Susceptible Nodes Research Articles

SPIR: The potential spreaders involved SIR model for information diffusion in social networks

The Susceptible–Infective–Removed (SIR) model is one of the most widely used models for the information diffusion research in social networks. Many researchers have devoted themselves to improving the classic SIR model in different aspects. However, on the one hand, the equations of these improved models are regarded as continuous functions, while the corresponding simulation experiments use discrete time, leading to the mismatch between numerical solutions got from mathematical method and experimental results obtained by simulating the spreading behaviour of each node. On the other hand, if the equations of these improved models are solved discretely, susceptible nodes will be calculated repeatedly, resulting in a big deviation from the actual value. In order to solve the above problem, this paper proposes a Susceptible–Potential–Infective–Removed (SPIR) model that analyses the diffusion process based on the discrete time according to simulation. Besides, this model also introduces a potential spreader set which solve the problem of repeated calculation effectively. To test the SPIR model, various experiments have been carried out from different angles on both artificial networks and real world networks. The Pearson correlation coefficient between numerical solutions of our SPIR equations and corresponding simulation results is mostly bigger than 0.95, which reveals that the proposed SPIR model is able to depict the information diffusion process with high accuracy.

Need for speed: An optimized gridding approach for spatially explicit disease simulations.

Numerical models for simulating outbreaks of infectious diseases are powerful tools for informing surveillance and control strategy decisions. However, large-scale spatially explicit models can be limited by the amount of computational resources they require, which poses a problem when multiple scenarios need to be explored to provide policy recommendations. We introduce an easily implemented method that can reduce computation time in a standard Susceptible-Exposed-Infectious-Removed (SEIR) model without introducing any further approximations or truncations. It is based on a hierarchical infection process that operates on entire groups of spatially related nodes (cells in a grid) in order to efficiently filter out large volumes of susceptible nodes that would otherwise have required expensive calculations. After the filtering of the cells, only a subset of the nodes that were originally at risk are then evaluated for actual infection. The increase in efficiency is sensitive to the exact configuration of the grid, and we describe a simple method to find an estimate of the optimal configuration of a given landscape as well as a method to partition the landscape into a grid configuration. To investigate its efficiency, we compare the introduced methods to other algorithms and evaluate computation time, focusing on simulated outbreaks of foot-and-mouth disease (FMD) on the farm population of the USA, the UK and Sweden, as well as on three randomly generated populations with varying degree of clustering. The introduced method provided up to 500 times faster calculations than pairwise computation, and consistently performed as well or better than other available methods. This enables large scale, spatially explicit simulations such as for the entire continental USA without sacrificing realism or predictive power.

Epidemic dynamics on weighted networks by adaptive rewiring

We study epidemic dynamics in weighted networks, where the susceptible nodes can change their infected neighbors by rewiring the edge. Two kinds of edges with different weights are considered and the effects of the edge density and the rewiring process on the epidemic properties are studied. The simulation results show that a fraction of infected nodes in the equilibrium state exhibit a nonmonotonic or monotonic behavior with the increase of the fraction of heavier weighted edges or the rewiring probability. Under the rewiring process, the system can evolve to a lower infected state in comparison with the result obtained in static network. The effect of rewiring different weighted edges on the epidemic behavior is also discussed.

Complex contagions with social reinforcement from different layers and neighbors

Researches about complex contagions on complex networks always neglect the reinforcement effect from different layers and neighbors simultaneously. In this paper we propose a non-Markovian model to describe complex contagions in which a susceptible node becoming adopted must take the social reinforcement from different layers and neighbors into consideration. Through extensive numerical simulations we find that the final adoption size will increase sharply with the information transmission probability at a large adoption threshold. In addition, for small values of adoption threshold, a few seeds could trigger a global contagion. However, there is a critical seed size below which the global contagion becomes impossible for large values of adoption threshold. Besides that, we develop an edge-based compartmental (EBC) theory to describe the proposed model, and it agrees well with numerical simulations.

Adaptive Mitigation of Multi-Virus Propagation: A Passivity-Based Approach

Malware propagation poses a growing threat to networked systems, such as computer networks and cyberphysical systems. Current approaches to defend against malware propagation are based on patching or filtering susceptible nodes at a fixed rate. When the propagation dynamics are unknown or uncertain, however, the static rate that is chosen may be either insufficient to remove all viruses or too high, incurring additional performance cost. In this paper, we formulate adaptive strategies for mitigating multiple malware epidemics when the propagation rate is unknown, using patching and filtering-based defense mechanisms. In order to identify conditions for ensuring that all viruses are asymptotically removed, we show that the malware propagation, patching, and filtering processes can be modeled as coupled passive dynamical systems. We prove that the patching rate required to remove all viruses is bounded above by the passivity index of the coupled system, and formulate the problem of selecting the minimum-cost mitigation strategy. Our results are evaluated through a numerical study.

Global dynamics of a network-based WSIS model for mobile malware propagation over complex networks

For understanding the influence of user security awareness on the long-term spreading behavior of malware over mobile networks, in this paper, we intensively study the global dynamics of a novel network-based epidemic model with weakly-protected and strongly-protected susceptible nodes. Both analytical and numerical results show that the global dynamics of the model is completely governed by a threshold value. Specifically, we prove that when the value is lower than one, the malware-free equilibrium is globally asymptotically stable and mobile malware will disappear. When the value is greater than one, mobile malware will persist on the network, and in the meantime there exists a unique malware equilibrium which is globally asymptotically stable under certain conditions. The obtained results improve and enrich some known ones. Interestingly, increasing the recovery rate of infected nodes can result in the increase of strongly-protected susceptible nodes and the decrease of the threshold value. The study has valuable guiding significance in effectively controlling mobile malware spread.

A cloud-assisted malware detection and suppression framework for wireless multimedia system in IoT based on dynamic differential game

As a novel dynamic network service infrastructure, Internet of Things (IoT) has gained remarkable popularity with obvious superiorities in the interoperability and real-time communication. Despite of the convenience in collecting information to provide the decision basis for the users, the vulnerability of embedded sensor nodes in multimedia devices makes the malware propagation a growing serious problem, which would harm the security of devices and their users financially and physically in wireless multimedia system (WMS). Therefore, many researches related to the malware propagation and suppression have been proposed to protect the topology and system security of wireless multimedia network. In these studies, the epidemic model is of great significance to the analysis of malware propagation. Considering the cloud and state transition of sensor nodes, a cloud-assisted model for malware detection and the dynamic differential game against malware propagation are proposed in this paper. Firstly, a SVM based malware detection model is constructed with the data sharing at the security platform in the cloud. Then the number of malware-infected nodes with physical infectivity to susceptible nodes is calculated precisely based on the attributes of WMS transmission. Then the state transition among WMS devices is defined by the modified epidemic model. Furthermore, a dynamic differential game and target cost function are successively derived for the Nash equilibrium between malware and WMS system. On this basis, a saddle-point malware detection and suppression algorithm is presented depending on the modified epidemic model and the computation of optimal strategies. Numerical results and comparisons show that the proposed algorithm can increase the utility of WMS efficiently and effectively.

A predator–prey model on the attacking behavior of malicious objects in wireless nanosensor networks

In this paper, we envision attacks of malicious objects and their transmission in wireless nanosensor networks (WNSNs) with the help of a predator–prey model. Here the prey consists of susceptible nanosensor nodes, susceptible nanosensor nodes with vaccination and infectious nanosensor nodes, whereas the predator consists of terminally infected nanosensor nodes. The dynamics of the system is analyzed at different equilibrium points to find the conditions for their stability. Impact of vaccination on the overall system is also analyzed. Extensive numerical simulation is performed to validate the vitality of the model developed. In short, the goal of this model is to determine whether WNSNs are able to survive against attacks of malicious objects or not and can vaccination play a key role in it.

Node-securing connectivity-based model to reduce infection spread in contaminated networks

Abstract Given a network with a set of contaminated and susceptible nodes, this article presents models for identifying a subset of susceptible nodes to secure (e.g., guard against infection, or remove from the network) such that the total number of nodes at risk of infection is minimized, subject to a limited budget for securing nodes. These models utilize a connectivity-based metric, in which a susceptible node is assumed to be at risk of infection if there exists a transmission path between it and any infected node, where no transmission path exists between two nodes if every path between them includes at least one secured node. The initial model presented, which is to the authors’ knowledge the first node-securing connectivity-based model for mitigating the spread of infection in contaminated networks, is then reformulated by use of a novel search space reduction algorithm. Computational testing is presented demonstrating the significant reductions in solution time achieved by the reformulated model.

The negative impact of understaging rectal cancer patients

BackgroundNeo-adjuvant chemoradiation followed by surgery and adjuvant therapy is standard treatment of clinical node positive rectal cancer. Understaging leads to delay in treatment with possible detrimental results. This study analyses effects of understaging stage III rectal cancer on long-term outcomes. MethodsA consecutive series of patients, operated on in MGH between 2004 and 2015 was included. Outcomes of non-neoadjuvantly treated clinical stage I patients who turned out to have pathological stage III disease and neoadjuvantly treated clinical stage III patients were retrospectively reviewed. The latter group was subdivided into patients who had persistent nodal disease (ypN+) and patients without positive lymph nodes after neoadjuvant treatment (ypN0). ResultsOf the 204 included patients, 30 had unexpected nodal disease on pathology. Clinical stage I-patients had higher rates of local recurrence, and rectal cancer and overall mortality than ypN0-patients. ConclusionUnderstaging stage III rectal cancer led to poorer oncologic outcomes, when compared to patients without positive lymph nodes on pathology after neoadjuvant. Future research should focus on identifying patients with treatment susceptible lymph node involvement.

Effects of individual popularity on information spreading in complex networks

In real world, human activities often exhibit preferential selection mechanism based on the popularity of individuals. However, this mechanism is seldom taken into account by previous studies about spreading dynamics on networks. Thus in this work, an information spreading model is proposed by considering the preferential selection based on individuals’ current popularity, which is defined as the number of individuals’ cumulative contacts with informed neighbors. A mean-field theory is developed to analyze the spreading model. Through systematically studying the information spreading dynamics on uncorrelated configuration networks as well as real-world networks, we find that the popularity preference has great impacts on the information spreading. On the one hand, the information spreading is facilitated, i.e., a larger final prevalence of information and a smaller outbreak threshold, if nodes with low popularity are preferentially selected. In this situation, the effective contacts between informed nodes and susceptible nodes are increased, and nodes almost have uniform probabilities of obtaining the information. On the other hand, if nodes with high popularity are preferentially selected, the final prevalence of information is reduced, the outbreak threshold is increased, and even the information cannot outbreak. In addition, the heterogeneity of the degree distribution and the structure of real-world networks do not qualitatively affect the results. Our research can provide some theoretical supports for the promotion of spreading such as information, health related behaviors, and new products, etc.

Takeover times for a simple model of network infection.

We study a stochastic model of infection spreading on a network. At each time step a node is chosen at random, along with one of its neighbors. If the node is infected and the neighbor is susceptible, the neighbor becomes infected. How many time steps does it take to completely infect a network of nodes, starting from a single infected node? An analogy to the classic “coupon collector” problem of probability theory reveals that the takeover time is dominated by extremal behavior, either when there are only a few infected nodes near the start of the process or a few susceptible nodes near the end. We show that for , the takeover time is distributed as a Gumbel distribution for the star graph, as the convolution of two Gumbel distributions for a complete graph and an Erdős-Rényi random graph, as a normal for a one-dimensional ring and a two-dimensional lattice, and as a family of intermediate skewed distributions for -dimensional lattices with (these distributions approach the convolution of two Gumbel distributions as approaches infinity). Connections to evolutionary dynamics, cancer, incubation periods of infectious diseases, first-passage percolation, and other spreading phenomena in biology and physics are discussed.

Explosive spreading on complex networks: The role of synergy.

In spite of the vast literature on spreading dynamics on complex networks, the role of local synergy, i.e., the interaction of elements that when combined produce a total effect greater than the sum of the individual elements, has been studied but only for irreversible spreading dynamics. Reversible spreading dynamics are ubiquitous but their interplay with synergy has remained unknown. To fill this knowledge gap, we articulate a model to incorporate local synergistic effect into the classical susceptible-infected-susceptible process, in which the probability for a susceptible node to become infected through an infected neighbor is enhanced when the neighborhood of the latter contains a number of infected nodes. We derive master equations incorporating the synergistic effect, with predictions that agree well with the numerical results. A striking finding is that when a parameter characterizing the strength of the synergy reinforcement effect is above a critical value, the steady-state density of the infected nodes versus the basic transmission rate exhibits an explosively increasing behavior and a hysteresis loop emerges. In fact, increasing the synergy strength can promote the spreading and reduce the invasion and persistence thresholds of the hysteresis loop. A physical understanding of the synergy promoting explosive spreading and the associated hysteresis behavior can be obtained through a mean-field analysis.

The propagation-weighted priority immunization strategy based on propagation tree

In this paper, we constructed the virus propagation tree for any infected node through improving the k-shell decomposition method. Supposing we determine the position of infected nodes, the root node of the propagation tree is an infected node and its children nodes are susceptible nodes. The virus can be diffused from the bottom to top along with the tree. Based on the analysis of the virus propagation tree, a propagation-weighted priority immunization strategy was proposed to vaccinate the influential nodes(the nodes are the several nodes of the most risky in the high-risk node and it is convenient for us to immune). The mathematical proof and the computer simulation on scale-free network are given. The results show that the propagation-weighted priority immunization is effective to prevent the virus from diffusing.

Epidemic spreading on evolving signed networks.

Most studies of disease spreading consider the underlying social network as obtained without the contagion, though epidemic influences people's willingness to contact others: A "friendly" contact may be turned to "unfriendly" to avoid infection. We study the susceptible-infected disease-spreading model on signed networks, in which each edge is associated with a positive or negative sign representing the friendly or unfriendly relation between its end nodes. In a signed network, according to Heider's theory, edge signs evolve such that finally a state of structural balance is achieved, corresponding to no frustration in physics terms. However, the danger of infection affects the evolution of its edge signs. To describe the coupled problem of the sign evolution and disease spreading, we generalize the notion of structural balance by taking into account the state of the nodes. We introduce an energy function and carry out Monte Carlo simulations on complete networks to test the energy landscape, where we find local minima corresponding to the so-called jammed states. We study the effect of the ratio of initial friendly to unfriendly connections on the propagation of disease. The steady state can be balanced or a jammed state such that a coexistence occurs between susceptible and infected nodes in the system.

Elementary proof of convergence to the mean-field model for the SIR process.

The susceptible-infected-recovered (SIR) model has been used extensively to model disease spread and other processes. Despite the widespread usage of this ordinary differential equation (ODE) based model which represents the mean-field approximation of the underlying stochastic SIR process on contact networks, only few rigorous approaches exist and these use complex semigroup and martingale techniques to prove that the expected fraction of the susceptible and infected nodes of the stochastic SIR process on a complete graph converges as the number of nodes increases to the solution of the mean-field ODE model. Extending the elementary proof of convergence for the SIS process introduced by Armbruster and Beck (IMA J Appl Math, doi: 10.1093/imamat/hxw010 , 2016) to the SIR process, we show convergence using only a system of three ODEs, simple probabilistic inequalities, and basic ODE theory. Our approach can also be generalized to many other types of compartmental models (e.g., susceptible-infected-recovered-susceptible (SIRS)) which are linear ODEs with the addition of quadratic terms for the number of new infections similar to the SI term in the SIR model.

A Way of Candidate Set Selection for Epidemic Routing in Delay Tolerate Networks

The radio propagation effect of fading or shadowing between mobile nodes has not been well considered in contact rate. One way to incorporate interference into contact rate is multiplied by probability that the infected node copies the message to the susceptible node during a contact in epidemic routing. In this paper, we develop a model to incorporate the wireless link reliability into contact rate. For application, we proposed a scheme for epidemical delay tolerate network with considering the effects based on the model. Simulation shows that proposed scheme has better performance.

Optimal Resource Allocation Over Time and Degree Classes for Maximizing Information Dissemination in Social Networks

We study the optimal control problem of allocating campaigning resources over the campaign duration and degree classes in a social network. Information diffusion is modeled as a Susceptible-Infected epidemic and direct recruitment of susceptible nodes to the infected (informed) class is used as a strategy to accelerate the spread of information. We formulate an optimal control problem for optimizing a net reward function, a linear combination of the reward due to information spread and cost due to application of controls. The time varying resource allocation and seeds for the epidemic are jointly optimized. A problem variation includes a fixed budget constraint. We prove the existence of a solution for the optimal control problem, provide conditions for uniqueness of the solution, and prove some structural results for the controls (e.g. controls are non-increasing functions of time). The solution technique uses Pontryagin's Maximum Principle and the forward-backward sweep algorithm (and its modifications) for numerical computations. Our formulations lead to large optimality systems with up to about 200 differential equations and allow us to study the effect of network topology (Erdos-Renyi/scale-free) on the controls. Results reveal that the allocation of campaigning resources to various degree classes depends not only on the network topology but also on system parameters such as cost/abundance of resources. The optimal strategies lead to significant gains over heuristic strategies for various model parameters. Our modeling approach assumes uncorrelated network, however, we find the approach useful for real networks as well. This work is useful in product advertising, political and crowdfunding campaigns in social networks.

Rescue of endemic states in interconnected networks with adaptive coupling.

We study the Susceptible-Infected-Susceptible model of epidemic spreading on two layers of networks interconnected by adaptive links, which are rewired at random to avoid contacts between infected and susceptible nodes at the interlayer. We find that the rewiring reduces the effective connectivity for the transmission of the disease between layers, and may even totally decouple the networks. Weak endemic states, in which the epidemics spreads when the two layers are interconnected but not in each layer separately, show a transition from the endemic to the healthy phase when the rewiring overcomes a threshold value that depends on the infection rate, the strength of the coupling and the mean connectivity of the networks. In the strong endemic scenario, in which the epidemics is able to spread on each separate network –and therefore on the interconnected system– the prevalence in each layer decreases when increasing the rewiring, arriving to single network values only in the limit of infinitely fast rewiring. We also find that rewiring amplifies finite-size effects, preventing the disease transmission between finite networks, as there is a non zero probability that the epidemics stays confined in only one network during its lifetime.

Evaluation of the resilience of air transportation network with adaptive capacity

ABSTRACTSecuring network resilience of air transportation system is essential to provide a stable level of service as one of major transport modes carrying international passengers and freights. In 2014, about 851 million passengers and 39 billion pounds of freights were delivered by over 9.5 million flights in the United States. As seen in Iceland volcano eruption in 2010, a deficiency of hub airports can bring a huge impact on the whole transport system and even on the world economy. So how the failure of individual node affects the overall network resilience is an important issue to study. Air transportation is known to be a scale-free network, which has few of hubs having high degree. So it is relatively robust against failure but vulnerable to targeted attack on a hub. There are numerous studies devoted to measure node vulnerability and evaluate network robustness; however, previous studies could not consider the node capacity for evaluating overall network performance. This study focuses on the network resilience, where the nodes are located in a real space and have a capacity to function. Using the data from Federal Aviation Administration, the simulation demonstrates and evaluates the resilience of the US air transportation network. This study proposes the indices of adaptative capacity for quantifying network resilience, which represent the ability of a network to replace an attacked node by other adjacent nodes. The simulation has two parts to measure the adaptive capacity of networks: under a single attack and a sustained attack. The results identify the susceptible nodes degrading the adaptive capacity of the network and evaluate each sub-network’s resilience in case of cascading node failures. Therefore, this study can help us to diagnose the vulnerable node and contribute the plan for improvement of network resilience.