Dynamic Geometric Graph Processes: Adjacency Operator Approach

• DOI: 10.1007/s00006-010-0228-6
• Publication Date: 2010-05-11T10:56:17Z
• Authors: René Schott; G. Stacey Staples
• Keywords: Unit cube; Adjacency list; Geometric graph theory; Unit square; Adjacency matrix; Random geometric graph; Geometric networks

The $d$-dimensional unit cube $[0; 1]^d$ is discretized to create a collection $V$ of vertices used to dene geometric graphs. Each subset of $V$ is uniquely associated with a geometric graph. Dening a dynamic random walk on the subsets of $V$ induces a walk 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 Cliord algebras and are used to recover information about the cycle structure and connected components of the $n$ graph of a sequence.