Online social networks (OSNs) facilitate the rapid and extensive spreading of rumors. While most existing methods for debunking rumors consider a solitary debunker, they overlook that rumor-mongering and debunking are interdependent and confrontational behaviors. In reality, a debunker must consider the impact of rumor-mongering behavior when making decisions. Moreover, a single rumor-debunking strategy is ineffective in addressing the complexity of the rumor environment in networks. Therefore, this article proposes a hybrid rumor-debunking approach that combines truth dissemination and regulatory measures based on the differential game theory under adversarial behaviors of rumor-mongering and debunking. Toward this end, we first establish a rumor propagation model using node-based modeling techniques that can be applied to any network structure. Next, we mathematically describe and analyze the processes of rumor-mongering and debunking. Finally, we validate the theoretical results of the proposed method through various comparative experiments, including comparisons with a random strategy, a uniform strategy, and single strategy models on real-world datasets collected from Facebook, Twitter, and YouTube. Furthermore, we harness two actual rumor events to estimate parameters and predict rumor propagation, thereby affirming the veracity and effectiveness of our rumor propagation model.

Identifying rumor sources in online social networks (OSNs) plays a crucial role in controlling the spread of rumors and mitigating their damage. However, the community structure of OSNs and timeliness of rumors increase the complexity of accurately characterizing rumor-spreading behavior, making it extremely challenging to identify the source of rumor in OSNs. Conventional studies on rumor source identification often overlook the community structure of OSNs and timeliness of rumors. This paper proposes a rumor sources identification framework that take these two aspects into consideration. First, we transfer the snapshots of a dynamic OSN into a community-structured dynamic OSN through 4 meticulously designed phases. Second, instead of considering a constant influence of rumors in traditional techniques, we introduce a time-varying dynamic propagation parameter to quantify the timeliness of rumor, and apply the propagation parameter to establish a microscopic rumor spreading model. This process addresses the issue of modeling the timeliness of rumor. Third, we adopt sensor-based observation techniques to collect the propagation information of rumor, and a hybrid sensor deployment strategy is designed to improve the efficiency of information collection. Fourth, we propose an algorithm for identifying single and multiple rumor sources in community-structured dynamic OSN, this algorithm initiates with the utilization of some criteria to analyze the gathered propagation information and estimate the suspicious communities that harbor the source of rumors. Subsequently, it applies maximum likelihood estimation method within each community to determine the ultimate source of the rumor. Experimental results on real-world and synthetic networks indicate that our method can accurately identify the real rumor sources. To the best of our knowledge, the proposed method is the first that can be used to identify rumor sources in community-structured dynamic OSNs.

Increasingly deployment of distributed renewable energy sources has driven the emergence of peer-to-peer (P2P) energy trading, which refers to trading energy directly among the end energy customers. This paper studies the feasibility and mechanism of establishing cooperation among different types of energy prosumers in a community microgrid environment with uncertainty. In particular, a coalitional game-based P2P energy trading framework is proposed, through which participating prosumers are enabled to form a coalition to share the energy trading profit by sufficiently considering the renewable energy and load uncertainties. To incentivize prosumers to participate in energy trading cooperation, some contract constraint rules are designed to ensure that the risk of the prosumer energy systems is controllable, and a theoretical analysis framework is presented to strictly prove the benefit of the prosumer coalition. A Shapely value-based payoff distribution scheme is designed to allocate profit for each prosumer in the coalition. Comprehensive numerical simulation is conducted to validate the proposed method.

This paper is dedicated to solving the problem of Advanced Persistent Threat (APT) attack and defense in the Industrial Internet of Things (IIoT). Due to the diversity of IIoT equipment and the inconsistency of protection capabilities, it is difficult for the existing uniform defense strategy and the random defense strategy to achieve ideal results. Considering that both attackers and defenders aim to achieve maximum benefits by paying the minimum cost, as well as the differences between devices, this paper proposes an equipment classification based differential game method for APT in IIoT. Firstly, all equipment is divided into two categories according to their protective capabilities. Secondly, the APT attack and defense process is mathematically described, and the corresponding differential game problem is formulated and analyzed theoretically. Finally, the theoretical results of this method are verified by various experiments, including the comparisons with the uniform defense strategy, the random defense strategy, and the latest model.

Industrial Internet of Things (IIoT) is vulnerable to advanced persistent threat (APT). In this article, we study a scenario in which APT is launched to attack IIoT devices. Considering the APTs lateral movement, a node-level state evolution model is established to calculate the probability of every device in an IIoT system to be compromised by APT. Based on this, a Stackelberg game model is proposed for the APT attacker and defender, which can accurately describe the gaming process. An effective computational approach is developed to obtain the potential Stackelberg equilibrium strategy pair of the game. Extensive case studies and comparison studies are conducted to validate the effectiveness of the proposed method.

The industrial internet of things (IIoT) is a key pillar of the intelligent society, integrating traditional industry with modern information technology to improve production efficiency and quality. However, the IIoT also faces serious challenges from advanced persistent threats (APTs), a stealthy and persistent method of attack that can cause enormous losses and damages. In this paper, we give the definition and development of APTs. Furthermore, we examine the types of APT attacks that each layer of the four-layer IIoT reference architecture may face and review existing defense techniques. Next, we use several models to model and analyze APT activities in IIoT to identify their inherent characteristics and patterns. Finally, based on a thorough discussion of IIoT security issues, we propose some open research topics and directions.

This paper proposes a new trading framework for enabling energy trading between end energy customers and a small-capacity Renewable Energy Plant (REP) in urban environments. To relieve the communication burden incurred by a large number of end customers who submit bids to purchase energy from the REP, the energy trading process is designed on a community basis. The end customers are grouped into multiple communities. With this approach, the proposed energy trading system coordinates the REP, the community energy management system in each community, and the local energy management system of each customer. The homomorphic encryption technology is integrated into the energy trading framework to enable the energy trading to be performed and settled without exposing individual customer's bidding energy purchase price and amount values. Extensive numerical simulations are conducted to validate the effectiveness of the proposed system.

In the absence of effective treatment for COVID-19, disease prevention and control have become a top priority across the world. However, the general lack of effective cooperation between communities makes it difficult to suppress the community spread of the global pandemic; hence repeated outbreaks of COVID-19 have become the norm. To address this problem, this paper considers community cooperation in disease monitoring and designs a joint epidemic monitoring mechanism, in which adjacent communities cooperate to enhance their monitoring capability. In this work, we formulate the epidemiological monitoring process as a coalitional game. Then, we propose a Shapley value-based payoffs distribution scheme for the coalitional game. A comprehensive analytical framework is developed to evaluate the advantages and sustainability of the cooperation between communities. Experimental results show that the proposed mechanism performs much better than the conventional non-cooperative monitoring design and can greatly increase each community's payoffs.

Backboned by smart meter networks, Advanced Metering Infrastructures (AMIs) play a critical role in smart grids. This paper studies a new False Data Injection Attack (FDIA) scenario targeting AMIs, in which the attacker injects and propagates computer worms (i.e., false data codes) to maliciously increase the readings of networked smart meters and create economic loss to the end customers. This paper establishes the false data code propagation and attack models in such a scenario; based on this, this paper proposes a differential game model for describing the attack and defense process for FDIA against AMIs. A computationally efficient algorithm is developed to solve the proposed differential model and obtain the potential Nash equilibrium (NE) strategy pair. Extensive numerical simulations are conducted to validate the effectiveness of the proposed method under different energy tariff structures.

Deploying Intrusion Detection Systems (IDSs) is an essential way to enhance the security of Multi-hop Clustered Wireless Sensor Networks (MCWSNs). The conventional IDS deployment architectural designs show limitations in ensuring the security of MCWSNs due to the limited monitoring range of the nodes. This paper proposes an efficient IDS deployment architecture for MCWSNs. The architecture is a hybrid design, in which the cluster heads and the sink collaboratively act as IDS agents to monitor the entire network and perform intrusion detection. Based on the new architecture, this paper proposes a resource allocation model to optimally allocate resources among the IDS agents so that the network's security metric can be maximized. A comprehensive analytical framework is proposed to analyze the optimality of the model's solution, and an efficient computational approach is developed to obtain the optimal resource allocation strategy. Extensive numerical simulation and comparison studies are conducted to validate the proposed method.

Real-world wireless sensor networks (WSNs) usually have dynamic topologies and are vulnerable to data tampering attacks, which may possibly lead to serious consequences. This paper addresses the issue of defending against data tampering attacks to dynamic WSNs. First, a hybrid diagnosis algorithm is proposed, which consists of three phases: data comparison phase, distributed diagnosis phase, and global diagnosis phase. In the distributed diagnosis phase, a majority voting mechanism and a multi-round diagnosis mechanism are introduced. As the phase may lead to inconsistent diagnosis results among different sensor nodes, we introduce a global diagnosis phase to overcome the shortcoming. Next, we evaluate the performance of the diagnosis algorithm, accompanied with a few examples. Finally, to achieve near perfect performance, some experiments on how to choose the parameters of the algorithm are provided. This work provides guidance for dealing with data tampering attacks in dynamic WSNs.

In this paper, we study the synchronizability of three kinds of dynamical weighted fractal networks (WFNs). These WFNs are weighted Cantor-dust networks, weighted Sierpinski networks and weighted Koch networks. We calculated some features of these WFNs, including average distance ([Formula: see text]), fractal dimension ([Formula: see text]), information dimension ([Formula: see text]), correlation dimension ([Formula: see text]). We analyze two representative types of synchronizable dynamical networks (the type-I and the type-II). There are two indexes ([Formula: see text] and [Formula: see text]) that can be used to characterize the synchronizability of the two types of dynamical network. Here, [Formula: see text] and [Formula: see text] are the minimum nonzero eigenvalue and the maximum eigenvalue of the Laplacian matrix of the network, respectively. We find that the larger scaling factor [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text] implies stronger synchronizability for the type-I dynamical WFNs.

A rumor about an entity can spread rapidly through online social networks (OSNs), which may lead to serious consequences. This article focuses on developing a cost-effective rumor-refuting strategy in the situation where the rumormonger is strategic. Based on a novel individual-based rumor spreading model, we estimate the expected net benefit of the rumormonger as well as the expected total loss of the rumor victim. On this basis, we reduce the original problem to a differential game-theoretic model. Next, we derive a system for solving the model. By solving the system, we get a dynamic strategy pair. Through extensive comparative experiments, we find that this strategy pair is effective in terms of the solution concept of Nash equilibrium. Hence, we conclude that the rumor-refuting strategy contained in this strategy pair is cost-effective. Finally, we examine the effect of the structure of the OSN on the cost effectiveness of the rumor-supporting and rumor-refuting strategies contained in the proposed strategy pair. To our knowledge, this is the first time the rumor-refuting problem is addressed under the assumption that the rumormonger is strategic.

Rumors have been widely spread in online social networks and they become a major concern in modern society. This paper is devoted to the design of a cost-effective rumor-containing scheme in online social networks through an optimal control approach. First, a new individual-based rumor spreading model is proposed, and the model considers the influence of the external environment on rumor spreading for the first time. Second, the cost-effectiveness is recommended to balance the loss caused by rumors against the cost of a rumor-containing scheme. On this basis, we reduce the original problem to an optimal control model. Next, we prove that this model is solvable, and we present the optimality system for the model. Finally, we show that the resulting rumor-containing scheme is cost-effective through extensive computer experiments.

Malicious code has posed a severe threat to modern society. Delivering antivirus program to networks is an important task of a cybersecurity company. As the bandwidth resource in a company is limited and precious, cybersecurity companies have to make a tradeoff between the impact(i.e. the economic loss) of malicious codes and the bandwidth assigned to transmit the antivirus programs. This paper addresses the malicious code and bandwidth tradeoff(MCBT) problem. By developing a novel malicious code and antivirus program interacting model, the total loss, which is the sum of the bandwidth usage fee and the economic loss, is quantified. On this basis, the MCBT problem is modelled as a constrained optimization problem that we refer to as the MCBT model, where the independent variable stands for bandwidth, and the objective function stands for the total loss. Some optimal bandwidth is determined by solving the MCBT model. Based on this, we propose a heuristic algorithm named DOWNHILL, which outperforms random strategies. Finally, the influence of some factors on the optimal bandwidth and the corresponding optimal total loss is uncovered through numerical simulations. To our knowledge, this is the first time the MCBT problem is treated in this way.

To cope with evolving computer viruses, antivirus programs must be periodically updated. Due to the limited network bandwidth, new virus patches are typically injected into a small subset of network nodes and then forwarded to the remaining nodes. A static patching strategy consists of a fixed patch injection rate and a fixed patch forwarding rate. This paper focuses on evaluating the performance of a static patching strategy. First, we introduce a novel autonomous node-level virus-patch propagation model to characterize the effect of a static patching strategy. Second, we show that the model is globally attracting, implying that regardless of the initial expected state of the network, the expected fraction of the infected nodes converges to the same value. Therefore, we use the asymptotic expected fraction of the infected nodes as the measure of performance of a static patching strategy. On this basis, we evaluate the performances of a few static patching strategies. Finally, we examine the influences of a few parameters on the performance of a static patching strategy. Our findings provide a significant guidance for restraining malware propagation.

Discount is a frequently used marketing tool. This paper is devoted to designing an effective dynamic discount pricing (DDP) strategy in competitive marketing. First, we introduce a competitive word-of-mouth (WOM) propagation model with DDP mechanism. On this basis, we model the original problem as an optimal control problem. Next, we derive the optimality system for the optimal control problem and, thereby, propose the concept of competitive DDP strategy. Finally, through comparative experiments, we find that the profit of a competitive DDP strategy is satisfactory. Our findings contribute to maximizing the marketing profit in the presence of commercial competitors.

To contain the prevalence of computer virus on a network, we have to continuously inject new patches into the network. As the limited communication bandwidth restricts the patch injection rate, we need to evaluate the performance of a patch injection rate in restraining computer infections. This paper focuses on the performance evaluation problem. We propose a virus–patch interacting model with patch injection mechanism, and show that this model admits a globally stable equilibrium. This result implies that the fraction of infected nodes will approach a common value. Therefore, we recommend the asymptotic fraction of infected nodes to serve as a measure of performance of the associated patch injection rate. We also examine the influence of different parameters on the asymptotic fraction of infected nodes. In particular, we find that patch injection is particularly effective for densely connected networks. This work takes the first step towards understanding the effectiveness of patch injection.

Weighted complex networks, especially scale-free networks, which characterize real-life systems better than non-weighted networks, have attracted considerable interest in recent years. Studies on the multifractality of weighted complex networks are still to be undertaken. In this paper, inspired by the concepts of Koch networks and Koch island, we propose a new family of weighted Koch networks, and investigate their multifractal behavior and topological properties. We find some key topological properties of the new networks: their vertex cumulative strength has a power-law distribution; there is a power-law relationship between their topological degree and weight strength; the networks have a high weighted clustering coefficient of 0.41004 (which is independent of the scaling factor c) in the limit of large generation t; the second smallest eigenvalue μ2 and the maximum eigenvalue μn are approximated by quartic polynomials of the scaling factor c for the general Laplacian operator, while μ2 is approximately a quartic polynomial of c and μn= 1.5 for the normalized Laplacian operator. Then, we find that weighted koch networks are both fractal and multifractal, their fractal dimension is influenced by the scaling factor c. We also apply these analyses to six real-world networks, and find that the multifractality in three of them are strong.

Locating influential nodes in temporal networks has attracted a lot of attention as data driven and diverse applications. Classic works either looked at analysing static networks or placed too much emphasis on the topological information but rarely highlighted the dynamics. In this paper, we take account the network dynamics and extend the concept of Dynamic-Sensitive centrality to temporal network. According to the empirical results on three real-world temporal networks and a theoretical temporal network for susceptible-infected-recovered (SIR) models, the temporal Dynamic-Sensitive centrality (TDC) is more accurate than both static versions and temporal versions of degree, closeness and betweenness centrality. As an application, we also use TDC to analyse the impact of time-order on spreading dynamics, we find that both topological structure and dynamics contribute the impact on the spreading influence of nodes, and the impact of time-order on spreading influence will be stronger when spreading rate b deviated from the epidemic threshold bc, especially for the temporal scale-free networks.