Optimal Graph Laplacian

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 systems, synchronization systems, and multi-agent systems. We consider the case where the network structure, i.e., the connection relationship of 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. However, the complexity of each iteration by interior point methods is $O(n^6)$, where $n$ is the number of nodes of the network. That is, if $n$ is large, interior point methods cannot solve our problem within a practical time. To resolve this issue, we propose a simple and efficient algorithm with the calculation complexity $O(n^2)$. Simulation experiments demonstrate that our method is useful to perform data-driven modeling of graph Laplacian dynamics.