Formal definition and entropy calculation of hierarchical attributed random graph

Abstract

Abstract For the representation of a complex object, the object is decomposed into several parts, and it is described by these decomposed parts and their relations. In general, the parts can be the primitive elements that can not be decomposed further, or still can be decomposed into their subparts. Therefore, the hierarchical description method is very natural and the hierarchical description is represented by a hierarchical graph whose vertices represent either primitive elements or graphs. These graohs also have vertices which contain primitive elements or graphs. When some uncertainty exists in the hierarchical description of a complex object either due to noise or minor deformation, a probabilistic description of the object ensemble is necessary. For this purpose, in this paper, we formally define the hierarchical attributed random graph and derive the equations for the entropy calculation of the hierarchical random graph.