The methods for generation of descriptors with applications to comparison of complex networks

This thesis concerns exact and inexact graph matching algorithms based on algebraic representsations of graphs, i.e., matrices or vectors. We focus on approximate graph matching with the use of descriptors, understood as pattern vectors associated to graphs. Several graph descriptors are presented, including ones known in literature and new ones created on the basis of various vertex-gathered information. The main issues connected with generation of graph pattern vectors are discussed, such as invariance under isomorphism and stability under slight perturbations in graph structure. We concentrate on algebraic descriptors, that is descriptors obtained from linear-algebra transformations of graph matrices, e.g., eigendecomposition or singular value decomposition. We also present graph pattern vectors generated on the basis of vertex relative importance measures such as PageRank or Hubs and Authorities algorithm. The methods for graph isomorphism, using vertex signatures to find correspondences between vertices of two graphs, are also described. The quality of graph matching is tested on artificial and real-world data. In experiments on artificial graphs we use graph database which consists of several families of directed graphs of different structure. In real-world data tests the groups of photos are transformed into graphs with the use of Harris corner detection and Delaunay triangulation. The new pattern vectors introduced in this thesis were also described in the article presented on Signal Processing Symposium, which was held on 24th-26th May 2007 at Warsaw University of Technology in Jachranka Village. This symposium was a part of the 20th IEEE-SPIE Symposium Photonics, Electronics and Web Engineering. The article is attached to this work. The rest of this work is organized as follows. First chapter presents an introduction with a short overview of graph matching methods and discussion of crucial issues connected with graph descriptors generation. In the chapter 2 we describe several matrix graph representations, being basis for various algebraic transformations and generation of descriptors. Later, in chapter 3 the key concepts of spectral graph theory are shown. This chapter aims to describe connections between structural properties of graphs and algebraic properties of their matrix representations. In the next part of this thesis (chapter 4) PageRank random walks model and its application for exact and approximate matching are described. Next we concentrate on graph descriptors based on spectral graph theory, including these obtained with eigendecomposition (chapter 5) and singular value decomposition (chapter 6). Afterwards descriptors based on vertex relative importance measures are presented (chapter 7). Finally, we show results of experiments on real world data (chapter 8) and make some conclusions (chapter 9). Lifelessness is merely an appearance, behind which unfamiliar forms of life lie concealed . . . Bruno Schulz