive opinion proportion grows with the density of authoritative nodes until the positive opinion prevails in the entire network. In particular, the relationship is linear in homogeneous topologies. Besides, it is also noteworthy that initial locations of the negative opinion source and authoritative nodes do not influence positive opinion proportion in homogeneous networks but have a significant impact on heterogeneous networks. The results are verified by numerical simulations and are helpful to understand the mechanism of two different opinions interacting with each other on online social networking sites.     

Analysis and models of bilateral investment treaties using a social networks approach

Bilateral investment treaties (BITs) are agreements between two countries for the reciprocal encouragement, promotion and protection of investments in each other’s territories by companies based in either country. Germany and Pakistan signed the first BIT in 1959 and since then, BITs are one of the most popular and widespread form of international agreement. In this work we study the proliferation of BITs using a social networks approach. We propose a network growth model that dynamically replicates the empirical topological characteristics of the BIT network.     

Planar unclustered scale-free graphs as models for technological and biological networks

Many real life networks present an average path length logarithmic with the number of nodes and a degree distribution which follows a power law. Often these networks have also a modular and self-similar structure and, in some cases — usually associated with topological restrictions — their clustering is low and they are almost planar. In this paper we introduce a family of graphs which share all these properties and are defined by two parameters. As their construction is deterministic, we obtain exact analytic expressions for relevant properties of the graphs including the degree distribution, degree correlation, diameter, and average distance, as a function of the two defining parameters. Thus, the graphs are useful to model some complex networks, in particular several families of technological and biological networks, and in the design of new practical communication algorithms in relation to their dynamical processes. They can also help understanding the underlying mechanisms that have produced their particular structure.     

Congestion analysis of traffic networks with direction-dependant heterogeneity

Traffic flow directionality and network weight asymmetry are widespread notions in traffic networks. This paper investigates the influence of direction-dependant heterogeneity on traffic congestion. To capture the effect of the link directionality and link weight asymmetry, the heterogeneity indexes of complex networks and the traffic flow model are introduced. The numerical results show that the critical value of heterogeneity determines congestion transition processes. The congestion degree increases with heterogeneity when the network heterogeneity is at a subcritical region. A network is more tolerant of congestion if the heterogeneity of the network is smaller or larger than the critical value. Furthermore, when heterogeneity reaches the critical value, the average number of accumulated vehicles arrives at the maximum and the traffic flow is under a serious congestion state. A significant improvement on the tolerance to congestion of traffic networks can be made if the network heterogeneity is controlled within a reasonable range.