Real Graphs Research Articles

Fast Continuous Subgraph Matching over Streaming Graphs via Backtracking Reduction

Streaming graphs are drawing increasing attention in both academic and industrial communities as many graphs in real applications evolve over time. Continuous subgraph matching (shorted as CSM) aims to report the incremental matches of a query graph in such streaming graphs. It involves two major steps, i.e., candidate maintenance and incremental match generation, to answer CSM. Throughout the course of continuous subgraph matching, incremental match generation backtracking over the search space dominates the total cost. However, most previous approaches focus on developing techniques for efficient candidate maintenance, while incremental match generation receives less attention despite its importance in CSM. Aiming to minimize the overall cost, we propose two techniques to reduce backtrackings in this paper. We present a cost-effective index CaLiG that yields tighter candidate maintenance, shrinking the search space of backtracking. In addition, we develop a novel incremental matching paradigm KSS that decomposes the query vertices into conditional kernel vertices and shell vertices. With the matches of kernel vertices, the incremental matches can be produced immediately by joining the candidates of shell vertices without any backtrackings. Benefiting from reduced backtrackings, the elapsed time of CSM decreases significantly. Extensive experiments over real graphs show that our method runs faster than the state-of-the-art algorithm orders of magnitude.

Personalized sampling graph collection with local differential privacy for link prediction

Link prediction (LP) is an attractive research problem on social network data. Yet, the link information may be leaked by the untrusted collector. As countermeasures, there are a few methods specially designed for link prediction under local differential privacy (LDP), which allow the third-party data collector collects user data for link prediction while protecting user connection privacy. In this paper, we propose a C ommunity-based graph collection with P ersonalized sampling R andomized R esponse (CPRR) which is a novel graph collection algorithm with LDP and reaches decent trade-off between sensitive link protection and link prediction performance. The proposed mechanism adopts a personalized sampling technique for each connection of a user and then utlizes the randomized response mechanism on the sampled subset. Based on the personalized sampling technique, we can reduce the injected noise in LDP. Meanwhile, considering the different edge distributions of the different regions on the original graph, we propose a community-based sampling strategy. Then, we prove that the proposed CPRR satisfies LDP. Through extensive experiments on several real-life graph datasets, we demonstrate that CPRR can achieve better results in balancing privacy protection and link prediction performance than the state-of-art baselines.

MiniGraph: Querying Big Graphs with a Single Machine

This paper presents MiniGraph, an out-of-core system for querying big graphs with a single machine. As opposed to previous single-machine graph systems, MiniGraph proposes a pipelined architecture to overlap I/O and CPU operations, and improves multi-core parallelism. It also introduces a hybrid model to support both vertex-centric and graph-centric parallel computations, to simplify parallel graph programming, speed up beyond-neighborhood computations, and parallelize computations within each subgraph. The model induces a two-level parallel execution model to explore both inter-subgraph and intra-subgraph parallelism. Moreover, MiniGraph develops new optimization techniques under its architecture. Using real-life graphs of different types, we show that MiniGraph is up to 76.1x faster than prior out-of-core systems, and performs better than some multi-machine systems that use up to 12 machines.

MaNIACS : Approximate Mining of Frequent Subgraph Patterns through Sampling

We present MaNIACS , a sampling-based randomized algorithm for computing high-quality approximations of the collection of the subgraph patterns that are frequent in a single, large, vertex-labeled graph, according to the Minimum Node Image-based (MNI) frequency measure. The output of MaNIACS comes with strong probabilistic guarantees, obtained by using the empirical Vapnik–Chervonenkis (VC) dimension, a key concept from statistical learning theory, together with strong probabilistic tail bounds on the difference between the frequency of a pattern in the sample and its exact frequency. MaNIACS leverages properties of the MNI-frequency to aggressively prune the pattern search space, and thus to reduce the time spent in exploring subspaces that contain no frequent patterns. In turn, this pruning leads to better bounds to the maximum frequency estimation error, which leads to increased pruning, resulting in a beneficial feedback effect. The results of our experimental evaluation of MaNIACS on real graphs show that it returns high-quality collections of frequent patterns in large graphs up to two orders of magnitude faster than the exact algorithm.

Computing Graph Edit Distance via Neural Graph Matching

Graph edit distance (GED) computation is a fundamental NP-hard problem in graph theory. Given a graph pair ( G 1 , G 2 ), GED is defined as the minimum number of primitive operations converting G 1 to G 2 . Early studies focus on search-based inexact algorithms such as A*-beam search, and greedy algorithms using bipartite matching due to its NP-hardness. They can obtain a sub-optimal solution by constructing an edit path (the sequence of operations that converts G 1 to G 2 ). Recent studies convert the GED between a given graph pair ( G 1 , G 2 ) into a similarity score in the range (0, 1) by a well designed function. Then machine learning models (mostly based on graph neural networks) are applied to predict the similarity score. They achieve a much higher numerical precision than the sub-optimal solutions found by classical algorithms. However, a major limitation is that these machine learning models cannot generate an edit path. They treat the GED computation as a pure regression task to bypass its intrinsic complexity, but ignore the essential task of converting G 1 to G 2 . This severely limits the interpretability and usability of the solution. In this paper, we propose a novel deep learning framework that solves the GED problem in a two-step manner: 1) The proposed graph neural network GEDGNN is in charge of predicting the GED value and a matching matrix; and 2) A post-processing algorithm based on k -best matching is used to derive k possible node matchings from the matching matrix generated by GEDGNN. The best matching will finally lead to a high-quality edit path. Extensive experiments are conducted on three real graph data sets and synthetic power-law graphs to demonstrate the effectiveness of our framework. Compared to the best result of existing GNN-based models, the mean absolute error (MAE) on GED value prediction decreases by 4.9% ~ 74.3%. Compared to the state-of-the-art searching algorithm Noah, the MAE on GED value based on edit path reduces by 53.6% ~ 88.1%.

A multi-scale graph embedding method via multiple corpora

Graph embedding aims at learning continuous vector representations for graphs which is crucial for graph analytics. Natural Language Process (NLP)-based graph embedding methods build corpus for graph data by treating substructures as words and then use NLP models to learn graph embeddings. However, the size difference and data redundancy among substructures are less explored in the built corpora. To mitigate this problem, we propose an unsupervised multi-scale graph embedding method. To be a specific, we first build multiple graph corpora for a graph dataset, where each corpus only contains substructures of specific granularity. Then, we extend a document embedding model to each graph corpus to obtain graph embeddings of different scales. At last, we obtain the final multi-scale embedding of a graph by pooling its multiple embeddings. Comprehensive experiments on real graph datasets indicate that the proposed method obtains competitive results with state-of-the-arts, and is superior to some classic graph kernels and graph embedding methods on six out of ten benchmark datasets.

Buckle Pose Estimation Using a Generative Adversarial Network

The buckle before the lens coating is still typically disassembled manually. The difference between the buckle and the background is small, while that between the buckles is large. This mechanical disassembly can also damage the lens. Therefore, it is important to estimate pose with high accuracy. This paper proposes a buckle pose estimation method based on a generative adversarial network. An edge extraction model is designed based on a segmentation network as the generator. Spatial attention is added to the discriminator to help it better distinguish between generated and real graphs. The generator thus generates delicate external contours and center edge lines with help from the discriminator. The external rectangle and the least square methods are used to determine the center position and deflection angle of the buckle, respectively. The center point and angle accuracies of the test datasets are 99.5% and 99.3%, respectively. The pixel error of the center point distance and the absolute error of the angle to the horizontal line are within 7.36 pixels and 1.98°, respectively. This method achieves the highest center point and angle accuracies compared to Hed, RCF, DexiNed, and PidiNet. It can meet practical requirements and boost the production efficiency of lens coatings.

A fast approximate method for k-edge connected component detection in graphs with high accuracy

Finding k-edge connected components (k-ECCs) is widely used to investigate the major cohesive structures within a graph. Traditional methods utilizing global minimum cut to split the graph iteratively are too slow to deal with graphs of a large size. Recently, approximate algorithms based on a decomposition framework have been proposed to speed up the process. However, these algorithms produce low-quality results in graphs containing vertex pairs that are k-connected but do not belong to a k-ECC. In this paper, we propose a decomposition framework that utilizes a local edge connectivity check method to obtain high-quality results in a shorter period. The proposed method utilizes an improved maximum flow algorithm to detect k-ECCs from local k-cores by merging more vertices in each iteration than existing methods. Moreover, a pruning strategy is applied in the decomposition framework to reduce unnecessary branches in the decomposition tree. Our experimental results on both synthetic and real graphs show that our proposed algorithm is faster than the state-of-the-art algorithms in most datasets and achieves effective results among the approximate algorithms with a 96% accuracy on our experimental datasets.

Efficient graph edit distance computation using isomorphic vertices

In this paper, we study the problem of graph edit distance (GED) computation. We empirically observe that real graph data contain many isomorphic substructures, which incur redundant computation. Based on this observation, we aim at reducing the cost of GED computation by avoiding redundant computation caused by isomorphic substructures. To detect isomorphic substructures, we precisely define a notion of vertex isomorphism and propose a dynamic programming algorithm that identifies isomorphic vertices in a graph. By taking advantage of isomorphic vertices, we develop an efficient GED computation algorithm. In experiments, we show that isomorphic vertices are effectively used in reducing the search space of GED computation, and as a result our approach improves the performance of GED computation.

О ПРИМЕНИМОСТИ ОДНОЙ МАРКОВСКОЙ МОДЕЛИ РАСПРОСТРАНЕНИЯ КОМПЬЮТЕРНЫХ ВИРУСОВ К E-MAIL-ГРАФАМ

Mathematical modeling of computer virus epidemics is the most important area of theoretical research in the field of information security. The objective of this paper is to analyze the applicability of the modified Reed-Frost model for epidemic analysis on real-life graphs. We compare the results calculated using the model with the results obtained during the simulation experiment with different epidemic parameters. As a result, the epidemic parameters, in which both methods are in good agreement with each other, have been identified.

On the Properties of Next Generation Wireless Backhaul

With the advent of 5G, cellular networks require a high number of base stations, possibly interconnected with wireless links, an evolution introduced in the last revision of 5G as the Integrated Access and Backhaul (IAB). Researchers are now working to optimize the complex topologies of the backhaul network, using synthetic models for the underlying visibility graph, i.e., the graph of possible connections between the base stations. The goal of this paper is to provide a novel methodology to generate visibility graphs starting from real data (and the data sets themselves together with the source code for their manipulation), in order to base the IAB design and optimization on assumptions that are as close as possible to reality. We introduce a GPU-based method to create visibility graphs from open data, we analyze the properties of the realistic visibility graphs, and we show that different geographic areas produce very different graphs. We run state-of-the-art algorithms to create wireless backhaul networks on top of visibility graphs, and we show that the results that exploit synthetic models are far from those that use our realistic graphs. Our conclusion is that the data-based approach we propose is essential to design mobile networks that work in a variety of real-world situations.

A Centralization Measure for Social Networks Assessment

This article presents a new centrality measure. The presented measure was compared with state-of-the-art measures and proved to combines two features, low computational complexity and high precision of calculations. The measure has been mathematical analyzed, and then, experiments on real-life graphs were conducted. The mathematical analysis showed many essential properties of the proposed measure. Statistical analysis of the conducted experiment showed that measure has precision the same as state-of-the-art measures with much lower computational complexity.

Learning attentive attribute-aware node embeddings in dynamic environments

Learning node embeddings is fundamental for numerous applications, such as link prediction and node classification. Node embeddings seek to learn a low-dimensional representation for each node in the graph. Many existing node representation learning methods for dynamic attributed graphs focus on preserving the temporal proximity of the nodes with relatively shallow models. However, real-life graphs are complex and usually exhibit evolutionary patterns of node attributes and graph structure. Therefore, the current state-of-the-art models fail to capture the information in the dynamic attributed graphs and settle for sub-optimal results. In this paper, we propose a novel model for embedding nodes in dynamic attributed graphs that captures the full extent of all relevant node information and the graph interrelations as well as graph evolutionary patterns. During model training, attribute-aware node embedding is learned using both graph and node properties in a dynamic context. Experiments demonstrate that our proposed method is superior to the state-of-the-art models in link prediction tasks. In addition, it introduces a novel way of learning richer representations by fully exploiting node attributes, graph structure, and evolutionary patterns in dynamic attributed graphs.

Quasi-Stable Coloring for Graph Compression

We propose quasi-stable coloring , an approximate version of stable coloring. Stable coloring, also called color refinement, is a well-studied technique in graph theory for classifying vertices, which can be used to build compact, lossless representations of graphs. However, its usefulness is limited due to its reliance on strict symmetries. Real data compresses very poorly using color refinement. We propose the first, to our knowledge, approximate color refinement scheme, which we call quasi-stable coloring. By using approximation, we alleviate the need for strict symmetry, and allow for a tradeoff between the degree of compression and the accuracy of the representation. We study three applications: Linear Programming, Max-Flow, and Betweenness Centrality, and provide theoretical evidence in each case that a quasi-stable coloring can lead to good approximations on the reduced graph. Next, we consider how to compute a maximal quasi-stable coloring: we prove that, in general, this problem is NP-hard, and propose a simple, yet effective algorithm based on heuristics. Finally, we evaluate experimentally the quasi-stable coloring technique on several real graphs and applications, comparing with prior approximation techniques.

In pursuit of the hidden features of GNN’s internal representations

We consider the problem of explaining Graph Neural Networks (GNNs). While most attempts aim at explaining the final decision of the model, we focus on the hidden layers to examine what the GNN actually captures and shed light on the hidden features built by the GNN. To that end, we first extract activation rules that identify sets of exceptionally co-activated neurons when classifying graphs in the same category. These rules define internal representations having a strong impact in the classification process. Then – this is the goal of the current paper – we interpret these rules by identifying a graph that is fully embedded in the related subspace identified by the rule. The graph search is based on a Monte Carlo Tree Search directed by a proximity measure between the graph embedding and the internal representation of the rule, as well as a realism factor that constrains the distribution of the labels of the graph to be similar to that observed on the dataset. Experiments including 6 real-world datasets and 3 baselines demonstrate that our method DISCERN generates realistic graphs of high quality which allows providing new insights into the respective GNN models.

A linear algorithm for semi-external cutnode computation

In the literature, many algorithms have been proposed for finding cutnodes on undirected graphs, since cutnodes are crucial to graph connectivity. Here, a cutnode of an undirected graph G is a node of G, whose deletion will cause a reachable pair of the other nodes in G to be unreachable. Currently, the difficulty of maintaining the entire G in the main memory makes researchers pay attention to compute cutnodes on semi-external memory model. This paper shows that traditional semi-external algorithms are limited by their unbounded time and I/O consumption, making them impractical when G is relatively large or complex. Thus, we propose a linear semi-external cutnode computation algorithm, named SECN. Assuming that G has n nodes and m edges, and B is the disk block size. SECN is the first that can find all the cutnodes of G in O(m+n) time and with O(mB) I/O cost on the semi-external memory model, as far as we know. SECN also has a smaller minimum memory space requirement than traditional algorithms. Our experimental evaluation conducted on both synthetic and real graphs confirms that SECN significantly outperforms existing algorithms (up to 103 times faster).

Efficient maximum k -plex computation over large sparse graphs

The k -plex model is a relaxation of the clique model by allowing every vertex to miss up to k neighbors. Designing exact and efficient algorithms for computing a maximum k -plex in a graph has been receiving increasing interest recently. However, the existing algorithms are still inefficient due to having major limitations. We in this paper design a new algorithm kPlexS for the maximum k -plex problem, with three novel contributions. Firstly, we propose a new framework for computing maximum k -plex over large sparse graphs, by iteratively extracting small dense subgraphs from it and then solving each of the extracted dense subgraphs by a branch-and-bound search. Secondly, we propose an efficient reduction algorithm CTCP to reduce the input graph size by exhaustively conducting vertex reduction and edge reduction. CTCP computes a smaller reduced graph and also has a lower time complexity than the existing techniques. Moreover, we iteratively invoke CTCP to reduce the input graph once a vertex has been processed and removed from it. Thirdly, we develop a branch-and-bound algorithm BBMatrix specifically targeting the dense subgraphs that are extracted from the input graph. BBMatrix represents its input graph by an adjacency matrix, and utilizes both first-order (i.e., individual vertices) and second-order information (i.e., pairs of vertices) for reduction and upper bounding. In addition, incremental techniques are proposed to efficiently apply the reduction and upper bounding during the recursion. Extensive empirical studies on large real graphs demonstrate that our algorithm kPlexS outperforms the state-of-the-art algorithms BnB, Maplex, and KpLeX.

Efficient index-free SimRank similarity search in large graphs by discounting path lengths

Link-based similarity search aims to find similar nodes for a given query node in a graph, which arises in numerous applications, including web spam detection, social network analysis and web search. Among existing methods, SimRank is a well-known similarity model, which provides an effective and trustful function for similarity search. A large amount of techniques on SimRank similarity search are devoted recently, which compute the similarity scores by traversing the paths between query and candidate nodes. However, the number of paths increases exponentially as path length increases, which makes the computation expensive and cannot support fast similarity search over large graphs. In this paper, we propose an efficient index-free SimRank similarity search approach, namely DisSim, which reduces the computational cost by discounting path length. We observe that SimRank could rapidly converge at a stable state and the results change little after a few of iterations. Based on the fast convergence, the similarity between nodes is defined as the SimRank score at the second iteration. For the computation of DisSim, we divide the similarity into one-step and two-step first-meeting probabilities. The one-step first-meeting probabilities are computed by path traverses from query to candidate nodes, which reduces computational cost by skipping unnecessary nodes. And the two-step first-meeting probabilities are computed by integrating the repeated parts of the paths. For further speeding up query processing, we develop a pruning algorithm, which prunes unpromising path traverses by setting a threshold, and the accuracy loss under threshold is given through mathematical analysis. Extensive experiments on real graphs demonstrate the performance of DisSim through comparing with the state-of-the-art algorithms.

OWGraMi: Efficient method for mining weighted subgraphs in a single graph

Recently, the problem of mining weighted subgraphs from a weighted single graph has become a vital issue because weighted graphs are generally used to restore, simulate or monitor many complex and large systems in which each object has a different role/level. This field has attracted the attention of numerous researchers, and within related studies Weighted Graph Mining (WeGraMi) can be considered as the state-of-the-art method. However, WeGraMi lacks a strategy to prune unweighted candidate subgraphs early in the process and needs a lot of time to compute the weight for all mined frequent subgraphs. In this paper, we optimize the WeGraMi algorithm with the use of two effective strategies in a new method, which we call Optimized Weighted Graph Mining (OWGraMi). Firstly, we use a strategy to prune all frequent edges which cannot reach the weighting threshold, and with this method we can decrease the number of unweighted candidates. Secondly, we reuse the weight of parent subgraphs when computing the weight for their child subgraphs, as this can reduce the running time for the mining process. On two real graph datasets, directed as well as undirected, our experiments show that both the running time and memory requirements for OWGraMi can be reduced significantly in comparison to the original algorithm.

Robustness modularity in complex networks.

A basic question in network community detection is how modular a given network is. This is usually addressed by evaluating the quality of partitions detected in the network. The Girvan-Newman (GN) modularity function is the standard way to make this assessment, but it has a number of drawbacks. Most importantly, it is not clearly interpretable, given that the measure can take relatively large values on partitions of random networks without communities. Here we propose a measure based on the concept of robustness: modularity is the probability to find trivial partitions when the structure of the network is randomly perturbed. This concept can be implemented for any clustering algorithm capable of telling when a group structure is absent. Tests on artificial and real graphs reveal that robustness modularity can be used to assess and compare the strength of the community structure of different networks. We also introduce two other quality functions: modularity difference, a suitably normalized version of the GN modularity, and information modularity, a measure of distance based on information compression. Both measures are strongly correlated with robustness modularity, but have lower time complexity, so they could be used on networks whose size makes the calculation of robustness modularity too costly.