Metric graphs are meaningful objects for modeling complex structures that arise in many real-world applications, such as road networks, river systems, earthquake faults, blood vessels, and filamentary structures in galaxies. To study metric graphs in the context of comparison, we are interested in determining the relative discriminative capabilities of two topology-based distances between a pair of arbitrary finite metric graphs: the persistence distortion distance and the intrinsic Cech distance. We explicitly show how to compute the intrinsic Cech distance between two metric graphs based solely on knowledge of the shortest systems of loops for the graphs. Our main theorem establishes an inequality between the intrinsic Cech and persistence distortion distances in the case when one of the graphs is a bouquet graph and the other is arbitrary. The relationship also holds when both graphs are constructed via wedge sums of cycles and edges.
Well-known methods for comparing graphs using distance measures include combinatorial (e.g., graph edit distance @cite_14 ) and spectral (e.g., eigenvalue decomposition @cite_4 ) approaches. Graph edit distance minimizes the cost of transforming one graph to another via a set of elementary operators such as node edge insertions deletions, while spectral approaches optimize objective functions based on properties of the graph spectra.
@cite_14 @cite_4
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Graph data have become ubiquitous and manipulating them based on similarity is essential for many applications. Graph edit distance is one of the most widely accepted measures to determine similarities between graphs and has extensive applications in the fields of pattern recognition, computer vision etc. Unfortunately, the problem of graph edit distance computation is NP-Hard in general. Accordingly, in this paper we introduce three novel methods to compute the upper and lower bounds for the edit distance between two graphs in polynomial time. Applying these methods, two algorithms AppFull and AppSub are introduced to perform different kinds of graph search on graph databases. Comprehensive experimental studies are conducted on both real and synthetic datasets to examine various aspects of the methods for bounding graph edit distance. Result shows that these methods achieve good scalability in terms of both the number of graphs and the size of graphs. The effectiveness of these algorithms also confirms the usefulness of using our bounds in filtering and searching of graphs. In this paper we generalize and improve the multiscale organization of graphs by introducing a new measure that quantifies the "closeness" between two nodes. The calculation of the measure is linear in the number of edges in the graph and involves just a small number of relaxation sweeps. A similar notion of distance is then calculated and used at each coarser level. We demonstrate the use of this measure in multiscale methods for several important combinatorial optimization problems and discuss the multiscale graph organization. Large probabilistic graphs arise in various domains spanning from social networks to biological and communication networks. An important query in these graphs is the k nearestneighbor query, which involves finding and reporting the k closest nodes to a specific node. This query assumes the existence of a measure of the “proximity” or the “distance” between any two nodes in the graph. To that end, we propose various novel distance functions that extend well known notions of classical graph theory, such as shortest paths and random walks. We argue that many meaningful distance functions are computationally intractable to compute exactly. Thus, in order to process nearest-neighbor queries, we resort to Monte Carlo sampling and exploit novel graph-transformation ideas and pruning opportunities. In our extensive experimental analysis, we explore the trade-offs of our approximation algorithms and demonstrate that they scale well on real-world probabilistic graphs with tens of millions of edges. Graphs are widely used to model complex data in many applications, such as bioinformatics, chemistry, social networks, pattern recognition, etc. A fundamental and critical query primitive is to efficiently search similar structures in a large collection of graphs. This paper studies the graph similarity queries with edit distance constraints. Existing solutions to the problem utilize fixed-size overlapping substructures to generate candidates, and thus become susceptible to large vertex degrees or large distance thresholds. In this paper, we present a partition-based approach to tackle the problem. By dividing data graphs into variable-size non-overlapping partitions, the edit distance constraint is converted to a graph containment constraint for candidate generation. We develop efficient query processing algorithms based on the new paradigm. A candidate pruning technique and an improved graph edit distance algorithm are also developed to further boost the performance. In addition, a cost-aware graph partitioning technique is devised to optimize the index. Extensive experiments demonstrate our approach significantly outperforms existing approaches. We introduce a new probabilistic proximity search algorithm for range and A"-nearest neighbor (A"-NN) searching in both coordinate and metric spaces. Although there exist solutions for these problems, they boil down to a linear scan when the space is intrinsically high dimensional, as is the case in many pattern recognition tasks. This, for example, renders the A"-NN approach to classification rather slow in large databases. Our novel idea is to predict closeness between elements according to how they order their distances toward a distinguished set of anchor objects. Each element in the space sorts the anchor objects from closest to farthest to it and the similarity between orders turns out to be an excellent predictor of the closeness between the corresponding elements. We present extensive experiments comparing our method against state-of-the-art exact and approximate techniques, both in synthetic and real, metric and nonmetric databases, measuring both CPU time and distance computations. The experiments demonstrate that our technique almost always improves upon the performance of alternative techniques, in some cases by a wide margin. Abstract Error-tolerant graph matching is a powerful concept that has various applications in pattern recognition and machine vision. In the present paper, a new distance measure on graphs is proposed. It is based on the maximal common subgraph of two graphs. The new measure is superior to edit distance based measures in that no particular edit operations together with their costs need to be defined. It is formally shown that the new distance measure is a metric. Potential algorithms for the efficient computation of the new measure are discussed. We consider the setting of Reeb graphs of piecewise linear functions and study distances between them that are stable, meaning that functions which are similar in the supremum norm ought to have similar Reeb graphs. We define an edit distance for Reeb graphs and prove that it is stable and universal, meaning that it provides an upper bound to any other stable distance. In contrast, via a specific construction, we show that the interleaving distance and the functional distortion distance on Reeb graphs are not universal. In mining information from very large graphs, processing time as well as system memory become computational bottlenecks as the properties of large graphs must be compared through each iteration of an algorithm. This is a particularly pronounced problem for complex properties. For example, distance metrics are used in many fundamental data mining algorithms including k-nearest neighbors for the classification task. Even the relatively efficient distance and similarity heuristics for large inputs, though, often require processing and memory well beyond linear with respect to the size of the input, and this rapidly becomes intractable with very large inputs. Complex properties such as the distance between two graphs can be extremely costly, but using samples of these large graphs to calculate the same properties proves to reduce memory requirements and processing time significantly without sacrificing quality of classification. Because the vast amount of Web data is easily and robustly represented with graphs, a data reduction technique that preserves the accuracy of mining algorithms on such inputs is important. The sampling techniques presented here show that very large graphs of Web content can be condensed into significantly smaller yet equally expressive graphs that lead to accurate but more efficient classification of Web content Reeb graphs are structural descriptors that capture shape properties of a topological space from the perspective of a chosen function. In this work, we define a combinatorial distance for Reeb graphs of orientable surfaces in terms of the cost necessary to transform one graph into another by edit operations. The main contributions of this paper are the stability property and the optimality of this edit distance. More precisely, the stability result states that changes in the Reeb graphs, measured by the edit distance, are as small as changes in the functions, measured by the maximum norm. The optimality result states that the edit distance discriminates Reeb graphs better than any other distance for Reeb graphs of surfaces satisfying the stability property. We consider the problem of assessing the similarity of 3D shapes using Reeb graphs from the standpoint of robustness under perturbations. For this purpose, 3D objects are viewed as spaces endowed with real-valued functions, while the similarity between the resulting Reeb graphs is addressed through a graph edit distance. The cases of smooth functions on manifolds and piecewise linear functions on polyhedra stand out as the most interesting ones. The main contribution of this paper is the introduction of a general edit distance suitable for comparing Reeb graphs in these settings. This edit distance promises to be useful for applications in 3D object retrieval because of its stability properties in the presence of noise. The emergence of real life graphs with billions of nodes poses significant challenges for managing and querying these graphs. One of the fundamental queries submitted to graphs is the shortest distance query. Online BFS (breadth-first search) and offline pre-computing pairwise shortest distances are prohibitive in time or space complexity for billion-node graphs. In this paper, we study the feasibility of building distance oracles for billion-node graphs. A distance oracle provides approximate answers to shortest distance queries by using a pre-computed data structure for the graph. Sketch-based distance oracles are good candidates because they assign each vertex a sketch of bounded size, which means they have linear space complexity. However, state-of-the-art sketch-based distance oracles lack efficiency or accuracy when dealing with big graphs. In this paper, we address the scalability and accuracy issues by focusing on optimizing the three key factors that affect the performance of distance oracles: landmark selection, distributed BFS, and answer generation. We conduct extensive experiments on both real networks and synthetic networks to show that we can build distance oracles of affordable cost and efficiently answer shortest distance queries even for billion-node graphs. Recent studies have shown that graph-based approaches are effective for semi-supervised learning. The key idea behind many graph-based approaches is to enforce the consistency between the class assignment of unlabeled examples and the pairwise similarity between examples. One major limitation with most graph-based approaches is that they are unable to explore dissimilarity or negative similarity. This is because the dissimilar relation is not transitive, and therefore is difficult to be propagated. Furthermore, negative similarity could result in unbounded energy functions, which makes most graph-based algorithms unapplicable. In this paper, we propose a new graph-based approach, termed as "mixed label propagation" which is able to effectively explore both similarity and dissimilarity simultaneously. In particular, the new framework determines the assignment of class labels by (1) minimizing the energy function associated with positive similarity, and (2) maximizing the energy function associated with negative similarity. Our empirical study with collaborative filtering shows promising performance of the proposed approach. Computational and memory costs restrict spectral techniques to rather small graphs, which is a serious limitation especially in video segmentation. In this paper, we propose the use of a reduced graph based on superpixels. In contrast to previous work, the reduced graph is reweighted such that the resulting segmentation is equivalent, under certain assumptions, to that of the full graph. We consider equivalence in terms of the normalized cut and of its spectral clustering relaxation. The proposed method reduces runtime and memory consumption and yields on par results in image and video segmentation. Further, it enables an efficient data representation and update for a new streaming video segmentation approach that also achieves state-of-the-art performance. Graph-based methods are a useful class of methods for improving the performance of unsupervised and semi-supervised machine learning tasks, such as clustering or information retrieval. However, the performance of existing graph-based methods is highly dependent on how well the affinity graph reflects the original data structure. We propose that multimedia such as images or videos consist of multiple separate components, and therefore more than one graph is required to fully capture the relationship between them. Accordingly, we present a new spectral method - the Feature Grouped Spectral Multigraph (FGSM) - which comprises the following steps. First, mutually independent subsets of the original feature space are generated through feature clustering. Secondly, a separate graph is generated from each feature subset. Finally, a spectral embedding is calculated on each graph, and the embeddings are scaled aggregated into a single representation. Using this representation, a variety of experiments are performed on three learning tasks - clustering, retrieval and recognition - on human action datasets, demonstrating considerably better performance than the state-of-the-art. The distance for a pair of vertices in a graph G is the length of the shortest path between them. The distance distribution for G specifies how many vertex pairs are at distance h, for all feasible values h. We study three fast randomized algorithms to approximate the distance distribution in large graphs. The Eppstein-Wang (EW) algorithm exploits sampling through a limited (logarithmic) number of Breadth-First Searches (BFSes). The Size-Estimation Framework (SEF) by Cohen employs random ranking and least-element lists to provide several estimators. Finally, the Approximate Neighborhood Function (ANF) algorithm by Palmer, Gibbons, and Faloutsos makes use of the probabilistic counting technique introduced by Flajolet and Martin, in order to estimate the number of distinct elements in a large multiset. We investigate how good is the approximation of the distance distribution, when the three algorithms are run in similar settings. The analysis of ANF derives from the results on the probabilistic counting method, while the one of sef is given by Cohen. For what concerns EW (originally designed for another problem), we extend its simple analysis in order to bound its error with high probability and to show its convergence. We then perform an experimental study on 30 real-world graphs, showing that our implementation of ew combines the accuracy of sef with the performance of ANF. Comparing two geometric graphs embedded in space is important in the field of transportation network analysis. Given street maps of the same city collected from different sources, researchers often need to know how and where they differ. However, the majority of current graph comparison algorithms are based on structural properties of graphs, such as their degree distribution or their local connectivity properties, and do not consider their spatial embedding. This ignores a key property of road networks since the similarity of travel over two road networks is intimately tied to the specific spatial embedding. Likewise, many current algorithms specific to street map comparison either do not provide quality guarantees or focus on spatial embeddings only. Motivated by road network comparison, we propose a new path-based distance measure between two planar geometric graphs that is based on comparing sets of travel paths generated over the graphs. Surprisingly, we are able to show that using paths of bounded link-length, we can capture global structural and spatial differences between the graphs. We show how to utilize our distance measure as a local signature in order to identify and visualize portions of high similarity in the maps. Finally, we present an experimental evaluation of our distance measure and its local signature on street map data from Berlin, Germany and Athens, Greece. Distance or similarity measures are essential to solve many pattern recognition problems such as classification, clustering, and retrieval problems. Various distance similarity measures that are applicable to compare two probability density functions, pdf in short, are reviewed and categorized in both syntactic and semantic relationships. A correlation coefficient and a hierarchical clustering technique are adopted to reveal similarities among numerous distance similarity measures. This paper presents a new spectral partitioning formulation which directly incorporates vertex size information by modifying the Laplacian of the graph. Modifying the Laplacian produces a generalized eigenvalue problem, which is reduced to the standard eigenvalue problem. Experiments show that the scaled ratio-cut costs of results on benchmarks with arbitrary vertex size improve by 22 when the eigenvectors of the Laplacian in the spectral partitioner KP are replaced by the eigenvectors of our modified Laplacian. The inability to handle vertex sizes in the spectral partitioning formulation has been a limitation in applying spectral partitioning in a multilevel setting. We investigate whether our new formulation effectively removes this limitation by combining it with a simple multilevel bottom-up clustering algorithm and an iterative improvement algorithm for partition refinement. Experiments show that in a multilevel setting where the spectral partitioner KP provides the initial partitions of the most contracted graph, using the modified Laplacian in place of the standard Laplacian is more efficient and more effective in the partitioning of graphs with arbitrary-size and unit-size vertices; average improvements of 17 and 18 are observed for graphs with arbitrary-size and unit-size vertices, respectively. Comparisons with other ratio-cut based partitioners on hypergraphs with unit-size as well as arbitrary-size vertices, show that the multilevel spectral partitioner produces either better results or almost identical results more efficiently. Inexact graph matching has been one of the significant research foci in the area of pattern analysis. As an important way to measure the similarity between pairwise graphs error-tolerantly, graph edit distance (GED) is the base of inexact graph matching. The research advance of GED is surveyed in order to provide a review of the existing literatures and offer some insights into the studies of GED. Since graphs may be attributed or non-attributed and the definition of costs for edit operations is various, the existing GED algorithms are categorized according to these two factors and described in detail. After these algorithms are analyzed and their limitations are identified, several promising directions for further research are proposed. Structured data including sets, sequences, trees and graphs, pose significant challenges to fundamental aspects of data management such as efficient storage, indexing, and similarity search. With the fast accumulation of graph databases, similarity search in graph databases has emerged as an important research topic. Graph similarity search has applications in a wide range of domains including cheminformatics, bioinformatics, sensor network management, social network management, and XML documents, among others. Most of the current graph indexing methods focus on subgraph query processing, i.e. determining the set of database graphs that contains the query graph and hence do not directly support similarity search. In data mining and machine learning, various graph kernel functions have been designed to capture the intrinsic similarity of graphs. Though successful in constructing accurate predictive and classification models for supervised learning, graph kernel functions have (i) high computational complexity and (ii) non-trivial difficulty to be indexed in a graph database. Our objective is to bridge graph kernel function and similarity search in graph databases by proposing (i) a novel kernel-based similarity measurement and (ii) an efficient indexing structure for graph data management. Our method of similarity measurement builds upon local features extracted from each node and their neighboring nodes in graphs. A hash table is utilized to support efficient storage and fast search of the extracted local features. Using the hash table, a graph kernel function is defined to capture the intrinsic similarity of graphs and for fast similarity query processing. We have implemented our method, which we have named G-hash, and have demonstrated its utility on large chemical graph databases. Our results show that the G-hash method achieves state-of-the-art performance for k-nearest neighbor (k-NN) classification. Most importantly, the new similarity measurement and the index structure is scalable to large database with smaller indexing size, faster indexing construction time, and faster query processing time as compared to state-of-the-art indexing methods such as C-tree, gIndex, and GraphGrep.