Product graph-based higher order contextual similarities for inexact subgraph matching

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this record Many algorithms formulate graph matching as an optimization of an objective function of pairwise quantification of nodes and edges of two graphs to be matched. Pairwise measurements usually consider local attributes but disregard contextual information involved in graph structures. We address this issue by proposing contextual similarities between pairs of nodes. This is done by considering the tensor product graph (TPG) of two graphs to be matched, where each node is an ordered pair of nodes of the operand graphs. Contextual similarities between a pair of nodes are computed by accumulating weighted walks (normalized pairwise similarities) terminating at the corresponding paired node in TPG. Once the contextual similarities are obtained, we formulate subgraph matching as a node and edge selection problem in TPG. We use contextual similarities to construct an objective function and optimize it with a linear programming approach. Since random walk formulation through TPG takes into account higher order information, it is not a surprise that we obtain more reliable similarities and better discrimination among the nodes and edges. Experimental results shown on synthetic as well as real benchmarks illustrate that higher order contextual similarities increase discriminating power and allow one to find approximate solutions to the subgraph matching problem.European Union Horizon 202

Application of Graph Theory in Computer Science

The field of mathematics have important roll in various fields. One of the important area in mathematics is Graph Theory. Which used in structural modeling in many area’s. The structural arrangements of various objects or technologies lead to new inventions and modification in the existing environment for enhancement in those field. The field of graph theory started from problem of Konigsberg bridge in 1735. This paper given an overview of the application of graph theory in heterogeneous field to some extent but mainly focuses on computer science application but uses graph theoretical concepts

Multiple graph matching and applications

En aplicaciones de reconocimiento de patrones, los grafos con atributos son en gran medida apropiados. Normalmente, los vértices de los grafos representan partes locales de los objetos i las aristas relaciones entre estas partes locales. No obstante, estas ventajas vienen juntas con un severo inconveniente, la distancia entre dos grafos no puede ser calculada en un tiempo polinómico. Considerando estas características especiales el uso de los prototipos de grafos es necesariamente omnipresente. Las aplicaciones de los prototipos de grafos son extensas, siendo las más habituales clustering, clasificación, reconocimiento de objetos, caracterización de objetos i bases de datos de grafos entre otras. A pesar de la diversidad de aplicaciones de los prototipos de grafos, el objetivo del mismo es equivalente en todas ellas, la representación de un conjunto de grafos. Para construir un prototipo de un grafo todos los elementos del conjunto de enteramiento tienen que ser etiquetados comúnmente. Este etiquetado común consiste en identificar que nodos de que grafos representan el mismo tipo de información en el conjunto de entrenamiento. Una vez este etiquetaje común esta hecho, los atributos locales pueden ser combinados i el prototipo construido. Hasta ahora los algoritmos del estado del arte para calcular este etiquetaje común mancan de efectividad o bases teóricas. En esta tesis, describimos formalmente el problema del etiquetaje global i mostramos una taxonomía de los tipos de algoritmos existentes. Además, proponemos seis nuevos algoritmos para calcular soluciones aproximadas al problema del etiquetaje común. La eficiencia de los algoritmos propuestos es evaluada en diversas bases de datos reales i sintéticas. En la mayoría de experimentos realizados los algoritmos propuestos dan mejores resultados que los existentes en el estado del arte.In pattern recognition, the use of graphs is, to a great extend, appropriate and advantageous. Usually, vertices of the graph represent local parts of an object while edges represent relations between these local parts. However, its advantages come together with a sever drawback, the distance between two graph cannot be optimally computed in polynomial time. Taking into account this special characteristic the use of graph prototypes becomes ubiquitous. The applicability of graphs prototypes is extensive, being the most common applications clustering, classification, object characterization and graph databases to name some. However, the objective of a graph prototype is equivalent to all applications, the representation of a set of graph. To synthesize a prototype all elements of the set must be mutually labeled. This mutual labeling consists in identifying which nodes of which graphs represent the same information in the training set. Once this mutual labeling is done the set can be characterized and combined to create a graph prototype. We call this initial labeling a common labeling. Up to now, all state of the art algorithms to compute a common labeling lack on either performance or theoretical basis. In this thesis, we formally describe the common labeling problem and we give a clear taxonomy of the types of algorithms. Six new algorithms that rely on different techniques are described to compute a suboptimal solution to the common labeling problem. The performance of the proposed algorithms is evaluated using an artificial and several real datasets. In addition, the algorithms have been evaluated on several real applications. These applications include graph databases and group-wise image registration. In most of the tests and applications evaluated the presented algorithms have showed a great improvement in comparison to state of the art applications

Pyramidal Stochastic Graphlet Embedding for Document Pattern Classification

This is the author accepted manuscript. The final version is available from IEEE via the DOI in this recordDocument pattern classification methods using graphs have received a lot of attention because of its robust representation paradigm and rich theoretical background. However, the way of preserving and the process for delineating documents with graphs introduce noise in the rendition of underlying data, which creates instability in the graph representation. To deal with such unreliability in representation, in this paper, we propose Pyramidal Stochastic Graphlet Embedding (PSGE). Given a graph representing a document pattern, our method first computes a graph pyramid by successively reducing the base graph. Once the graph pyramid is computed, we apply Stochastic Graphlet Embedding (SGE) for each level of the pyramid and combine their embedded representation to obtain a global delineation of the original graph. The consideration of pyramid of graphs rather than just a base graph extends the representational power of the graph embedding, which reduces the instability caused due to noise and distortion. When plugged with support vector machine, our proposed PSGE has outperformed the state-of-The-art results in recognition of handwritten words as well as graphical symbols.European Union Horizon 2020Ministerio de Educación, Cultura y Deporte, SpainRamon y Cajal FellowshipCERCA Program/Generalitat de Cataluny

Bayesian graph edit distance

This paper describes a novel framework for comparing and matching corrupted relational graphs. The paper develops the idea of edit-distance originally introduced for graph-matching by Sanfeliu and Fu [1]. We show how the Levenshtein distance can be used to model the probability distribution for structural errors in the graph-matching problem. This probability distribution is used to locate matches using MAP label updates. We compare the resulting graph-matching algorithm with that recently reported by Wilson and Hancock. The use of edit-distance offers an elegant alternative to the exhaustive compilation of label dictionaries. Moreover, the method is polynomial rather than exponential in its worst-case complexity. We support our approach with an experimental study on synthetic data and illustrate its effectiveness on an uncalibrated stereo correspondence problem. This demonstrates experimentally that the gain in efficiency is not at the expense of quality of match

Musings on Symbol Recognition

This paper does not pretend to be yet another survey on symbol recognition methods. It will rather try to take a step back, look at the main efforts done in that area throughout the years and propose some interesting directions to investigate

Hierarchical stochastic graphlet embedding for graph-based pattern recognition

This is the final version. Available on open access from Springer via the DOI in this recordDespite being very successful within the pattern recognition and machine learning community, graph-based methods are often unusable with many machine learning tools. This is because of the incompatibility of most of the mathematical operations in graph domain. Graph embedding has been proposed as a way to tackle these difficulties, which maps graphs to a vector space and makes the standard machine learning techniques applicable for them. However, it is well known that graph embedding techniques usually suffer from the loss of structural information. In this paper, given a graph, we consider its hierarchical structure for mapping it into a vector space. The hierarchical structure is constructed by topologically clustering the graph nodes, and considering each cluster as a node in the upper hierarchical level. Once this hierarchical structure of graph is constructed, we consider its various configurations of its parts, and use stochastic graphlet embedding (SGE) for mapping them into vector space. Broadly speaking, SGE produces a distribution of uniformly sampled low to high order graphlets as a way to embed graphs into the vector space. In what follows, the coarse-to-fine structure of a graph hierarchy and the statistics fetched through the distribution of low to high order stochastic graphlets complements each other and include important structural information with varied contexts. Altogether, these two techniques substantially cope with the usual information loss involved in graph embedding techniques, and it is not a surprise that we obtain more robust vector space embedding of graphs. This fact has been corroborated through a detailed experimental evaluation on various benchmark graph datasets, where we outperform the state-of-the-art methods.European Union Horizon 2020Ministerio de Educación, Cultura y Deporte, SpainGeneralitat de Cataluny

AEDNet: Adaptive Edge-Deleting Network For Subgraph Matching

Subgraph matching is to find all subgraphs in a data graph that are isomorphic to an existing query graph. Subgraph matching is an NP-hard problem, yet has found its applications in many areas. Many learning-based methods have been proposed for graph matching, whereas few have been designed for subgraph matching. The subgraph matching problem is generally more challenging, mainly due to the different sizes between the two graphs, resulting in considerable large space of solutions. Also the extra edges existing in the data graph connecting to the matched nodes may lead to two matched nodes of two graphs having different adjacency structures and often being identified as distinct objects. Due to the extra edges, the existing learning based methods often fail to generate sufficiently similar node-level embeddings for matched nodes. This study proposes a novel Adaptive Edge-Deleting Network (AEDNet) for subgraph matching. The proposed method is trained in an end-to-end fashion. In AEDNet, a novel sample-wise adaptive edge-deleting mechanism removes extra edges to ensure consistency of adjacency structure of matched nodes, while a unidirectional cross-propagation mechanism ensures consistency of features of matched nodes. We applied the proposed method on six datasets with graph sizes varying from 20 to 2300. Our evaluations on six open datasets demonstrate that the proposed AEDNet outperforms six state-of-the-arts and is much faster than the exact methods on large graphs

Approximating the minimum hub cover problem on planar graphs

We study an approximation algorithm with a performance guarantee to solve a new NP-hard optimization problem on planar graphs. The problem, which is referred to as the minimum hub cover problem, has recently been introduced to the literature to improve query processing over large graph databases. Planar graphs also arise in various graph query processing applications, such as; biometric identification, image classification, object recognition, and so on. Our algorithm is based on a well-known graph decomposition technique that partitions the graph into a set of outerplanar graphs and provides an approximate solution with a proven performance ratio. We conduct a comprehensive computational experiment to investigate the empirical performance of the algorithm. Computational results demonstrate that the empirical performance of the algorithm surpasses its guaranteed performance. We also apply the same decomposition approach to develop a decomposition-based heuristic, which is much more efficient than the approximation algorithm in terms of computation time. Computational results also indicate that the efficacy of the decomposition-based heuristic in terms of solution quality is comparable to that of the approximation algorithm