Algebraic connectivity of interdependent networks
Introduction
In the last decades, there has been a significant advance in understanding the structure and functioning of complex networks [1], [2]. Statistical models of networks are now widely used to describe a broad range of complex systems, from networks of human contacts to interactions amongst proteins. In particular, emerging phenomena of a population of dynamically interacting units has always fascinated scientists. Dynamic phenomena are ubiquitous in nature and play a key role within various contexts in sociology [3], and technology [4]. To date, the problem of how the structural properties of a network influences the convergence and stability of its synchronized states has been extensively investigated and discussed, both numerically and theoretically [5], [6], [7], [8], [9], with special attention given to networks of coupled oscillators [10], [11], [12], [13].
In the present work, we focus on the second smallest eigenvalue of a graph’s Laplacian matrix, also called algebraic connectivity. This metric plays an important role on, among others, synchronization of coupled oscillators, network robustness, consensus problems, belief propagation, graph partitioning, and distributed filtering in sensor networks[14], [15], [16], [17], [18]. For example, the time it takes to synchronize Kuramoto oscillators upon any network scales with the inverse of [19], [20], [21], [22]. In other words, larger values of enable synchronization in both discrete and continuous-time systems, even in the presence of transmission delays [23], [24]. As a second application, graphs with “small” algebraic connectivity have a relatively clean bisection, i.e. the smaller , the fewer links must be removed to generate a bipartition [25]. Furthermore, we illustrate the role of the algebraic connectivity in the diffusion dynamic process. For the sake of simplicity, we model the diffusive dynamics as a commodity exchange governed by the following differential equation: where represents the commodity or the state of the th component, its neighbors, and the Laplacian matrix, as further defined in Section 2. The equilibrium state is that in which all gradients in (1) reach zero, thus the rate of the slowest exponential decay (of the deviation from the equilibrium) is proportional to the algebraic connectivity [26]. Hence, the higher the algebraic connectivity of the matrix, the smaller the “proper time”.
Despite the latest advances in the research on synchronization and graph spectra, current research methods mostly focus on individual networks treated as isolated systems. In reality, complex systems are seldom isolated. For example, a power grid and a communication network may strongly depend on each other. A power station depends on a communication node for information, whereas a communication node depends on a power station for electricity [20]; similarly, a pathogen may spread from one species to another. Much effort has been devoted to predict cascading effects in such interdependent networks [27], [28], [29]: the largest connected component has been shown to exhibit a spectacular phase transition after a critical number of faults is reached. Quite recently, a novel approach has been introduced by resorting to the spectral analysis of interdependent networks. By means of the graph spectra, the epidemic thresholds of interdependent networks have been estimated, and absolute boundaries have been provided [27]. These scenarios motivated us to study the influence of interdependent networks on diffusive processes via their spectral properties.
In this work, we show analytically and numerically how the algebraic connectivity of interdependent networks experiences a phase transition upon the addition (or removal) of a sufficient number of interlinks between two identical networks. As a direct consequence, the proper time of a diffusion process on top of the NoN system is not affected by interlink additions, as long as the number of interlinks is higher than a critical threshold. The location of the described transition depends on the link addition strategy, as well as on the algebraic connectivity of the single networks. Gomez et al. [30] applied perturbation theory to approximate lower bounds for in a multiplex scenario, for which they conclude that interdependent networks speed up diffusion processes. Although the latter study hints the existence of a sudden shift, their authors over-sighted the existence of a phase transition, which we fully characterize via mean-field theory and by investigating additional spectral properties for different values of . In particular, we observe that the phase transition is reflected in spectral partitioning algorithms, as illustrated in Fig. 1.
This paper is structured as follows. Section 2 introduces some required terminology, the Laplacian matrix, and its corresponding spectra. Sections 3.1 Exact results for mean-field theory, 3.2 Approximating provide some analytical results for the algebraic connectivity of interdependent networks, based on both mean-field approach and perturbation theory, respectively. Our models are able to predict the fraction of links that will cause the algebraic connectivity transition. Finally, Section 4 validates our previous results through extensive numerical results. This section also exposes results on regular, random, small-world, and scale-free networks. Conclusions are drawn in Section 5.
Section snippets
Graph theory basics
A graph is composed by a set of nodes interconnected by a set of links . Suppose one has two networks and , each with a set of nodes and a set of links respectively. For simplicity, in the following we will suppose any dependence relation to be symmetric, i.e. all networks are undirected.
The global system resulting from the connection of the two networks is a network with nodes and “intralinks” plus a number of “interlinks” joining
Analytical results
This section introduces two independent analytical approaches to compute for the interdependent graph setup described in the previous section. The two approaches are based on mean-field theory and perturbation theory, which span Sections 3.1 Exact results for mean-field theory, 3.2 Approximating, respectively. For a small number of added interlinks, both of the proposed theories are in agreement with each other, which validates our analysis.
Simulations
The previous section provided two analytical means to estimate the dependence of on the topology of the interdependence links, namely mean field and perturbation theory. In this section, we will introduce four model networks to test the predictability and the limits of our predictions.
Conclusions
Our contribution beacons a significant starting point to the understanding of diffusion driven dynamics on interdependent networks. Having in mind synchronization applications, this work focuses on the algebraic connectivity of interdependent networks. We provided evidence that upon increasing the number of interlinks between two originally isolated networks, the algebraic connectivity experiences a phase transition. That is, there exists a critical number of diagonal interlinks beyond which
Acknowledgments
This research has been partly supported by the European project MOTIA (Grant JLS-2009-CIPS-AG-C1-016); the EU Research Framework Programme 7 via the CONGAS project (Grant FP7-ICT 317672); and the EU Network of Excellence EINS (Grant FP7-ICT 288021).
References (36)
From kuramoto to crawford: exploring the onset of synchronization in populations of coupled oscillators
Physica D
(2000)- et al.
Droop-controlled inverters are Kuramoto oscillators
- et al.
Synchronization in complex networks
Phys. Rep.
(2008) - et al.
Average consensus in networks of multi-agents with both switching topology and coupling time-delay
Physica A
(2008) - et al.
Emergence of scaling in random networks
Science
(1999) - et al.
Collective dynamics of small world networks
Nature
(1998) The N-intertwined SIS epidemic network model
Computing
(2011)- et al.
A structure preserving model for power system stability analysis, power apparatus and systems
IEEE Transactions on PAS-
(1981) - et al.
Synchronization in small-world systems
Phys. Rev. Lett.
(2002) - et al.
Synchronization of networks with prescribed degree distributions
IEEE Transactions on Circuits and Systems-I
(2006)
Laplacian spectra and synchronization processes on complex networks
Synchronization in scale-free dynamical networks: robustness and fragility, circuits and systems i: fundamental theory and applications
IEEE Transactions on
Master stability functions for synchronized coupled systems
Phys. Rev. Lett.
The Kuramoto model: a simple paradigm for synchronization phenomena
Rev. Modern Phys.
Ultrafast consensus in small-world networks
Cited by (59)
Spectral properties of token graphs
2024, Linear Algebra and Its ApplicationsSpectral clustering methods for multiplex networks
2019, Physica A: Statistical Mechanics and its ApplicationsThe algebraic connectivity of graphs with given circumference
2019, Theoretical Computer ScienceCitation Excerpt :Many applications in the real world are closely related to the algebraic connectivity. It has been observed that the algebraic connectivity plays an important role on synchronization of coupled oscillators, network robustness, consensus problems, belief propagation, and distributed filtering in sensor networks [10,16,22,25,28,29]. As an example, the algebraic connectivity quantifies the speed of convergence of consensus algorithms.
Temperature dependent network stability in simple alcohols and pure water: The evolution of Laplace spectra
2019, Journal of Molecular LiquidsCitation Excerpt :Here, we wished to minimize possible size effects by simulating larger systems, containing thousands of molecules (see Section 2.2). It has already been shown that the presence of near-zero eigenvalues generally indicates the existence of strong communities, or nearly disconnected components [10,14]. The so-called Fiedler eigenvalue has been extensively studied in diverse application areas, including biology, neuron science, spectral clustering technique, etc. … [18–22,43–45].
On algebraic connectivity of directed scale-free networks
2018, Journal of the Franklin InstituteCitation Excerpt :Analyzing the spectral properties of complex networks has attracted much attention, in particular the second smallest eigenvalue of the associated Laplacian matrices [7,10–16]. This special eigenvalue is referred to as the algebraic connectivity of networks [17–20], and is known as an important measure for the diffusion speed of many diffusion processes over networks (e.g. consensus, synchronization, information/innovation spreading, epidemics) [15,19]. In [10] it was demonstrated that undirected small-world networks have much higher algebraic connectivity than regular networks.
Comparing multilayer brain networks between groups: Introducing graph metrics and recommendations
2018, NeuroImageCitation Excerpt :Multilayer networks can show non-trivial properties that are not merely the result of the sum of its layers (Kivelä et al., 2014; Nicosia et al., 2013; Sahneh et al., 2015). This approach has been applied effectively to several networks, such as social networks, transportation networks, and synthetic networks, demonstrating that empirical systems can be better understood when the influence of interacting networks are considered (De Domenico et al., 2013; Granell et al., 2013; Hernández et al., 2014). Multilayer network approaches have recently been introduced to the field of neuroscience (Brookes et al., 2016; Buldú and Porter, 2017; Crofts et al., 2016; De Domenico et al., 2016; Tewarie et al., 2016b; Yu et al., 2017), where different layers can correspond to different frequency-band specific networks or networks from different modalities.