Pairs Of Subgraphs Research Articles

Graph Anomaly Detection via Multi-Scale Contrastive Learning Networks with Augmented View

Graph anomaly detection (GAD) is a vital task in graph-based machine learning and has been widely applied in many real-world applications. The primary goal of GAD is to capture anomalous nodes from graph datasets, which evidently deviate from the majority of nodes. Recent methods have paid attention to various scales of contrastive strategies for GAD, i.e., node-subgraph and node-node contrasts. However, they neglect the subgraph-subgraph comparison information which the normal and abnormal subgraph pairs behave differently in terms of embeddings and structures in GAD, resulting in sub-optimal task performance. In this paper, we fulfill the above idea in the proposed multi-view multi-scale contrastive learning framework with subgraph-subgraph contrast for the first practice. To be specific, we regard the original input graph as the first view and generate the second view by graph augmentation with edge modifications. With the guidance of maximizing the similarity of the subgraph pairs, the proposed subgraph-subgraph contrast contributes to more robust subgraph embeddings despite of the structure variation. Moreover, the introduced subgraph-subgraph contrast cooperates well with the widely-adopted node-subgraph and node-node contrastive counterparts for mutual GAD performance promotions. Besides, we also conduct sufficient experiments to investigate the impact of different graph augmentation approaches on detection performance. The comprehensive experimental results well demonstrate the superiority of our method compared with the state-of-the-art approaches and the effectiveness of the multi-view subgraph pair contrastive strategy for the GAD task. The source code is released at https://github.com/FelixDJC/GRADATE.

Characterizing Forbidden Pairs for the Edge-Connectivity of a Connected Graph to Be Its Minimum Degree

Let H be a class of given graphs. A graph G is said to be H-free if G contains no induced copies of H for any H∈H. In this article, we characterize all connected subgraph pairs {R,S} guranteeing the edge-connectivity of a connected {R,S}-free graph to have the same minimum degree. Our result is a supplement of Wang et al. Furthermore, we obtain a relationship of forbidden sets when those general parameters have the recurrence relation.

Mining Topological Dependencies of Recurrent Congestion in Road Networks

The discovery of spatio-temporal dependencies within urban road networks that cause Recurrent Congestion (RC) patterns is crucial for numerous real-world applications, including urban planning and the scheduling of public transportation services. While most existing studies investigate temporal patterns of RC phenomena, the influence of the road network topology on RC is often overlooked. This article proposes the ST-Discovery algorithm, a novel unsupervised spatio-temporal data mining algorithm that facilitates effective data-driven discovery of RC dependencies induced by the road network topology using real-world traffic data. We factor out regularly reoccurring traffic phenomena, such as rush hours, mainly induced by the daytime, by modelling and systematically exploiting temporal traffic load outliers. We present an algorithm that first constructs connected subgraphs of the road network based on the traffic speed outliers. Second, the algorithm identifies pairs of subgraphs that indicate spatio-temporal correlations in their traffic load behaviour to identify topological dependencies within the road network. Finally, we rank the identified subgraph pairs based on the dependency score determined by our algorithm. Our experimental results demonstrate that ST-Discovery can effectively reveal topological dependencies in urban road networks.

Graph partitioning and graph neural network based hierarchical graph matching for graph similarity computation

Graph similarity computation aims to predict a similarity score between one pair of graphs to facilitate downstream applications, such as finding the most similar chemical compounds similar to a query compound or Fewshot 3D Action Recognition. Recently, some graph similarity computation models based on neural networks have been proposed, which are either based on graph-level interaction or node-level comparison. However, when the number of nodes in the graph increases, it will inevitably bring about reduced representation ability or high computation cost.Motivated by this observation, we propose a graph partitioning and graph neural network-based model, called PSimGNN, to effectively resolve this issue. Specifically, each of the input graphs is partitioned into a set of subgraphs to extract the local structural features directly. Next, a novel graph neural network with an attention mechanism is designed to map each subgraph into an embedding vector. Some of these subgraph pairs are automatically selected for node-level comparison to supplement the subgraph-level embedding with fine-grained information. Finally, coarse-grained interaction information among subgraphs and fine-grained comparison information among nodes in different subgraphs are integrated to predict the final similarity score. Experimental results on graph datasets with different graph sizes demonstrate that PSimGNN outperforms state-of-the-art methods in graph similarity computation tasks using approximate Graph Edit Distance (GED) as the graph similarity metric.

Top-k overlapping densest subgraphs: approximation algorithms and computational complexity

A central problem in graph mining is finding dense subgraphs, with several applications in different fields, a notable example being identifying communities. While a lot of effort has been put in the problem of finding a single dense subgraph, only recently the focus has been shifted to the problem of finding a set of densest subgraphs. An approach introduced to find possible overlapping subgraphs is the Top-k-Overlapping Densest Subgraphs problem. Given an integer k ge 1 and a parameter lambda > 0, the goal of this problem is to find a set of k dense subgraphs that may share some vertices. The objective function to be maximized takes into account the density of the subgraphs, the parameter lambda and the distance between each pair of subgraphs in the solution. The Top-k-Overlapping Densest Subgraphs problem has been shown to admit a frac{1}{10}-factor approximation algorithm. Furthermore, the computational complexity of the problem has been left open. In this paper, we present contributions concerning the approximability and the computational complexity of the problem. For the approximability, we present approximation algorithms that improve the approximation factor to frac{1}{2}, when k is smaller than the number of vertices in the graph, and to frac{2}{3}, when k is a constant. For the computational complexity, we show that the problem is NP-hard even when k=3.

Organic skeleton correspondence using part arrangements

Shape correspondence between semantically similar organic shapes with large shape variations is a difficult problem in shape analysis. Since part geometries are no longer similar, we claim that the challenge is to extract and compare prominent shape substructures, which are recurring part arrangements among semantically related shapes. Our main premise is that the challenge can be solved more efficiently on curve skeleton graphs of shapes, which provide a concise abstraction of shape geometry and structure. Instead of directly searching exponentially many skeleton subgraphs, our method extracts the intrinsic reflectional symmetry axis of the skeleton to guide the generation of subgraphs as part arrangements. For any two subgraphs from two skeletons, their orientations are aligned and their pose variations are normalized for matching. Finally, the matchings of all subgraph pairs are evaluated and accumulated to the skeletal feature node correspondences. The comparison results with the state-of-the-art work show that our method significantly improves the efficiency and accuracy of the semantic correspondence between a variety of shapes.

Geometric discriminative deep features for traffic image analysis

Traffic image analysis is an important application in intelligent transportation. For local features’ robustness to image variances, such as scale changes and occlusions, they are widely used in image classification. However, how to integrate these local features for modeling traffic images optimally is still a crucial challenge. In this paper, a novel deep learning method, geometric discriminative feature fusion (GDFF), is proposed to tackle this problem. First, we use a variety of data sets to train the general convolutional neural network (CNN), which is used to extract the features of the training and test set after deep level. Deep architecture makes it possible for people to learn more abstract and internal features that are robust to changes in viewpoint and illumination. It can fuse image geometric related local features, such as local regions’ RGB histograms, into high level discriminative features, which can be used for better classifying complex scene images. Our framework’s central task is to build a structural kernel, called discriminative topological kernel. Firstly, we segment the traffic images into several regions and use a region connected graph (RCG) to model regions location relationships. We use frequent sub graph mining algorithm to mine all frequent sub structures (topologies) occurs in all training RCGs. And a selection algorithm is designed to select the k qualified topologies from the entire mined frequent topologies. We call these selected topologies geometric feature fusers, which are both high discriminative and low redundant structures in all training RCGs. Finally, given a pair of RCGs and to each geometric fuser, we extract all pairs of sub graphs sharing the same topology and calculate distance between them. All k distances are accumulated for the final kernel. The experimental result demonstrates the effectiveness of our method.

Arbitrary Overlap Constraints in Graph Packing Problems

In earlier versions of the community discovering problem, the overlap between communities was restricted by a simple count upper-bound. In this paper, we introduce the [Formula: see text]-Packing with [Formula: see text]-Overlap problem to allow for more complex constraints in the overlap region than those previously studied. Let [Formula: see text] be all possible subsets of vertices of [Formula: see text] each of size at most [Formula: see text], and [Formula: see text] be a function. The [Formula: see text]-Packing with [Formula: see text]-Overlap problem seeks at least [Formula: see text] induced subgraphs in a graph [Formula: see text] subject to: (i) each subgraph has at most [Formula: see text] vertices and obeys a property [Formula: see text], and (ii) for any pair [Formula: see text], with [Formula: see text], [Formula: see text] (i.e., the pair [Formula: see text] does not conflict). We also consider a variant that arises in clustering applications: each subgraph of a solution must contain a set of vertices from a given collection of sets [Formula: see text], and no pair of subgraphs may share vertices from the sets of [Formula: see text]. In addition, we propose similar formulations for packing hypergraphs. We give an [Formula: see text] algorithm for our problems where [Formula: see text] is the parameter and [Formula: see text] and [Formula: see text] are constants, provided that: (i) [Formula: see text] is computable in polynomial time in [Formula: see text] and (ii) the function [Formula: see text] satisfies specific conditions. Specifically, [Formula: see text] is hereditary, applicable only to overlapping subgraphs, and computable in polynomial time in [Formula: see text] and [Formula: see text]. Motivated by practical applications we give several examples of [Formula: see text] functions which meet those conditions.

Four forbidden subgraph pairs for hamiltonicity of 3-connected graphs

For non-negative integers i, j and k, we denote the generalized net as N i,j,k , which is a triangle with disjoint paths of length i, j and k, attached to distinct vertices of the triangle. In this paper, we prove that every 3-connected {K 1,3,N 8−i,i,1}-free graph is hamiltonian, where 1 ≤ i ≤ 4.

A branch-and-price-and-cut method for computing an optimal bramble

Given an undirected graph, a bramble is a set of connected subgraphs (called bramble elements) such that every pair of subgraphs either contains a common node, or such that an edge (i,j) exists with node i belonging to one subgraph and node j belonging to the other. In this paper we examine the problem of finding the bramble number of a graph, along with a set of bramble elements that yields this number. The bramble number is the largest cardinality of a minimum hitting set over all bramble elements on this graph. A graph with bramble number k has a treewidth of k−1. We provide a branch-and-price-and-cut method that generates columns corresponding to bramble elements, and rows corresponding to hitting sets. We then examine the computational efficacy of our algorithm on a randomly generated data set.

A Network of Networks Perspective on Global Trade.

Mutually intertwined supply chains in contemporary economy result in a complex network of trade relationships with a highly non-trivial topology that varies with time. In order to understand the complex interrelationships among different countries and economic sectors, as well as their dynamics, a holistic view on the underlying structural properties of this network is necessary. This study employs multi-regional input-output data to decompose 186 national economies into 26 industry sectors and utilizes the approach of interdependent networks to analyze the substructure of the resulting international trade network for the years 1990–2011. The partition of the network into national economies is observed to be compatible with the notion of communities in the sense of complex network theory. By studying internal versus cross-subgraph contributions to established complex network metrics, new insights into the architecture of global trade are obtained, which allow to identify key elements of global economy. Specifically, financial services and business activities dominate domestic trade whereas electrical and machinery industries dominate foreign trade. In order to further specify each national sector’s role individually, (cross-)clustering coefficients and cross-betweenness are obtained for different pairs of subgraphs. The corresponding analysis reveals that specific industrial sectors tend to favor distinct directionality patterns and that the cross-clustering coefficient for geographically close country pairs is remarkably high, indicating that spatial factors are still of paramount importance for the organization of trade patterns in modern economy. Regarding the evolution of the trade network’s substructure, globalization is well-expressed by trends of several structural characteristics (e.g., link density and node strength) in the interacting network framework. Extreme events, such as the financial crisis 2008/2009, are manifested as anomalies superimposed to these trends. The marked reorganization of trade patterns, associated with this economic crisis in comparison to “normal” annual fluctuations in the network structure is traced and quantified by a new widely applicable generalization of the Hamming distance to weighted networks.

Characterizing Heavy Subgraph Pairs for Pancyclicity

Earlier results originating from Bedrossian's PhD Thesis focus on characterizing pairs of forbidden subgraphs that imply hamiltonian properties. Instead of forbidding certain induced subgraphs, here we relax the requirements by imposing Ore-type degree conditions on the induced subgraphs. In particular, adopting the terminology introduced by Cada, for a graph $$G$$ G on $$n$$ n vertices and a fixed graph $$H$$ H , we say that $$G$$ G is $$H$$ H - $$o_1$$ o 1 -heavy if every induced subgraph of $$G$$ G isomorphic to $$H$$ H contains two nonadjacent vertices with degree sum at least $$n+1$$ n + 1 in $$G$$ G . For a family $${\mathcal {H}}$$ H of graphs, $$G$$ G is called $${\mathcal {H}}$$ H - $$o_{1}$$ o 1 -heavy if $$G$$ G is $$H$$ H - $$o_1$$ o 1 -heavy for every $$H\in \mathcal {H}$$ H ? H . In this paper we characterize all connected graphs $$R$$ R and $$S$$ S other than $$P_3$$ P 3 (the path on three vertices) such that every 2-connected $$\{R,S\}$$ { R , S } - $$o_1$$ o 1 -heavy graph is either a cycle or pancyclic, thereby extending previous results on forbidden subgraph conditions for pancyclicity and on heavy subgraph conditions for hamiltonicity.

Coupling Graphs, Efficient Algorithms and B-Cell Epitope Prediction.

Coupling graphs are newly introduced in this paper to meet many application needs particularly in the field of bioinformatics. A coupling graph is a two-layer graph complex, in which each node from one layer of the graph complex has at least one connection with the nodes in the other layer, and vice versa. The coupling graph model is sufficiently powerful to capture strong and inherent associations between subgraph pairs in complicated applications. The focus of this paper is on mining algorithms of frequent coupling subgraphs and bioinformatics application. Although existing frequent subgraph mining algorithms are competent to identify frequent subgraphs from a graph database, they perform poorly on frequent coupling subgraph mining because they generate many irrelevant subgraphs. We propose a novel graph transformation technique to transform a coupling graph into a generic graph. Based on the transformed coupling graphs, existing graph mining methods are then utilized to discover frequent coupling subgraphs. We prove that the transformation is precise and complete and that the restoration is reversible. Experiments carried out on a database containing 10,511 coupling graphs show that our proposed algorithm reduces the mining time very much in comparison with the existing subgraph mining algorithms. Moreover, we demonstrate the usefulness of frequent coupling subgraphs by applying our algorithm to make accurate predictions of epitopes in antibody-antigen binding.

Parameterized Algorithms for the H-Packing with t-Overlap Problem

We introduce the k-H-Packing with t-Overlap problem to formalize the problem of discovering overlapping communities in real networks. More precisely, in the k-H-Packing with t-Overlap problem, we search in a graph G for at least k subgraphs each isomorphic to a graph H such that any pair of subgraphs shares at most t vertices. In contrast with previous work where communities are disjoint, we regulate the overlap through a variable t. Our focus is on the parameterized complexity of the k-H-Packing with t-Overlap problem. Here, we provide a new technique for this problem generalizing the crown decomposition technique [2]. Using our global rule, we achieve a kernel with size bounded by 2(rk r) for the k-Kr-Packing with (r 2)Overlap problem. That is, when H is a clique of size r and t = r 2. In addition, we introduce the first parameterized algorithm for the kH-Packing with t-Overlap problem when H is an arbitrary graph of size r. Our algorithm combines a bounded search tree with a greedy localization technique and runs in time O(r rk k (r t 1)k+2 n r ), where n = |V (G)|, r = |V (H)|, and t < r. Finally, we apply this search tree algorithm to the kernel obtained for the k-Kr-Packing with (r 2)-Overlap problem, and we show that this approach is faster than applying a brute-force algorithm in the kernel. In all our results, r and t are constants.

Heavy subgraph pairs for traceability of block-chains

A graph is called traceable if it contains a Hamilton path, i.e., a path containing all its vertices. Let G be a graph on n vertices. We say that an induced subgraph of G is o-1-heavy if it contains two nonadjacent vertices which satisfy an Ore-type degree condition for traceability, i.e., with degree sum at least n−1 in G. A block-chain is a graph whose block graph is a path, i.e., it is either a P1, P2, or a 2-connected graph, or a graph with at least one cut vertex and exactly two end-blocks. Obviously, every traceable graph is a block-chain, but the reverse does not hold. In this paper we characterize all the pairs of connected o-1-heavy graphs that guarantee traceability of block-chains. Our main result is a common extension of earlier work on degree sum conditions, forbidden subgraph conditions and heavy subgraph conditions for traceability.

Two Forbidden Subgraph Pairs for Hamiltonicity of 3-Connected Graphs

For integers i, j, k with $${i\geq j\geq k\geq 0}$$ , let N i, j, k be the graph obtained by identifying end vertices of three disjoint paths of lengths i, j, k to the vertices of a triangle. In this paper, we show that every 3-connected {K 1,3, N i, 7-i, 2}-free graph is hamiltonian, where $${i \in \{4,5\}}$$ . This result is sharp in the sense that no one of the numbers i, 7?i and 2 in N i, 7-i, 2 can be replaced by a larger number.

On the maximum positive semi-definite nullity and the cycle matroid of graphs

Let G =( V,E )b e ag raph withV = {1,2,...,n}, in which we allow parallel edges but no loops, and let S+(G) be the set of all positive semi-definite n × n matrices A =( ai,j )w ith ai,j =0i fij and i and j are non-adjacent, ai,jfij and i and j are connected by exactly one edge, and ai,j ∈ R if i = j or i and j are connected by parallel edges. The maximum positive semi-definite nullity of G, denoted by M+(G), is the maximum nullity attained by any matrix A ∈S +(G). A k-separation of G is a pair of subgraphs (G1,G2) such that V (G1) ∪ V (G2 )= V , E(G1) ∪ E(G2 )= E, E(G1) ∩ E(G2 )= ∅ and |V (G1) ∩ V (G2)| = k .W henG has a k-separation (G1,G2 )w ithk ≤ 2, we give a formula for the maximum positive semi-definite nullity of G in terms of G1,G2 ,a nd in case ofk = 2, also two other specified graphs. For a graph G ,l etcG denote the number of components in G. As a corollary of the result on k-separations with k ≤ 2, we obtain that M+(G) − cG = M+(G � ) − cG for graphs G and Gthat have isomorphic cycle matroids.

The Identification of Similarities between Biological Networks: Application to the Metabolome and Interactome

The increasing interest in systems biology has resulted in extensive experimental data describing networks of interactions (or associations) between molecules in metabolism, protein–protein interactions and gene regulation. Comparative analysis of these networks is central to understanding biological systems. We report a novel method (PHUNKEE: Pairing subgrapHs Using NetworK Environment Equivalence) by which similar subgraphs in a pair of networks can be identified. Like other methods, PHUNKEE explicitly considers the graphical form of the data and allows for gaps. However, it is novel in that it includes information about the context of the subgraph within the adjacent network. We also explore a new approach to quantifying the statistical significance of matching subgraphs. We report similar subgraphs in metabolic pathways and in protein–protein interaction networks. The most similar metabolic subgraphs were generally found to occur in processes central to all life, such as purine, pyrimidine and amino acid metabolism. The most similar pairs of subgraphs found in the protein–protein interaction networks of Drosophila melanogaster and Saccharomyces cerevisiae also include central processes such as cell division but, interestingly, also include protein sub–networks involved in pre-mRNA processing. The inclusion of network context information in the comparison of protein interaction networks increased the number of similar subgraphs found consisting of proteins involved in the same functional process. This could have implications for the prediction of protein function.

Sufficient conditions for [formula omitted]-optimality of graphs with small conditional diameter

A restricted edge-cut S of a connected graph G is an edge-cut such that G − S has no isolated vertex. The restricted edge-connectivity λ ′ ( G ) is the minimum cardinality over all restricted edge-cuts. A graph is said to be λ ′ -optimal if λ ′ ( G ) = ξ ( G ) , where ξ ( G ) denotes the minimum edge-degree of G defined as ξ ( G ) = min { d ( u ) + d ( v ) − 2 : u v ∈ E ( G ) } . The P -diameter of G measures how far apart a pair of subgraphs satisfying a given property P can be, and hence it generalizes the standard concept of diameter. In this paper we prove two kind of results, according to which property P is chosen. First, let D 1 (resp. D 2 ) be the P -diameter where P is the property that the corresponding subgraphs have minimum degree at least one (resp. two). We prove that a graph with odd girth g is λ ′ -optimal if D 1 ⩽ g − 2 and D 2 ⩽ g − 5 . For even girth we obtain a similar result. Second, let F ⊂ V ( G ) with | F | = δ − 1 , δ ⩾ 2 , being the minimum degree of G. Using the property Q of being vertices of G − F we prove that a graph with girth g ∉ { 4 , 6 , 8 } is λ ′ -optimal if this Q -diameter is at most 2 ⌈ ( g − 3 ) / 2 ⌉ .

Thep-intersection number of a complete bipartite graph and orthogonal double coverings of a clique

Thep-intersection graph of a collection of finite sets {S i } i=1 n is the graph with vertices 1, ...,n such thati, j are adjacent if and only if |S i ∩S j |≥p. Thep-intersection number of a graphG, herein denoted θ p (G), is the minimum size of a setU such thatG is thep-intersection graph of subsets ofU. IfG is the complete bipartite graphK n,n andp≥2, then θ p (K n, n )≥(n 2+(2p−1)n)/p. Whenp=2, equality holds if and only ifK n has anorthogonal double covering, which is a collection ofn subgraphs ofK n , each withn−1 edges and maximum degree 2, such that each pair of subgraphs shares exactly one edge. By construction,K n has a simple explicit orthogonal double covering whenn is congruent modulo 12 to one of {1, 2, 5, 7, 10, 11}.