Dynamic Geometric Graph Processes: Adjacency Operator Approach

Abstract

A random geometric graph is constructed by randomly choosing a set of points in the unit cube [0, 1] d and connecting two points by an edge if their Euclidean distance is at most some fixed distance r > 0. Graphs of this type are of particular interest as models of wireless networks. In this paper, the d-dimensional unit cube [0, 1] d is discretized to create a collection V of vertices used to define geometric graphs. With r fixed, dynamic random walks are defined on the subsets of V, resulting in dynamic random walks on the collection of geometric graphs in the discretized cube. These walks naturally model addition-deletion networks and can be visualized as walks on hypercubes with loops. Adjacency operators are constructed using subalgebras of Clifford algebras and are used to recover information about the cycle structure and connected components of graphs in the sequence.