Efficient network immunization under limited knowledge

Beyond network centrality: individual-level behavioral traits for predicting information superspreaders in social media

Understanding the heterogeneous role of individuals in large-scale information spreading is essential to manage online behavior as well as its potential offline consequences. To this end, most existing studies from diverse research domains focus on the disproportionate role played by highly connected 'hub' individuals. However, we demonstrate here that information superspreaders in online social media are best understood and predicted by simultaneously considering two individual-level behavioral traits: influence and susceptibility. Specifically, we derive a nonlinear network-based algorithm to quantify individuals' influence and susceptibility from multiple spreading event data. By applying the algorithm to large-scale data from Twitter and Weibo, we demonstrate that individuals' estimated influence and susceptibility scores enable predictions of future superspreaders above and beyond network centrality, and reveal new insights into the network positions of the superspreaders.

Avalanche-size distribution of Cayley tree

Attacks on networks is a very important issue in developing strategies of eradicating spreads of malicious phenomena in networks, such as epidemics and fake information. This field of research is referred to as networks immunization. The traditional approach to evaluating the effectiveness of attacks on networks focuses on measuring macro parameters related to the entire attack, such as the critical probability of a percolation occurrence in the network [Formula: see text] and the relative size of the largest component in the network, known as the giant component, but not considering the attack on a micro perspective, which is the analysis of node removals, during an attack, themselves, their characteristics and results. In this paper we present and apply the last method of focusing on the micro scale of an attack. Based on the theory of percolation in networks, we analyze the phenomenon of an avalanche which results due to a single node removal from a network. An avalanche is a state in which a removal of a single node from the giant component of a network leads to the disconnection of additional nodes. This process significantly contributes to the fragmentation (immunization) of the network, comparing to the impact of the initial node removal alone. Specifically, we focus on the size parameter of an avalanche, which is the number of nodes that are disconnected from the giant component due to a single node removal. Relating to a random attack on a network of the type of Cayley tree, we derive analytically the distribution of the sizes of avalanches that occur during the entire attack on it, until the network is dismantled (immunized) and the attack is terminated.

First encounters on Watts-Strogatz networks and Barabási-Albert networks

The Watts-Strogatz networks are important models that interpolate between regular lattices and random graphs, and Barabási-Albert networks are famous models that explain the origin of the scale-free networks. Here, we consider the first encounters between two particles (e.g., prey A and predator B) embedded in the Watts-Strogatz networks and the Barabási-Albert networks. We address numerically the mean first-encounter time (MFET) while the two particles are moving and the mean first-passage time (MFPT) while the prey is fixed, aiming at uncovering the impact of the prey's motion on the encounter time, and the conditions where the motion of the prey would accelerate (or slow) the encounter between the two particles. Different initial conditions are considered. In the case where the two particles start independently from sites that are selected randomly from the stationary distribution, on the Barabási-Albert networks, the MFET is far less than the MFPT, and the impact of prey's motion on the encounter time is enormous, whereas, on the Watts-Strogatz networks (including Erdős-Rényi random networks), the MFET is about 0.5-1 times the MFPT, and the impact of prey's motion on the encounter time is relatively small. We also consider the case where prey A starts from a fixed site and the predator starts from a randomly drawn site and present the conditions where the motion of the prey would accelerate (or slow) the encounter between the two particles. The relation between the MFET (or MFPT) and the average path length is also discussed.

Coupled spreading between information and epidemics on multiplex networks with simplicial complexes

The way of information diffusion among individuals can be quite complicated, and it is not only limited to one type of communication, but also impacted by multiple channels. Meanwhile, it is easier for an agent to accept an idea once the proportion of their friends who take it goes beyond a specific threshold. Furthermore, in social networks, some higher-order structures, such as simplicial complexes and hypergraph, can describe more abundant and realistic phenomena. Therefore, based on the classical multiplex network model coupling the infectious disease with its relevant information, we propose a novel epidemic model, in which the lower layer represents the physical contact network depicting the epidemic dissemination, while the upper layer stands for the online social network picturing the diffusion of information. In particular, the upper layer is generated by random simplicial complexes, among which the herd-like threshold model is adopted to characterize the information diffusion, and the unaware-aware-unaware model is also considered simultaneously. Using the microscopic Markov chain approach, we analyze the epidemic threshold of the proposed epidemic model and further check the results with numerous Monte Carlo simulations. It is discovered that the threshold model based on the random simplicial complexes network may still cause abrupt transitions on the epidemic threshold. It is also found that simplicial complexes may greatly influence the epidemic size at a steady state.

Network resilience of non-hub nodes failure under memory and non-memory based attacks with limited information

Previous studies on network robustness mainly concentrated on hub node failures with fully known network structure information. However, hub nodes are often well protected and not accessible to damage or malfunction in a real-world networked system. In addition, one can only gain insight into limited network connectivity knowledge due to large-scale properties and dynamic changes of the network itself. In particular, two different aggression patterns are present in a network attack: memory based attack, in which failed nodes are not attacked again, or non-memory based attack; that is, nodes can be repeatedly attacked. Inspired by these motivations, we propose an attack pattern with and without memory based on randomly choosing n non-hub nodes with known connectivity information. We use a network system with the Poisson and power-law degree distribution to study the network robustness after applying two failure strategies of non-hub nodes. Additionally, the critical threshold 1 - _p and the size of the giant component S are determined for a network configuration model with an arbitrary degree distribution. The results indicate that the system undergoes a continuous second-order phase transition subject to the above attack strategies. We find that 1 - _p gradually tends to be stable after increasing rapidly with n. Moreover, the failure of non-hub nodes with a higher degree is more destructive to the network system and makes it more vulnerable. Furthermore, from comparing the attack strategies with and without memory, the results highlight that the system shows better robustness under a non-memory based attack relative to memory based attacks for n > 1. Attacks with memory can block the system's connectivity more efficiently, which has potential applications in real-world systems. Our model sheds light on network resilience under memory and non-memory based attacks with limited information attacks and provides valuable insights into designing robust real-world systems.

Percolation of interdependent networks with limited knowledge

Real-world networks are often not isolated and the interdependence between different networks in a complex system is as important as the topological connectivity within individual networks. We develop a theoretical framework to study the robustness of interdependent networks under attacks with limited knowledge. A node may be attacked if it is the most connected node among a given number of potential victims. This number is referred to as the attacker's knowledge level, which joins the two ends, namely, the random failure with zero knowledge and the intentional attack with full knowledge of the network. We introduce percolation models with attacks over one layer and two layers as well as mixed site-bond percolation. Along with the discontinuous phase transition, we show the existence of a critical knowledge level, which indicates a transition of network robustness under the competition between connectivity and interdependence. It is unraveled that interdependent networks can be extremely fragile to the extent that a random failure on two layers would be more deleterious than a targeted attack with full knowledge over one layer. Moreover, we find that a balanced distribution of attack knowledge on both layers tends to be most destructive if the total knowledge is a conserved quantity.

A sampling-guided unsupervised learning method to capture percolation in complex networks

The use of machine learning methods in classical and quantum systems has led to novel techniques to classify ordered and disordered phases, as well as uncover transition points in critical phenomena. Efforts to extend these methods to dynamical processes in complex networks is a field of active research. Network-percolation, a measure of resilience and robustness to structural failures, as well as a proxy for spreading processes, has numerous applications in social, technological, and infrastructural systems. A particular challenge is to identify the existence of a percolation cluster in a network in the face of noisy data. Here, we consider bond-percolation, and introduce a sampling approach that leverages the core-periphery structure of such networks at a microscopic scale, using onion decomposition, a refined version of the k-core. By selecting subsets of nodes in a particular layer of the onion spectrum that follow similar trajectories in the percolation process, percolating phases can be distinguished from non-percolating ones through an unsupervised clustering method. Accuracy in the initial step is essential for extracting samples with information-rich content, that are subsequently used to predict the critical transition point through the confusion scheme, a recently introduced learning method. The method circumvents the difficulty of missing data or noisy measurements, as it allows for sampling nodes from both the core and periphery, as well as intermediate layers. We validate the effectiveness of our sampling strategy on a spectrum of synthetic network topologies, as well as on two real-word case studies: the integration time of the US domestic airport network, and the identification of the epidemic cluster of COVID-19 outbreaks in three major US states. The method proposed here allows for identifying phase transitions in empirical time-varying networks.

Natural Gas Scarcity Risk in the Belt and Road Economies Based on Complex Network and Multi-Regional Input-Output Analysis

Natural gas scarcity poses a significant risk to the global economy. The risk of production loss due to natural gas scarcity can be transferred to downstream economies through globalized supply chains. Therefore, it is important to quantify and analyze how natural gas scarcity in some regions affects the Belt and Road (B&R) economies. The embodied natural gas scarcity risks (EGSRs) of B&R economies are assessed and the EGSR transmission network is constructed. The built network shows a small-world nature. This illustrates that any interruption in key countries will quickly spread to neighboring countries, potentially affecting the global economy. The top countries, including Turkey, China, Ukraine, and India are identified in EGSR exports, which also have relatively high values of closeness centrality. The findings illustrate that the shortage of natural gas supply in these countries may have a significant impact on downstream countries or sectors and the resulting economic losses spread rapidly. These countries are critical to the resilience of the B&R economies to natural gas scarcity. The top nations, including Turkmenistan, Macedonia, and Georgia are also identified in EGSR imports, highlighting their vulnerability to natural gas scarcity. Further, the community analysis of the network provides a fresh perspective for formulating fair and reasonable allocation policies of natural gas resources and minimizing the large-scale spread of economic losses caused by natural gas scarcity.

Phase transition behavior of finite clusters under localized attack

Most previous studies focused on the giant component to explore the structural robustness of complex networks under malicious attacks. As an important failure mode, localized attacks (LA) can excellently describe the local failure diffusion mechanism of many real scenarios. However, the phase transition behavior of finite clusters, as important network components, has not been clearly understood yet under LA. Here, we develop a percolation framework to theoretically and simulatively study the phase transition behavior of functional nodes belonging to the finite clusters of size greater than or equal to s(s=2,3,…) under LA in this paper. The results reveal that random network exhibits second-order phase transition behavior, the critical threshold p_c increases significantly with increasing s, and the network becomes vulnerable. In particular, we find a new general scaling relationship with the critical exponent δ=-2 between the fraction of finite clusters and s. Furthermore, we apply the theoretical framework to some real networks and predict the phase transition behavior of finite clusters in real networks after they face LA. The framework and results presented in this paper are helpful to promote the design of more critical infrastructures and inspire new insights into studying phase transition behaviors for finite clusters in the network.

Effects of topological characteristics on rhythmic states of the D-dimensional Kuramoto model in complex networks

Synchronization is a ubiquitous phenomenon in engineering and natural ecosystems. While the dynamics of synchronization modeled by the Kuramoto model are commonly studied in two dimensions and the state of dynamic units is characterized by a scalar angle variable, we studied the Kuramoto model generalized to D dimensions in the framework of a complex network and utilized the local synchronous order parameter between the agent and its neighbors as the controllable variable to adjust the coupling strength. Here, we reported that average connectivity of networks affects the time-dependent, rhythmic, cyclic state. Importantly, we found that the level of heterogeneity of networks governs the rhythmic state in the transition process. The analytical treatment for observed scenarios in a D-dimensional Kuramoto model at D=3 was provided. These results offered a platform for a better understanding of time-dependent swarming and flocking dynamics in nature.

Modelling Artificial Immunization Processes to Counter Cyberthreats

This paper looks at the problem of cybersecurity in modern cyber-physical and information systems and proposes an immune-like approach to the information security of modern complex systems. This approach is based on the mathematical modeling in information security—in particular, the use of immune methods to protect several critical system nodes from a predetermined range of attacks, and to minimize the success of an attack on the system. The methodological approach is to systematize the tasks, means and modes of immunization to describe how modern systems can counter the spread of computer attacks. The main conclusions and recommendations are that using an immunization approach will not only improve the security of systems, but also define principles for building systems that are resistant to cyber attacks. The immunization approach enables a symmetrical response to an intruder in a protected system to be produced rapidly. This symmetry provides a step-by-step neutralization of all stages of a cyber attack, which, combined with the accumulation of knowledge of the attacker’s actions, allows a base of defensive responses to be generated for various cyber attack scenarios. The theoretical conclusions are supported by practical experiments describing real-world scenarios for the use of immunization tools to protect against cyber threats.

Dynamical Recovery of Complex Networks under a Localised Attack

In real systems, some damaged nodes can spontaneously become active again when recovered from themselves or their active neighbours. However, the spontaneous dynamical recovery of complex networks that suffer a local failure has not yet been taken into consideration. To model this recovery process, we develop a framework to study the resilience behaviours of the network under a localised attack (LA). Since the nodes’ state within the network affects the subsequent dynamic evolution, we study the dynamic behaviours of local failure propagation and node recoveries based on this memory characteristic. It can be found that the fraction of active nodes switches back and forth between high network activity and low network activity, which leads to the spontaneous emergence of phase-flipping phenomena. These behaviours can be found in a random regular network, Erdős-Rényi network and Scale-free network, which shows that these three types of networks have the same or different resilience behaviours under an LA and random attack. These results will be helpful for studying the spontaneous recovery real systems under an LA. Our work provides insight into understanding the recovery process and a protection strategy of various complex systems from the perspective of damaged memory.

Improving robustness in interdependent networks under intentional attacks by optimizing intra-link allocation

The interdependent network is particularly vulnerable to attacks on high degree nodes; therefore, the improvement of its robustness under intentional attacks has become an important topic. In this paper, we put forward a new metric to quantify the robustness of interdependent networks against intentional attacks and develop an improved simulated annealing algorithm (ISAA) to maximize this metric by optimizing the allocation of intra-links in subnetworks. Based on the comparison between the ISAA and existing algorithms, it is found that the algorithm presented in this paper is more effective to enhance the robustness of an interdependent scale-free network (ISFN). By applying the ISAA to ISFNs with different coupling preferences, there is a key finding that the robustness of the optimized ISFN is significantly stronger than that of the original ISFN. In particular, for cases of disassortative and random couplings, no sudden collapse occurs in optimized ISFNs. According to the analysis of the degree and the clustering coefficient, we find that the subnetwork of the optimized ISFN exhibits an onion-like structure. In addition, the ISFN whose robustness is enhanced to resist the attacks on high degree nodes is still robust to the intentional attacks concerning the betweenness and PageRank.

Immunization of networks with limited knowledge and temporary immunity

Modern view of network resilience and epidemic spreading has been shaped by percolation tools from statistical physics, where nodes and edges are removed or immunized randomly from a large-scale network. In this paper, we produce a theoretical framework for studying targeted immunization in networks, where only n nodes can be observed at a time with the most connected one among them being immunized and the immunity it has acquired may be lost subject to a decay probability ρ. We examine analytically the percolation properties as well as scaling laws, which uncover distinctive characters for Erdős-Rényi and power-law networks in the two dimensions of n and ρ. We study both the case of a fixed immunity loss rate as well as an asymptotic total loss scenario, paving the way to further understand temporary immunity in complex percolation processes with limited knowledge.

Percolation of attack with tunable limited knowledge

Percolation models shed a light on network integrity and functionality and have numerous applications in network theory. This paper studies a targeted percolation (α model) with incomplete knowledge where the highest degree node in a randomly selected set of n nodes is removed at each step, and the model features a tunable probability that the removed node is instead a random one. A "mirror image" process (β model) in which the target is the lowest degree node is also investigated. We analytically calculate the giant component size, the critical occupation probability, and the scaling law for the percolation threshold with respect to the knowledge level n under both models. We also derive self-consistency equations to analyze the k-core organization including the size of the k core and its corona in the context of attacks under tunable limited knowledge. These percolation models are characterized by some interesting critical phenomena and reveal profound quantitative structure discrepancies between Erdős-Rényi networks and power-law networks.

Epidemics with containment measures

We propose a tractable epidemic model that includes containment measures. In the absence of containment measures, the epidemics spread exponentially fast whenever the infectivity rate is positive λ>0. The containment measures are modeled by considering a time-dependent modulation of the bare infectivity λ leading to effective infectivity that decays in time for each infected individual, mimicking, for instance, the combined effect of the asymptomatic onset of the disease, testing policies, and quarantine. We consider a wide range of temporal kernels for effective infectivity, and we investigate the effect of the considered containment measures. We find that not all kernels are able to push the epidemic dynamics below the epidemic threshold with some containment measures only able to reduce the rate of the exponential growth of newly infected individuals. We also propose a pandemic model caused by a growing number of separated foci.