Algebraic connectivity of interdependent networks

Martı́n-Hernández, Javier; Wang, H.; Mieghem, Piet Van; D’Agostino, Gregorio

doi:10.1016/j.physa.2014.02.043

Cited by 64 publications

(75 citation statements)

References 34 publications

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“…[1,−1] and the Fiedler cut distinguishes the layers [13][14][15]. For c > c * , due to the multiplicity of λ 2 (L), there is a corresponding Fiedler eigenspace.…”

Section: If and Only If C Cmentioning

confidence: 99%

“…Sahneh et al compute the exact value analytically [14]. Martin-Hernandez et al analyze the algebraic connectivity and Fiedler vector of multiplex structures, with the addition of a number of interlayer links in two configurations; diagonal (one-to-one) and random [15]. They show that for the first case, algebraic connectivity saturates after adding a sufficient number of links.…”

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confidence: 99%

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Maximizing algebraic connectivity in interconnected networks

Shakeri¹,

Albin²,

Sahneh³

et al. 2016

Phys. Rev. E

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Algebraic connectivity, the second eigenvalue of the Laplacian matrix, is a measure of node and link connectivity on networks. When studying interconnected networks it is useful to consider a multiplex model, where the component networks operate together with interlayer links among them. In order to have a well-connected multilayer structure, it is necessary to optimally design these interlayer links considering realistic constraints. In this work, we solve the problem of finding an optimal weight distribution for one-to-one interlayer links under budget constraint. We show that for the special multiplex configurations with identical layers, the uniform weight distribution is always optimal. On the other hand, when the two layers are arbitrary, increasing the budget reveals the existence of two different regimes. Up to a certain threshold budget, the second eigenvalue of the supra-Laplacian is simple, the optimal weight distribution is uniform, and the Fiedler vector is constant on each layer. Increasing the budget past the threshold, the optimal weight distribution can be nonuniform. The interesting consequence of this result is that there is no need to solve the optimization problem when the available budget is less than the threshold, which can be easily found analytically. DOI: 10.1103/PhysRevE.93.030301Real-world networks are often connected together and therefore influence each other [1]. Robust design of interdependent networks is critical to allow uninterrupted flow of information, power, and goods in spite of possible errors and attacks [2][3][4]. The second eigenvalue of the Laplacian matrix, λ 2 (L), is a good measure of network robustness [5]. Fiedler shows that algebraic connectivity increases by adding links [6]. Moreover, it is harder to bisect a network with higher algebraic connectivity [7]. The second eigenvalue of the Laplacian matrix is also a measure of the speed of mixing for a Markov process on a network [8]. Boyd et al. maximize the mixing rate by assigning optimum link weights in the setting of a single layer (see Refs. [9,10]).For multiplex networks (see Fig. 1), a natural question is the following. Given fixed network layers, how should the weights be assigned to interlayer links in order to maximize algebraic connectivity?The behavior of λ 2 , in the case of identical weights, i.e., with a fixed coupling weight p for every interlayer link, has been studied recently. networks by choosing from a set of interlayer links with predefined weights [16].In this Rapid Communication we remove the constraint of identical interlinking weights and pose the problem of finding the maximum algebraic connectivity for a one-toone interconnected structure between different layers in the presence of limited resources. We show that up to the threshold budget p * N (where p * is the same threshold studied in Refs. [11,13,14]) the uniform distribution of identical weights is actually optimal. For larger budgets, the optimal distribution of weights is generally not uniform.Let G = (V,E) represent a network a...

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“…[1,−1] and the Fiedler cut distinguishes the layers [13][14][15]. For c > c * , due to the multiplicity of λ 2 (L), there is a corresponding Fiedler eigenspace.…”

Section: If and Only If C Cmentioning

confidence: 99%

mentioning

confidence: 99%

Maximizing algebraic connectivity in interconnected networks

Shakeri¹,

Albin²,

Sahneh³

et al. 2016

Phys. Rev. E

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“…Hernandez et al [21] found the complete spectra of interconnected networks with identical components. Sole-Ribalta et al [22] studied the interconnection of more than two networks with an arbitrary oneto-one correspondence structure.…”

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confidence: 99%

Exact coupling threshold for structural transition reveals diversified behaviors in interconnected networks

2015

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An interconnected network features a structural transition between two regimes [F. Radicchi and A. Arenas, Nat. Phys. 9, 717 (2013)]: 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) Fig. 1, where the interconnection strength between the layers is parametrized by a coupling weight p > 0. Radicchi and Arenas [16] demonstrated the existence of a structural transition point p * . Depending on the coupling weight p between the two networks, the collective interconnected network can function in two regimes: if p < p * , the two networks are structurally distinguishable; whereas if p > p * , they behave as a whole.While studying diffusion processes on the same type of interconnected network in Fig. 1, Gomez et al. [14] observed superdiffusion: for sufficiently large p, the diffusion in the interconnected network takes place faster than in either of the networks separately. Superdiffusion arises due to the synergistic effect of the network interconnection and exemplifies a characteristic phenomenon in interconnected networks. Placement of the introduction point of superdiffusion with respect to the critical point p * is missing in the literature.Whereas the existence of a critical transition p * was reported in [16], here, we determine the exact coupling threshold p * . Our exact solution illuminates the role of each individual * Corresponding author: faryad@ksu.edu network component and their combined configuration on the structural transition phenomena and uncovers unexpected behaviors. Specifically, we show structural transition is not a necessary condition for achieving superdiffusion. Indeed, superdiffusion can be achieved for a coupling weight p even below the structural transition threshold p * , which is surprising because, intuitively, synergy is not expected if the network components are functioning distinctly. Moreover, we observe that the structural transition disappears when one of the network components has vanishing algebraic connectivity [18][19][20], as is the case for a class of scalefree networks. Therefore, components of such interconnected network topologies become indistinguishable despite very weak coupling between them.Spectral analysis plays a key role in understanding interconnected networks. Hernandez et al. [21] found the complete spectra of interconnected networks with identical components. studied the interconnection of more than two networks with an arbitrary oneto-one correspondence structure. employed eigenvalue interlacing [18] to provide bounds for the Laplacian spectra of an interconnected network with a general interconnection pattern. In addition, in a similar context of structural transition as [16], D'Agostino [24] showed that adding interconnection links among networks causes st...

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“…Finally, it is informative to look at the quantum entropy of M. S q (M) shows a clear peak after p * and before p (see the inset of Fig. 4), i.e., in the region after the transition observed in [7,14] and before the one we have introduced here. In fact, by studying the sign of the derivative of S q , it can be proven that the quantum entropy must have a peak before p .…”

Section: The Aggregate-equivalent Multiplex Networkmentioning

confidence: 91%

Characterization of multiple topological scales in multiplex networks through supra-Laplacian eigengaps

Cozzo

Moreno

2016

Phys. Rev. E

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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.

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Algebraic connectivity of interdependent networks

Cited by 64 publications

References 34 publications

Maximizing algebraic connectivity in interconnected networks

Maximizing algebraic connectivity in interconnected networks

Exact coupling threshold for structural transition reveals diversified behaviors in interconnected networks

Characterization of multiple topological scales in multiplex networks through supra-Laplacian eigengaps

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