Weak percolation on multiplex networks

Hidden multiscale organization and robustness of real multiplex networks

Hidden geometry enables the investigation of complex networks at different scales. Extending this framework to multiplex networks, we uncover a different kind of mesoscopic organization in real multiplex systems, named clan, a group of nodes that preserve local geometric arrangements across layers. Furthermore, we reveal the intimate relationship between the unfolding of clan structure and mutual percolation against targeted attacks, leading to an ambivalent role of clans: making a system fragile yet less prone to complete shattering. Finally, we confirm the correlation between the multiscale nature of geometric organization and the overall robustness. Our findings expand the significance of hidden geometry in network function, while also highlighting potential pitfalls in evaluating and controlling catastrophic failure of multiplex systems.

The fundamental benefits of multiplexity in ecological networks

A tipping point presents perhaps the single most significant threat to an ecological system as it can lead to abrupt species extinction on a massive scale. Climate changes leading to the species decay parameter drifts can drive various ecological systems towards a tipping point. We investigate the tipping-point dynamics in multi-layer ecological networks supported by mutualism. We unveil a natural mechanism by which the occurrence of tipping points can be delayed by multiplexity that broadly describes the diversity of the species abundances, the complexity of the interspecific relationships, and the topology of linkages in ecological networks. For a double-layer system of pollinators and plants, coupling between the network layers occurs when there is dispersal of pollinator species. Multiplexity emerges as the dispersing species establish their presence in the destination layer and have a simultaneous presence in both. We demonstrate that the new mutualistic links induced by the dispersing species with the residence species have fundamental benefits to the well-being of the ecosystem in delaying the tipping point and facilitating species recovery. Articulating and implementing control mechanisms to induce multiplexity can thus help sustain certain types of ecosystems that are in danger of extinction as the result of environmental changes.

Group percolation in interdependent networks with reinforcement network layer

In many real-world interdependent network systems, nodes often work together to form groups, which can enhance robustness to resist risks. However, previous group percolation models are always of a first-order phase transition, regardless of the group size distribution. This motivates us to investigate a generalized model for group percolation in interdependent networks with a reinforcement network layer to eliminate collapse. Some backup devices that are equipped for a density ρ of reinforced nodes constitute the reinforcement network layer. For each group, we assume that at least one node of the group can function in one network and a node in another network depends on the group to function. We find that increasing the density ρ of reinforcement nodes and the size S of the dependency group can significantly enhance the robustness of interdependent networks. Importantly, we find the existence of a hybrid phase transition behavior and propose a method for calculating the shift point of percolation types. The most interesting finding is the exact universal solution to the minimal density _ρ of reinforced nodes (or the minimum group size _S) to prevent abrupt collapse for Erdős-Rényi, scale-free, and regular random interdependent networks. Furthermore, we present the validity of the analytic solutions for a triple point _ρ (or _S ), the corresponding phase transition point _p , and second-order phase transition points _p in interdependent networks. These findings might yield a broad perspective for designing more resilient interdependent infrastructure networks.

Targeting attack hypergraph networks

In modern systems, from brain neural networks to social group networks, pairwise interactions are not sufficient to express higher-order relationships. The smallest unit of their internal function is not composed of a single functional node but results from multiple functional nodes acting together. Therefore, researchers adopt the hypergraph to describe complex systems. The targeted attack on random hypergraph networks is still a problem worthy of study. This work puts forward a theoretical framework to analyze the robustness of random hypergraph networks under the background of a targeted attack on nodes with high or low hyperdegrees. We discovered the process of cascading failures and the giant connected cluster (GCC) of the hypergraph network under targeted attack by associating the simple mapping of the factor graph with the hypergraph and using percolation theory and generating function. On random hypergraph networks, we do Monte-Carlo simulations and find that the theoretical findings match the simulation results. Similarly, targeted attacks are more effective than random failures in disintegrating random hypergraph networks. The threshold of the hypergraph network grows as the probability of high hyperdegree nodes being deleted increases, indicating that the network's resilience becomes more fragile. When considering real-world scenarios, our conclusions are validated by real-world hypergraph networks. These findings will help us understand the impact of the hypergraph's underlying structure on network resilience.

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].

Higher-order percolation processes on multiplex hypergraphs

Higher-order interactions are increasingly recognized as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraphs as well as simplicial complexes capture the higher-order interactions of complex systems and allow us to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensemble where each layer is characterized by hyperedges of given cardinality. We observe that in presence of the structural cutoff the ensemble of multiplex hypergraphs can be mapped to an ensemble of multiplex bipartite networks. We reveal the relation between higher-order percolation processes in random multiplex hypergraphs, interdependent percolation of multiplex networks, and K-core percolation. The structural correlations of the random multiplex hypergraphs are shown to have a significant effect on their percolation properties. The wide range of critical behaviors observed for higher-order percolation processes on multiplex hypergraphs elucidates the mechanisms responsible for the emergence of discontinuous transition and uncovers interesting critical properties which can be applied to the study of epidemic spreading and contagion processes on higher-order networks.

Enhanced robustness of single-layer networks with redundant dependencies

Dependency links in single-layer networks offer a convenient way of modeling nonlocal percolation effects in networked systems where certain pairs of nodes are only able to function together. We study the percolation properties of the weak variant of this model: Nodes with dependency neighbors may continue to function if at least one of their dependency neighbors is active. We show that this relaxation of the dependency rule allows for more robust structures and a rich variety of critical phenomena, as percolation is not determined strictly by finite dependency clusters. We study Erdős-Rényi and random scale-free networks with an underlying Erdős-Rényi network of dependency links. We identify a special "cusp" point above which the system is always stable, irrespective of the density of dependency links. We find continuous and discontinuous hybrid percolation transitions, separated by a tricritical point for Erdős-Rényi networks. For scale-free networks with a finite degree cutoff we observe the appearance of a critical point and corresponding double transitions in a certain range of the degree distribution exponent. We show that at a special point in the parameter space, where the critical point emerges, the giant viable cluster has the unusual critical singularity S-S_{c}∝(p-p_{c})^{1/4}. We study the robustness of networks where connectivity degrees and dependency degrees are correlated and find that scale-free networks are able to retain their high resilience for strong enough positive correlation, i.e., when hubs are protected by greater redundancy.

Exotic critical behavior of weak multiplex percolation

We describe the critical behavior of weak multiplex percolation, a generalization of percolation to multiplex or interdependent networks. A node can determine its active or inactive status simply by referencing neighboring nodes. This is not the case for the more commonly studied generalization of percolation to multiplex networks, the mutually connected clusters, which requires an interconnecting path within each layer between any two vertices in the giant mutually connected component. We study the emergence of a giant connected component of active nodes under the weak percolation rule, finding several nontypical phenomena. In two layers, the giant component emerges with a continuous phase transition, but with quadratic growth above the critical threshold. In three or more layers, a discontinuous hybrid transition occurs, similar to that found in the giant mutually connected component. In networks with asymptotically powerlaw degree distributions, defined by the decay exponent γ, the discontinuity vanishes but at γ=1.5 in three layers, more generally at γ=1+1/(M-1) in M layers.

The "weak" interdependence of infrastructure systems produces mixed percolation transitions in multilayer networks

Previous studies of multilayer network robustness model cascading failures via a node-to-node percolation process that assumes "strong" interdependence across layers-once a node in any layer fails, its neighbors in other layers fail immediately and completely with all links removed. This assumption is not true of real interdependent infrastructures that have emergency procedures to buffer against cascades. In this work, we consider a node-to-link failure propagation mechanism and establish "weak" interdependence across layers via a tolerance parameter α which quantifies the likelihood that a node survives when one of its interdependent neighbors fails. Analytical and numerical results show that weak interdependence produces a striking phenomenon: layers at different positions within the multilayer system experience distinct percolation transitions. Especially, layers with high super degree values percolate in an abrupt manner, while those with low super degree values exhibit both continuous and discontinuous transitions. This novel phenomenon we call mixed percolation transitions has significant implications for network robustness. Previous results that do not consider cascade tolerance and layer super degree may be under- or over-estimating the vulnerability of real systems. Moreover, our model reveals how nodal protection activities influence failure dynamics in interdependent, multilayer systems.

Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions

This work presents exact expressions for size distributions of weak and multilayer connected components in two generalizations of the configuration model: networks with directed edges and multiplex networks with an arbitrary number of layers. The expressions are computable in a polynomial time and, under some restrictions, are tractable from the asymptotic theory point of view. If first partial moments of the degree distribution are finite, the size distribution for two-layer connected components in multiplex networks exhibits an exponent -3/2 in the critical regime, whereas the size distribution of weakly connected components in directed networks exhibits two critical exponents -1/2 and -3/2.

Eradicating catastrophic collapse in interdependent networks via reinforced nodes

In interdependent networks, it is usually assumed, based on percolation theory, that nodes become nonfunctional if they lose connection to the network giant component. However, in reality, some nodes, equipped with alternative resources, together with their connected neighbors can still be functioning after disconnected from the giant component. Here, we propose and study a generalized percolation model that introduces a fraction of reinforced nodes in the interdependent networks that can function and support their neighborhood. We analyze, both analytically and via simulations, the order parameter-the functioning component-comprising both the giant component and smaller components that include at least one reinforced node. Remarkably, it is found that, for interdependent networks, we need to reinforce only a small fraction of nodes to prevent abrupt catastrophic collapses. Moreover, we find that the universal upper bound of this fraction is 0.1756 for two interdependent Erdős-Rényi (ER) networks: regular random (RR) networks and scale-free (SF) networks with large average degrees. We also generalize our theory to interdependent networks of networks (NONs). These findings might yield insight for designing resilient interdependent infrastructure networks.

Clustering determines the dynamics of complex contagions in multiplex networks

We present the mathematical analysis of generalized complex contagions in a class of clustered multiplex networks. The model is intended to understand spread of influence, or any other spreading process implying a threshold dynamics, in setups of interconnected networks with significant clustering. The contagion is assumed to be general enough to account for a content-dependent linear threshold model, where each link type has a different weight (for spreading influence) that may depend on the content (e.g., product, rumor, political view) that is being spread. Using the generating functions formalism, we determine the conditions, probability, and expected size of the emergent global cascades. This analysis provides a generalization of previous approaches and is especially useful in problems related to spreading and percolation. The results present nontrivial dependencies between the clustering coefficient of the networks and its average degree. In particular, several phase transitions are shown to occur depending on these descriptors. Generally speaking, our findings reveal that increasing clustering decreases the probability of having global cascades and their size, however, this tendency changes with the average degree. There exists a certain average degree from which on clustering favors the probability and size of the contagion. By comparing the dynamics of complex contagions over multiplex networks and their monoplex projections, we demonstrate that ignoring link types and aggregating network layers may lead to inaccurate conclusions about contagion dynamics, particularly when the correlation of degrees between layers is high.

Message passing theory for percolation models on multiplex networks with link overlap

Multiplex networks describe a large variety of complex systems, including infrastructures, transportation networks, and biological systems. Most of these networks feature a significant link overlap. It is therefore of particular importance to characterize the mutually connected giant component in these networks. Here we provide a message passing theory for characterizing the percolation transition in multiplex networks with link overlap and an arbitrary number of layers M. Specifically we propose and compare two message passing algorithms that generalize the algorithm widely used to study the percolation transition in multiplex networks without link overlap. The first algorithm describes a directed percolation transition and admits an epidemic spreading interpretation. The second algorithm describes the emergence of the mutually connected giant component, that is the percolation transition, but does not preserve the epidemic spreading interpretation. We obtain the phase diagrams for the percolation and directed percolation transition in simple representative cases. We demonstrate that for the same multiplex network structure, in which the directed percolation transition has nontrivial tricritical points, the percolation transition has a discontinuous phase transition, with the exception of the trivial case in which all the layers completely overlap.

Recovery of Interdependent Networks

Recent network research has focused on the cascading failures in a system of interdependent networks and the necessary preconditions for system collapse. An important question that has not been addressed is how to repair a failing system before it suffers total breakdown. Here we introduce a recovery strategy for nodes and develop an analytic and numerical framework for studying the concurrent failure and recovery of a system of interdependent networks based on an efficient and practically reasonable strategy. Our strategy consists of repairing a fraction of failed nodes, with probability of recovery γ, that are neighbors of the largest connected component of each constituent network. We find that, for a given initial failure of a fraction 1 - p of nodes, there is a critical probability of recovery above which the cascade is halted and the system fully restores to its initial state and below which the system abruptly collapses. As a consequence we find in the plane γ - p of the phase diagram three distinct phases. A phase in which the system never collapses without being restored, another phase in which the recovery strategy avoids the breakdown, and a phase in which even the repairing process cannot prevent system collapse.

Multiplex networks with heterogeneous activities of the nodes

In multiplex networks with a large number of layers, the nodes can have different activities, indicating the total number of layers in which the nodes are present. Here we model multiplex networks with heterogeneous activity of the nodes and we study their robustness properties. We introduce a percolation model where nodes need to belong to the giant component only on the layers where they are active (i.e., their degree on that layer is larger than zero). We show that when there are enough nodes active only in one layer, the multiplex becomes more resilient and the transition becomes continuous. We find that multiplex networks with a power-law distribution of node activities are more fragile if the distribution of activity is broader. We also show that while positive correlations between node activity and degree can enhance the robustness of the system, the phase transition may become discontinuous, making the system highly unpredictable.

Multiple tipping points and optimal repairing in interacting networks

Systems composed of many interacting dynamical networks-such as the human body with its biological networks or the global economic network consisting of regional clusters-often exhibit complicated collective dynamics. Three fundamental processes that are typically present are failure, damage spread and recovery. Here we develop a model for such systems and find a very rich phase diagram that becomes increasingly more complex as the number of interacting networks increases. In the simplest example of two interacting networks we find two critical points, four triple points, ten allowed transitions and two 'forbidden' transitions, as well as complex hysteresis loops. Remarkably, we find that triple points play the dominant role in constructing the optimal repairing strategy in damaged interacting systems. To test our model, we analyse an example of real interacting financial networks and find evidence of rapid dynamical transitions between well-defined states, in agreement with the predictions of our model.

Growing multiplex networks with arbitrary number of layers

This paper focuses on the problem of growing multiplex networks. Currently, the results on the joint degree distribution of growing multiplex networks present in the literature pertain to the case of two layers and are confined to the special case of homogeneous growth and are limited to the state state (that is, the limit of infinite size). In the present paper, we first obtain closed-form solutions for the joint degree distribution of heterogeneously growing multiplex networks with arbitrary number of layers in the steady state. Heterogeneous growth means that each incoming node establishes different numbers of links in different layers. We consider both uniform and preferential growth. We then extend the analysis of the uniform growth mechanism to arbitrary times. We obtain a closed-form solution for the time-dependent joint degree distribution of a growing multiplex network with arbitrary initial conditions. Throughout, theoretical findings are corroborated with Monte Carlo simulations. The results shed light on the effects of the initial network on the transient dynamics of growing multiplex networks and takes a step towards characterizing the temporal variations of the connectivity of growing multiplex networks, as well as predicting their future structural properties.

Bootstrap percolation on spatial networks

Bootstrap percolation is a general representation of some networked activation process, which has found applications in explaining many important social phenomena, such as the propagation of information. Inspired by some recent findings on spatial structure of online social networks, here we study bootstrap percolation on undirected spatial networks, with the probability density function of long-range links' lengths being a power law with tunable exponent. Setting the size of the giant active component as the order parameter, we find a parameter-dependent critical value for the power-law exponent, above which there is a double phase transition, mixed of a second-order phase transition and a hybrid phase transition with two varying critical points, otherwise there is only a second-order phase transition. We further find a parameter-independent critical value around -1, about which the two critical points for the double phase transition are almost constant. To our surprise, this critical value -1 is just equal or very close to the values of many real online social networks, including LiveJournal, HP Labs email network, Belgian mobile phone network, etc. This work helps us in better understanding the self-organization of spatial structure of online social networks, in terms of the effective function for information spreading.

Measuring and modeling correlations in multiplex networks

The interactions among the elementary components of many complex systems can be qualitatively different. Such systems are therefore naturally described in terms of multiplex or multilayer networks, i.e., networks where each layer stands for a different type of interaction between the same set of nodes. There is today a growing interest in understanding when and why a description in terms of a multiplex network is necessary and more informative than a single-layer projection. Here we contribute to this debate by presenting a comprehensive study of correlations in multiplex networks. Correlations in node properties, especially degree-degree correlations, have been thoroughly studied in single-layer networks. Here we extend this idea to investigate and characterize correlations between the different layers of a multiplex network. Such correlations are intrinsically multiplex, and we first study them empirically by constructing and analyzing several multiplex networks from the real world. In particular, we introduce various measures to characterize correlations in the activity of the nodes and in their degree at the different layers and between activities and degrees. We show that real-world networks exhibit indeed nontrivial multiplex correlations. For instance, we find cases where two layers of the same multiplex network are positively correlated in terms of node degrees, while other two layers are negatively correlated. We then focus on constructing synthetic multiplex networks, proposing a series of models to reproduce the correlations observed empirically and/or to assess their relevance.

Threshold cascades with response heterogeneity in multiplex networks

Threshold cascade models have been used to describe the spread of behavior in social networks and cascades of default in financial networks. In some cases, these networks may have multiple kinds of interactions, such as distinct types of social ties or distinct types of financial liabilities; furthermore, nodes may respond in different ways to influence from their neighbors of multiple types. To start to capture such settings in a stylized way, we generalize a threshold cascade model to a multiplex network in which nodes follow one of two response rules: some nodes activate when, in at least one layer, a large enough fraction of neighbors is active, while the other nodes activate when, in all layers, a large enough fraction of neighbors is active. Varying the fractions of nodes following either rule facilitates or inhibits cascades. Near the inhibition regime, global cascades appear discontinuously as the network density increases; however, the cascade grows more slowly over time. This behavior suggests a way in which various collective phenomena in the real world could appear abruptly yet slowly.

Giant components in directed multiplex networks

We describe the complex global structure of giant components in directed multiplex networks that generalizes the well-known bow-tie structure, generic for ordinary directed networks. By definition, a directed multiplex network contains vertices of one type and directed edges of m different types. In directed multiplex networks, we distinguish a set of different giant components based on the existence of directed paths of different types between their vertices such that for each type of edges, the paths run entirely through only edges of that type. If, in particular, m=2, we define a strongly viable component as a set of vertices in which for each type of edges each two vertices are interconnected by at least two directed paths in both directions, running through the edges of only this type. We show that in this case, a directed multiplex network contains in total nine different giant components including the strongly viable component. In general, the total number of giant components is 3^{m}. For uncorrelated directed multiplex networks, we obtain exactly the size and the emergence point of the strongly viable component and estimate the sizes of other giant components.

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.

Nonlinear growth and condensation in multiplex networks

Different types of interactions coexist and coevolve to shape the structure and function of a multiplex network. We propose here a general class of growth models in which the various layers of a multiplex network coevolve through a set of nonlinear preferential attachment rules. We show, both numerically and analytically, that by tuning the level of nonlinearity these models allow us to reproduce either homogeneous or heterogeneous degree distributions, together with positive or negative degree correlations across layers. In particular, we derive the condition for the appearance of a condensed state in which one node in each layer attracts an extensive fraction of all the edges.

Multiple resource demands and viability in multiplex networks

Many complex systems demand manifold resources to be supplied from distinct channels to function properly, e.g., water, gas, and electricity for a city. Here, we study a model for viability of such systems demanding more than one type of vital resource be produced and distributed by resource nodes in multiplex networks. We found a rich variety of behaviors such as discontinuity, bistability, and hysteresis in the fraction of viable nodes with respect to the density of networks and the fraction of resource nodes. Our result suggests that viability in multiplex networks is not only exposed to the risk of abrupt collapse but also suffers excessive complication in recovery.