Percolation in real multiplex networks

Applying the Stackelberg game to assess critical infrastructure vulnerability: Based on a general multi-layer network model

Critical infrastructure systems (CIS) are closely related to human life. Attacks against CIS occur frequently, making accurate and effective protection of CIS essential. Vulnerability assessment is the primary issue to achieving this goal. The interconnected characteristic of CIS means that it is best represented by a multi-layer network, but a uniform model is absent. Game theory offers a suitable framework for researching intelligent confrontation. Previous research combining game theory and network science mainly focuses on a single-layer network and lacks a comprehensive assessment that combines qualitative and quantitative aspects of vulnerability. In this paper, we apply the Stackelberg game to the multi-layer network and comprehensively assess vulnerability based on game equilibrium to realize accurate protection. We first present a method for constructing a general model of the multi-layer network and introduce a multi-layer weighted factor to extend topological attribute metrics. Then, we design a Stackelberg game model for the multi-layer network. Furthermore, we qualitatively analyze the impact of multi-layer network characteristics on vulnerability and propose a method to quantify vulnerability. Experiments show that the vulnerability of the multi-layer network is greatly influenced by the multi-layer weighted factor, single-layer network type, and inter-layer coupling method. The quantitative value of network vulnerability does not rely entirely on the topology attributes on which the attack and defense costs depend but also relates to the resources available. Our work provides an adaptive model for CIS and gives a new approach to developing accurate protection based on comprehensive vulnerability assessment, which deserves further study.

Robustness Analysis of Multilayer Infrastructure Networks Based on Incomplete Information Stackelberg Game: Considering Cascading Failures

The growing importance of critical infrastructure systems (CIS) makes maintaining their normal operation against deliberate attacks such as terrorism a significant challenge. Combining game theory and complex network theory provides a framework for analyzing CIS robustness in adversarial scenarios. Most existing studies focus on single-layer networks, while CIS are better modeled as multilayer networks. Research on multilayer network games is limited, lacking methods for constructing incomplete information through link hiding and neglecting the impact of cascading failures. We propose a multilayer network Stackelberg game model with incomplete information considering cascading failures (MSGM-IICF). First, we describe the multilayer network model and define the multilayer node-weighted degree. Then, we present link hiding rules and a cascading failure model. Finally, we construct MSGM-IICF, providing methods for calculating payoff functions from the different perspectives of attackers and defenders. Experiments on synthetic and real-world networks demonstrate that link hiding improves network robustness without considering cascading failures. However, when cascading failures are considered, they become the primary factor determining network robustness. Dynamic capacity allocation enhances network robustness, while changes in dynamic costs make the network more vulnerable. The proposed method provides a new way of analyzing the robustness of diverse CIS, supporting resilient CIS design.

Distinguishing between different percolation regimes in noisy dynamic networks with an application to epileptic seizures

In clinical neuroscience, epileptic seizures have been associated with the sudden emergence of coupled activity across the brain. The resulting functional networks-in which edges indicate strong enough coupling between brain regions-are consistent with the notion of percolation, which is a phenomenon in complex networks corresponding to the sudden emergence of a giant connected component. Traditionally, work has concentrated on noise-free percolation with a monotonic process of network growth, but real-world networks are more complex. We develop a class of random graph hidden Markov models (RG-HMMs) for characterizing percolation regimes in noisy, dynamically evolving networks in the presence of edge birth and edge death. This class is used to understand the type of phase transitions undergone in a seizure, and in particular, distinguishing between different percolation regimes in epileptic seizures. We develop a hypothesis testing framework for inferring putative percolation mechanisms. As a necessary precursor, we present an EM algorithm for estimating parameters from a sequence of noisy networks only observed at a longitudinal subsampling of time points. Our results suggest that different types of percolation can occur in human seizures. The type inferred may suggest tailored treatment strategies and provide new insights into the fundamental science of epilepsy.

Robustness measurement of multiplex networks based on graph spectrum

Networks can provide effective representations of the relationships between elements in complex systems through nodes and links. On this basis, relationships between multiple systems are often characterized as multilayer networks (or networks of networks). As a typical representative, a multiplex network is often used to describe a system in which there are many replaceable or dependent relationships among elements in different layers. This paper studies robustness measures for different types of multiplex networks by generalizing the natural connectivity calculated from the graph spectrum. Experiments on model and real multiplex networks show a close correlation between the robustness of multiplex networks consisting of connective or dependent layers and the natural connectivity of aggregated networks or intersections between layers. These indicators can effectively measure or estimate the robustness of multiplex networks according to the topology of each layer. Our findings shed new light on the design and protection of coupled complex systems.

Hidden transition in multiplex networks

Weak multiplex percolation generalizes percolation to multi-layer networks, represented as networks with a common set of nodes linked by multiple types (colors) of edges. We report a novel discontinuous phase transition in this problem. This anomalous transition occurs in networks of three or more layers without unconnected nodes, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P(0)\,=\,0$$\end{document}P(0)=0. Above a critical value of a control parameter, the removal of a tiny fraction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta$$\end{document}Δ of nodes or edges triggers a failure cascade which ends either with the total collapse of the network, or a return to stability with the system essentially intact. The discontinuity is not accompanied by any singularity of the giant component, in contrast to the discontinuous hybrid transition which usually appears in such problems. The control parameter is the fraction of nodes in each layer with a single connection, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi \,=\,P(1)$$\end{document}Π=P(1). We obtain asymptotic expressions for the collapse time and relaxation time, above and below the critical point \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _c$$\end{document}Πc, respectively. In the limit \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \rightarrow 0$$\end{document}Δ→0 the total collapse for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi \,>\,\Pi _\text {c}$$\end{document}Π>Πc takes a time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T \propto 1/(\Pi -\Pi _\text {c})$$\end{document}T∝1/(Π-Πc), while there is an exponential relaxation below \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _\text {c}$$\end{document}Πc with a relaxation time \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau \propto 1/[\Pi _\text {c}-\Pi ]$$\end{document}τ∝1/[Πc-Π].

Robustness of Air Transportation as Complex Networks:Systematic Review of 15 Years of Research and Outlook into the Future

Air transportation systems are an important part of the critical infrastructure in our connected world. Accordingly, a better understanding and improvements in the resilience of the overall air transportation system are essential to the well-functioning of our society and overall sustainability of human beings. In the literature, network science is increasingly used to better understand the resilience dynamics of air transportation. Given the wide application of tools for network science and the importance of designing resilient air transportation systems, a rich body of studies has emerged in recent years. This review paper synthesizes the related literature that has been published throughout the last 15 years regarding the robustness of air transportation systems. The contributions of this work consist of two major elements. The first part provides a comprehensive discussion and cross-comparison of the reported results. We cover several major topics, including node importance identification, failure versus attack profiles, recovery and improvement techniques, and networks of networks approaches. The second part of this paper complements the review of aggregated findings by elaborating on a future agenda for robust air transportation research. Our survey-style overview hopefully contributes toward a better understanding of the state of the art in this research area, and, in turn, to the improvement of future air transportation resilience and sustainability.

Identification of COVID-19 Spreaders Using Multiplex Networks Approach

In this work, we present a methodology to identify COVID-19 spreaders using the analysis of the relationship between socio-cultural and economic characteristics with the number of infections and deaths caused by the COVID-19 virus in different countries. For this, we analyze the information of each country using the complex networks approach, specifically by analyzing the spreaders countries based on the separator set in 5-layer multiplex networks. The results show that, we obtain a classification of the countries based on their numerical values in socioeconomics, population, Gross Domestic Product (GDP), health and air connections; where, in the spreader set there are those countries that have high, medium or low values in the different characteristics; however, the aspect that all the countries belonging to the separator set share is a high value in air connections.

Universality class of explosive percolation in Barabási-Albert networks

In this work, we study explosive percolation (EP) in Barabási-Albert (BA) network, in which nodes are born with degree k = m, for both product rule (PR) and sum rule (SR) of the Achlioptas process. For m = 1 we find that the critical point t_c = 1 which is the maximum possible value of the relative link density t; Hence we cannot have access to the other phase like percolation in one dimension. However, for m > 1 we find that t_c decreases with increasing m and the critical exponents ν, α, β and γ for m > 1 are found to be independent not only of the value of m but also of PR and SR. It implies that they all belong to the same universality class like EP in the Erdös-Rényi network. Besides, the critical exponents obey the Rushbrooke inequality α + 2β + γ ≥ 2 but always close to equality.

    PACS numbers: 61.43.Hv, 64.60.Ht, 68.03.Fg, 82.70.Dd.

Optimal percolation on multiplex networks

Optimal percolation is the problem of finding the minimal set of nodes whose removal from a network fragments the system into non-extensive disconnected clusters. The solution to this problem is important for strategies of immunization in disease spreading, and influence maximization in opinion dynamics. Optimal percolation has received considerable attention in the context of isolated networks. However, its generalization to multiplex networks has not yet been considered. Here we show that approximating the solution of the optimal percolation problem on a multiplex network with solutions valid for single-layer networks extracted from the multiplex may have serious consequences in the characterization of the true robustness of the system. We reach this conclusion by extending many of the methods for finding approximate solutions of the optimal percolation problem from single-layer to multiplex networks, and performing a systematic analysis on synthetic and real-world multiplex networks.

The interdependent network of gene regulation and metabolism is robust where it needs to be

Despite being highly interdependent, the major biochemical networks of the living cell-the networks of interacting genes and of metabolic reactions, respectively-have been approached mostly as separate systems so far. Recently, a framework for interdependent networks has emerged in the context of statistical physics. In a first quantitative application of this framework to systems biology, here we study the interdependent network of gene regulation and metabolism for the model organism Escherichia coli in terms of a biologically motivated percolation model. Particularly, we approach the system's conflicting tasks of reacting rapidly to (internal and external) perturbations, while being robust to minor environmental fluctuations. Considering its response to perturbations that are localized with respect to functional criteria, we find the interdependent system to be sensitive to gene regulatory and protein-level perturbations, yet robust against metabolic changes. We expect this approach to be applicable to a range of other interdependent networks.Although networks of interacting genes and metabolic reactions are interdependent, they have largely been treated as separate systems. Here the authors apply a statistical framework for interdependent networks to E. coli, and show that it is sensitive to gene and protein perturbations but robust against metabolic changes.

Fluctuations in percolation of sparse complex networks

We study the role of fluctuations in percolation of sparse complex networks. To this end we consider two random correlated realizations of the initial damage of the nodes and we evaluate the fraction of nodes that are expected to remain in the giant component of the network in both cases or just in one case. Our framework includes a message-passing algorithm able to predict the fluctuations in a single network, and an analytic prediction of the expected fluctuations in ensembles of sparse networks. This approach is applied to real ecological and infrastructure networks and it is shown to characterize the expected fluctuations in their response to external damage.

Optimizing diffusion in multiplexes by maximizing layer dissimilarity

Diffusion in a multiplex depends on the specific link distribution between the nodes in each layer, but also on the set of the intralayer and interlayer diffusion coefficients. In this work we investigate, in a quantitative way, the efficiency of multiplex diffusion as a function of the topological similarity among multiplex layers. This similarity is measured by the distance between layers, taken among the pairs of layers. Results are presented for a simple two-layer multiplex, where one of the layers is held fixed, while the other one can be rewired in a controlled way in order to increase or decrease the interlayer distance. The results indicate that, for fixed values of all intra- and interlayer diffusion coefficients, a large interlayer distance generally enhances the global multiplex diffusion, providing a topological mechanism to control the global diffusive process. For some sets of networks, we develop an algorithm to identify the most sensitive nodes in the rewirable layer, so that changes in a small set of connections produce a drastic enhancement of the global diffusion of the whole multiplex system.