Moment-Based Analysis of Synchronization in Small-World Networks of Oscillators
📝 Abstract
In this paper, we investigate synchronization in a small-world network of coupled nonlinear oscillators. This network is constructed by introducing random shortcuts in a nearest-neighbors ring. The local stability of the synchronous state is closely related with the support of the eigenvalue distribution of the Laplacian matrix of the network. We introduce, for the first time, analytical expressions for the first three moments of the eigenvalue distribution of the Laplacian matrix as a function of the probability of shortcuts and the connectivity of the underlying nearest-neighbor coupled ring. We apply these expressions to estimate the spectral support of the Laplacian matrix in order to predict synchronization in small-world networks. We verify the efficiency of our predictions with numerical simulations.
💡 Analysis
In this paper, we investigate synchronization in a small-world network of coupled nonlinear oscillators. This network is constructed by introducing random shortcuts in a nearest-neighbors ring. The local stability of the synchronous state is closely related with the support of the eigenvalue distribution of the Laplacian matrix of the network. We introduce, for the first time, analytical expressions for the first three moments of the eigenvalue distribution of the Laplacian matrix as a function of the probability of shortcuts and the connectivity of the underlying nearest-neighbor coupled ring. We apply these expressions to estimate the spectral support of the Laplacian matrix in order to predict synchronization in small-world networks. We verify the efficiency of our predictions with numerical simulations.
📄 Content
In recent years, systems of dynamical nodes interconnected through a complex network have attracted a good deal of attention [20]. Biological and chemical networks, neural networks, social and economic networks [9], the power grid, the Internet and the World Wide Web [8] are examples of the wide range of applications that motivate this interest (see also [15], [4] and references therein). Several modeling approaches can be found in the literature [8], [22], [1]. In this paper, we focus our attention on the so-called smallworld phenomenon and a model proposed by Newman and Strogatz to replicate this phenomenon.
Once the network is modeled, one is usually interested in two types of problems. The first involves structural properties of the model. The second involves the performance of dynamical processes run on those networks. In the latter direction, the performance of random walks [12], Markov processes [6], gossip algorithms [5], consensus in a network of agents [16], [10], or synchronization of oscillators [21], [17], are very well reported in the literature. These dynamical processes are mostly studied in the traditional context of deterministic networks of relatively small size and/or regular structure. Even though many noteworthy results have been achieved for large-scale probabilistic networks [13]- [2], there is substantial reliance on numerical simulations.
The eigenvalue spectrum of an undirected graph contains a great deal of information about structural and dynamical properties [7]. In particular, we focus our attention on the spectrum of the (combinatorial) Laplacian matrix uniquely associated with an undirected graph [3]. This spectrum {preciado,jadbabai}@seas.upenn.edu contains useful information about, for example, the number of spanning trees, or the stability of synchronization of a network of oscillators. We analyze the low-order moments of the Kirchhoff matrix spectrum corresponding to smallworld networks.
The paper is organized as follows. In Section II, we review the master stability function approach. In Section III, we derive closed-form expressions for the low-order moments of the Laplacian eigenvalue distribution associated with a probabilistic small-world network. Our expressions are valid for networks of asymptotically large size. Section IV applies our results to the problem of synchronization of a probabilistic small-world network of oscillators. The numerical results in this section corroborate our predictions.
In this section we review the master-stability-function (MSF) approach, proposed by Pecora and Carrol in [17], to study local stability of synchronization in networks of nonlinear oscillators. Using this approach, we reduce the problem of studying local stability of synchronization to the algebraic problem of studying the spectral support of the Laplacian matrix of the network. First, we introduce some needed graph-theoretical background.
In the case of a network with symmetrical connections, undirected graphs provide a proper description of the network topology. An undirected graph G consists of a set of N nodes or vertices, denoted by V = {v 1 , …, v n }, and a set of edges E, where E ∈ V × V . In our case, (v i , v j ) ∈ E implies (v j , v i ) ∈ E, and this pair corresponds to a single edge with no direction; the vertices v i and v j are called adjacent vertices (denoted by v i ∼ v j ) and are incident to the edge (v i , v j ). We only consider simple graphs (i.e., undirected graphs that have no self-loops, so v i = v j for an edge (v i , v j ), and no more than one edge between any two different vertices). A walk on G of length k from v 0 to v k is an ordered set of vertices
The degree d i of a vertex v i is the number of edges incident to it. The degree sequence of G is the list of degrees, usually given in non-increasing order. The clustering coefficient, introduced in [22], is a measure of the number of triangles in a given graph, where a triangle is defined by the set of edges {(i, j) , ( j, k) , (k, i)} such that i ∼ j ∼ k ∼ i. Specifically, we define clustering as the total number of triangles in a graph, T (G) , divided by the number of triangles in a complete (allto-all) graph with N vertices, i.e., the coefficient is equal to T (G) N 3 .
There are several choices for such a representation. For example, the adjacency matrix of an undirected graph G, denoted by A(G) = [a i j ], is defined entry-wise by a i j = 1 if nodes i and j are adjacent, and a i j = 0 otherwise. (Note that a ii = 0 for simple graphs.) Notice also that the degree d i can be written as d i = ∑ N j=1 a i j . We can arrange the degrees on the diagonal of a diagonal matrix to yield the degree matrix, D = diag (d i ). The Laplacian matrix (also called Kirchhoff matrix, or combinatorial Laplacian matrix) is defined in terms of the degree and adjacency matrices as L(G) = D(G) -A(G). For undirected graphs, L(G) is a symmetric positive semidefinite matrix [3]. Consequently, it has a full s
This content is AI-processed based on ArXiv data.