Isospectral Reductions of Dynamical Networks

We present a general and flexible procedure which allows for the reduction (or expansion) of any dynamical network while preserving the spectrum of the network’s adjacency matrix. Computationally, this process is simple and easily implemented for the analysis of any network. Moreover, it is possible to isospectrally reduce a network with respect to any network characteristic including centrality, betweenness, etc. This procedure also establishes new equivalence relations which partition all dynamical networks into spectrally equivalent classes. Here, we present general facts regarding isospectral network transformations which we then demonstrate in simple examples. Overall, our procedure introduces new possibilities for the analysis of networks in ways that are easily visualized.

💡 Research Summary

The paper introduces a mathematically rigorous yet computationally straightforward framework for reducing (or expanding) dynamical networks while exactly preserving the spectrum of their adjacency matrices. The core idea is to partition the vertex set V into a chosen subset S (the “reduction set”) and its complement R = V \ S, then to eliminate the vertices in S by replacing their influence on R with a rational function of the spectral parameter λ. Concretely, the adjacency matrix A is written in block form

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