Cavity analysis on the robustness of random networks against targeted attacks: Influences of degree-degree correlations

Gaussian level-set percolation on complex networks

We present a solution of the problem of level-set percolation for multivariate Gaussians defined in terms of weighted graph Laplacians on complex networks. It is achieved using a cavity or message passing approach, which allows one to obtain the essential ingredient required for the solution, viz. a self-consistent determination of locally varying percolation probabilities. The cavity solution can be evaluated both for single large instances of locally treelike graphs, and in the thermodynamic limit of random graphs of finite mean degree in the configuration model class. The critical level h_{c} of the percolation transition is obtained through the condition that the largest eigenvalue of a weighted version B of a nonbacktracking matrix satisfies λ_{max}(B)|_{h_{c}}=1. We present level-dependent distributions of local percolation probabilities for Erdős-Rényi networks and and for networks with degree distributions described by power laws. We find that there is a strong correlation between marginal single-node variances of a massless multivariate Gaussian and local percolation probabilities at a given level h, which is nearly perfect at negative values h, but weakens as h↗0 for the system with power-law degree distribution, and generally also for negative values of h, if the multivariate Gaussian acquires a nonzero mass. The theoretical analysis simplifies in the case of random regular graphs with uniform edge weights of the weighted graph Laplacian of the system and uniform mass parameter of the Gaussian field. An asymptotic analysis reveals that for edge weights K=K(c)≡1 the critical percolation threshold h_{c} decreases to 0, as the degree c of the random regular graph diverges. For K=(c)=1/c, however, the critical percolation threshold h_{c} is shown to diverge as c→∞.

Level-set percolation of Gaussian random fields on complex networks

We provide an explicit solution of the problem of level-set percolation for multivariate Gaussians defined in terms of weighted graph Laplacians on complex networks. The solution requires an analysis of the heterogeneous microstructure of the percolation problem, i.e., a self-consistent determination of locally varying percolation probabilities. This is achieved using a cavity or message passing approach. It can be evaluated, both for single large instances of locally treelike graphs, and in the thermodynamic limit of random graphs of finite mean degree in the configuration model class.

Belief propagation on networks with cliques and chordless cycles

It is well known that tree-based theories can describe the properties of undirected clustered networks with extremely accurate results [S. Melnik et al., Phys. Rev. E 83, 036112 (2011)10.1103/PhysRevE.83.036112]. It is reasonable to suggest that a motif-based theory would be superior to a tree one, since additional neighbor correlations are encapsulated in the motif structure. In this paper, we examine bond percolation on random and real world networks using belief propagation in conjunction with edge-disjoint motif covers. We derive exact message passing expressions for cliques and chordless cycles of finite size. Our theoretical model gives good agreement with Monte Carlo simulation and offers a simple, yet substantial improvement on traditional message passing, showing that this approach is suitable to study the properties of random and empirical networks.

Synergistic epidemic spreading in correlated networks

We investigate the effect of degree correlation on a susceptible-infected-susceptible (SIS) model with a nonlinear cooperative effect (synergy) in infectious transmissions. In a mean-field treatment of the synergistic SIS model on a bimodal network with tunable degree correlation, we identify a discontinuous transition that is independent of the degree correlation strength unless the synergy is absent or extremely weak. Regardless of synergy (absent or present), a positive and negative degree correlation in the model reduces and raises the epidemic threshold, respectively. For networks with a strongly positive degree correlation, the mean-field treatment predicts the emergence of two discontinuous jumps in the steady-state infected density. To test the mean-field treatment, we provide approximate master equations of the present model. We quantitatively confirm that the approximate master equations agree with not only all qualitative predictions of the mean-field treatment but also corresponding Monte Carlo simulations.

Fate of articulation points and bredges in percolation

We investigate the statistics of articulation points and bredges (bridge edges) in complex networks in which bonds are randomly removed in a percolation process. Because of the heterogeneous structure of a complex network, the probability of a node to be an articulation point or the probability of an edge to be a bredge will not be homogeneous across the network. We therefore analyze full distributions of articulation point probabilities as well as bredge probabilities, using a message-passing or cavity approach to the problem. Our methods allow us to obtain these distributions both for large single instances of networks and for ensembles of networks in the configuration model class in the thermodynamic limit, through a single unified approach. We also evaluate deconvolutions of these distributions according to degrees of the node or the degrees of both adjacent nodes in the case of bredges. We obtain closed form expressions for the large mean degree limit of Erdős-Rényi networks. Moreover, we reveal and are able to rationalize a significant amount of structure in the evolution of articulation point and bredge probabilities in response to random bond removal. We find that full distributions of articulation point and bredge probabilities in real networks and in their randomized counterparts may exhibit significant differences even where average articulation point and bredge probabilities do not. We argue that our results could be exploited in a variety of applications, including approaches to network dismantling or to vaccination and islanding strategies to prevent the spread of epidemics or of blackouts in process networks.

The responsiveness of criminal networks to intentional attacks: Disrupting darknet drug trade

Physical, technological, and social networks are often at risk of intentional attack. Despite the wide-spanning importance of network vulnerability, very little is known about how criminal networks respond to attacks or whether intentional attacks affect criminal activity in the long-run. To assess criminal network responsiveness, we designed an empirically-grounded agent-based simulation using population-level network data on 16,847 illicit drug exchanges between 7,295 users of an active darknet drug market and statistical methods for simulation analysis. We consider three attack strategies: targeted attacks that delete structurally integral vertices, weak link attacks that delete large numbers of weakly connected vertices, and signal attacks that saturate the network with noisy signals. Results reveal that, while targeted attacks are effective when conducted at a large-scale, weak link and signal attacks deter more potential drug transactions and buyers when only a small portion of the network is attacked. We also find that intentional attacks affect network behavior. When networks are attacked, actors grow more cautious about forging ties, connecting less frequently and only to trustworthy alters. Operating in tandem, these two processes undermine long-term network robustness and increase network vulnerability to future attacks.

Heterogeneity in outcomes of repeated instances of percolation experiments

We investigate the heterogeneity of outcomes of repeated instances of percolation experiments in complex networks using a message-passing approach to evaluate heterogeneous, node-dependent probabilities of belonging to the giant or percolating cluster, i.e., the set of mutually connected nodes whose size scales linearly with the size of the system. We evaluate these both for large finite single instances and for synthetic networks in the configuration model class in the thermodynamic limit. For the latter, we consider both Erdős-Rényi and scale-free networks as examples of networks with narrow and broad degree distributions, respectively. For real-world networks we use an undirected version of a Gnutella peer-to-peer file-sharing network with N=62568 nodes as an example. We derive the theory for multiple instances of both uncorrelated and correlated percolation processes. For the uncorrelated case, we also obtain a closed-form approximation for the large mean degree limit of Erdős-Rényi networks.

Fault tolerance of random graphs with respect to connectivity: Mean-field approximation for semidense random graphs

The fault tolerance of random graphs with unbounded degrees with respect to connectivity is investigated, which relates to the reliability of wireless sensor networks with unreliable relay nodes. The model evaluates the network breakdown probability that a graph is disconnected after stochastic node removal. To establish a mean-field approximation for the model, we propose the cavity method for finite systems. The analysis enables us to obtain an approximation formula for random graphs with any number of nodes and an arbitrary degree distribution. In addition, its asymptotic analysis reveals that the phase transition occurs in semidense random graphs whose average degree grows logarithmically. These results, which are supported by numerical simulations, coincide with the mathematical results, indicating successful predictions by the mean-field approximation for unbounded but not dense random graphs.

Complete synchronization of the global coupled dynamical network induced by Poisson noises

The different Poisson noise-induced complete synchronization of the global coupled dynamical network is investigated. Based on the stability theory of stochastic differential equations driven by Poisson process, we can prove that Poisson noises can induce synchronization and sufficient conditions are established to achieve complete synchronization with probability 1. Furthermore, numerical examples are provided to show the agreement between theoretical and numerical analysis.

Structural instability of large-scale functional networks

We study how large functional networks can grow stably under possible cascading overload failures and evaluated the maximum stable network size above which even a small-scale failure would cause a fatal breakdown of the network. Employing a model of cascading failures induced by temporally fluctuating loads, the maximum stable size n_max has been calculated as a function of the load reduction parameter r that characterizes how quickly the total load is reduced during the cascade. If we reduce the total load sufficiently fast (r ≥ r_c), the network can grow infinitely. Otherwise, n_max is finite and increases with r. For a fixed r(< r_c), n_max for a scale-free network is larger than that for an exponential network with the same average degree. We also discuss how one detects and avoids the crisis of a fatal breakdown of the network from the relation between the sizes of the initial network and the largest component after an ordinarily occurring cascading failure.

Robustness analysis of bimodal networks in the whole range of degree correlation

We present an exact analysis of the physical properties of bimodal networks specified by the two peak degree distribution fully incorporating the degree-degree correlation between node connections. The structure of the correlated bimodal network is uniquely determined by the Pearson coefficient of the degree correlation, keeping its degree distribution fixed. The percolation threshold and the giant component fraction of the correlated bimodal network are analytically calculated in the whole range of the Pearson coefficient from -1 to 1 against two major types of node removal, which are the random failure and the degree-based targeted attack. The Pearson coefficient for next-nearest-neighbor pairs is also calculated, which always takes a positive value even when the correlation between nearest-neighbor pairs is negative. From the results, it is confirmed that the percolation threshold is a monotonically decreasing function of the Pearson coefficient for the degrees of nearest-neighbor pairs increasing from -1 and 1 regardless of the types of node removal. In contrast, the node fraction of the giant component for bimodal networks with positive degree correlation rapidly decreases in the early stage of random failure, while that for bimodal networks with negative degree correlation remains relatively large until the removed node fraction reaches the threshold. In this sense, bimodal networks with negative degree correlation are more robust against random failure than those with positive degree correlation.

Percolation transitions in the survival of interdependent agents on multiplex networks, catastrophic cascades, and solid-on-solid surface growth

We present an efficient algorithm for simulating percolation transitions of mutually supporting viable clusters on multiplex networks (also known as "catastrophic cascades on interdependent networks"). This algorithm maps the problem onto a solid-on-solid-type model. We use this algorithm to study interdependent agents on duplex Erdös-Rényi (ER) networks and on lattices with dimensions 2, 3, 4, and 5. We obtain surprising results in all these cases, and we correct statements in the literature for ER networks and for two-dimensional lattices. In particular, we find that d=4 is the upper critical dimension and that the percolation transition is continuous for d≤4 but-at least for d≠3-not in the universality class of ordinary percolation. For ER networks we verify that the cluster statistics is exactly described by mean-field theory but find evidence that the cascade process is not. For d=5 we find a first-order transition as for ER networks, but we find also that small clusters have a nontrivial mass distribution that scales at the transition point. Finally, for d=2 with intermediate-range dependency links we propose a scenario that differs from that proposed in W. Li et al. [Phys. Rev. Lett. 108, 228702 (2012)].

Robustness of oscillatory behavior in correlated networks

Understanding network robustness against failures of network units is useful for preventing large-scale breakdowns and damages in real-world networked systems. The tolerance of networked systems whose functions are maintained by collective dynamical behavior of the network units has recently been analyzed in the framework called dynamical robustness of complex networks. The effect of network structure on the dynamical robustness has been examined with various types of network topology, but the role of network assortativity, or degree–degree correlations, is still unclear. Here we study the dynamical robustness of correlated (assortative and disassortative) networks consisting of diffusively coupled oscillators. Numerical analyses for the correlated networks with Poisson and power-law degree distributions show that network assortativity enhances the dynamical robustness of the oscillator networks but the impact of network disassortativity depends on the detailed network connectivity. Furthermore, we theoretically analyze the dynamical robustness of correlated bimodal networks with two-peak degree distributions and show the positive impact of the network assortativity.

Cavity-based robustness analysis of interdependent networks: influences of intranetwork and internetwork degree-degree correlations

We develop a methodology for analyzing the percolation phenomena of two mutually coupled (interdependent) networks based on the cavity method of statistical mechanics. In particular, we take into account the influence of degree-degree correlations inside and between the networks on the network robustness against targeted (random degree-dependent) attacks and random failures. We show that the developed methodology is reduced to the well-known generating function formalism in the absence of degree-degree correlations. The validity of the developed methodology is confirmed by a comparison with the results of numerical experiments. Our analytical results indicate that the robustness of the interdependent networks depends on both the intranetwork and internetwork degree-degree correlations in a nontrivial way for both cases of random failures and targeted attacks.

Robustness of cooperation on scale-free networks under continuous topological change

In this paper, we numerically investigate the robustness of cooperation clusters in prisoner's dilemma played on scale-free networks, where the network topologies change by continuous removal and addition of nodes. Each removal and addition can be either random or intentional. We therefore have four different strategies in changing network topology: random removal and random addition (RR), random removal and preferential addition (RP), targeted removal and random addition (TR), and targeted removal and preferential addition (TP). We find that cooperation clusters are most fragile against TR, while they are most robust against RP, even for large values of the temptation coefficient for defection. The effect of the degree mixing pattern of the network is not the primary factor for the robustness of cooperation under continuous change in network topology, which is quite different from the cases observed in static networks. Cooperation clusters become more robust as the number of links of hubs occupied by cooperators increase. Our results might infer the fact that a huge variety of individuals is needed for maintaining global cooperation in social networks in the real world where each node representing an individual is constantly removed and added.

Observability transitions in correlated networks

Yang, Wang, and Motter [Phys. Rev. Lett. 109, 258701 (2012)] analyzed a model for network observability transitions in which a sensor placed on a node makes the node and the adjacent nodes observable. The size of the connected components comprising the observable nodes is a major concern of the model. We analyze this model in random heterogeneous networks with degree correlation. With numerical simulations and analytical arguments based on generating functions, we find that negative degree correlation makes networks more observable. This result holds true both when the sensors are placed on nodes one by one in a random order and when hubs preferentially receive the sensors. Finally, we numerically optimize networks with a fixed degree sequence with respect to the size of the largest observable component. Optimized networks have negative degree correlation induced by the resulting hub-repulsive structure; the largest hubs are rarely connected to each other, in contrast to the rich-club phenomenon of networks.

Robustness of onionlike correlated networks against targeted attacks

Recently, it was found by Schneider et al. [Proc. Natl. Acad. Sci. USA 108, 3838 (2011)], using simulations, that scale-free networks with "onion structure" are very robust against targeted high degree attacks. The onion structure is a network where nodes with almost the same degree are connected. Motivated by this work, we propose and analyze, based on analytical considerations, an onionlike candidate for a nearly optimal structure against simultaneous random and targeted high degree node attacks. The nearly optimal structure can be viewed as a set of hierarchically interconnected random regular graphs,the degrees and populations of whose nodes are specified by the degree distribution. This network structure exhibits an extremely assortative degree-degree correlation and has a close relationship to the "onion structure." After deriving a set of exact expressions that enable us to calculate the critical percolation threshold and the giant component of a correlated network for an arbitrary type of node removal, we apply the theory to the cases of random scale-free networks that are highly vulnerable against targeted high degree node removal. Our results show that this vulnerability can be significantly reduced by implementing this onionlike type of degree-degree correlation without much undermining the almost complete robustness against random node removal. We also investigate in detail the robustness enhancement due to assortative degree-degree correlation by introducing a joint degree-degree probability matrix that interpolates between an uncorrelated network structure and the onionlike structure proposed here by tuning a single control parameter. The optimal values of the control parameter that maximize the robustness against simultaneous random and targeted attacks are also determined. Our analytical calculations are supported by numerical simulations.

Belief-propagation algorithm and the Ising model on networks with arbitrary distributions of motifs

We generalize the belief-propagation algorithm to sparse random networks with arbitrary distributions of motifs (triangles, loops, etc.). Each vertex in these networks belongs to a given set of motifs (generalization of the configuration model). These networks can be treated as sparse uncorrelated hypergraphs in which hyperedges represent motifs. Here a hypergraph is a generalization of a graph, where a hyperedge can connect any number of vertices. These uncorrelated hypergraphs are treelike (hypertrees), which crucially simplifies the problem and allows us to apply the belief-propagation algorithm to these loopy networks with arbitrary motifs. As natural examples, we consider motifs in the form of finite loops and cliques. We apply the belief-propagation algorithm to the ferromagnetic Ising model with pairwise interactions on the resulting random networks and obtain an exact solution of this model. We find an exact critical temperature of the ferromagnetic phase transition and demonstrate that with increasing the clustering coefficient and the loop size, the critical temperature increases compared to ordinary treelike complex networks. However, weak clustering does not change the critical behavior qualitatively. Our solution also gives the birth point of the giant connected component in these loopy networks.