In recent years, the advent of new sensor technologies and social network infrastructure has provided huge opportunities and challenges for analyzing data recorded on such networks. For analyzing data recorded on regular lattices, computational harmonic analysis tools such as the Fourier and wavelet transforms have well-developed theories and proven track records of success. It is therefore quite important to extend such tools from the classical setting of regular lattices to the more general setting of graphs and networks. In this article, we first review basics of Laplacian matrices of a graph whose eigenpairs are often interpreted as the frequencies and the Fourier basis vectors on a given graph. We point out, however, that such an interpretation is misleading unless the underlying graph is unweighted path or cycle. We then discuss our recent effort of constructing multiscale basis dictionaries on a graph including the Hierarchical Graph Laplacian Eigenbasis Dictionary and the Generalized Haar-Walsh Wavelet Packet Dictionary, which are viewed as the generalization of the classical hierarchical block DCTs and the Haar-Walsh wavelet packets for the graph setting.

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Periodical cicadas are known for their perfectly synchronized prime number emergences of either 13 or 17 year period. One of th eexplanations for the prime periodicity is that the prime periods are selected to prevent cicadas from resonating with predators with submultiple periods. This paper considers this hypothesis by investigating a population model for periodical predator and prey. The study shows that if the periods of the two periodical species are not coprime, then the predator cannot resist the invasion of the prey. On the other hand, if the periods are coprime, then the predator can eradicate the prey. These results suggest that prime number periods are disadvantageous under periodic predation pressure. Numerical investigations illustrate a non-coprime case where both populations coexist.

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