The stochastic evolution of rumors within a population

We study rumor propagation process with incubation and constant immigration. We take into account a deterministic rumor spreading model and demonstrate the persistence of a rumor when the basic reproduction number is greater than one. Due to the presence of a randomness in the influence that the incubators exert on ignorants, we extrapolate the deterministic rumor model to a stochastic one by using a stochastic coefficient for the term representing the latter influence within the system. The existence and boundedness of both local and global solutions are demonstrated. We prove the uniqueness of these solutions. Conditions of extinction is also established. We perform numerical simulations to verify our stochastic model. The present work can assist decision takers in the analysis of the dynamical evolution of rumors in a given society as well as in the study of information dissemination strategies.     

Analyzing the dynamics of a stochastic rumor propagation model incorporating media coverage

In the age of information globalization, research on the mechanism of propagation will help mitigate the bad influence of rumors. Based on the classical rumor propagation model, this paper further analyzes the internal mechanism of the stochastic rumor propagation model incorporating media coverage with white noise. We investigate the existence of a unique global positive solution to the model and study the dynamic properties of the solutions around the rumor-free and local equilibrium points of the deterministic model. Furthermore, we establish sufficient conditions for the existence of traversal static distribution in the model. Numerical simulation shows that the role of media coverage is crucial to reduce the rumor propagation scale. The larger the coverage rate is, the smaller the rumor propagation scale is.     

Dynamic analysis of a rumor propagation model with Lévy noise

In this paper, we are concerned with a rumor propagation model with L vy noise. We first prove that there exists a positive global solution. Then, the asymptotic behaviors around the rumor‐free equilibrium and rumor‐epidemic equilibrium are obtained. Lastly, simulations verify our results.     

Quasirandom Rumor Spreading on Expanders

Randomized rumor spreading is an efficient way to distribute information in networks. Recently, a quasirandom version of this protocol has been proposed. It was proven that it works equally well or even better in many settings.In this work, we exhibit a natural expansion property for networks, which ensures that quasirandom rumor spreading informs all nodes of the network in logarithmic time with high probability. This expansion property is satisfied, among others, by many expander graphs, random regular graphs, and Erdős-Rényi random graphs.     

Dynamical Analysis and Control Strategies of Rumor Spreading Models in Both Homogeneous and Heterogeneous Networks

In recent years, rumor propagation in social networks attracts more researchers’ attention. In this paper, we have established I2S2R rumor spreading models in both homogeneous networks and heterogeneous networks considering the effect of time delay. In the homogeneous network model, we obtain the basic reproduction number by means of the next-generation matrix. Besides, the local stability and the global stability of the equilibrium points are discussed by linearization approach of nonlinear systems and Lyapunov function. In the heterogeneous network model, we calculate the basic reproduction number through algebraic method. In addition, Lyapunov functional method and Lasalle invariance principle are applied to study the stability of equilibrium points in the complex network model. Further, we put forward some useful strategies to control the spreading of rumor based on the complex network theory. Finally, we take advantage of numerical simulations to verify the theory above and come up with necessary conclusions.

     

Spatiotemporal dynamics analysis and optimal control method for an SI reaction-diffusion propagation model

In the new social media era, it is becoming increasingly important to explore the propagation rules for rumors in social networks. This article is concerned with investigating a diffusive susceptible-infected rumor propagation model with a nonlinear propagation function in a spatially heterogeneous environment. We establish the uniform persistence and analyze the asymptotic behavior of the rumor-spreading steady state for the spatially heterogeneous model when one of the diffusion coefficients tends to zero. Moreover, to better reflect the effect of a time delay on the process of rumor propagation, we establish a spatially homogeneous model with a time delay and prove the existence and local stability of the corresponding equilibrium point. Furthermore, the optimal control in the spatially homogeneous environment case is derived. Finally, several numerical simulations are performed to verify the theoretical results in both spatially heterogeneous and spatially homogeneous systems.     

Randomized broadcast in networks

In this paper we study the rate at which a rumor spreads through an undirected graph. This study has two important applications in distributed computation: in simple, robust and efficient broadcast protocols, and in the maintenance of replicated databases.     

The complexity of social groups and social systems described by graph structures

This paper presents methodology which permits the complete ranking of nondirected graphs (NDG's) on an attribute labelled ‘complexity.’ The technique applies to both small and large systems as might arise in studies of group or organization behavior. The methodology extends to cover the complexity of directed graphs (DG's) and permits the detailed specification of individual and group behavior.For the NDG an abstract automaton representing the participants' interaction or communications function is sited at each node. Each automaton is constructed so its internal complexity is sufficient to realize the minimal social action (e.g. transmission of a rumor and the path followed by the rumor) within the framework of the NDG. It is shown that the complexity of each node automaton depends upon the order of the graph, the degree of the node and the longest path parameter of the graph. The combined complexity of node automata constitutes the complexity of the NDG. The complexity of a DG is specified as a composition of complexities computed for the associated NDG and logical devices which produce the observed behavior. Illustrative examples pertaining to the committee-subcommittee problem and to organizational structures are presented.     

图的存活率与消防员问题

消防员问题可视为传染病、火灾、谣言、计算机病毒等传播的一个简化模型.假设一把火在一个图的某个点或多个点燃起,消防员选择若干个未着火的顶点进行防护,然后火蔓延到前一步着火点的未燃邻点.当火不再蔓延时整个过程结束.消防员问题自1995年提出以来引起了人们的广泛关注.本文简述了与消防员问题相关的最近研究进展,包括算法复杂性、...     

Optimal control of a birth and death epidemic process

We employ a birth and death process to describe the spread of an infectious disease through a closed population. Control of the epidemic can be effected at any instant by varying the birth and death rates to represent quarantine and medical care programs. An optimal strategy is one which minimizes the expected discounted losses and costs resulting from the epidemic process and the control programs over an infinite horizon. We formulate the problem as a continuous-time Markov decision model. Then we present conditions ensuring that optimal quarantine and medical care program levels are nonincreasing functions of the number of infectives in the population. We also analyze the dependence of the optimal strategy on the model parameters. Finally, we present an application of the model to the control of a rumor.     

Surviving rate of graphs and Firefighter Problem

The Firefighter Problem on a graph can be viewed as a simplified model of the spread of contagion, fire, rumor, computer virus, etc. The fire breaks out at one or more vertices in a graph at the first round, and the fire-fighter chooses some vertices to protect. The fire spreads to all non-protected neighbors at the beginning of each time-step. The process stops when the fire can no longer spread. The Firefighter Problem has attracted considerable attention since it was introduced in 1995. In this paper we provide a survey on recent research progress of this field, including algorithms and complexity, Fire-fighter Problem for special graphs (finite and infinite) and digraphs, surviving rate and burning number of graphs. We also collect some open problems and possible research subjects.     

A social network model with proximity prestige property

In general, many real-world networks not only possess scale-free and high clustering coefficient properties, but also have a fast information transmission capability. However, the existing network models are unable to well present the intrinsic fast information transmission feature. The initial infected nodes and the network topology are two factors that affect the information transmission capability. By using preferential attachment to high proximity prestige nodes and triad formation, we provide a proximity prestige network model, which has scale-free property and high clustering coefficient. Simulation results further indicate that the new model also possesses tunable information transmission capability archived by adjusting its parameters. Moreover, comparing with the BA scale-free network, the proximity prestige network PPNet05 achieves a higher transmission capability when messages travel based on SIR and SIS models. Our conclusions are directed to possible applications in rumor or information spreading mechanisms.     

Predator–Prey Dynamics on Infinite Trees: A Branching Random Walk Approach

We are interested in predator–prey dynamics on infinite trees, which can informally be seen as particular two-type branching processes where individuals may die (or be infected) only after their parent dies (or is infected). We study two types of such dynamics: the chase–escape process, introduced by Kordzakhia with a variant by Bordenave who sees it as a rumor propagation model, and the birth-and-assassination process, introduced by Aldous and Krebs. We exhibit a coupling between these processes and branching random walks killed at the origin. This sheds new light on the chase–escape and birth-and-assassination processes, which allows us to recover by probabilistic means previously known results and also to obtain new results. For instance, we find the asymptotic behavior of the tail of the number of infected individuals in both the subcritical and critical regimes for the chase–escape process and show that the birth-and-assassination process ends almost surely at criticality.     

Randomized rumor spreading in poorly connected small‐world networks

The Push‐Pull protocol is a well‐studied round‐robin rumor spreading protocol defined as follows: initially a node knows a rumor and wants to spread it to all nodes in a network quickly. In each round, every informed node sends the rumor to a random neighbor, and every uninformed node contacts a random neighbor and gets the rumor from her if she knows it. We analyze the behavior of this protocol on random ‐trees, a class of power law graphs, which are small‐world and have large clustering coefficients, built as follows: initially we have a ‐clique. In every step a new node is born, a random ‐clique of the current graph is chosen, and the new node is joined to all nodes of the ‐clique. When is fixed, we show that if initially a random node is aware of the rumor, then with probability after rounds the rumor propagates to nodes, where is the number of nodes and is any slowly growing function. Since these graphs have polynomially small conductance, vertex expansion and constant treewidth, these results demonstrate that Push‐Pull can be efficient even on poorly connected networks. On the negative side, we prove that with probability the protocol needs at least rounds to inform all nodes. This exponential dichotomy between time required for informing almost all and all nodes is striking. Our main contribution is to present, for the first time, a natural class of random graphs in which such a phenomenon can be observed. Our technique for proving the upper bound successfully carries over to a closely related class of graphs, the random ‐Apollonian networks, for which we prove an upper bound of rounds for informing nodes with probability when is fixed. Here, © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 185–208, 2016     

Mining social networks using wave propagation

With the development of modern technology(communication, transportation, etc.), many new social networks have formed and influenced our life. The research of mining these new social networks has been used in many aspects. But compared with traditional networks, these new social networks are usually very large. Due to the complexity of the latter, few model can be adapted to mine them effectively. In this paper, we try to mine these new social networks using Wave Propagation process and mainly discuss two applications of our model, solving Message Broadcasting problem and Rumor Spreading problem. Our model has the following advantages: (1) We can simulate the real networks message transmitting process in time since we include a time factor in our model. (2) Our Message Broadcasting algorithm can mine the underlying relationship of real networks and represent some clustering properties. (3) We also provide an algorithm to detect social network and find the rumor makers. Complexity analysis shows our algorithms are scalable for large social network and stable analysis proofs our algorithms are stable.     

Optimal control of information epidemics modeled as Maki Thompson rumors

We model the spread of information in a homogeneously mixed population using the Maki Thompson rumor model. We formulate an optimal control problem, from the perspective of single campaigner, to maximize the spread of information when the campaign budget is fixed. Control signals, such as advertising in the mass media, attempt to convert ignorants and stiflers into spreaders. We show the existence of a solution to the optimal control problem when the campaigning incurs non-linear costs under the isoperimetric budget constraint. The solution employs Pontryagin’s Minimum Principle and a modified version of forward backward sweep technique for numerical computation to accommodate the isoperimetric budget constraint. The techniques developed in this paper are general and can be applied to similar optimal control problems in other areas.We have allowed the spreading rate of the information epidemic to vary over the campaign duration to model practical situations when the interest level of the population in the subject of the campaign changes with time. The shape of the optimal control signal is studied for different model parameters and spreading rate profiles. We have also studied the variation of the optimal campaigning costs with respect to various model parameters. Results indicate that, for some model parameters, significant improvements can be achieved by the optimal strategy compared to the static control strategy. The static strategy respects the same budget constraint as the optimal strategy and has a constant value throughout the campaign horizon. This work finds application in election and social awareness campaigns, product advertising, movie promotion and crowdfunding campaigns.     

Modelling ship operational reliability over a mission under regular inspections

A ship is required to operate for a fixed mission period. Should a critical item of equipment fail at sea, the ship is subject to a costly event with potentially high risk to ship and crew. Given warning of a pending defect, the ship can try to return to port under its own power and thus attempt to avoid an at sea failure. Defects which lead to a failure are detected by inspection, and the task is to select the appropriate frequency of inspection to balance the number of occasions that a ship fails at sea and the number of preventive inspection based returns to port during a mission to correct a defect. The modelling entails using the delay time concept. Expressions are established for the expected number of preventive and failure returns over a mission, and an example given of a cost based balance to select an optimal inspection period. Although addressing ship reliability, the model has relevance to the mission reliability of any repairable equipment with remote main repair facilities.     

On Martingale Approximation of Adapted Processes

We show that the existence of a martingale approximation of a stationary process depends on the choice of the filtration. There exists a stationary linear process which has a martingale approximation with respect to the natural filtration, but no approximation with respect to a larger filtration with respect to which it is adapted and regular. There exists a stationary process adapted, regular, and having a martingale approximation with respect to a given filtration but not (regular and having a martingale approximation) with respect to the natural filtration.