Optimal graph Laplacian

Abstract

This paper provides a construction method of the nearest graph Laplacian to a matrix identified from measurement data of graph Laplacian dynamics that include biochemical, synchronization, and multi-agent systems. We consider the case in which the network structure, i.e., the relationship between edges of a given graph, is known. A problem of finding the nearest graph Laplacian is formulated as a convex optimization problem. Thus, our problem can be solved using interior point methods. Moreover, we can also implement the alternating direction method of multipliers (ADMM) for our problem. However, the complexities of each iteration due to the use of interior point methods and ADMM are larger than O(n6) and O(n3), respectively, where n is the number of nodes of the network. That is, if n is large, interior point methods and ADMM cannot solve our problem within a practical time. To resolve this issue, we propose a simple and efficient algorithm with calculation complexity O(n2). The simulation experiments demonstrate that our method is useful to perform data-driven modeling of graph Laplacian dynamics.