Heterogeneity Index for Directed Graphs

Although there are a number of existing measures for quantifying the structural properties of undirected graphs, there are relatively few corresponding measures for directed graphs. To fill this gap in the literature, in this paper, we explore how to extend Estrada’s heterogeneity index from undirected to directed graphs and define an analogous heterogeneity measure for directed graphs. From the perspective of object recognition, this measure opens up the possibility of using directed graphs (such as nearest neighbour graphs) to represent the arrangement of object features. This type of representation is potentially more discriminating than an undirected graph. We show how our new heterogeneity measure can be used to characterize k-nearest neighbour graphs representing the arrangement of object features extracted from objects in the COIL-20 database. We compare the performance of this measure with the original Estrada’s heterogeneity index. Finally we achieve the conclusion that our measure gives a better characterization performance.