Clustering-Based Model Reduction of Laplacian Dynamics With Weakly Connected Topology

Abstract

This article studies the structure-preserving model reduction of Laplacian dynamics, which represent weakly connected directed networks with diffusive couplings. The notion of clusterability is introduced to guarantee a bounded reduction error, and a clustering algorithm is then proposed to partition the nodes into clusters, such that the nodes in each cluster form a connected subgraph of the original network. Then, a reduced-order model, which is established using the generalized balanced form of the original network, preserves the weakly connection structure and consensus property. Finally, the effectiveness of the proposed approach is illustrated by a numerical example.

Model Reduction Methods for Complex Network Systems

“…Dissimilarity-Based Clustering. For generic network systems, we may resort to a dissimilarity-based clustering approach presented in e.g., (69,29,28,62). In line with data classification or pattern recognition in the other domains, dissimilarity-based clustering for dynamic networks starts with a proper metric that quantifies the difference between any pair of nodes (subsystems) in a network.…”

“…Then, clusterability is defined between two nodes i, j if they satisfy Equation 18. Clusterability of all nodes in each cluster then guarantees the stability of the error η(s) − η(s) (62).…”

11 Reduced-order modeling of large-scale network systems

“…For linear time-variant networks, nodal dissimilarity can be always defined as the transfer from the external inputs to the vertex states. This mechanism of dissimilarity-based clustering is applicable to different types of dynamical networks; see, e. g., [12,19,19,17] for more results on second-order networks, directed networks, and controlled power networks. For nonlinear networks, DC gain, a function of input amplitude, can be considered [48], in which model reduction aggregates state variables having similar DC gains.…”

“…Then clustering algorithms, e. g., hierarchical clustering and K-means clustering, can be adapted to group nodes in such a way that nodes in the same cluster are more similar to each other than to those in other clusters [12,63]. Subsequent research in [12,22,17,19] shows that the dissimilarity-based clustering method can also be extended to second-order networks, controlled power networks, and directed networks. In [25,24], a framework is presented on how to build a reduced-order model from a given clustering.…”

Coarse-graining and reconstruction for Markov matrices

“…Implicitly, the involved operators (maybe in a modified form) have been used in literature (see e.g. [ChS20,PMK06]) and we recall technical results from [MiS20] in Section 2. However, to the author's knowledge no structural study has been done so far.…”