Maximizing algebraic connectivity in interconnected networks

Abstract

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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