and eigenvalues of the Laplacian decays with increasing iteration count. Thus, it is difficult to devise a fast and distributed method for clustering graphs based on the heat equation.

In [@Cit:cluster1; @Cit:cluster2], a novel algorithm based on the idea of permanent excitation of the eigenvectors of $\I-\L$ using dynamical systems theory is constructed. In a theme similar to Mark Kac’s question “Can one hear the shape of a drum?” [@DrumShape], it was demonstrated that by evolving the wave equation in the graph, nodes can “hear” the eigenvectors of the graph Laplacian using only local information. Moreover, it was shown, both theoretically and on examples, that the wave equation based algorithm is orders of magnitude faster than random walk based approaches for graphs with large mixing times. The overall idea of the wave equation based approach is to simulate, in a distributed fashion, the propagation of a wave through the graph and capture the frequencies at which the graph “resonates”. In other words, it was shown that by using these frequencies one can compute the eigenvectors of the Laplacian, thus clustering the graph.

The wave equation based clustering approach can be described as follows. Analogous to the heat equation case (Eq. 