We show that a network can self-organize its structure in a completely distributed manner in order to optimize its synchronizability whilst satisfying the local constraints: non-negativity of edge weights, and maximum weighted degree of nodes. A novel multilayer approach is presented which uses a distributed strategy to estimate two spectral functions of the graph Laplacian, the algebraic connectivity λ2 and the eigenratio r = λn/λ2. These local estimates are then used to evolve the edge weights so as to maximize λ2, or minimize r and, hence, achieve an optimal structure.

SUMMARYFunctions of eigenvalues of the graph Laplacian matrix L, especially the extremal non-trivial eigenvalues, the algebraic connectivity 2 and the spectral radius n , have been shown to be important in determining the performance in a host of consensus and synchronisation applications. In this paper, we focus on formulating an entirely distributed control law for the control of edge weights in an undirected graph to solve a constrained optimisation problem involving these extremal eigenvalues.As an objective for the distributed control law, edge weights must be found that minimise the spectral radius of the graph Laplacian, thereby maximising the robustness of the network to time delays under a simple linear consensus protocol. To constrain the problem, we use both local weight constraints that weights must be non-negative, and a global connectivity constraint, maintaining a designated minimum algebraic connectivity. This ensures that the network remains sufficiently well connected.The distributed control law is formulated as a multilayer strategy, using three layers of successive distributed estimation. Adequate timescale separation between the layers is of paramount importance for the proper functioning of the system, and we derive conditions under which the distributed system converges as we would expect for the centralised control or optimisation system to converge.

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms Distributed Adaptive Optimization and Control of Network StructuresLouis Kempton, Guido Herrmann and Mario di Bernardo Abstract-In this paper we present a generic distributed weight adaptation framework to optimize some network observables of interest. We focus on the algebraic connectivity λ2, the spectral radius λn, the synchronizability λn/λ2, or the total effective graph resistance Ω of undirected weighted networks, and describe distributed systems for the estimation of these functions and their derivatives for on-line adaptation of the edge weights.