Distributed optimisation and control of graph Laplacian eigenvalues for robust consensus via an adaptive multi-layer strategy

Functions of eigenvalues of the graph Laplacian matrix L, especially the extremal non-trivial eigenvalues, the algebraic connectivity λ2 and the spectral radius λn, 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 in the simple linear consensus protocol [1]. 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 multi-layer strategy, using three layers of successive distributed estimation. Adequate time-scale 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.