Suppose we have a virus or one competing idea product that propagates over a multiple profile (e.g., social) network. Can we predict what proportion of the network will actually get "infected" (e.g., spread the idea or buy the competing product), when the nodes of the network appear to have different sensitivity based on their profile? For example, if there are two profiles @math and @math in a network and the nodes of profile @math and profile @math are susceptible to a highly spreading virus with probabilities @math and @math respectively, what percentage of both profiles will actually get infected from the virus at the end? To reverse the question, what are the necessary conditions so that a predefined percentage of the network is infected? We assume that nodes of different profiles can infect one another and we prove that under realistic conditions, apart from the weak profile (great sensitivity), the stronger profile (low sensitivity) will get infected as well. First, we focus on cliques with the goal to provide exact theoretical results as well as to get some intuition as to how a virus affects such a multiple profile network. Then, we move to the theoretical analysis of arbitrary networks. We provide bounds on certain properties of the network based on the probabilities of infection of each node in it when it reaches the steady state. Finally, we provide extensive experimental results that verify our theoretical results and at the same time provide more insight on the problem.
The majority of the presented models have studied a single epidemic in a single topology whereas in later work @cite_6 , multiple virus models are introduced. All of them require that the network is fair-play, which means that all nodes have exactly the same behavior towards the viruses. In @cite_6 , an @math model was used with two competing viruses that infect nodes of arbitrary topologies where the nodes are mutually immune. The main result is that for any topology the stronger virus (above threshold) survives and wipes out the weaker one ("winner takes all"). In later work, the condition for mutual immunity is removed and the focus is on the conditions where nodes are infected from both viruses @cite_12 . In both cases, the model used is SIS-like whereas in @cite_8 the authors provide a generalized model that includes the majority of known epidemiological models with appropriate parameter definition in discrete time. All the aforementioned models, are applied to simple networks whereas recent work @cite_20 , refers to competing viruses in a composite network and conjectures that the stronger one will prevail over the weaker. Once again, the problem is formulated by a non-linear dynamical system.
@cite_20 @cite_12 @cite_6 @cite_8
2398538542 1914370899 2124486211 2171031021 2025392770 2964074424 2963120360 2152769037 2112494680 2040330999 2118820208 2102365618 1914027636 1987511126 2068322213 2086916865 2130482496 2092418988 2038195874 2047311061
The spread of epidemics and malware is commonly modeled by diffusion processes on networks. Protective interventions such as vaccinations or installing anti-virus software are used to contain their spread. Typically, each node in the network has to decide its own strategy of securing itself, and its benefit depends on which other nodes are secure, making this a natural game-theoretic setting. There has been a lot of work on network security game models, but most of the focus has been either on simplified epidemic models or homogeneous network structure. We develop a new formulation for an epidemic containment game, which relies on the characterization of the SIS model in terms of the spectral radius of the network. We show in this model that pure Nash equilibria (NE) always exist, and can be found by a best response strategy. We analyze the complexity of finding NE, and derive rigorous bounds on their costs and the Price of Anarchy or PoA (the ratio of the cost of the worst NE to the optimum social cost) in general graphs as well as in random graph models. In particular, for arbitrary power-law graphs with exponent β > 2, we show that the PoA is bounded by O(T2(β -1))), where T = γ α is the ratio of the recovery rate to the transmission rate in the SIS model. We prove that this bound is tight up to a constant factor for the Chung-Lu random power-law graph model. We study the characteristics of Nash equilibria empirically in different real communication and infrastructure networks, and find that our analytical results can help explain some of the empirical observations. Given a contact network that changes over time (say, day vs night connectivity), and the SIS (susceptible infected susceptible, flu like) virus propagation model, what can we say about its epidemic threshold? That is, can we determine when a small infection will "take-off" and create an epidemic? Consequently then, which nodes should we immunize to prevent an epidemic? This is a very real problem, since, e.g. people have different connections during the day at work, and during the night at home. Static graphs have been studied for a long time, with numerous analytical results. Time-evolving networks are so hard to analyze, that most existing works are simulation studies [5]. Specifically, our contributions in this paper are: (a) we formulate the problem by approximating it by a Non-linear Dynamical system (NLDS), (b) we derive the first closed formula for the epidemic threshold of time-varying graphs under the SIS model, and finally (c) we show the usefulness of our threshold by presenting efficient heuristics and evaluate the effectiveness of our methods on synthetic and real data like the MIT reality mining graphs. The strong analogy between biological viruses and their computational counterparts has motivated the authors to adapt the techniques of mathematical epidemiology to the study of computer virus propagation. In order to allow for the most general patterns of program sharing, a standard epidemiological model is extended by placing it on a directed graph and a combination of analysis and simulation is used to study its behavior. The conditions under which epidemics are likely to occur are determined, and, in cases where they do, the dynamics of the expected number of infected individuals are examined as a function of time. It is concluded that an imperfect defense against computer viruses can still be highly effective in preventing their widespread proliferation, provided that the infection rate does not exceed a well-defined critical epidemic threshold. > Suppose we have two competing ideas products viruses, that propagate over a social or other network. Suppose that they are strong virulent enough, so that each, if left alone, could lead to an epidemic. What will happen when both operate on the network? Earlier models assume that there is perfect competition: if a user buys product 'A' (or gets infected with virus 'X'), she will never buy product 'B' (or virus 'Y'). This is not always true: for example, a user could install and use both Firefox and Google Chrome as browsers. Similarly, one type of flu may give partial immunity against some other similar disease. In the case of full competition, it is known that 'winner takes all,' that is the weaker virus product will become extinct. In the case of no competition, both viruses survive, ignoring each other. What happens in-between these two extremes? We show that there is a phase transition: if the competition is harsher than a critical level, then 'winner takes all;' otherwise, the weaker virus survives. These are the contributions of this paper (a) the problem definition, which is novel even in epidemiology literature (b) the phase-transition result and (c) experiments on real data, illustrating the suitability of our results. We propose a simple game for modeling containment of the spread of viruses in a graph of n nodes. Each node must choose to either install anti-virus software at some known cost C, or risk infection and a loss L if a virus that starts at a random initial point in the graph can reach it without being stopped by some intermediate node. We prove many game theoretic properties of the model, including an easily applied characterization of Nash equilibria, culminating in our showing that a centralized solution can give a much better total cost than an equilibrium solution. Though it is NP-hard to compute such a social optimum, we show that the problem can be reduced to a previously unconsidered combinatorial problem that we call the sum-of-squares partition problem. Using a greedy algorithm based on sparse cuts, we show that this problem can be approximated to within a factor of O(log1.5 n). Understanding the dynamics of computer virus (malware, worm) in cyberspace is an important problem that has attracted a fair amount of attention. Early investigations for this purpose adapted biological epidemic models, and thus inherited the so-called homogeneity assumption that each node is equally connected to others. Later studies relaxed this often unrealistic homogeneity assumption, but still focused on certain power-law networks. Recently, researchers investigated epidemic models in arbitrary networks (i.e., no restrictions on network topology). However, all these models only capture push-based infection, namely that an infectious node always actively attempts to infect its neighboring nodes. Very recently, the concept of pull-based infection was introduced but was not treated rigorously. Along this line of research, the present article investigates push- and pull-based epidemic spreading dynamics in arbitrary networks, using a nonlinear dynamical systems approach. The article advances the state-of-the-art as follows: (1) It presents a more general and powerful sufficient condition (also known as epidemic threshold in the literature) under which the spreading will become stable. (2) It gives both upper and lower bounds on the global mean infection rate, regardless of the stability of the spreading. (3) It offers insights into, among other things, the estimation of the global mean infection rate through localized monitoring of a small constant number of nodes, without knowing the values of the parameters. Theoretical modeling of computer virus worm epidemic dynamics is an important problem that has attracted many studies. However, most existing models are adapted from biological epidemic ones. Although biological epidemic models can certainly be adapted to capture some computer virus spreading scenarios (especially when the so-called homogeneity assumption holds), the problem of computer virus spreading is not well understood because it has many important perspectives that are not necessarily accommodated in the biological epidemic models. In this article, we initiate the study of such a perspective, namely that of adaptive defense against epidemic spreading in arbitrary networks. More specifically, we investigate a nonhomogeneous Susceptible-Infectious-Susceptible (SIS) model where the model parameters may vary with respect to time. In particular, we focus on two scenarios we call semi-adaptive defense and fully adaptive defense, which accommodate implicit and explicit dependency relationships between the model parameters, respectively. In the semi-adaptive defense scenario, the model’s input parameters are given; the defense is semi-adaptive because the adjustment is implicitly dependent upon the outcome of virus spreading. For this scenario, we present a set of sufficient conditions (some are more general or succinct than others) under which the virus spreading will die out; such sufficient conditions are also known as epidemic thresholds in the literature. In the fully adaptive defense scenario, some input parameters are not known (i.e., the aforementioned sufficient conditions are not applicable) but the defender can observe the outcome of virus spreading. For this scenario, we present adaptive control strategies under which the virus spreading will die out or will be contained to a desired level. We propose a simple game for modeling containment of the spread of viruses in a graph of n nodes. Each node must choose to either install anti-virus software at some known cost C, or risk infection and a loss L if a virus that starts at a random initial point in the graph can reach it without being stopped by some intermediate node. The goal of individual nodes is to minimize their individual expected cost. We prove many game theoretic properties of the model, including an easily applied characterization of Nash equilibria, culminating in our showing that allowing selfish users to choose Nash equilibrium strategies is highly undesirable, because the price of anarchy is an unacceptable Θ(n) in the worst case. This shows in particular that a centralized solution can give a much better total cost than an equilibrium solution. Though it is NP-hard to compute such a social optimum, we show that the problem can be reduced to a previously unconsidered combinatorial problem that we call the sum-of-squares partition problem. Using a greedy algorithm based on sparse cuts, we show that this problem can be approximated to within a factor of O(log2 n), giving the same approximation ratio for the inoculation game. How will a virus propagate in a real networkq How long does it take to disinfect a network given particular values of infection rate and virus death rateq What is the single best node to immunizeq Answering these questions is essential for devising network-wide strategies to counter viruses. In addition, viral propagation is very similar in principle to the spread of rumors, information, and “fads,” implying that the solutions for viral propagation would also offer insights into these other problem settings. We answer these questions by developing a nonlinear dynamical system (NLDS) that accurately models viral propagation in any arbitrary network, including real and synthesized network graphs. We propose a general epidemic threshold condition for the NLDS system: we prove that the epidemic threshold for a network is exactly the inverse of the largest eigenvalue of its adjacency matrix. Finally, we show that below the epidemic threshold, infections die out at an exponential rate. Our epidemic threshold model subsumes many known thresholds for special-case graphs (e.g., Erdos--Renyi, BA powerlaw, homogeneous). We demonstrate the predictive power of our model with extensive experiments on real and synthesized graphs, and show that our threshold condition holds for arbitrary graphs. Finally, we show how to utilize our threshold condition for practical uses: It can dictate which nodes to immunize; it can assess the effects of a throttling policy; it can help us design network topologies so that they are more resistant to viruses. Besides the epidemic threshold, the recently proposed viral conductance @j by [11] may be regarded as an additional characterizer of the viral robustness of a network, that measures the overall ease in which viruses can spread in a particular network. Motivated to explain observed features of the viral conductance @j in simulations [29], we have analysed this metric in depth using the N-intertwined SIS epidemic model, that upper bounds the real infection probability in any network and, hence, provides safe-side bounds on which network protection can be based. Our study here derives a few exact results for @j, a number of different lower and upper bounds for @j with variable accuracy. We also extend the theory of the N-intertwined SIS epidemic model, by deducing formal series expansions of the steady-state fraction of infected nodes for any graph and any effective infection rate, that result in a series for the viral conductance @j. Though approximate, we illustrate here that the N-intertwined SIS epidemic model is so far the only SIS model on networks that is analytically tractable, and valuable to provide first order estimates of the epidemic impact in networks. Finally, inspired by the analogy between virus spread and synchronization of coupled oscillators in a network, we propose the synchronizability as the analogue of the viral conductance. The influence of the network characteristics on the virus spread is analyzed in a new-the N -intertwined Markov chain-model, whose only approximation lies in the application of mean field theory. The mean field approximation is quantified in detail. The N -intertwined model has been compared with the exact 2N-state Markov model and with previously proposed ldquohomogeneousrdquo or ldquolocalrdquo models. The sharp epidemic threshold tauc , which is a consequence of mean field theory, is rigorously shown to be equal to tauc = 1 (lambdamax(A)) , where lambdamax(A) is the largest eigenvalue-the spectral radius-of the adjacency matrix A . A continued fraction expansion of the steady-state infection probability at node j is presented as well as several upper bounds. If a false rumor propagates via Twitter, while the truth propagates between friends in Facebook, which one will prevail? This question captures the essence of the problem we address here. We study the intertwined propagation of two competing "memes" (or viruses, rumors, products etc.) in a composite network. A key novelty is the use of a composite network, which in its simplest model is defined as a single set of nodes with two distinct types of edges interconnecting them. Each meme spreads across the composite network in accordance to an SIS-like propagation model (a flu-like infection-recovery). To study the epidemic behavior of our system, we formulate it as a non-linear dynamic system (NLDS). We develop a metric for each meme that is based on the eigenvalue of an appropriately constructed matrix and argue that this metric plays a key role in determining the "winning" meme. First, we prove that our metric determines the tipping point at which both memes become extinct eventually. Second, we conjecture that the meme with the strongest metric will most likely prevail over the other, and we show evidence of that via simulations in both real and synthetic composite networks. Our work is among the first to study the interplay between two competing memes in composite networks. Many network phenomena are well modeled as spreads of epidemics through a network. Prominent examples include the spread of worms and email viruses, and, more generally, faults. Many types of information dissemination can also be modeled as spreads of epidemics. In this paper we address the question of what makes an epidemic either weak or potent. More precisely, we identify topological properties of the graph that determine the persistence of epidemics. In particular, we show that if the ratio of cure to infection rates is larger than the spectral radius of the graph, then the mean epidemic lifetime is of order log n, where n is the number of nodes. Conversely, if this ratio is smaller than a generalization of the isoperimetric constant of the graph, then the mean epidemic lifetime is of order e sup na , for a positive constant a. We apply these results to several network topologies including the hypercube, which is a representative connectivity graph for a distributed hash table, the complete graph, which is an important connectivity graph for BGP, and the power law graph, of which the AS-level Internet graph is a prime example. We also study the star topology and the Erdos-Renyi graph as their epidemic spreading behaviors determine the spreading behavior of power law graphs. The effect of virus spreading in a telecommunication network, where a certain curing strategy is deployed, can be captured by epidemic models. In the N-intertwined model proposed and studied in [1], [2], the probability of each node to be infected depends on the curing and infection rate of its neighbors. In this paper, we consider the case where all infection rates are equal and different values of curing rates can be deployed within a given budget, in order to minimize the overall infection of the network. We investigate this difficult optimization together with a related problem where the curing budget must be minimized within a given level of network infection. Some properties of these problems are derived and several solution algorithms are proposed. These algorithms are compared on two real world network instances, while Erdos-Renyi graphs and some special graphs such as the cycle, the star, the wheel and the complete bipartite graph are also addressed. In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In the LOCAL model, this is quite simple: each node broadcasts all of its information in each round, and the number of rounds required will be equal to the diameter of the underlying communication graph. In the GOSSIP model, each node must independently choose a single neighbor to contact, and the lack of global information makes it difficult to make any sort of principled choice. As such, researchers have focused on the uniform gossip algorithm, in which each node independently selects a neighbor uniformly at random. When the graph is well-connected, this works quite well. In a string of beautiful papers, researchers proved a sequence of successively stronger bounds on the number of rounds required in terms of the conductance φ and graph size n, culminating in a bound of O(φ-1 log n). In this paper, we show that a fairly simple modification of the protocol gives an algorithm that solves the information dissemination problem in at most O(D + polylog (n)) rounds in a network of diameter D, with no dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of D, which applies even in the LOCAL model. In fact, we prove that something stronger is true: any algorithm that requires T rounds in the LOCAL model can be simulated in O(T + polylog(n)) rounds in the GOSSIP model. We thus prove that these two models of distributed computation are essentially equivalent. We consider the problem of controlling the propagation of an epidemic outbreak in an arbitrary contact network by distributing vaccination resources throughout the network. We analyze a networked version of the Susceptible-Infected-Susceptible (SIS) epidemic model when individuals in the network present different levels of susceptibility to the epidemic. In this context, controlling the spread of an epidemic outbreak can be written as a spectral condition involving the eigenvalues of a matrix that depends on the network structure and the parameters of the model. We study the problem of finding the optimal distribution of vaccines throughout the network to control the spread of an epidemic outbreak. We propose a convex framework to find cost-optimal distribution of vaccination resources when different levels of vaccination are allowed.We illustrate our approaches with numerical simulations in a real social network. We study the process of the spread of an infection among mobile nodes moving on a finite, grid based map. A random walk and a novel adversarial model are considered as two extreme cases of node mobility. With N nodes, we present analytical and simulation results for both mobility models for a square grid map with size √G × √G. A key finding is that with random mobility the total time to infect all nodes decreases with N while with an adversarial model we observe a reverse trend. Specifically, the random case results in a total infection time of Θ(GlogGlogN (N) as opposed to the adversarial case where the total infection time is found to be Θ(√(Glog(N). We also explore the possibility of emulating such an infection process as a mobile interaction game with wireless sensor motes, and the above results are complimented by traces obtained from an empirical study with humans as players in an outdoor field. We consider the problem of finding the graph on which an epidemic spreads, given only the times when each node gets infected. While this is a problem of central importance in several contexts -- offline and online social networks, e-commerce, epidemiology -- there has been very little work, analytical or empirical, on finding the graph. Clearly, it is impossible to do so from just one epidemic; our interest is in learning the graph from a small number of independent epidemics. For the classic and popular "independent cascade" epidemics, we analytically establish sufficient conditions on the number of epidemics for both the global maximum-likelihood (ML) estimator, and a natural greedy algorithm to succeed with high probability. Both results are based on a key observation: the global graph learning problem decouples into n local problems -- one for each node. For a node of degree d, we show that its neighborhood can be reliably found once it has been infected O(d2 log n) times (for ML on general graphs) or O(d log n) times (for greedy on trees). We also provide a corresponding information-theoretic lower bound of Ω(d log n); thus our bounds are essentially tight. Furthermore, if we are given side-information in the form of a super-graph of the actual graph (as is often the case), then the number of epidemic samples required -- in all cases -- becomes independent of the network size n. The Internet has a very complex connectivity recently modeled by the class of scale-free networks. This feature, which appears to be very efficient for a communications network, favors at the same time the spreading of computer viruses. We analyze real data from computer virus infections and find the average lifetime and persistence of viral strains on the Internet. We define a dynamical model for the spreading of infections on scale-free networks, finding the absence of an epidemic threshold and its associated critical behavior. This new epidemiological framework rationalizes data of computer viruses and could help in the understanding of other spreading phenomena on communication and social networks. Given a network of who-contacts-whom or who-links-to-whom, will a contagious virus product meme spread and ‘take over’ (cause an epidemic) or die out quickly? What will change if nodes have partial, temporary or permanent immunity? The epidemic threshold is the minimum level of virulence to prevent a viral contagion from dying out quickly and determining it is a fundamental question in epidemiology and related areas. Most earlier work focuses either on special types of graphs or on specific epidemiological cascade models. We are the first to show the G2-threshold (twice generalized) theorem, which nicely de-couples the effect of the topology and the virus model. Our result unifies and includes as special case older results and shows that the threshold depends on the first eigenvalue of the connectivity matrix, (a) for any graph and (b) for all propagation models in standard literature (more than 25, including H.I.V.). Our discovery has broad implications for the vulnerability of real, complex networks and numerous applications, including viral marketing, blog dynamics, influence propagation, easy answers to ‘what-if’ questions, and simplified design and evaluation of immunization policies. We also demonstrate our result using extensive simulations on real networks, including on one of the biggest available social-contact graphs containing more than 31 million interactions among more than 1 million people representing the city of Portland, Oregon, USA.