# Adaptive Graph Partition Methods for Structured Graphs

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## Abstract

Graphs can be models for many real-world systems, where nodes indicate the entities and edges indicate the pairwise connections in between. In various cases, it is important to detect informative subsets of nodes such that the nodes within the subsets are ’closer’ to each other. For example, in a cellular network, determining appropriate node subsets can reduce the operation costs. A subset is usually called a cluster. This leads to the graph clustering problem. Furthermore, plenty of systems in the real world are changing over time, and consequently, graphs as models vary with time as well. It is thus also important to update the clusters when the graph changes.

In this thesis work, we studied two problems from the cellular network background. We needed to partition graphs that have certain structures and cluster their nodes to minimize certain cost functions. In the ﬁrst problem, we partitioned a bipartite graph by minimizing the so-called MinMaxCut cost function, while in the second problem, we partitioned a structured graph by minimizing the so-called Modiﬁed-MinMaxCut cost function. The structural property of the graph is incorporated in deﬁning this new cost function. The solutions we proposed are under the framework of spectral clustering, where one relies on the eigenvectors of the graph matrices, e.g., the Laplacian matrix or the adjacency matrix, and any clustering algorithm, e.g., K-means, to partition nodes into disjoint clusters.

Furthermore, for the time-variant graph, we decomposed the problem into two steps. First, we transformed the variations in the graph topology into perturbations to the graph matrices. Then we transformed the update of the clusters into an update of the (generalized) eigenvectors of these graph matrices. We utilized matrix perturbation theory to update the generalized eigenvectors and then update the clusters. Our simulations showed that on synthetic data, the proposed method can eﬃciently track the eigenvectors and the clusters generated by the updated eigenvectors have almost the same cost function value as that of exact computation.