Graph Quantization

Vector quantization(VQ) is a lossy data compression technique from signal processing, which is restricted to feature vectors and therefore inapplicable for combinatorial structures. This contribution presents a theoretical foundation of graph quantization (GQ) that extends VQ to the domain of attributed graphs. We present the necessary Lloyd-Max conditions for optimality of a graph quantizer and consistency results for optimal GQ design based on empirical distortion measures and stochastic optimization. These results statistically justify existing clustering algorithms in the domain of graphs. The proposed approach provides a template of how to link structural pattern recognition methods other than GQ to statistical pattern recognition.Comment: 24 pages; submitted to CVI

Properties of the Sample Mean in Graph Spaces and the Majorize-Minimize-Mean Algorithm

One of the most fundamental concepts in statistics is the concept of sample mean. Properties of the sample mean that are well-defined in Euclidean spaces become unwieldy or even unclear in graph spaces. Open problems related to the sample mean of graphs include: non-existence, non-uniqueness, statistical inconsistency, lack of convergence results of mean algorithms, non-existence of midpoints, and disparity to midpoints. We present conditions to resolve all six problems and propose a Majorize-Minimize-Mean (MMM) Algorithm. Experiments on graph datasets representing images and molecules show that the MMM-Algorithm best approximates a sample mean of graphs compared to six other mean algorithms

Geometry of Graph Edit Distance Spaces

In this paper we study the geometry of graph spaces endowed with a special class of graph edit distances. The focus is on geometrical results useful for statistical pattern recognition. The main result is the Graph Representation Theorem. It states that a graph is a point in some geometrical space, called orbit space. Orbit spaces are well investigated and easier to explore than the original graph space. We derive a number of geometrical results from the orbit space representation, translate them to the graph space, and indicate their significance and usefulness in statistical pattern recognition

An interdisciplinary survey of network similarity methods

Comparative graph and network analysis play an important role in both systems biology and pattern recognition, but existing surveys on the topic have historically ignored or underserved one or the other of these fields. We present an integrative introduction to the key objectives and methods of graph and network comparison in each field, with the intent of remaining accessible to relative novices in order to mitigate the barrier to interdisciplinary idea crossover. To guide our investigation, and to quantitatively justify our assertions about what the key objectives and methods of each field are, we have constructed a citation network containing 5,793 vertices from the full reference lists of over two hundred relevant papers, which we collected by searching Google Scholar for ten different network comparison-related search terms. We investigate its basic statistics and community structure, and frame our presentation around the papers found to have high importance according to five different standard centrality measures

Subgraph Similarity Search in Large Graphs

One of the major challenges in applications related to social networks, computational biology, collaboration networks etc., is to efficiently search for similar patterns in their underlying graphs. These graphs are typically noisy and contain thousands of vertices and millions of edges. In many cases, the graphs are unlabeled and the notion of similarity is also not well defined. We study the problem of searching an induced subgraph in a large target graph that is most similar to the given query graph. We assume that the query graph and target graph are undirected and unlabeled. We use graphlet kernels \cite{shervashidze2009efficient} to define graph similarity. Graphlet kernels are known to perform better than other kernels in different applications. Our algorithm maps topological neighborhood information of vertices in the query and target graphs to vectors. These local topological informations are then combined to find a target subgraph having highly similar global topology with the given query graph. We tested our algorithm on several real world networks such as facebook network, google plus network, youtube network, amazon network etc. Most of them contain thousands of vertices and million edges. Our algorithm is able to detect highly similar matches when queried in these networks. Our multi-threaded implementation takes about one second to find the match on a 32 core machine, excluding the time for one time preprocessing. Computationally expensive parts of our algorithm can be further scaled to standard parallel and distributed frameworks like map-reduce

Graph Kernels based on High Order Graphlet Parsing and Hashing

Graph-based methods are known to be successful in many machine learning and pattern classification tasks. These methods consider semi-structured data as graphs where nodes correspond to primitives (parts, interest points, segments, etc.) and edges characterize the relationships between these primitives. However, these non-vectorial graph data cannot be straightforwardly plugged into off-the-shelf machine learning algorithms without a preliminary step of -- explicit/implicit -- graph vectorization and embedding. This embedding process should be resilient to intra-class graph variations while being highly discriminant. In this paper, we propose a novel high-order stochastic graphlet embedding (SGE) that maps graphs into vector spaces. Our main contribution includes a new stochastic search procedure that efficiently parses a given graph and extracts/samples unlimitedly high-order graphlets. We consider these graphlets, with increasing orders, to model local primitives as well as their increasingly complex interactions. In order to build our graph representation, we measure the distribution of these graphlets into a given graph, using particular hash functions that efficiently assign sampled graphlets into isomorphic sets with a very low probability of collision. When combined with maximum margin classifiers, these graphlet-based representations have positive impact on the performance of pattern comparison and recognition as corroborated through extensive experiments using standard benchmark databases.Comment: arXiv admin note: substantial text overlap with arXiv:1702.0015

Stable and Informative Spectral Signatures for Graph Matching

In this paper, we consider the approximate weighted graph matching problem and introduce stable and informative first and second order compatibility terms suitable for inclusion into the popular integer quadratic program formulation. Our approach relies on a rigorous analysis of stability of spectral signatures based on the graph Laplacian. In the case of the first order term, we derive an objective function that measures both the stability and informativeness of a given spectral signature. By optimizing this objective, we design new spectral node signatures tuned to a specific graph to be matched. We also introduce the pairwise heat kernel distance as a stable second order compatibility term; we justify its plausibility by showing that in a certain limiting case it converges to the classical adjacency matrix-based second order compatibility function. We have tested our approach on a set of synthetic graphs, the widely-used CMU house sequence, and a set of real images. These experiments show the superior performance of our first and second order compatibility terms as compared with the commonly used ones.Comment: final version for CVPR201

Avoiding Unnecessary Information Loss: Correct and Efficient Model Synchronization Based on Triple Graph Grammars

Model synchronization, i.e., the task of restoring consistency between two interrelated models after a model change, is a challenging task. Triple Graph Grammars (TGGs) specify model consistency by means of rules that describe how to create consistent pairs of models. These rules can be used to automatically derive further rules, which describe how to propagate changes from one model to the other or how to change one model in such a way that propagation is guaranteed to be possible. Restricting model synchronization to these derived rules, however, may lead to unnecessary deletion and recreation of model elements during change propagation. This is inefficient and may cause unnecessary information loss, i.e., when deleted elements contain information that is not represented in the second model, this information cannot be recovered easily. Short-cut rules have recently been developed to avoid unnecessary information loss by reusing existing model elements. In this paper, we show how to automatically derive (short-cut) repair rules from short-cut rules to propagate changes such that information loss is avoided and model synchronization is accelerated. The key ingredients of our rule-based model synchronization process are these repair rules and an incremental pattern matcher informing about suitable applications of them. We prove the termination and the correctness of this synchronization process and discuss its completeness. As a proof of concept, we have implemented this synchronization process in eMoflon, a state-of-the-art model transformation tool with inherent support of bidirectionality. Our evaluation shows that repair processes based on (short-cut) repair rules have considerably decreased information loss and improved performance compared to former model synchronization processes based on TGGs.Comment: 33 pages, 20 figures, 3 table

Sublinear Models for Graphs

This contribution extends linear models for feature vectors to sublinear models for graphs and analyzes their properties. The results are (i) a geometric interpretation of sublinear classifiers, (ii) a generic learning rule based on the principle of empirical risk minimization, (iii) a convergence theorem for the margin perceptron in the sublinearly separable case, and (iv) the VC-dimension of sublinear functions. Empirical results on graph data show that sublinear models on graphs have similar properties as linear models for feature vectors

Graph Kernels: State-of-the-Art and Future Challenges

Graph-structured data are an integral part of many application domains, including chemoinformatics, computational biology, neuroimaging, and social network analysis. Over the last two decades, numerous graph kernels, i.e. kernel functions between graphs, have been proposed to solve the problem of assessing the similarity between graphs, thereby making it possible to perform predictions in both classification and regression settings. This manuscript provides a review of existing graph kernels, their applications, software plus data resources, and an empirical comparison of state-of-the-art graph kernels.Comment: Accepted by Foundations and Trends in Machine Learning, 202