14,436 research outputs found
Exact Coupling Threshold for Structural Transition in Interconnected Networks
Interconnected networks are mathematical representation of systems where two
or more simple networks are coupled to each other. Depending on the coupling
weight between the two components, the interconnected network can function in
two regimes: one where the two networks are structurally distinguishable, and
one where they are not. The coupling threshold--denoting this structural
transition--is one of the most crucial concepts in interconnected networks.
Yet, current information about the coupling threshold is limited. This letter
presents an analytical expression for the exact value of the coupling threshold
and outlines network interrelation implications
Structural transition in interdependent networks with regular interconnections
Networks are often made up of several layers that exhibit diverse degrees of
interdependencies. A multilayer interdependent network consists of a set of
graphs that are interconnected through a weighted interconnection matrix , where the weight of each inter-graph link is a non-negative real number . Various dynamical processes, such as synchronization, cascading failures
in power grids, and diffusion processes, are described by the Laplacian matrix
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 , being the identity matrix, it has
been shown that there exists a structural transition at some critical coupling,
. This transition is such that dynamical processes are separated into
two regimes: if , the network acts as a whole; whereas when , the network operates as if the graphs encoding the layers were isolated. In
this paper, we extend and generalize the structural transition threshold to a regular interconnection matrix (constant row and column sum).
Specifically, we provide upper and lower bounds for the transition threshold in interdependent networks with a regular interconnection matrix
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 in terms of the minimum cut and show, through
a counter-example, 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 constrains imposed by multiplex networks.Comment: 13 pages, APS format. Submitted for publicatio
Towards real-world complexity: an introduction to multiplex networks
Many real-world complex systems are best modeled by multiplex networks of
interacting network layers. The multiplex network study is one of the newest
and hottest themes in the statistical physics of complex networks. Pioneering
studies have proven that the multiplexity has broad impact on the system's
structure and function. In this Colloquium paper, we present an organized
review of the growing body of current literature on multiplex networks by
categorizing existing studies broadly according to the type of layer coupling
in the problem. Major recent advances in the field are surveyed and some
outstanding open challenges and future perspectives will be proposed.Comment: 20 pages, 10 figure
Layer degradation triggers an abrupt structural transition in multiplex networks
Network robustness is a central point in network science, both from a
theoretical and a practical point of view. In this paper, we show that layer
degradation, understood as the continuous or discrete loss of links' weight,
triggers a structural transition revealed by an abrupt change in the algebraic
connectivity of the graph. Unlike traditional single layer networks, multiplex
networks exist in two phases, one in which the system is protected from link
failures in some of its layers and one in which all the system senses the
failure happening in one single layer. We also give the exact critical value of
the weight of the intra-layer links at which the transition occurs for
continuous layer degradation and its relation to the value of the coupling
between layers. This relation allows us to reveal the connection between the
transition observed under layer degradation and the one observed under the
variation of the coupling between layers.Comment: 8 pages, and 8 figures in Revtex style. Submitted for publicatio
Disease Localization in Multilayer Networks
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 SIS and SIR dynamics, as well as upper and lower bounds for
the disease prevalence in the steady state for the SIS scenario. Using the
quasi-stationary 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. 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 verified the barrier effect, i.e., for
three-layer configuration, when the layer with the largest 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.Comment: Revised version. 25 pages and 18 figure
Multiple structural transitions in interacting networks
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
Layer-switching cost and optimality in information spreading on multiplex networks
We study a model of information spreading on multiplex networks, in which
agents interact through multiple interaction channels (layers), say online vs.\
offline communication layers, subject to layer-switching cost for transmissions
across different interaction layers. The model is characterized by the
layer-wise path-dependent transmissibility over a contact, that is dynamically
determined dependently on both incoming and outgoing transmission layers. We
formulate an analytical framework to deal with such path-dependent
transmissibility and demonstrate the nontrivial interplay between the
multiplexity and spreading dynamics, including optimality. It is shown that the
epidemic threshold and prevalence respond to the layer-switching cost
non-monotonically and that the optimal conditions can change in abrupt
non-analytic ways, depending also on the densities of network layers and the
type of seed infections. Our results elucidate the essential role of
multiplexity that its explicit consideration should be crucial for realistic
modeling and prediction of spreading phenomena on multiplex social networks in
an era of ever-diversifying social interaction layers.Comment: 15 pages, 7 figure
Multilayer Networks in a Nutshell
Complex systems are characterized by many interacting units that give rise to
emergent behavior. A particularly advantageous way to study these systems is
through the analysis of the networks that encode the interactions among the
system's constituents. During the last two decades, network science has
provided many insights in natural, social, biological and technological
systems. However, real systems are more often than not interconnected, with
many interdependencies that are not properly captured by single layer networks.
To account for this source of complexity, a more general framework, in which
different networks evolve or interact with each other, is needed. These are
known as multilayer networks. Here we provide an overview of the basic
methodology used to describe multilayer systems as well as of some
representative dynamical processes that take place on top of them. We round off
the review with a summary of several applications in diverse fields of science.Comment: 16 pages and 3 figures. Submitted for publicatio
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