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Networks are often made up of several layers that exhibit diverse degrees of interdependencies. An interdependent network consists of a set of graphs G that are interconnected through a weighted interconnection matrix B, where the weight of each intergraph link is a non-negative real number p. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix Q characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix B=pI, I being the identity matrix, it has been shown that there exists a structural transition at some critical coupling p*. This transition is such that dynamical processes are separated into two regimes: if p>p*, the network acts as a whole; whereas when p<p*, the network operates as if the graphs encoding the layers were isolated. In this paper, we extend and generalize the structural transition threshold p* to a regular interconnection matrix B (constant row and column sum). Specifically, we provide upper and lower bounds for the transition threshold p* in interdependent networks with a regular interconnection matrix B and derive the exact transition threshold for special scenarios using the formalism of quotient graphs. Additionally, we discuss the physical meaning of the transition threshold p* in terms of the minimum cut and show, through a counterexample, that the structural transition does not always exist. Our results are one step forward on the characterization of more realistic multilayer networks and might be relevant for systems that deviate from the topological constraints imposed by multiplex networks.

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... The eigen-pair of eigenvalue 2p and eigenvector (u, −u) holds for directed multilayer networks, which is previously found in undirected multilayered networks [12,21]. The joint effect of the Laplacian matrices Q 1 and Q 2 for each layer, encoded in the matrix ΔQ, determines the deviation of the convergence rate of the whole system from that of the integrated multilayer. ...

The multilayer network framework has served to describe and uncover a number of novel and unforeseen physical behaviors and regimes in interacting complex systems. However, the majority of existing studies are built on undirected multilayer networks while most complex systems in nature exhibit directed interactions. Here, we propose a framework to analyze diffusive dynamics on multilayer networks consisting of at least one directed layer. We rigorously demonstrate that directionality in multilayer networks can fundamentally change the behavior of diffusive dynamics: from monotonic (in undirected systems) to non-monotonic diffusion with respect to the interlayer coupling strength. Moreover, for certain multilayer network configurations, the directionality can induce a unique superdiffusion regime for intermediate values of the interlayer coupling, wherein the diffusion is even faster than that corresponding to the theoretical limit for undirected systems, i.e. the diffusion in the integrated network obtained from the aggregation of each layer. We theoretically and numerically show that the existence of superdiffusion is fully determined by the directionality of each layer and the topological overlap between layers. We further provide a formulation of multilayer networks displaying superdiffusion. Our results highlight the significance of incorporating the interacting directionality in multilevel networked systems and provide a framework to analyze dynamical processes on interconnected complex systems with directionality. © 2021 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft

... The eigen-pair of eigenvalue 2p and eigenvector (u, −u) holds for directed multilayer networks, which is previously found in undirected multilayered networks [11,21]. The joint effect of the Laplacian matrices Q 1 and Q 2 for each layer, encoded in the matrix ∆Q, determines the deviation of the convergence rate of the whole system from that of the integrated multilayer. ...

The multilayer network framework has served to describe and uncover a number of novel and unforeseen physical behaviors and regimes in interacting complex systems. However, the majority of existing studies are built on undirected multilayer networks while most complex systems in nature exhibit directed interactions. Here, we propose a framework to analyze diffusive dynamics on multilayer networks consisting of at least one directed layer. We rigorously demonstrate that directionality in multilayer networks can fundamentally change the behavior of diffusive dynamics: from monotonic (in undirected systems) to non-monotonic diffusion with respect to the interlayer coupling strength. Moreover, for certain multilayer network configurations, the directionality can induce a unique superdiffusion regime for intermediate values of the interlayer coupling, wherein the diffusion is even faster than that corresponding to the theoretical limit for undirected systems, i.e., the diffusion in the integrated network obtained from the aggregation of each layer. We theoretically and numerically show that the existence of superdiffusion is fully determined by the directionality of each layer and the topological overlap between layers. We further provide a formulation of multilayer networks displaying superdiffusion. Our results highlight the significance of incorporating the interacting directionality in multilevel networked systems and provide a framework to analyze dynamical processes on interconnected complex systems with directionality.

The second smallest eigenvalue of the Laplacian matrix is determinative in characterizing many network properties and is known as algebraic connectivity. In this paper, we investigate the problem of maximizing algebraic connectivity in multilayer networks by allocating interlink weights subject to a budget while allowing arbitrary interconnections. For budgets below a threshold, we identify an upper-bound for maximum algebraic connectivity which is independent of interconnections pattern and is reachable with satisfying a certain regularity condition. For efficient numerical approaches in regions of no analytical solution, we cast the problem into a convex framework that explores the problem from several perspectives and, particularly, transforms into a graph embedding problem that is easier to interpret and related to the optimum diffusion phase. Allowing arbitrary interconnections entails regions of multiple transitions, giving more diverse diffusion phases with respect to one-to-one interconnection case. When there is no limitation on the interconnections pattern, we derive several analytical results characterizing the optimal weights by individual Fiedler vectors. We use the ratio of algebraic connectivity and the layer sizes to explain the results. Finally, we study the placement of a limited number of interlinks by greedy heuristics, using the Fiedler vector components of each layer.

Many real-world systems can be modeled as interconnected multilayer networks, namely a set of networks interacting with each other. Here we present a perturbative approach to study the properties of a general class of interconnected networks as inter-network interactions are established. We reveal multiple structural transitions for the algebraic connectivity of such systems, between regimes in which each network layer keeps its independent identity or drives diffusive processes over the whole system, thus generalizing previous results reporting a single transition point. Furthermore we show that, at first order in perturbation theory, the growth of the algebraic connectivity of each layer depends only on the degree configuration of the interaction network (projected on the respective Fiedler vector), and not on the actual interaction topology. Our findings can have important implications in the design of robust interconnected networked system, particularly in the presence of network layers whose integrity is more crucial for the functioning of the entire system. We finally show results of perturbation theory applied to the adjacency matrix of the interconnected network, which can be useful to characterize percolation processes on such systems.

Using an information theoretic point of view, we investigate how a dynamics acting on a network can be coarse grained through the use of graph partitions. Specifically, we are interested in how aggregating the state space of a Markov process according to a partition impacts on the thus obtained lower-dimensional dynamics. We highlight that for a dynamics on a particular graph there may be multiple coarse grained descriptions that capture different, incomparable features of the original process. For instance, a coarse graining induced by one partition may be commensurate with a time-scale separation in the dynamics, while another coarse graining may correspond to a different lower-dimensional dynamics that preserves the Markov property of the original process. Taking inspiration from the literature of Computational Mechanics, we find that a convenient tool to summarise and visualise such dynamical properties of a coarse grained model (partition) is the entrogram. The entrogram gathers certain information-theoretic measures, which quantify how information flows across time steps. These information theoretic quantities include the entropy rate, as well as a measure for the memory contained in the process, i.e., how well the dynamics can be approximated by a first order Markov process. We use the entrogram to investigate how specific macro-scale connection patterns in the state-space transition graph of the original dynamics result in desirable properties of coarse grained descriptions. We thereby provide a fresh perspective on the interplay between structure and dynamics in networks, and the process of partitioning from an information theoretic perspective. We focus on networks that may be approximated by both a core-periphery or a clustered organization, and highlight that each of these coarse grained descriptions can capture different aspects of a Markov process acting on the network.

An interconnected network features a structural transition between two regimes [F. Radicchi and A. Arenas, Nat. Phys. 9, 717 (2013)1745-247310.1038/nphys2761]: one where the network components are structurally distinguishable and one where the interconnected network functions as a whole. Our exact solution for the coupling threshold uncovers network topologies with unexpected behaviors. Specifically, we show conditions that superdiffusion, introduced by Gómez et al. [Phys. Rev. Lett. 110, 028701 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.028701], can occur despite the network components functioning distinctly. Moreover, we find that components of certain interconnected network topologies are indistinguishable despite very weak coupling between them.

We present a continuous formulation of epidemic spreading on multilayer networks using a tensorial representation, extending the models of monoplex networks to this context. We derive analytical expressions for the epidemic threshold of the susceptible-infected-susceptible (SIS) and susceptible-infected-recovered dynamics, as well as upper and lower bounds for the disease prevalence in the steady state for the SIS scenario. Using the quasistationary state method, we numerically show the existence of disease localization and the emergence of two or more susceptibility peaks, which are characterized analytically and numerically through the inverse participation ratio. At variance with what is observed in single-layer networks, we show that disease localization takes place on the layers and not on the nodes of a given layer. Furthermore, when mapping the critical dynamics to an eigenvalue problem, we observe a characteristic transition in the eigenvalue spectra of the supra-contact tensor as a function of the ratio of two spreading rates: If the rate at which the disease spreads within a layer is comparable to the spreading rate across layers, the individual spectra of each layer merge with the coupling between layers. Finally, we report on an interesting phenomenon, the barrier effect; i.e., for a three-layer configuration, when the layer with the lowest eigenvalue is located at the center of the line, it can effectively act as a barrier to the disease. The formalism introduced here provides a unifying mathematical approach to disease contagion in multiplex systems, opening new possibilities for the study of spreading processes.

Network representations are useful for describing the structure of a large
variety of complex systems. Although most studies of networks suppose that
nodes are connected by only a single type of edge, most real and engineered
systems are multiplex because they include multiple subsystems and layers of
connectivity. This new paradigm has attracted a great deal of attention and one
challenge is to characterize multilayer networks both structurally and
dynamically. One way to address the latter is to study the spectral properties
of such networks. Here, we show that by using the framework of graph quotients,
a series of rigorous results for both the adjacency and Laplacian matrices of
multilayer networks can be proven. Specifically, we derive relationships
between the eigenvalue spectra of multilayer networks and their most natural
aggregates and show the dynamical implications of working with either of the
two representations. Our work thus contributes to the study of dynamical
processes whose critical properties are determined by the spectral properties
of the underlying network.

Most real and engineered systems include multiple subsystems and layers of
connectivity, and it is important to take such features into account to try to
improve our understanding of these systems. It is thus necessary to generalize
"traditional" network theory by developing (and validating) a framework and
associated tools to study multilayer systems in a comprehensive fashion. The
origins of such efforts date back several decades and arose in multiple
disciplines, and now the study of multilayer networks has become one of the
most important directions in network science. In this paper, we discuss the
history of multilayer networks (and related concepts) and review the exploding
body of work on such networks. To unify the disparate terminology in the large
body of recent work, we discuss a general framework for multilayer networks,
construct a dictionary of terminology to relate the numerous existing concepts
to each other, and provide a thorough discussion that compares, contrasts, and
translates between related notions such as multilayer networks, multiplex
networks, interdependent networks, networks of networks, and many others. We
also survey and discuss existing data sets that can be represented as
multilayer networks. We review attempts to generalize single-layer-network
diagnostics to multilayer networks. We also discuss the rapidly expanding
research on multilayer-network models and notions like community detection,
connected components, tensor decompositions, and various types of dynamical
processes on multilayer networks. We conclude with a summary and an outlook.

Most real-world networks are not isolated. In order to function fully, they are interconnected with other networks, and this interconnection influences their dynamic processes. For example, when the spread of a disease involves two species, the dynamics of the spread within each species (the contact network) differs from that of the spread between the two species (the interconnected network). We model two generic interconnected networks using two adjacency matrices, A and B, in which A is a 2N×2N matrix that depicts the connectivity within each of two networks of size N, and B a 2N×2N matrix that depicts the interconnections between the two. Using an N-intertwined mean-field approximation, we determine that a critical susceptible-infected-susceptible (SIS) epidemic threshold in two interconnected networks is 1/λ_{1}(A+αB), where the infection rate is β within each of the two individual networks and αβ in the interconnected links between the two networks and λ_{1}(A+αB) is the largest eigenvalue of the matrix A+αB. In order to determine how the epidemic threshold is dependent upon the structure of interconnected networks, we analytically derive λ_{1}(A+αB) using a perturbation approximation for small and large α, the lower and upper bound for any α as a function of the adjacency matrix of the two individual networks, and the interconnections between the two and their largest eigenvalues and eigenvectors. We verify these approximation and boundary values for λ_{1}(A+αB) using numerical simulations, and determine how component network features affect λ_{1}(A+αB). We note that, given two isolated networks G_{1} and G_{2} with principal eigenvectors x and y, respectively, λ_{1}(A+αB) tends to be higher when nodes i and j with a higher eigenvector component product x_{i}y_{j} are interconnected. This finding suggests essential insights into ways of designing interconnected networks to be robust against epidemics.

Our current world is linked by a complex mesh of networks where information,
people and goods flow. These networks are interdependent each other, and
present structural and dynamical features different from those observed in
isolated networks. While examples of such "dissimilar" properties are becoming
more abundant, for example diffusion, robustness and competition, it is not yet
clear where these differences are rooted in. Here we show that the composition
of independent networks into an interconnected network of networks undergoes a
structurally sharp transition as the interconnections are formed. Depending of
the relative importance of inter and intra-layer connections, we find that the
entire interdependent system can be tuned between two regimes: in one regime,
the various layers are structurally decoupled and they act as independent
entities; in the other regime, network layers are indistinguishable and the
whole system behave as a single-level network. We analytically show that the
transition between the two regimes is discontinuous even for finite size
networks. Thus, any real-world interconnected system is potentially at risk of
abrupt changes in its structure that may reflect in new dynamical properties.

We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-Laplacian matrix, which consists of a dimensional lifting of the Laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the structure of eigenvectors and eigenvalues of the complete network in terms of the spectral properties of the individual layers. The spectrum of the supra-Laplacian allows us to understand the physics of diffusionlike processes on top of multiplex networks.

In social settings, individuals interact through webs of relationships. Each
individual is a node in a complex network (or graph) of interdependencies and
generates data, lots of data. We label the data by its source, or formally
stated, we index the data by the nodes of the graph. The resulting signals
(data indexed by the nodes) are far removed from time or image signals indexed
by well ordered time samples or pixels. DSP, discrete signal processing,
provides a comprehensive, elegant, and efficient methodology to describe,
represent, transform, analyze, process, or synthesize these well ordered time
or image signals. This paper extends to signals on graphs DSP and its basic
tenets, including filters, convolution, z-transform, impulse response, spectral
representation, Fourier transform, frequency response, and illustrates DSP on
graphs by classifying blogs, linear predicting and compressing data from
irregularly located weather stations, or predicting behavior of customers of a
mobile service provider.

The smart grid of the future, while expected to affect all areas of the electric power system, from generation, to transmission, to distribution, cannot function without an extensive data communication system. Smart grid has the potential to support high levels of distributed generation (DG); however the current standards governing the interconnection of DG do not allow the implementation of several applications which may be beneficial to the grid. This paper discusses some of the smart grid applications, and estimates the communication requirements of a medium data intensive smart grid device. Two issues that will become very important with the spread of DG are DG Islanding and DG Availability. For each issue, we propose data communication enabled solutions and enhancements.

We investigate the consequence of failures, occurring on the electrical grid, on a telecommunication network. We have focused on the Italian electrical transmission network and the backbone of the internet network for research (GARR). Electrical network has been simulated using the DC power flow method; data traffic on GARR by a model of the TCP/IP basic features. The status of GARR nodes has been related to the power level of the (geographically) neighbouring electrical nodes (if the power level of a node is lower than a threshold, all communication nodes depending on it are switched off). The electrical network has been perturbed by lines removal: the consequent re-dispatching reduces the power level in all nodes. This reduces the number of active GARR nodes and, thus, its Quality of Service (QoS). Averaging over many configurations of perturbed electrical network, we have correlated the degradation of the electrical network with that of the communication network. Results point to a sizeable amplification of the effects of faults on the electrical network on the communication network, also in the case of a moderate coupling between the two networks.

Modern network-like systems are usually coupled in such a way that
failures in one network can affect the entire system. In
infrastructures, biology, sociology, and economy, systems are
interconnected and events taking place in one system can propagate to
any other coupled system. Recent studies on such coupled systems show
that the coupling increases their vulnerability to random failure.
Properties for interdependent networks differ significantly from those
of single-network systems. In this article, these results are reviewed
and the main properties discussed.

Complex networks have been studied intensively for a decade, but research still focuses on the limited case of a single, non-interacting network. Modern systems are coupled together and therefore should be modelled as interdependent networks. A fundamental property of interdependent networks is that failure of nodes in one network may lead to failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. A dramatic real-world example of a cascade of failures ('concurrent malfunction') is the electrical blackout that affected much of Italy on 28 September 2003: the shutdown of power stations directly led to the failure of nodes in the Internet communication network, which in turn caused further breakdown of power stations. Here we develop a framework for understanding the robustness of interacting networks subject to such cascading failures. We present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Surprisingly, a broader degree distribution increases the vulnerability of interdependent networks to random failure, which is opposite to how a single network behaves. Our findings highlight the need to consider interdependent network properties in designing robust networks.

We consider the joint optimization of sensor placement and transmission structure for data gathering, where a given number of nodes need to be placed in a field such that the sensed data can be reconstructed at a sink within specified distortion bounds while minimizing the energy consumed for communication. We assume that the nodes use joint entropy coding based on explicit communication between sensor nodes, and consider both maximum and average distortion bounds. The optimization is complex since it involves an interplay between the spaces of possible transmission structures given radio reachability limitations, and feasible placements satisfying distortion bounds. We address this problem by first looking at the simplified problem of optimal placement in the one-dimensional case. An analytical solution is derived for the case when there is a simple aggregation scheme, and numerical results are provided for the cases when joint entropy encoding is used. We use the insight from our 1-D analysis to extend our results to the 2-D case, and show that our algorithm for two-dimensional placement and transmission structure provides significant power benefit over a commonly used combination of uniformly random placement and shortest path trees. Categories and Subject Descriptors:

Reducing the complexity of large systems described as complex networks is key to understanding them and a crucial issue is to know which properties of the initial system are preserved in the reduced one. Here we use random walks to design a coarse graining scheme for complex networks. By construction the coarse graining preserves the slow modes of the walk, while reducing significantly the size and the complexity of the network. In this sense our coarse graining allows us to approximate large networks by smaller ones, keeping most of their relevant spectral properties.

The notion that our nation's critical infrastructures are highly
interconnected and mutually dependent in complex ways, both physically
and through a host of information and communications technologies
(so-called "cyberbased systems"), is more than an abstract, theoretical
concept. As shown by the 1998 failure of the Galaxy 4 telecommunications
satellite, the prolonged power crisis in California, and many other
recent infrastructure disruptions, what happens to one infrastructure
can directly and indirectly affect other infrastructures, impact large
geographic regions and send ripples throughout the national a global
economy. This article presents a conceptual framework for addressing
infrastructure interdependencies that could serve as the basis for
further understanding and scholarship in this important area. We use
this framework to explore the challenges and complexities of
interdependency. We set the stage for this discussion by explicitly
defining the terms infrastructure, infrastructure dependencies, and
infrastructure interdependencies and introducing the fundamental concept
of infrastructures as complex adaptive systems. We then focus on the
interrelated factors and system conditions that collectively define the
six dimensions. Finally, we discuss some of the research challenges
involved in developing, applying, and validating modeling and simulation
methodologies and tools for infrastructure interdependency
analysis

Determining a set of “important” nodes in a network constitutes a basic endeavor in network science. Inspired by electrical flows in a resistor network, we propose the best conducting node j in a graph G as the minimizer of the diagonal element Qjj† of the pseudoinverse matrix Q† of the weighted Laplacian matrix of the graph G. We propose a new graph metric that complements the effective graph resistance RG and that specifies the heterogeneity of the nodal spreading capacity in a graph. Various formulas and bounds for the diagonal element Qjj† are presented. Finally, we compute the pseudoinverse matrix of the Laplacian of star, path, and cycle graphs and derive an expansion and lower bound of the effective graph resistance RG based on the complement of the graph G.

Various real-world networks interact with and depend on each other. The design of the interconnection between interacting networks is one of the main challenges to achieve a robust interdependent network. Due to cost considerations, network providers are inclined to interconnect nodes that are geographically close. Accordingly, we propose two topologies, the random geographic graph and the relative neighborhood graph, for the design of interconnection in interdependent networks that incorporates the geographic location of nodes. Differing from the one-to-one interconnection studied in the literature, one node in one network can depend on an arbitrary number of nodes in the other network. We derive the average number of interdependent links for the two topologies, which enables their comparison. For the two topologies, we evaluate the impact of the interconnection structure on the robustness of interdependent networks against cascading failures. The two topologies are assessed on the real-world coupled Italian Internet and the electric transmission network. Finally, we propose the derivative of the largest mutually connected component with respect to the fraction of failed nodes as a robustness metric. This robustness metric quantifies the damage of the network introduced by a small fraction of initial failures well before the critical fraction of failures at which the whole network collapses.

Neuronal oscillations exist across a broad frequency spectrum, and are thought to provide a mechanism of interaction between spatially separated brain regions. Since ongoing mental activity necessitates the simultaneous formation of multiple networks, it seems likely that the brain employs interactions within multiple frequency bands, as well as cross-frequency coupling, to support such networks. Here, we propose a multi-layer network framework that elucidates this pan-spectral picture of network interactions. Our network consists of multiple layers (frequency-band specific networks) that influence each other via inter-layer (cross-frequency) coupling. Applying this model to MEG resting-state data and using envelope correlations as connectivity metric, we demonstrate strong dependency between within layer structure and inter-layer coupling, indicating that networks obtained in different frequency bands do not act as independent entities. More specifically, our results suggest that frequency band specific networks are characterized by a common structure seen across all layers, superimposed by layer specific connectivity, and inter-layer coupling is most strongly associated with this common mode. Finally, using a biophysical model, we demonstrate that there are two regimes of multi-layer network behaviour; one in which different layers are independent and a second in which they operate highly dependent. Results suggest that the healthy human brain operates at the transition point between these regimes, allowing for integration and segregation between layers. Overall, our observations show that a complete picture of global brain network connectivity requires integration of connectivity patterns across the full frequency spectrum.

Multilayer networks have been the subject of intense research during the last few years, as they represent better the interdependent nature of many real world systems. Here, we address the question of describing the three different structural phases in which a multiplex network might exist. We show that each phase can be characterized by the presence of gaps in the spectrum of the supra-Laplacian of the multiplex network. We therefore unveil the existence of different topological scales in the system, whose relation characterizes each phase. Moreover, by capitalizing on the coarse-grained representation that is given in terms of quotient graphs, we explain the mechanisms that produce those gaps as well as their dynamical consequences.

A general two-layer network consists of two networks G1 and G2, whose interconnection pattern is specified by the interconnectivity matrix B. We deduce desirable properties of B from a dynamic process point of view. Many dynamic processes are described by the Laplacian matrix Q. A regular topological structure of the interconnectivity matrix B (constant row and column sum) enables the computation of a nontrivial eigenmode (eigenvector and eigenvalue) of Q. The latter eigenmode is independent from G1 and G2. Such a regularity in B, associated to equitable partitions, suggests design rules for the construction of interconnected networks and is deemed crucial for the interconnected network to show intriguing behavior, as discovered earlier for the special case where B=wI refers to an individual node to node interconnection with interconnection strength w. Extensions to a general m-layer network are also discussed.

We consider a model for the diffusion of epidemics in a population that is partitioned into local communities. In particular, assuming a mean-field approximation, we analyze a continuous-time susceptible-infected-susceptible (SIS) model that has appeared recently in the literature. The probability by which an individual infects individuals in its own community is different from the probability of infecting individuals in other communities. The aim of the model, compared to the standard, nonclustered one, is to provide a compact description for the presence of communities of local infection where the epidemic process is faster compared to the rate at which it spreads across communities. Ultimately, it provides a tool to express the probability of epidemic outbreaks in the form of a metastable infection probability. In the proposed model, the spatial structure of the network is encoded by the adjacency matrix of clusters, i.e., the connections between local communities, and by the vector of the sizes of local communities. Thus, the existence of a nontrivial metastable occupancy probability is determined by an epidemic threshold which depends on the clusters' size and on the intercommunity network structure.

We develop a theoretical framework for the study of epidemic-like social
contagion in large scale social systems. We consider the most general setting
in which different communication platforms or categories form multiplex
networks. Specifically, we propose a contact-based information spreading model,
and show that the critical point of the multiplex system associated to the
active phase is determined by the layer whose contact probability matrix has
the largest eigenvalue. The framework is applied to a number of different
situations, including a real multiplex system. Finally, we also show that when
the system through which information is disseminating is inherently multiplex,
working with the graph that results from the aggregation of the different
layers is flawed.

Civil infrastructures are vital elements of a nation's economy and quality of life. They represent a massive capital investment, and, at the same time, are an economic engine of enormous power. Modern economies rely on the ability to move goods, people, and information safely and reliably. Consequently, it is of the utmost importance to government, business, and the public at-large that the flow of services provided by a nation's infrastructure continues unimpeded in the face of a broad range of natural and man-made hazards. In a continuous search for increased efficiency, our way of life is increasingly dependent on tightly coupled, highly sophisticated networks of transportation, electric power, and telecommunications systems from which essentially all redundancy has been removed. These systems become vulnerable to failure simply through their inherent complexity – and although failure may be predictable – its mode and mechanisms are not. The terrorist attacks of September 11 provided ample and horrific evidence of a previously unimaginable complex system failure. Passenger throughput and airport security, seemingly unrelated to structural performance, were critical in producing the most devastating building collapse in history. Therefore, from a comprehensive hazard mitigation standpoint, it is necessary to look beyond the first-order effects of an event and instead seek to understand the perturbed behaviours of a complex, ''system of systems''. Making these systems inherently more resilient and reliable will require more than just improved engineering and technology – complex socio-technological systems also have critical human and institutional elements that need to be integrated into design and operational procedures.

Seminal paper on algebraic connectivity of a network

The second smallest eigenvalue of the Laplacian matrix, also known as the algebraic connectivity, is a remarkable measure to unfold the robustness of complex networks. In this paper we study the asymptotic behavior of the algebraic connectivity in the Erd˝os-Rényi random graph, the simplest model to describe a complex network. We estimate analytically the mean and the variance of the algebraic connectivity by approximating it with the minimum nodal degree. The resulting estimate improves a known expression for the asymptotic behavior of the algebraic connectivity (19). Simulations emphasize the accuracy of the analytical estimation. Further, we study the algebraic connectivity in relation to the graph's robustness to node and link failures, i.e. the number of nodes and links that have to be removed in order to disconnect a graph, called the node and the link connectivity. Extensive simulations show that the node, the edge connectivity and the minimal nodal degree converge to an identical distribution already at small graph sizes.

A study of failures in interconnected networks highlights the vulnerability of tightly coupled infrastructures and shows the need to consider mutually dependent network properties in designing resilient systems.

We study synchronization in an array of coupled nonlinear systems with delay and nonreciprocal time-varying coupling and present synchronization criteria which generalize previous synchronization results. We show that the array synchronizes when the nondelay coupling term is cooperative and large enough. Furthermore, we show that the synchronization criteria are related to several matrix quantities describing the coupling topology. These quantities can be considered as generalizations of the concept of algebraic connectivity to directed graphs.

- F. Harary

Robustness of interdependent networks: The case of communication networks and the power grid

- M Parandehgheibi
- E Modiano

M. Parandehgheibi and E. Modiano, Robustness of interdependent networks: The case of communication networks and the
power grid, in 2013 IEEE Global Communications Conference
(GLOBECOM) (IEEE, Atlanta, 2013), pp. 2164-2169.