Analysis of Network Ensembles

Abstract

Subject of this dissertation is the assessment of graph similarity. The application context and ultimate aim is the analysis of network ensembles, i.e. collections of networks, in the sense of identifying structure among them, e.g. groups of highly similar networks. Structure is in this context understood as some form of regularity or description of the similarities among the considered networks.As an illustration, consider a collection of two types of networks, where networks of the same type are very similar, while networks of different types are very dissimilar. These two groups form some kind of similarity that is of interest when the ensemble is the object to be analyzed.Consequently, graphs are in this situation the elementary entities and the main interest is the measurement of structural similarities between them.The interest in graphs as opposed to e.g. vectors as basic objects is motivated by their descriptive capabilities: some objects, e.g. electric circuits, social networks, comprehend important structural properties that can be expressed directly by modeling them as graphs. They have also found to be a powerful description mechanism for objects that do not incorporate an obvious relational structure as for example in image recognition.Using graphs to describe objects leads to sets or collections of graphs on which problems of supervised and unsupervised learning are to be solved. A fundamental prerequisite in such approaches is the ability to compare the elementary objects, i.e. assess similarity or dissimilarity between them. For a number supervised and unsupervised learning algorithms a similarity or distance on the objects of analysis is even the sole prerequisite for their application, a prominent example given by support vector machines (c.f.Vapnik(1998)). Motivated by these considerations, three approaches for assessing and measuring similarity between graphs are developed.