Optimal Graph Laplacian

TL;DR

This paper introduces an efficient method to find the nearest graph Laplacian matrix from measurement data, enabling practical data-driven modeling of complex network dynamics.

Contribution

A novel, computationally efficient algorithm with $O(n^2)$ complexity for constructing the nearest graph Laplacian from data, improving over traditional interior point methods.

Findings

The proposed algorithm accurately approximates the graph Laplacian from data.

Simulation results confirm the method's effectiveness in modeling network dynamics.

The approach is practical for large-scale networks due to its low complexity.

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 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.