Hypercube Graph Representations and Fuzzy Measures of Graph Properties

We describe a novel hypercube graph representation for labeled graphs with arbitrary edge weights in the interval [0, 1]. This representation admits graphical models for weighted adjacency matrices, which are useful in a number of real world applications wherein the strength of connections between graph nodes is important. It enables us to bring to bear a full arsenal of fuzzy set theoretic measures such as fuzzy subsethood, entropy, completeness, and mutual subsethood to the description of graphs. Our hypercube representation also provides a direct similarity metric between pairs of graphs, which is particularly useful for external comparisons among sets of graphs. The unitary complement of this similarity metric in turn provides a distance metric between two graphs, thus enabling us to perform vector processing operations on graphs, e.g., clustering, change detection, hypothesis testing as to the independence of two graphs, feature extraction for neural network and/or statistical classifiers, and antecedent specification for fuzzy mappings. We derive the probability mass function of this metric for two independent random graphs. The hypercube graph representation finds applications in problems where we are dealing with labeled graphs, e.g., computer networks, social networks, graphical information retrieval, and data fusion problems involving virtual networks of events. Of special interest are labeled graphs with fixed vertices whose edges and their corresponding weights vary over time, as well as graphs that evolve in time by the addition of new vertices and edges.