Static and dynamic behavior of multiplex networks under interlink strength variation

Abstract

-It has recently been suggested [1] that in a two-level multiplex network, a gradual change in the value of the"interlayer" strength p can provoke an abrupt structural transition. The critical point p * at which this happens is system-dependent. In this article, we show in a similar way as in [2] that this is a consequence of the graph Laplacian formalism used in [1]. We calculate the evolution of p * as a function of system size for ER and RR networks. We investigate the behavior of structural measures and dynamical processes of a two-level system as a function of p, by Monte-Carlo simulations, for simple particle diffusion and for reaction-diffusion systems. We find that as p increases there is a smooth transition from two separate networks to a single one. We cannot find any abrupt change in static or dynamic behavior of the underlying system.Introduction. -In the past several years, single networks have been extensively studied [3-8] both regarding their structure and also regarding different interactions between their nodes. In real world systems, however, there may be more than one type of relationship for the same collection of objects constituting the network. Consider, for example, the communication and power networks in a given geographical region. In this case, a failure in some power station will affect not only the functionality of the power grid, but also the routing system for all computers that uses electrical power to sustain its functionality. Thus, more recently [9], effort has been applied to the so called "interdependent" or "interconnected" networks, meaning a system of two or more networks that are linked together. Several articles have been published investigating the properties of these systems [9,10] as well as the evolution of dynamical processes on them [11,12]. In their simplest form, they consist of two single networks having their nodes connected in a one-to-one configuration with each "interlink" having the same strength p (in the range of 0 < p < ∞). A first question of interest is how does the variation of the parameter p affect the global structural properties of the entire system. To this end, one can make use of the Laplacian matrix. According to graph theory, given a graph of N nodes, the adjacency matrix A is defined as:

The space of genotypes is a network of networks: implications for evolutionary and extinction dynamics

“…In natural populations, this means that extinction can occur as the critical threshold is approached, even if the environment is not changing faster than it did at earlier times 32 , and even if the rate of change diminishes but does not halt. An open question of potential relevance for the complex networks community is the nature of this transition in a network-of-networks context 19 , 58 – 61 , chiefly if it is continuous or discontinuous in the limit of infinite genotype spaces. In case it is a truly critical transition, it would be important to know about the existence of universal exponents, independent of details of the fitness landscape, characterizing for example the time to equilibrium or the maximum genotypic response.…”

Dynamical leaps due to microscopic changes in multilayer networks

“…It has been noted that the existence of the point of inflection p c might follow from linear algebra arguments [34]. At this point it is also not certain whether p c tends to zero in the limit of N → ∞, or whether it is indeed significant in describing the observed changes in the dynamic timescales [35].This work extends the analysis of multiplex timescales to cover the effect of the specific type of inter-layer coupling. Instead of gradually tuning up the intensity of inter-layer links p, we now switch on the inter-layer links at unit intensity one by one.…”

“…It has been noted that the existence of the point of inflection p c might follow from linear algebra arguments [34]. At this point it is also not certain whether p c tends to zero in the limit of N → ∞, or whether it is indeed significant in describing the observed changes in the dynamic timescales [35].…”

Adaptive Finite-Time Mixed Interlayer Synchronization of Two-Layer Complex Networks with Time-Varying Coupling Delay

“…Thus, multiplex networks, proposed by Mucha et al [1], would be more appropriate for describing systems in the real-world than traditional (single-layer) complex networks. In the past few years, many efforts have been made to investigate various problems of multiplex networks, such as topological structure, dynamic behavior [2], synchronizability [3,4], spectral property [5], diffusion process [6], and synchronization [7][8][9][10].…”