Graph Distance Measures Research Articles

Comparison of Graph Distance Measures for Movie Similarity Using a Multilayer Network Model.

Graph distance measures have emerged as an effective tool for evaluating the similarity or dissimilarity between graphs. Recently, there has been a growing trend in the application of movie networks to analyze and understand movie stories. Previous studies focused on computing the distance between individual characters in narratives and identifying the most important ones. Unlike previous techniques, which often relied on representing movie stories through single-layer networks based on characters or keywords, a new multilayer network model was developed to allow a more comprehensive representation of movie stories, including character, keyword, and location aspects. To assess the similarities among movie stories, we propose a methodology that utilizes a multilayer network model and layer-to-layer distance measures. We aim to quantify the similarity between movie networks by verifying two aspects: (i) regarding many components of the movie story and (ii) quantifying the distance between their corresponding movie networks. We tend to explore how five graph distance measures reveal the similarity between movie stories in two aspects: (i) finding the order of similarity among movies within the same genre, and (ii) classifying movie stories based on genre. We select movies from various genres: sci-fi, horror, romance, and comedy. We extract movie stories from movie scripts regarding character, keyword, and location entities to perform this. Then, we compute the distance between movie networks using different methods, such as the network portrait divergence, the network Laplacian spectra descriptor (NetLSD), the network embedding as matrix factorization (NetMF), the Laplacian spectra, and D-measure. The study shows the effectiveness of different methods for identifying similarities among various genres and classifying movies across different genres. The results suggest that the efficiency of an approach on a specific network type depends on its capacity to capture the inherent network structure of that type. We propose incorporating the approach into movie recommendation systems.

Temporal segmentation of EEG based on functional connectivity network structure

In the study of brain functional connectivity networks, it is assumed that a network is built from a data window in which activity is stationary. However, brain activity is non-stationary over sufficiently large time periods. Addressing the analysis electroencephalograph (EEG) data, we propose a data segmentation method based on functional connectivity network structure. The goal of segmentation is to ensure that within a window of analysis, there is similar network structure. We designed an intuitive and flexible graph distance measure to quantify the difference in network structure between two analysis windows. This measure is modular: a variety of node importance indices can be plugged into it. We use a reference window versus sliding window comparison approach to detect changes, as indicated by outliers in the distribution of graph distance values. Performance of our segmentation method was tested in simulated EEG data and real EEG data from a drone piloting experiment (using correlation or phase-locking value as the functional connectivity strength metric). We compared our method under various node importance measures and against matrix-based dissimilarity metrics that use singular value decomposition on the connectivity matrix. The results show the graph distance approach worked better than matrix-based approaches; graph distance based on partial node centrality was most sensitive to network structural changes, especially when connectivity matrix values change little. The proposed method provides EEG data segmentation tailored for detecting changes in terms of functional connectivity networks. Our study provides a new perspective on EEG segmentation, one that is based on functional connectivity network structure differences.

Study State Dynamics of Team Passing Networks in Soccer Games

ABSTRACT Complex networks have been widely used in studying collective behaviours in soccer sports, such as examining tactical strategies, recognizing team characteristics, and discovering topological determinants for high team performance. The passing network of a team evolves and displays different temporal patterns, that are strongly linked to team status, tactical strategies, attacking/defending transitions, etc. Nevertheless, existing research has not illuminated the state dynamics of team passing networks, whereas similar methods have been extensively used in examining the dynamical brain networks constructed from human brain neuroimage data. This study aims to investigate the state dynamics of team passing networks in soccer sports. The introduced method incorporates multiple techniques, including sliding time window, network modeling, graph distance measure, clustering, and cluster validation. The final match of the FIFA World Cup 2018 was taken as an example, and the state dynamics of teams Croatia and France were analyzed respectively. Additionally, the effects of the time windows and graph distance measures on the results were briefly discussed. This study presents a novel outlook on examining the dynamics of team passing networks, as it facilitates the recognition of important team states or state transitions in soccer and other team ball-passing sports for further analysis.

Graph distance measures based on Balaban indices

Dehmer et al. [Interrelations of graph distance measures based on topological indices, PLoS One 9 (2014) e94985] explored the graph measures based on some topological indices and demonstrated some useful properties of them. In this work, we survey the graph measures based on the Balaban index of a graph, and prove some extremal properties on them for several classes of graphs. Moreover, through numerical calculation and comparison, we find the graph measures based on Balaban index of a graph behave better than that based on Randić index in some special cases.

Models of similarity in complex networks.

The analysis of networks describing many social, economic, technological, biological and other systems has attracted a lot of attention last decades. Since most of these complex systems evolve over time, there is a need to investigate the changes, which appear in the system, in order to assess the sustainability of the network and to identify stable periods. In the literature, there have been developed a large number of models that measure the similarity among the networks. There also exist some surveys, which consider a limited number of similarity measures and then perform their correlation analysis, discuss their properties or assess their performances on synthetic benchmarks or real networks. The aim of the article is to extend these studies. The article considers 39 graph distance measures and compares them on simple graphs, random graph models and real networks. The author also evaluates the performance of the models in order to identify which of them can be applied to large networks. The results of the study reveal some important aspects of existing similarity models and provide a better understanding of their advantages and disadvantages. The major finding of the work is that many graph similarity measures of different nature are well correlated and that some comprehensive methods are well agreed with simple models. Such information can be used for the choice of appropriate similarity measure as well as for further development of new models for similarity assessment in network structures.

Graph Matching with Distance-Preserving Crossover

Graph models are fundamental to any kind of application on structured real-world problems. Any comparison between graphs by a graph distance measure requires the solution of the inexact graph matching problem, which constitutes a hard combinatorial optimization problem. An inexact matching problem includes in its formulation robustness to any type of perturbation, such as, for instance, noise, inherently present in real-world environments. In this paper, we introduce the concept of distance-preserving crossover operators for genetic algorithms for this task. For large graphs, our algorithm outperforms any state-of-the-art approximate algorithm—in particular, genetic algorithms with alternative crossover operators, which are to the best of our knowledge currently limited to no more than 50 nodes. We use a two-level local search heuristic to further enhance the results, pushing the limits to up to 300 nodes: a first local search step is directly integrated into the crossover operator; another one is applied independently during offspring generation.

Detecting sequences of system states in temporal networks

Many time-evolving systems in nature, society and technology leave traces of the interactions within them. These interactions form temporal networks that reflect the states of the systems. In this work, we pursue a coarse-grained description of these systems by proposing a method to assign discrete states to the systems and inferring the sequence of such states from the data. Such states could, for example, correspond to a mental state (as inferred from neuroimaging data) or the operational state of an organization (as inferred by interpersonal communication). Our method combines a graph distance measure and hierarchical clustering. Using several empirical data sets of social temporal networks, we show that our method is capable of inferring the system’s states such as distinct activities in a school and a weekday state as opposed to a weekend state. We expect the methods to be equally useful in other settings such as temporally varying protein interactions, ecological interspecific interactions, functional connectivity in the brain and adaptive social networks.

Properties of graph distance measures by means of discrete inequalities

Abstract In this paper, we investigate graph distance measures based on topological graph measures. Those measures can be used to measure the structural distance between graphs. When studying the scientific literature, one is aware that measuring distance/similarity between graphs meaningfully has been intricate. We demonstrate that our measures are well-defined and prove bounds for investigating their value domain. Also, we generate numerical results and demonstrate that the measures have useful properties.

Efficient graph pattern matching framework for network-based in-vehicle fault detection

Abnormal messages propagated from faulty operations in an in-vehicle network may severely harm a vehicular system, but they cannot be easily detected when their information is not known in advance. To support an efficient detection of faulty message patterns propagated in the in-vehicle network, this paper presents a novel graph pattern matching framework built upon a message log-driven graph modeling. Our framework models the unknown operation of the in-vehicle network as a query graph and the reference database of normal operations as data graphs. Given a query graph and data graphs, we determine whether the query graph represents normal or fault operation by using the distance measure on the data graphs. The analysis of the faulty message propagation requires to consider the sequence of events in the distance measure, and thus, using the conventional graph distance measures can generate false negatives due to the lack of consideration of the sequence relationships among the events. We therefore propose a novel distance metric based on the maximum common subgraph (MCS) between two graphs and the sequence numbers of messages, which works robustly even for the abnormal faulty patterns and can avoid false negatives in large databases. Since the problem of MCS computation is NP-hard, we also propose two efficient filtering techniques, one based on the lower bound of the MCS distance for a polynomial-time approximation and the other based on edge pruning. Experiments performed on real and synthetic datasets to assess our framework show that ours significantly outperforms the previously existing methods in terms of both performance and accuracy of query responses.

A comparative analysis of new graph distance measures and graph edit distance

The problem of determining the structural similarity or distance between graphs got considerable attention during the last decades; consequently, various similarity and distance measures for networks have already been investigated. Dehmer et al. studied a new distance measure for graphs, which is based on topological indices. An interesting problem is to compare our class of measures with the well-known graph edit distance, which has been studied extensively and often serves as a benchmark measure; but note that GED is generally NP-hard. In this paper, we compare the edit distance with our graph distance measures; in order to do so we use some well-known topological indices such as graph energy, Wiener index, Randić index and graph entropy. By using several special graph classes numerical results reveal that the graph distance measure based on graph energy approximates GED well. This fact could trigger an important research direction for studying graph edit distance and other comparative network measures.

Counterexamples to conjectures on graph distance measures based on topological indexes

In this paper we disprove three conjectures from Dehmer et al. (2014) on graph distance measures based on topological indices by providing explicit classes of trees that do not satisfy proposed inequalities. The constructions are based on the families of trees that have the same Wiener index, graph energy or Randić index – but different degree sequences.

New inequalities for network distance measures by using graph spectra

Eigenvalues of graph-theoretical matrices often reflect structure properties of networks (graphs) meaningfully. In this paper, we explore inequalities for graph distance measures which are based on topological indices. Some of these indices are based on eigenvalues of graph-theoretical matrices. We here consider the adjacency matrix, the Laplacian matrix and signless Laplacian matrix. Besides proving the inequalities, we discuss the usefulness of these measures and state some conjectures.

Graph distance measures based on topological indices revisited

Graph distance measures based on topological indices have been already explored by Dehmer et al. Also, inequalities for those graph distance measures have been proved. In this paper, we continue studying such comparative graph measures based on the well-known Wiener index, graph energy and Randić index, respectively. We prove extremal properties of the graph distance measures for some special classes of graphs. To demonstrate useful properties of the measures, we also discuss numerical results. To conclude the paper we state some open problems.

Attributed graph distance measure for automatic detection of attention deficit hyperactive disordered subjects.

Attention Deficit Hyperactive Disorder (ADHD) is getting a lot of attention recently for two reasons. First, it is one of the most commonly found childhood disorders and second, the root cause of the problem is still unknown. Functional Magnetic Resonance Imaging (fMRI) data has become a popular tool for the analysis of ADHD, which is the focus of our current research. In this paper we propose a novel framework for the automatic classification of the ADHD subjects using their resting state fMRI (rs-fMRI) data of the brain. We construct brain functional connectivity networks for all the subjects. The nodes of the network are constructed with clusters of highly active voxels and edges between any pair of nodes represent the correlations between their average fMRI time series. The activity level of the voxels are measured based on the average power of their corresponding fMRI time-series. For each node of the networks, a local descriptor comprising of a set of attributes of the node is computed. Next, the Multi-Dimensional Scaling (MDS) technique is used to project all the subjects from the unknown graph-space to a low dimensional space based on their inter-graph distance measures. Finally, the Support Vector Machine (SVM) classifier is used on the low dimensional projected space for automatic classification of the ADHD subjects. Exhaustive experimental validation of the proposed method is performed using the data set released for the ADHD-200 competition. Our method shows promise as we achieve impressive classification accuracies on the training (70.49%) and test data sets (73.55%). Our results reveal that the detection rates are higher when classification is performed separately on the male and female groups of subjects.

Interrelations of graph distance measures based on topological indices.

In this paper, we derive interrelations of graph distance measures by means of inequalities. For this investigation we are using graph distance measures based on topological indices that have not been studied in this context. Specifically, we are using the well-known Wiener index, Randić index, eigenvalue-based quantities and graph entropies. In addition to this analysis, we present results from numerical studies exploring various properties of the measures and aspects of their quality. Our results could find application in chemoinformatics and computational biology where the structural investigation of chemical components and gene networks is currently of great interest.

Comparison of Different Graph Distance Metrics for Semantic Text Based Classification

Nowadays semantic information of text is used largely for text classification task instead of bag-of-words approaches. This is due to having some limitations of bag of word approaches to represent text appropriately for certain kind of documents. On the other hand, semantic information can be represented through feature vectors or graphs. Among them, graph is normally better than traditional feature vector due to its powerful data structure. However, very few methodologies exist in the literature for semantic representation of graph. Error tolerant graph matching techniques such as graph similarity measures can be utilised for text classification. However, the techniques like Maximum Common Subgraph (mcs) and Minimum Common Supergraph (MCS) for graph similarity measures are computationally NP-hard problem. In the present paper summarized texts are used during extraction of semantic information to make it computationally faster. The semantic information of texts are represented through the discourse representation structures and later transformed into graphs. Five different graph distance measures based on Maximum Common Subgraph (mcs) and Minimum Common Supergraph (MCS) are used with k-NN classifier to evaluate text classification task. The text documents are taken from Reuters21578 text database distributed over 20 classes. Ten documents of each class for both training and testing purpose are used in the present work. From the results, it has been observed that the techniques have more or less equivalent potential to do text classification and as good as traditional bag-of-words approaches.

Efficient processing of graph similarity search

A graph similarity search is to find a set of graphs from a graph database that are similar to a given query graph. Existing works solve this problem by first defining a similarity measure between two graphs, and then presenting a filtering mechanism that reduces the number of candidate graphs. The candidate graphs are then verified by performing expensive graph search operations such as finding maximum common subgraphs. Existing works, however, do not report some similar graphs from a graph database while dissimilar graphs are not discarded during the filtering phase. To overcome this problem, in this paper, we first present a graph distance measure that can identify hidden but similar graphs that could not be discovered by previous graph distance measures. We then devise a series of filtering and validation rules to discard and identify non-matching and definitely-matching graphs, respectively, by calculating lower and upper bounds of the distance between a query and a data graph. To execute these filtering and validation rules efficiently during runtime, an index structure is also proposed. Lastly, a verification algorithm that verifies candidate graphs according to our graph distance measure is presented. Experiments on real datasets show that our approach can efficiently and effectively perform graph similarity search by significantly reducing the number of candidate graphs that must be verified, and by returning similar graphs.

On the subgraph epimorphism problem

In this paper we study the problem of deciding the existence of a subgraph epimorphism between two graphs. Our interest in this variant of graph matching problem stems from the study of model reductions in systems biology, where large systems of biochemical reactions can be naturally represented by bipartite digraphs of species and reactions. In this setting, model reduction can be formalized as the existence of a sequence of vertex deletion and merge operations that transforms a first reaction graph into a second graph. This problem is in turn equivalent to the existence of a subgraph (corresponding to delete operations) epimorphism (i.e. surjective homomorphism, corresponding to merge operations) from the first graph to the second. In this paper, we study theoretical properties of subgraph epimorphisms in general directed graphs. We first characterize subgraph epimorphisms (SEPI), subgraph isomorphisms (SISO) and graph epimorphisms (EPI) in terms of graph transformation operations. Then we study the graph distance measures induced by these transformations. We show that they define metrics on graphs and compare them. On the algorithmic side, we show that the SEPIexistence problem is NP-complete by reduction of SAT and present a constraint satisfaction algorithm that has been successfully used to solve practical SEPIproblems on a large benchmark of reaction graphs from systems biology.

Spanning tree separation reveals community structure in networks

We present a simple, intuitive, and effective approach for network clustering. It is based on basic concepts of linear algebra such as efficient calculation of spanning trees, and can be implemented in a few lines of code. We introduce the node separation measure spanning tree separation (STS) and the corresponding graph distance measure spanning tree vector similarity distance (STVSD). We demonstrate that the STS is a link salience measure able to identify the backbone of networks. The STVSD is used to reveal the hierarchical community structure of networks. We show that it, together with the clustering quality measure partition density, is on a par with the best graph or network clustering methods known, in terms of both quality and efficiency. In perspective, we note that our approach could also handle weighted and directed networks and could be used for identification of overlapping communities.

Structure-based graph distance measures of high degree of precision

In recent years, evaluating graph distance has become more and more important in a variety of real applications and many graph distance measures have been proposed. Among all of those measures, structure-based graph distance measures have become the research focus due to their independence of the definition of cost functions. However, existing structure-based graph distance measures have low degree of precision because only node and edge information of graphs are employed in these measures. To improve the precision of graph distance measures, we define substructure abundance vector (SAV) to capture more substructure information of a graph. Furthermore, based on SAV, we propose unified graph distance measures which are generalization of the existing structure-based graph distance measures. In general, the unified graph distance measures can evaluate graph distance in much finer grain. We also show that unified graph distance measures based on occurrence mapping and some of their variants are metrics. Finally, we apply the unified graph distance metric and its variants to the population evolution analysis and construct distance graphs of marker networks in three populations, which reflect the single nucleotide polymorphism (SNP) linkage disequilibrium (LD) differences among these populations.