In this paper we present a method for constructing a generative prototype for a set of graphs by adopting a minimum description length approach. The method is posed in terms of learning a generative supergraph model from which the new samples can be obtained by an appropriate sampling mechanism. We commence by constructing a probability distribution for the occurrence of nodes and edges over the supergraph. We encode the complexity of the supergraph using an approximate Von Neumann entropy. A variant of the EM algorithm is developed to minimize the description length criterion in which the structure of the supergraph and the node correspondences between the sample graphs and the supergraph are treated as missing data. To generate new graphs, we assume that the nodes and edges of graphs arise under independent Bernoulli distributions and sample new graphs according to their node and edge occurrence probabilities. Empirical evaluations on real-world databases demonstrate the practical utility of the proposed algorithm and show the effectiveness of the generative model for the tasks of graph classification, graph clustering and generating new sample graphs.

Riesen and Bunke recently proposed a novel dissimilarity based approach for embedding graphs into a vector space. One drawback of their approach is the computational cost graph edit operations required to compute the dissimilarity for graphs. In this paper we explore whether the Jensen-Shannon divergence can be used as a means of computing a fast similarity measure between a pair of graphs. We commence by computing the Shannon entropy of a graph associated with a steady state random walk. We establish a family of prototype graphs by using an information theoretic approach to construct generative graph prototypes. With the required graph entropies and a family of prototype graphs to hand, the Jensen-Shannon divergence between a sample graph and a prototype graph can be computed. It is defined as the Jensen-Shannon between the pair of separate graphs and a composite structure formed by the pair of graphs. The required entropies of the graphs can be efficiently computed, the proposed graph embedding using the Jensen-Shannon divergence avoids the burdensome graph edit operation. We explore our approach on several graph datasets abstracted from computer vision and bioinformatics databases.

In this chapter, we compare and contrast two approaches to the problem of embedding non-Euclidean data, namely geometric and structure preserving embedding. Under the first heading, we explore how spherical embedding can be used to embed data onto the surface of sphere of optimal radius. Here we explore both elliptic and hyperbolic geometries, i. e., positive and negative curvatures. Our results on synthetic and real data show that the elliptic embedding performs well under noisy conditions and can deliver low-distortion embeddings for a wide variety of datasets. Hyperbolic data seems to be much less common (at least in our datasets) and is more difficult to accurately embed. Under the second heading, we show how the Ihara zeta function can be used to embed hypergraphs in a manner which reflects their underlying relational structure. Specifically, we show how a polynomial characterization derived from the Ihara zeta function leads to an embedding which captures the prime cycle structure of the hypergraphs.

In this paper we propose a prototype size selection method for a set of sample graphs. Our first contribution is to show how approximate set coding can be extended from the vector to graph domain. With this framework to hand we show how prototype selection can be posed as optimizing the mutual information between two partitioned sets of sample graphs. We show how the resulting method can be used for prototype graph size selection. In our experiments, we apply our method to a real-world dataset and investigate its performance on prototype size selection tasks.

Generative models are commonly used in statistical pattern recognition to describe the probability distributions of patterns in a vector space. In recent years, sustained by the wide range of mathematical tools available in vector space, many algorithms for constructing generative models have been developed. Compared with the advanced development of the generative model for vectors, the development of a generative model for graphs has had less progress. In this thesis, we aim to solve the problem of constructing the generative model for graphs using information theory. Given a set of sample graphs, the generative model for the graphs we aim to construct should be able to not only capture the structural variation of the sample graphs, but to also allow new graphs which share similar properties with the original graphs to be generated. In this thesis, we pose the problem of constructing a generative model for graphs as that of constructing a supergraph structure for the graphs. In Chapter 3, we describe a method of constructing a supergraph-based generative model given a set of sample graphs. By adopting the a posteriori probability developed in a graph matching problem, we obtain a probabilistic framework which measures the likelihood of the sample graphs, given the structure of the supergraph and the correspondence information between the nodes of the sample graphs and those of the supergraph. The supergraph we aim to obtain is one which maximizes the likelihood of the sample graphs. The supergraph is represented here by its adjacency matrix, and we develop a variant of the EM algorithm to locate the adjacency matrix that maximizes the likelihood of the sample graphs. Experimental evaluations demonstrate that the constructed supergraph performs well on classifying graphs. In Chapter 4, we aim to develop graph characterizations that can be used to measure the complexity of graphs. The first graph characterization developed is the von Neumann entropy of a graph associated with its normalized Laplacian matrix. This graph characterization is defined by the eigenvalues of the normalized Laplacian matrix, therefore it is a member of the graph invariant characterization. By applying some transformations, we also develop a simplified form of the von Neumann entropy, which can be expressed in terms of the node degree statistics of the graphs. Experimental results reveal that effectiveness of the two graph characterizations. Our third contribution is presented in Chapter 5, where we use the graph characterization developed in Chapter 4 to measure the supergraph complexity and we develop a novel framework for learning a supergraph using the minimum description length criterion. We combine the Jensen-Shanon kernel with our supergraph construction and this provides us with a way of measuring graph similarity. Moreover, we also develop a method of sampling new graphs from the supergraph. The supergraph we present in this chapter is a generative model which can fulfil the tasks of graph classification, graph clustering, and of generating new graphs. We experiment with both the COIL and “Toy” datasets to illustrate the utility of our generative model. Finally, in Chapter 6, we propose a method of selecting prototype graphs of the most appropriate size from candidate prototypes. The method works by partitioning the sample graphs into two parts and approximating their hypothesis space using the partition functions. From the partition functions, the mutual information between the two sets is defined. The prototype which gives the highest mutual information is selected.

In this paper we explore how the von Neumann entropy can be used as a measure of graph complexity. We also develop a simplified form for the von Neumann entropy of a graph that can be computed in terms of node degree statistics. We compare the resulting complexity with Estrada's heterogeneity index which measures the heterogeneity of the node degree across a graph and reveal a new link between Estrada's index and the commute time on a graph. Finally, we explore how the von Neumann entropy can be used in conjunction with thermodynamic depth. This measure has been shown to overcome problems associated with iso-spectrality encountered when using complexity measures based on spectral graph theory. Our experimental evaluation of the simplified von Neumann entropy explores (a) the accuracy of the underlying approximation, (b) a comparison with alternative graph characterizations, and (c) the application of the entropy-based thermodynamic depth to characterize protein-protein interaction networks. Crown Copyright (c) 2012 Published by Elsevier B.V. All rights reserved.

In this paper, we aim to present a principled approach to the problem of depth-based complexity characterisation of graphs. Our idea is to decompose graphs into substructures of increasing size, and then to measure the complexity of these substructures using Shannon entropy or von-Neumann entropy. We commence by identifying the dominant vertex in a graph. From the dominant vertex, we construct subgraphs of increasing K layers, so-called semidiameter subgraphs. We then measure how the entropy varies with increasing K layer semidiameter subgraphs. We construct a vector of subgraph entropies for each graph, a depth-based complexity trace, and then perform graph clustering in the principal components space of the vectors. We explore our approach on both synthetic data and datasets from the domain of bioinformatics.

In this paper we present a method of sampling from a probabilistic generative model for a set of graphs. Our method is based on the assumption that the nodes and edges of graphs arise under independent Bernoulli distributions. We sample graphs from the generative model according to the node and edge occurrence probabilities. We explain the construction of our generative model and then compute the node and edge occurrence probabilities which allow us to formulate a sampling procedure. We demonstrate experimentally to what extent the graphs sampled by our method reproduce the salient properties of the graphs in the original training sample.

In this paper we show how to approximate the von Neumann entropy associated with the Laplacian eigenspectrum of graphs and exploit it as a characteristic for the clustering and classification of graphs. We commence from the von Neumann entropy and approximate it by replacing the Shannon entropy by its quadratic counterpart. We then show how the quadratic entropy can be expressed in terms of a series of permutation invariant traces. This leads to a simple approximate form for the entropy in terms of the elements of the adjacency matrix which can be evaluated in quadratic time. We use this approximate expression for the entropy as a unary characteristic for graph clustering. Experiments on real world data illustrate the effectiveness of the method.

We present a method for constructing a generative model for sets of graphs by adopting a minimum description length approach. The method is posed in terms of learning a generative supergraph model from which the new samples can be obtained by an appropriate sampling mechanism. We commence by constructing a probability distribution for the occurrence of nodes and edges over the supergraph. We encode the complexity of the supergraph using the von-Neumann entropy. A variant of EM algorithm is developed to minimize the description length criterion in which the node correspondences between the sample graphs and the supergraph are treated as missing data.The maximization step involves updating both the node correspondence information and the structure of supergraph using graduated assignment. Empirical evaluations on real data reveal the practical utility of our proposed algorithm and show that our generative model gives good graph classification results.

For most HVS(Human Visual System) perceptual models, the JND (Just Noticeable Difference) values in highly-textured image regions have little difference with those in edge areas. This is not consistent with the characteristics of human vision. In this paper, an improved method is introduced to give a better content-based perceptual mask than traditional ones using the arrangement of scale-invariant feature points. It could decrease the JND values in edge areas of those traditional masks so that they have an obvious difference with values in highly textured areas. Experimental results show the advantages of this improved approach visually, and the enhancement of the invisibility of watermarks.

In this paper we explore and compare two contrasting graph characterizations. The first of these is Estrada's heterogeneity index, which measures the heterogeneity of the node degree across a graph. Our second measure is the the von Neumann entropy associated with the Laplacian eigenspectrum of graphs. Here we show how to approximate the von Neumann entropy by replacing the Shannon entropy by its quadratic counterpart. This quadratic entropy can be expressed in terms of a series of permutation invariant traces, which can be computed from the node degrees in quadratic time. We compare experimentally the effectiveness of the approximate expression for the entropy with the heterogeneity index.

This talk focusses on work aimed at developing a principled probabilistic and information theoretic framework for learning generative models of relational structure. The aim is develop methods that can be used to learn models that can capture the variability present in graph-structures used to represent shapes or arrangements of shape-primitives in images. Here nodes represent the parts of shape-primitives representing an object, and the the edges represent the relationships which prevail between the parts. The aim is to learn the relationships from examples. Of course such structures can exhibit variability in the arrangement of parts, and the data used in training can be subject to uncertainty. It hence represents a demanding learning problem, for which there is limited available methodology.

We present a method for constructing a generative model for sets of graphs by adopting a minimum description length approach. The method is posed in terms of learning a generative supergraph model from which the new samples can be obtained by an appropriate sampling mechanism. We commence by constructing a probability distribution for the occurrence of nodes and edges over the supergraph. We encode the complexity of the supergraph using the von-Neumann entropy. A variant of EM algorithm is developed to minimize the description length criterion in which the node correspondences between the sample graphs and the supergraph are treated as missing data.The maximization step involves updating both the node correspondence information and the structure of supergraph using graduated assignment. In the experimental part, we demonstrate the practical utility of our proposed algorithm and show that our generative model gives good graph classification results. Besides, we show how to perform graph clustering with Jensen-Shannon kernel and generate new sample graphs.

This paper describes a method for constructing a generative model for sets of graphs. The method is posed in terms of learning a supergraph from which the samples can be obtained by edit operations. We construct a probability distribution for the occurrence of nodes and edges over the supergraph. We use the EM algorithm to learn both the structure of the supergraph and the correspondences between the nodes of the sample graphs and those of the supergraph, which are treated as missing data. In the experimental evaluation of the method, we a) prove that our supergraph learning method can lead to an optimal or suboptimal supergraph, and b) show that our proposed generative model gives good graph classification results.

This paper discusses the analysis, design, and implementation of a Web based database system capable of handling on-line graduate study applications. The aim of the system is to provide a simple, user-friendly interface and a secure on-line database for submitting, retrieving, and sharing application data through Internet, hence to speed up the application process. We chose to base on the Web for many reasons including user familiarity, broad availability, low distribution cost, and minimal development time. By means of experimenting the system, we describe some design challenges of a Web based database application, such as authentication, access control, and security issues, and how we choose to address these challenges and build the system efficiently and effectively