Journal articles: 'Large graph'

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Ji, Shengwei, Chenyang Bu, Lei Li, and Xindong Wu. "Local Graph Edge Partitioning." ACM Transactions on Intelligent Systems and Technology 12, no. 5 (October 31, 2021): 1–25. http://dx.doi.org/10.1145/3466685.

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Graph edge partitioning, which is essential for the efficiency of distributed graph computation systems, divides a graph into several balanced partitions within a given size to minimize the number of vertices to be cut. Existing graph partitioning models can be classified into two categories: offline and streaming graph partitioning models. The former requires global graph information during the partitioning, which is expensive in terms of time and memory for large-scale graphs. The latter creates partitions based solely on the received graph information. However, the streaming model may result in a lower partitioning quality compared with the offline model. Therefore, this study introduces a Local Graph Edge Partitioning model, which considers only the local information (i.e., a portion of a graph instead of the entire graph) during the partitioning. Considering only the local graph information is meaningful because acquiring complete information for large-scale graphs is expensive. Based on the Local Graph Edge Partitioning model, two local graph edge partitioning algorithms—Two-stage Local Partitioning and Adaptive Local Partitioning—are given. Experimental results obtained on 14 real-world graphs demonstrate that the proposed algorithms outperform rival algorithms in most tested cases. Furthermore, the proposed algorithms are proven to significantly improve the efficiency of the real graph computation system GraphX.

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Burch, Michael. "Visual analytics of large dynamic digraphs." Information Visualization 16, no. 3 (August 3, 2016): 167–78. http://dx.doi.org/10.1177/1473871616661194.

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In this article, we investigate the problem of visually representing and analyzing large dynamic directed graphs that consist of many vertices, edges, and time steps. With this work we do not primarily focus on graph details but more on achieving an overview about long graph sequences with the major focus to be scalable in vertex, edge, and time dimensions. To reach this goal, we first map each graph to a bipartite layout with vertices in the same order for each graph supporting a preservation of the viewer’s mental map. A sequence of graphs is placed in a left-to-right reading direction. To further reduce link crossings, we draw partial links with user-definable lengths and finally apply edge splatting as a concept to emphasize graph structures by color coding the generated density fields. Time-varying visual patterns can be recognized by inspecting the changes in the color coding in certain regions in the display. We illustrate the usefulness of the approach in two case studies investigating call graphs changing during software development with 21 releases which is a rather short graph sequence but contains several thousand vertices and edges. Visual scalability in the time dimension is shown with more than 1000 graphs from a dynamic social network dataset consisting of face-to-face contacts acquired during the Hypertext 2009 conference recorded by radio-frequency identification badges.

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Aristoff, David, and Charles Radin. "Emergent Structures in Large Networks." Journal of Applied Probability 50, no. 3 (September 2013): 883–88. http://dx.doi.org/10.1239/jap/1378401243.

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We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to the emergent multipartite structure, separated by a phase transition from a region of disordered graphs. An essential feature is the formalism of graph limits as developed by Lovász et al. for dense random graphs.

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Aristoff, David, and Charles Radin. "Emergent Structures in Large Networks." Journal of Applied Probability 50, no. 03 (September 2013): 883–88. http://dx.doi.org/10.1017/s0021900200009918.

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We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to the emergent multipartite structure, separated by a phase transition from a region of disordered graphs. An essential feature is the formalism of graph limits as developed by Lovász et al. for dense random graphs.

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Wichianpaisarn, Tanawat, and Chariya Uiyyasathian. "Graphs with large clique-chromatic numbers." Discrete Mathematics, Algorithms and Applications 07, no. 04 (December 2015): 1550055. http://dx.doi.org/10.1142/s179383091550055x.

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The clique-chromatic number of a graph [Formula: see text], [Formula: see text], is the least number of colors on [Formula: see text] without a monocolored maximal clique of size at least two. If [Formula: see text] is triangle-free, [Formula: see text]; we then consider only graphs with a triangle. Unlike the chromatic number, the clique-chromatic number of a graph is not necessary to be at least those of its subgraphs. Thus, for any family of graphs [Formula: see text], the boundedness of [Formula: see text][Formula: see text] has been investigated. Many families of graphs are proved to have a bounded set of clique-chromatic numbers. In literature, only few families of graphs are shown to have an unbounded set of clique-chromatic numbers, for instance, the family of line graphs. This paper gives another family of graphs with such an unbounded set. These graphs are obtained by the well-known Mycielski’s construction with a certain property of the initial graph.

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Wong, Pak Chung, Harlan Foote, Patrick Mackey, George Chin, Heidi Sofia, and Jim Thomas. "A Dynamic Multiscale Magnifying Tool for Exploring Large Sparse Graphs." Information Visualization 7, no. 2 (April 17, 2008): 105–17. http://dx.doi.org/10.1057/palgrave.ivs.9500177.

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We present an information visualization tool, known as GreenMax, to visually explore large small-world graphs with up to a million graph nodes on a desktop computer. A major motivation for scanning a small-world graph in such a dynamic fashion is the demanding goal of identifying not just the well-known features but also the unknown–known and unknown–unknown features of the graph. GreenMax uses a highly effective multilevel graph drawing approach to pre-process a large graph by generating a hierarchy of increasingly coarse layouts that later support the dynamic zooming of the graph. This paper describes the graph visualization challenges, elaborates our solution, and evaluates the contributions of GreenMax in the larger context of visual analytics on large small-world graphs. We report the results of two case studies using GreenMax and the results support our claim that we can use GreenMax to locate unexpected features or structures behind a graph.

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Ferber, Asaf, Kyle Luh, and Oanh Nguyen. "Embedding large graphs into a random graph." Bulletin of the London Mathematical Society 49, no. 5 (July 10, 2017): 784–97. http://dx.doi.org/10.1112/blms.12066.

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Ma, Yuliang, Ye Yuan, Meng Liu, Guoren Wang, and Yishu Wang. "Graph simulation on large scale temporal graphs." GeoInformatica 24, no. 1 (November 30, 2019): 199–220. http://dx.doi.org/10.1007/s10707-019-00381-y.

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Wagenpfeil, Stefan, Binh Vu, Paul Mc Kevitt, and Matthias Hemmje. "Fast and Effective Retrieval for Large Multimedia Collections." Big Data and Cognitive Computing 5, no. 3 (July 22, 2021): 33. http://dx.doi.org/10.3390/bdcc5030033.

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The indexing and retrieval of multimedia content is generally implemented by employing feature graphs. These graphs typically contain a significant number of nodes and edges to reflect the level of detail in feature detection. A higher level of detail increases the effectiveness of the results, but also leads to more complex graph structures. However, graph traversal-based algorithms for similarity are quite inefficient and computationally expensive, especially for large data structures. To deliver fast and effective retrieval especially for large multimedia collections and multimedia big data, an efficient similarity algorithm for large graphs in particular is desirable. Hence, in this paper, we define a graph projection into a 2D space (Graph Code) and the corresponding algorithms for indexing and retrieval. We show that calculations in this space can be performed more efficiently than graph traversals due to the simpler processing model and the high level of parallelization. As a consequence, we demonstrate experimentally that the effectiveness of retrieval also increases substantially, as the Graph Code facilitates more levels of detail in feature fusion. These levels of detail also support an increased trust prediction, particularly for fused social media content. In our mathematical model, we define a metric triple for the Graph Code, which also enhances the ranked result representations. Thus, Graph Codes provide a significant increase in efficiency and effectiveness, especially for multimedia indexing and retrieval, and can be applied to images, videos, text and social media information.

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El Moussawi, Adnan, Nacera Bennacer Seghouani, and Francesca Bugiotti. "BGRAP: Balanced GRAph Partitioning Algorithm for Large Graphs." Journal of Data Intelligence 2, no. 2 (June 2021): 116–35. http://dx.doi.org/10.26421/jdi2.2-2.

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The definition of effective strategies for graph partitioning is a major challenge in distributed environments since an effective graph partitioning allows to considerably improve the performance of large graph data analytics computations. In this paper, we propose a multi-objective and scalable Balanced GRAph Partitioning (\algo) algorithm, based on Label Propagation (LP) approach, to produce balanced graph partitions. \algo defines a new efficient initialization procedure and different objective functions to deal with either vertex or edge balance constraints while considering edge direction in graphs. \algo is implemented of top of the open source distributed graph processing system Giraph. The experiments are performed on various graphs with different structures and sizes (going up to 50.6M vertices and 1.9B edges) while varying the number of partitions. We evaluate \algo using several quality measures and the computation time. The results show that \algo (i) provides a good balance while reducing the cuts between the different computed partitions (ii) reduces the global computation time, compared to LP-based algorithms.

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LINIAL, NATHAN, JIŘÍ MATOUŠEK, OR SHEFFET, and GÁBOR TARDOS. "Graph Colouring with No Large Monochromatic Components." Combinatorics, Probability and Computing 17, no. 4 (July 2008): 577–89. http://dx.doi.org/10.1017/s0963548308009140.

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For a graph G and an integer t we let mcct(G) be the smallest m such that there exists a colouring of the vertices of G by t colours with no monochromatic connected subgraph having more than m vertices. Let be any non-trivial minor-closed family of graphs. We show that mcc2(G) = O(n2/3) for any n-vertex graph G ∈ . This bound is asymptotically optimal and it is attained for planar graphs. More generally, for every such , and every fixed t we show that mcct(G)=O(n2/(t+1)). On the other hand, we have examples of graphs G with no Kt+3 minor and with mcct(G)=Ω(n2/(2t−1)).It is also interesting to consider graphs of bounded degrees. Haxell, Szabó and Tardos proved mcc2(G) ≤ 20000 for every graph G of maximum degree 5. We show that there are n-vertex 7-regular graphs G with mcc2(G)=Ω(n), and more sharply, for every ϵ > 0 there exists cϵ > 0 and n-vertex graphs of maximum degree 7, average degree at most 6 + ϵ for all subgraphs, and with mcc2(G) ≥ cϵn. For 6-regular graphs it is known only that the maximum order of magnitude of mcc2 is between $\sqrt n$ and n.We also offer a Ramsey-theoretic perspective of the quantity mcct(G).

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Et. al., M. Sailaja,. "Ensemble Distributed Search-FSGM-CRD Compressed Cache Algorithm for Large Datasets." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 2 (April 11, 2021): 2854–58. http://dx.doi.org/10.17762/turcomat.v12i2.2317.

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Frequent sub-graph mining (FSM) is a alternative of frequent pattern mining where patterns are graphs. Among the entities, graph based representation is utilized to effectively represent the complex relationships. Various graph mining techniques are developed from the past many years, most the challenging tasks in graph mining is frequent sub-graph mining (FSM). In FSM many of the existing algorithms consider only graph based structure, the relationships based on entities involved and strength is not considered. It is very important to handle the complex and huge data. There is very huge demand in distributed computational approaches. In this paper, An Ensemble Distributed Search-FSGM-CRD Compressed Cache Algorithm is developed and implemented to find frequent sub graphs

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Yang, Jianye, Wu Yao, and Wenjie Zhang. "Keyword Search on Large Graphs: A Survey." Data Science and Engineering 6, no. 2 (March 31, 2021): 142–62. http://dx.doi.org/10.1007/s41019-021-00154-4.

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AbstractWith the prevalence of Internet access and online services, various big graphs are generated in many real applications (e.g., online social networks and knowledge graphs). An important task on analyzing and mining these graphs is keyword search. Essentially, given a graph G and query Q associated with a set of keywords, the keyword search aims to find a substructure (e.g., rooted tree or subgraph) S in G such that nodes in S collectively cover part of or all keywords in Q, and in the meanwhile, S is optimal on some user specified semantics. Keyword search on graphs can be applied in many real-life applications, such as point-of-interests recommendation and web search facility. In spite of the great importance of graph keyword search, we, however, notice that the latest survey on this topic is far out of date. Consequently, there is prompt need to conduct a comprehensive survey in this research direction. Motivated by this, in this survey, we systematically review graph keyword search studies by classifying the existing works into different categories according to the specific problem definition. This survey aims to provide the researchers a comprehensive understanding of existing graph keyword search solutions.

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YUSTER, RAPHAEL. "Dense Graphs With a Large Triangle Cover Have a Large Triangle Packing." Combinatorics, Probability and Computing 21, no. 6 (September 27, 2012): 952–62. http://dx.doi.org/10.1017/s0963548312000235.

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It is well known that a graph with m edges can be made triangle-free by removing (slightly less than) m/2 edges. On the other hand, there are many classes of graphs which are hard to make triangle-free, in the sense that it is necessary to remove roughly m/2 edges in order to eliminate all triangles.We prove that dense graphs that are hard to make triangle-free have a large packing of pairwise edge-disjoint triangles. In particular, they have more than m(1/4+cβ) pairwise edge-disjoint triangles where β is the density of the graph and c ≥ is an absolute constant. This improves upon a previous m(1/4−o(1)) bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. We conjecture that such graphs have an asymptotically optimal triangle packing of size m(1/3−o(1)).We extend our result from triangles to larger cliques and odd cycles.

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Duan, Yucong, Lixu Shao, and Gongzhu Hu. "Specifying Knowledge Graph with Data Graph, Information Graph, Knowledge Graph, and Wisdom Graph." International Journal of Software Innovation 6, no. 2 (April 2018): 10–25. http://dx.doi.org/10.4018/ijsi.2018040102.

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Knowledge graphs have been widely adopted, in large part owing to their schema-less nature. It enables knowledge graphs to grow seamlessly and allows for new relationships and entities as needed. A knowledge graph is a graph constructed by representing each item, entity and user as nodes, and linking those nodes that interact with each other via edges. Knowledge graphs have abundant natural semantics and can contain various and more complete information. It is an expression mechanism close to natural language. However, we still lack a unified definition and standard expression form of knowledge graph. The authors propose to clarify the expression of knowledge graph as a whole. They clarify the architecture of knowledge graph from data, information, knowledge, and wisdom aspects respectively. The authors also propose to specify knowledge graph in a progressive manner as four basic forms including data graph, information graph, knowledge graph and wisdom graph.

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Leonard, Lorne, Alan M. MacEachren, and Kamesh Madduri. "Graph-based visual analysis for large-scale hydrological modeling." Information Visualization 16, no. 3 (August 9, 2016): 205–16. http://dx.doi.org/10.1177/1473871616661868.

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This article reports on the development and application of a visual analytics approach to big data cleaning and integration focused on very large graphs, constructed in support of national-scale hydrological modeling. We explain why large graphs are required for hydrology modeling and describe how we create two graphs using continental United States heterogeneous national data products. The first smaller graph is constructed by assigning level-12 hydrological unit code watersheds as nodes. Creating and cleaning graphs at this scale highlight the issues that cannot be addressed without high-resolution datasets and expert intervention. Expert intervention, aided with visual analytical tools, is necessary to address edge directions at the second graph scale: subdividing continental United States streams as edges (851,265,305) and nodes (683,298,991) for large-scale hydrological modeling. We demonstrate how large graph workflows are created and are used for automated analysis to prepare the user interface for visual analytics. We explain the design of the visual interface using a watershed case study and then discuss how the visual interface is used to engage the expert user to resolve data and graph issues.

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Hora, Akihito. "Central Limit Theorems and Asymptotic Spectral Analysis on Large Graphs." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 02 (April 1998): 221–46. http://dx.doi.org/10.1142/s0219025798000144.

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Regarding the adjacency matrix of a graph as a random variable in the framework of algebraic or noncommutative probability, we discuss a central limit theorem in which the size of a graph grows in several patterns. Various limit distributions are observed for some Cayley graphs and some distance-regular graphs. To obtain the central limit theorem of this type, we make combinatorial analysis of mixed moments of noncommutative random variables on one hand, and asymptotic analysis of spectral structure of the graph on the other hand.

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Lin, Zhe, Fan Zhang, Xuemin Lin, Wenjie Zhang, and Zhihong Tian. "Hierarchical core maintenance on large dynamic graphs." Proceedings of the VLDB Endowment 14, no. 5 (January 2021): 757–70. http://dx.doi.org/10.14778/3446095.3446099.

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The model of k -core and its decomposition have been applied in various areas, such as social networks, the world wide web, and biology. A graph can be decomposed into an elegant k -core hierarchy to facilitate cohesive subgraph discovery and network analysis. As many real-life graphs are fast evolving, existing works proposed efficient algorithms to maintain the coreness value of every vertex against structure changes. However, the maintenance of the k -core hierarchy in existing studies is not complete because the connections among different k -cores in the hierarchy are not considered. In this paper, we study hierarchical core maintenance which is to compute the k -core hierarchy incrementally against graph dynamics. The problem is challenging because the change of hierarchy may be large and complex even for a slight graph update. In order to precisely locate the area affected by graph dynamics, we conduct in-depth analyses on the structural properties of the hierarchy, and propose well-designed local update techniques. Our algorithms significantly outperform the baselines on runtime by up to 3 orders of magnitude, as demonstrated on 10 real-world large graphs.

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Su, Jing, Hongyu Wang, and Bing Yao. "On Elegant Labelling and Magic Labelling of Large-Scale Graphs." Discrete Dynamics in Nature and Society 2022 (March 28, 2022): 1–10. http://dx.doi.org/10.1155/2022/6301674.

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In this paper, we deduce the equivalence relationship among strongly c-elegant labelling, super-edge magic total labelling, edge antimagic total labelling, and super t , 1 -magical labelling. We study some properties of the graph with a strongly c-elegant labelling. On the basis of small-scale graphs with strongly c-elegant labelling, several types of large-scale graphs are constructed through graph operations, and we further prove the existence of their strongly c-elegant labelling. In addition, we also define a transformation of strongly c-elegant labelling, which provides a method for the transformation between several strongly c-elegant labellings of a graph.

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Krivelevich, Michael. "Ks-Free Graphs Without Large Kr-Free Subgraphs." Combinatorics, Probability and Computing 3, no. 3 (September 1994): 349–54. http://dx.doi.org/10.1017/s0963548300001243.

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The main result of this paper is that for every 2 ≤ r < s, and n sufficiently large, there exist graphs of order n, not containing a complete graph on s vertices, in which every relatively not too small subset of vertices spans a complete graph on r vertices. Our results improve on previous results of Bollobás and Hind.

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Wade, Andrew R. "Explicit laws of large numbers for random nearest-neighbour-type graphs." Advances in Applied Probability 39, no. 2 (June 2007): 326–42. http://dx.doi.org/10.1239/aap/1183667613.

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Under the unifying umbrella of a general result of Penrose and Yukich (Annals of Applied Probability13 (2003), 277-303) we give laws of large numbers (in the Lp sense) for the total power-weighted length of several nearest-neighbour-type graphs on random point sets in ℝd, d ∈ ℕ. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest-neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour graph.

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Serratosa, Francesc. "A Methodology to Generate Attributed Graphs with a Bounded Graph Edit Distance for Graph-Matching Testing." International Journal of Pattern Recognition and Artificial Intelligence 32, no. 11 (July 24, 2018): 1850038. http://dx.doi.org/10.1142/s0218001418500386.

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This paper presents a methodology for generating pairs of attributed graphs with a lower and upper- bounded graph edit distance (GED). It is independent of the type of attributes on nodes and edges. The algorithm is composed of three steps: randomly generating a graph, generating another graph as a sub-graph of the first, and adding structural and semantic noise to both. These graphs, together with their bounded distances, can be used to manufacture synthetic databases of large graphs. The exact GED between large graphs cannot be obtained for runtime reasons since it has to be computed through an optimal algorithm with an exponential computational cost. Through this database, we can test the behavior of the known or new sub-optimal error-tolerant graph-matching algorithms against a lower and an upper bound GED on large graphs, even though we do not have the true distance. It is not clear how the error induced by the use of sub-optimal algorithms grows with problem size. Thus, with this methodology, we can generate graph databases and analyze if the current assumption that we can extrapolate algorithms’ behavior from matching small graphs to large graphs is correct or not. We also show that with some restrictions, the methodology returns the optimal GED in a quadratic time and that it can also be used to generate graph databases to test exact sub-graph isomorphism algorithms.

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Mathieu, Claire, and Michel de Rougemont. "Large very dense subgraphs in a stream of edges." Network Science 9, no. 4 (December 2021): 403–24. http://dx.doi.org/10.1017/nws.2021.17.

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AbstractWe study the detection and the reconstruction of a large very dense subgraph in a social graph with n nodes and m edges given as a stream of edges, when the graph follows a power law degree distribution, in the regime when $m=O(n. \log n)$ . A subgraph S is very dense if it has $\Omega(|S|^2)$ edges. We uniformly sample the edges with a Reservoir of size $k=O(\sqrt{n}.\log n)$ . Our detection algorithm checks whether the Reservoir has a giant component. We show that if the graph contains a very dense subgraph of size $\Omega(\sqrt{n})$ , then the detection algorithm is almost surely correct. On the other hand, a random graph that follows a power law degree distribution almost surely has no large very dense subgraph, and the detection algorithm is almost surely correct. We define a new model of random graphs which follow a power law degree distribution and have large very dense subgraphs. We then show that on this class of random graphs we can reconstruct a good approximation of the very dense subgraph with high probability. We generalize these results to dynamic graphs defined by sliding windows in a stream of edges.

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Razi, Adeel, Mohamed L. Seghier, Yuan Zhou, Peter McColgan, Peter Zeidman, Hae-Jeong Park, Olaf Sporns, Geraint Rees, and Karl J. Friston. "Large-scale DCMs for resting-state fMRI." Network Neuroscience 1, no. 3 (October 2017): 222–41. http://dx.doi.org/10.1162/netn_a_00015.

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This paper considers the identification of large directed graphs for resting-state brain networks based on biophysical models of distributed neuronal activity, that is, effective connectivity. This identification can be contrasted with functional connectivity methods based on symmetric correlations that are ubiquitous in resting-state functional MRI (fMRI). We use spectral dynamic causal modeling (DCM) to invert large graphs comprising dozens of nodes or regions. The ensuing graphs are directed and weighted, hence providing a neurobiologically plausible characterization of connectivity in terms of excitatory and inhibitory coupling. Furthermore, we show that the use of Bayesian model reduction to discover the most likely sparse graph (or model) from a parent (e.g., fully connected) graph eschews the arbitrary thresholding often applied to large symmetric (functional connectivity) graphs. Using empirical fMRI data, we show that spectral DCM furnishes connectivity estimates on large graphs that correlate strongly with the estimates provided by stochastic DCM. Furthermore, we increase the efficiency of model inversion using functional connectivity modes to place prior constraints on effective connectivity. In other words, we use a small number of modes to finesse the potentially redundant parameterization of large DCMs. We show that spectral DCM—with functional connectivity priors—is ideally suited for directed graph theoretic analyses of resting-state fMRI. We envision that directed graphs will prove useful in understanding the psychopathology and pathophysiology of neurodegenerative and neurodevelopmental disorders. We will demonstrate the utility of large directed graphs in clinical populations in subsequent reports, using the procedures described in this paper.

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Wade, Andrew R. "Explicit laws of large numbers for random nearest-neighbour-type graphs." Advances in Applied Probability 39, no. 02 (June 2007): 326–42. http://dx.doi.org/10.1017/s0001867800001786.

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Under the unifying umbrella of a general result of Penrose and Yukich (Annals of Applied Probability 13 (2003), 277-303) we give laws of large numbers (in the L p sense) for the total power-weighted length of several nearest-neighbour-type graphs on random point sets in ℝ d , d ∈ ℕ. Some of these results are known; some are new. We give limiting constants explicitly, where previously they have been evaluated in less generality or not at all. The graphs we consider include the k-nearest-neighbours graph, the Gabriel graph, the minimal directed spanning forest, and the on-line nearest-neighbour graph.

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Rao, Bapuji, and Sarojananda Mishra. "A New Approach to Community Graph Partition Using Graph Mining Techniques." International Journal of Rough Sets and Data Analysis 4, no. 1 (January 2017): 75–94. http://dx.doi.org/10.4018/ijrsda.2017010105.

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Knowledge extraction is very much possible from the community graph using graph mining techniques. The authors have studied the related definitions of graph partition in terms of both mathematical as well as computational aspects. To derive knowledge from a particular sub-community graph of a large community graph, the authors start partitioning the large community graph into smaller sub-community graphs. Thus, the knowledge extraction from the sub-community graph becomes easier and faster. The proposed approach of partition is done by detection of edges among the community members of dissimilar community. By studying existing techniques followed by different researchers, the authors propose a new and simple algorithm for partitioning the community graph into sub-community graphs using graph mining techniques. Finally, the authors have considered a benchmark dataset as example which verifies the strength and easiness of the proposed algorithm.

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Liu, Yang, Wei Wei, and Heyang Xu. "Engineering Bi-Connected Component Overlay for Maximum-Flow Parallel Acceleration in Large Sparse Graph." Xibei Gongye Daxue Xuebao/Journal of Northwestern Polytechnical University 36, no. 5 (October 2018): 955–62. http://dx.doi.org/10.1051/jnwpu/20183650955.

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Network maximum flow problem is important and basic in graph theory, and one of its research directions is maximum-flow acceleration in large-scale graph. Existing acceleration strategy includes graph contraction and parallel computation, where there is still room for improvement:(1) The existing two acceleration strategies are not fully integrated, leading to their limited acceleration effect; (2) There is no sufficient support for computing multiple maximum-flow in one graph, leading to a lot of redundant computation. (3)The existing preprocessing methods need to consider node degrees and capacity constraints, resulting in high computational complexity. To address above problems, we identify the bi-connected components in a given graph and build an overlay, which can help split the maximum-flow problem into several subproblems and then solve them in parallel. The algorithm only uses the connectivity in the graph and has low complexity. The analyses and experiments on benchmark graphs indicate that the method can significantly shorten the calculation time in large sparse graphs.

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Bennett, Patrick, Andrzej Dudek, and Shira Zerbib. "Large triangle packings and Tuza’s conjecture in sparse random graphs." Combinatorics, Probability and Computing 29, no. 5 (July 22, 2020): 757–79. http://dx.doi.org/10.1017/s0963548320000115.

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AbstractThe triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

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Coppini, Fabio, Helge Dietert, and Giambattista Giacomin. "A law of large numbers and large deviations for interacting diffusions on Erdős–Rényi graphs." Stochastics and Dynamics 20, no. 02 (July 10, 2019): 2050010. http://dx.doi.org/10.1142/s0219493720500100.

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We consider a class of particle systems described by differential equations (both stochastic and deterministic), in which the interaction network is determined by the realization of an Erdős–Rényi graph with parameter [Formula: see text], where [Formula: see text] is the size of the graph (i.e. the number of particles). If [Formula: see text], the graph is the complete graph (mean field model) and it is well known that, under suitable hypotheses, the empirical measure converges as [Formula: see text] to the solution of a PDE: a McKean–Vlasov (or Fokker–Planck) equation in the stochastic case, or a Vlasov equation in the deterministic one. It has already been shown that this holds for rather general interaction networks, that include Erdős–Rényi graphs with [Formula: see text], and properly rescaling the interaction to account for the dilution introduced by [Formula: see text]. However, these results have been proven under strong assumptions on the initial datum which has to be chaotic, i.e. a sequence of independent identically distributed random variables. The aim of our contribution is to present results — Law of Large Numbers and Large Deviation Principle — assuming only the convergence of the empirical measure of the initial condition.

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Tian, Yanjia, and Xiang Feng. "Large Margin Graph Embedding-Based Discriminant Dimensionality Reduction." Scientific Programming 2021 (December 3, 2021): 1–12. http://dx.doi.org/10.1155/2021/2934362.

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Discriminant graph embedding-based dimensionality reduction methods have attracted more and more attention over the past few decades. These methods construct an intrinsic graph and penalty graph to preserve the intrinsic geometry structures of intraclass samples and separate the interclass samples. However, the marginal samples cannot be accurately characterized only by penalty graphs since they treat every sample equally. In practice, these marginal samples often influence the classification performance, which needs to be specially tackled. In this study, the near neighbors’ hypothesis margin of marginal samples has been further maximized to separate the interclass samples and improve the discriminant ability by integrating intrinsic graph and penalty graph. A novel discriminant dimensionality reduction named LMGE-DDR has been proposed. Several experiments on public datasets have been conducted to verify the effectiveness of the proposed LMGE-DDR such as ORL, Yale, UMIST, FERET, CMIU-PIE09, and AR. LMGE-DDR performs better than other compared methods, and the corresponding standard deviation of LMGE-DDR is smaller than others. This demonstrates that the evaluation method verifies the effectiveness of the introduced method.

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Pacher, Dominic, Robert Binna, and Günther Specht. "Optimizing large knowledge networks in spatial computers." Knowledge Engineering Review 31, no. 4 (September 2016): 367–90. http://dx.doi.org/10.1017/s0269888916000187.

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AbstractThis paper presents a novel concept of a Spatially Aware Graph Store, which realizes a Graph Store on top of a spatial computer architecture to manage graphs in one, two or three physical dimensions. In this environment, the physical distance between graph nodes strongly affects graph traversal performance. Consequently, a Spatially Aware Graph Store needs to minimize these distances to operate efficiently. We show that this minimization can be achieved in two ways. First, by increasing the dimensionality of the spatial computer and second by applying optimization methods. For the latter, this work introduces a novel Mid Point Optimization method to quickly optimize large real-world knowledge networks by rearranging nodes in a way that distances between linked nodes are reduced. In addition, a Local Optimization method is subsequently applied to refine the result. Finally, the Node Decomposition method is presented that splits nodes with many edges into several smaller nodes to achieve a further reduction of distances between linked nodes.Our results show that the overall distances between nodes can be reduced by three orders of magnitude for 3D in comparison to one-dimensional (1D) Spatially Aware Graph Stores. The suggested Mid Point Optimization method achieves a reduction by another order of magnitude. In a 3D spatial computer, Local Optimization is capable of reducing distances by another 20%. However, in 1D and 2D spatial computers it becomes a prohibitive time consuming method. Finally, the Node Decomposition enables an additional distance reduction by 40% in Scale Free Graph Data sets.

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Cheng, Hong, and Jeffrey Xu Yu. "Clustering Large Attributed Graph." Journal of Information Processing 20, no. 4 (2012): 806–13. http://dx.doi.org/10.2197/ipsjjip.20.806.

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Duan, Shi-Jie, and Feng Li. "Exact Wiener Index of the Direct Product of a Path and a Wheel Graph." Mathematical Problems in Engineering 2022 (February 10, 2022): 1–6. http://dx.doi.org/10.1155/2022/1077549.

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The direct product is one of the most important methods to construct large-scale graphs using existing small-scale graphs, and the topological structure parameters of the constructed large-scale graphs can be derived from small-scale graphs. For a simple undirected graph G , its Wiener index W G is defined as the sum of the distances between all different unordered pairs of vertices in the graph. Path is one of the most common and useful graphs, and it is found in almost all virtual and real networks; wheel graph is a kind of graph with good properties and convenient construction. In this paper, the exact value of the Wiener index of the direct product of a path and a wheel graph is given, and the obtained Wiener index is only derived from the orders of the two factor graphs.

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Xu, Jin, Yu Zhong, and Bo Peng. "Parallel k-Way Partitioning Approach for Large Graphs." Advanced Materials Research 912-914 (April 2014): 1309–12. http://dx.doi.org/10.4028/www.scientific.net/amr.912-914.1309.

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With the emergence of large social networks, such as Facebook and Twitter, graphs with millions to billions vertices are common. Instead of processing the network within a single machine, all the applications related are intended to be done in a distributed way using a cluster of commodity machines. In this paper, we study the parallel graph partitioning problem, which is the fundamental operation for large graphs. With the help of Hadoop/MapReduce, we propose aparallel k-way partitioningapproach. Unlike the previous ones, which require enough memory to keep the whole graph data within, our novel approach breaks such limitations. Also, due to the distributed nature, it is easy to integrate our partitioning approach into existed parallel platforms. We conduct extensive experiments on real graphs and synthetic graphs. All the experimental results prove the effectiveness and efficiency of our approach.

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Nikoghosyan, Zh G. "Graph Invariants and Large Cycles: A Survey." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–11. http://dx.doi.org/10.1155/2011/206404.

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Graph invariants provide a powerful analytical tool for investigation of abstract substructures of graphs. This paper is devoted to large cycle substructures, namely, Hamilton, longest and dominating cycles and some generalized cycles including Hamilton and dominating cycles as special cases. In this paper, we have collected 36 pure algebraic relations between basic (initial) graph invariants ensuring the existence of a certain type of large cycles. These simplest kind of relations having no forerunners in the area actually form a source from which nearly all possible hamiltonian results (including well-known Ore's theorem, Posa's theorem, and many other generalizations) can be developed further by various additional new ideas, generalizations, extensions, restrictions, and structural limitations.

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Cordella, L. P., P. Foggia, C. Sansone, and M. Vento. "A (sub)graph isomorphism algorithm for matching large graphs." IEEE Transactions on Pattern Analysis and Machine Intelligence 26, no. 10 (October 2004): 1367–72. http://dx.doi.org/10.1109/tpami.2004.75.

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Cai, Zhuang, Kang Zhang, and Dong-Ni Hu. "Visualizing large graphs by layering and bundling graph edges." Visual Computer 35, no. 5 (April 30, 2018): 739–51. http://dx.doi.org/10.1007/s00371-018-1509-7.

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Aridhi, Sabeur, Alberto Montresor, and Yannis Velegrakis. "BLADYG: A Graph Processing Framework for Large Dynamic Graphs." Big Data Research 9 (September 2017): 9–17. http://dx.doi.org/10.1016/j.bdr.2017.05.003.

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Zheng, Weiguo, Lei Zou, Xiang Lian, Dong Wang, and Dongyan Zhao. "Efficient Graph Similarity Search Over Large Graph Databases." IEEE Transactions on Knowledge and Data Engineering 27, no. 4 (April 1, 2015): 964–78. http://dx.doi.org/10.1109/tkde.2014.2349924.

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Yuan, Ye, Guoren Wang, Lei Chen, and Haixun Wang. "Graph similarity search on large uncertain graph databases." VLDB Journal 24, no. 2 (December 9, 2014): 271–96. http://dx.doi.org/10.1007/s00778-014-0373-y.

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Vengerovsky, V. "Eigenvalue Distribution of a Large Weighted Bipartite Random Graph." Zurnal matematiceskoj fiziki, analiza, geometrii 10, no. 2 (June 25, 2014): 240–55. http://dx.doi.org/10.15407/mag10.02.240.

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Jarrar, Mustafa, and Anton Deik. "The Graph Signature." International Journal on Semantic Web and Information Systems 11, no. 2 (April 2015): 36–65. http://dx.doi.org/10.4018/ijswis.2015040102.

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Querying large data graphs has brought the attention of the research community. Many solutions were proposed, such as Oracle Semantic Technologies, Virtuoso, RDF3X, and C-Store, among others. Although such approaches have shown good performance in queries with medium complexity, they perform poorly when the complexity of the queries increases. In this paper, the authors propose the Graph Signature Index, a novel and scalable approach to index and query large data graphs. The idea is that they summarize a graph and instead of executing the query on the original graph, they execute it on the summaries. The authors' experiments with Yago (16M triples) have shown that e.g., a query with 4 levels costs 62 sec using Oracle but it only costs about 0.6 sec with their index. Their index can be implemented on top of any Graph database, but they chose to implement it as an extension to Oracle on top of the SEM_MATCH table function. The paper also introduces disk-based versions of the Trace Equivalence and Bisimilarity algorithms to summarize data graphs, and discusses their complexity and usability for RDF graphs.

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Zhou, Jiang, Lizhu Sun, Hongmei Yao, and Changjiang Bu. "On the nullity of connected graphs with least eigenvalue at least -2." Applicable Analysis and Discrete Mathematics 7, no. 2 (2013): 250–61. http://dx.doi.org/10.2298/aadm130710014z.

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Let L (resp. L+) be the set of connected graphs with least adjacency eigenvalue at least -2 (resp. larger than -2). The nullity of a graph G, denoted by ?(G), is the multiplicity of zero as an eigenvalue of the adjacency matrix of G. In this paper, we give the nullity set of L+ and an upper bound on the nullity of exceptional graphs. An expression for the nullity of generalized line graphs is given. For G ? L, if ?(G) is sufficiently large, then G is a proper generalized line graph (G is not a line graph).

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Hàn, Hiêp, Troy Retter, Vojtêch Rödl, and Mathias Schacht. "Ramsey-type numbers involving graphs and hypergraphs with large girth." Combinatorics, Probability and Computing 30, no. 5 (April 12, 2021): 722–40. http://dx.doi.org/10.1017/s0963548320000383.

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AbstractErdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle Ck. The existence of such graphs H was confirmed by the third author and Ruciński.We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(Ck; r) is the r-colour Ramsey number for the cycle Ck) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic Ck Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.

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Huang, Xueqin, Xianqiang Zhu, Xiang Xu, Qianzhen Zhang, and Ailin Liang. "Parallel Learning of Dynamics in Complex Systems." Systems 10, no. 6 (December 15, 2022): 259. http://dx.doi.org/10.3390/systems10060259.

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Dynamics always exist in complex systems. Graphs (complex networks) are a mathematical form for describing a complex system abstractly. Dynamics can be learned efficiently from the structure and dynamics state of a graph. Learning the dynamics in graphs plays an important role in predicting and controlling complex systems. Most of the methods for learning dynamics in graphs run slowly in large graphs. The complexity of the large graph’s structure and its nonlinear dynamics aggravate this problem. To overcome these difficulties, we propose a general framework with two novel methods in this paper, the Dynamics-METIS (D-METIS) and the Partitioned Graph Neural Dynamics Learner (PGNDL). The general framework combines D-METIS and PGNDL to perform tasks for large graphs. D-METIS is a new algorithm that can partition a large graph into multiple subgraphs. D-METIS innovatively considers the dynamic changes in the graph. PGNDL is a new parallel model that consists of ordinary differential equation systems and graph neural networks (GNNs). It can quickly learn the dynamics of subgraphs in parallel. In this framework, D-METIS provides PGNDL with partitioned subgraphs, and PGNDL can solve the tasks of interpolation and extrapolation prediction. We exhibit the universality and superiority of our framework on four kinds of graphs with three kinds of dynamics through an experiment.

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Sonobe, Tomohiro. "An Efficient Monte Carlo Approach to Compute PageRank for Large Graphs on a Single PC." Foundations of Computing and Decision Sciences 41, no. 1 (March 1, 2016): 29–43. http://dx.doi.org/10.1515/fcds-2016-0002.

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AbstractThis paper describes a novel Monte Carlo based random walk to compute PageRanks of nodes in a large graph on a single PC. The target graphs of this paper are ones whose size is larger than the physical memory. In such an environment, memory management is a difficult task for simulating the random walk among the nodes. We propose a novel method that partitions the graph into subgraphs in order to make them fit into the physical memory, and conducts the random walk for each subgraph. By evaluating the walks lazily, we can conduct the walks only in a subgraph and approximate the random walk by rotating the subgraphs. In computational experiments, the proposed method exhibits good performance for existing large graphs with several passes of the graph data.

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Yang, Yajun, Zhongfei Li, Xin Wang, and Qinghua Hu. "Finding the Shortest Path with Vertex Constraint over Large Graphs." Complexity 2019 (February 19, 2019): 1–13. http://dx.doi.org/10.1155/2019/8728245.

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Graph is an important complex network model to describe the relationship among various entities in real applications, including knowledge graph, social network, and traffic network. Shortest path query is an important problem over graphs and has been well studied. This paper studies a special case of the shortest path problem to find the shortest path passing through a set of vertices specified by user, which is NP-hard. Most existing methods calculate all permutations for given vertices and then find the shortest one from these permutations. However, the computational cost is extremely expensive when the size of graph or given set of vertices is large. In this paper, we first propose a novel exact heuristic algorithm in best-first search way and then give two optimizing techniques to improve efficiency. Moreover, we propose an approximate heuristic algorithm in polynomial time for this problem over large graphs. We prove the ratio bound is 3 for our approximate algorithm. We confirm the efficiency of our algorithms by extensive experiments on real-life datasets. The experimental results validate that our algorithms always outperform the existing methods even though the size of graph or given set of vertices is large.

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Leal, José. "Visualization of path patterns in semantic graphs." Computer Science and Information Systems 17, no. 1 (2020): 229–52. http://dx.doi.org/10.2298/csis180717038l.

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Graphs with a large number of nodes and edges are difficult to visualize. Semantic graphs add to the challenge since their nodes and edges have types and this information must be mirrored in the visualization. A common approach to cope with this difficulty is to omit certain nodes and edges, displaying sub-graphs of smaller size. However, other transformations can be used to summarize semantic graphs and this research explores a particular one, both to reduce the graph?s size and to focus on its path patterns. A-graphs are a novel kind of graph designed to highlight path patterns using this kind of summarization. They are composed of a-nodes connected by a-edges, and these reflect respectively edges and nodes of the semantic graph. A-graphs trade the visualization of nodes and edges by the visualization of graph path patterns involving typed edges. Thus, they are targeted to users that require a deep understanding of the semantic graph it represents, in particular of its path patterns, rather than to users wanting to browse the semantic graph?s content. A-graphs help programmers querying the semantic graph or designers of semantic measures interested in using it as a semantic proxy. Hence, a-graphs are not expected to compete with other forms of semantic graph visualization but rather to be used as a complementary tool. This paper provides a precise definition both of a-graphs and of the mapping of semantic graphs into a-graphs. Their visualization is obtained with a-graphs diagrams. A web application to visualize and interact with these diagrams was implemented to validate the proposed approach. Diagrams of well-known semantic graphs are presented to illustrate the use of agraphs for discovering path patterns in different settings, such as the visualization of massive semantic graphs, the codification of SPARQL or the definition of semantic measures. The validation with large semantic graphs is the basis for a discussion on the insights provided by a-graphs on large semantic graphs: the difference between a-graphs and ontologies, path pattern visualization using a-graphs and the challenges posed by large semantic graphs.

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JOHANNSEN, DANIEL, MICHAEL KRIVELEVICH, and WOJCIECH SAMOTIJ. "Expanders Are Universal for the Class of All Spanning Trees." Combinatorics, Probability and Computing 22, no. 2 (January 3, 2013): 253–81. http://dx.doi.org/10.1017/s0963548312000533.

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A graph is calleduniversalfor a given graph class(or, equivalently,-universal) if it contains a copy of every graph inas a subgraph. The construction of sparse universal graphs for various classeshas received a considerable amount of attention. There is particular interest in tight-universal graphs, that is, graphs whose number of vertices is equal to the largest number of vertices in a graph from. Arguably, the most studied case is that whenis some class of trees. In this work, we are interested in(n,Δ), the class of alln-vertex trees with maximum degree at most Δ. We show that everyn-vertex graph satisfying certain natural expansion properties is(n,Δ)-universal. Our methods also apply to the case when Δ is some function ofn. Since random graphs are known to be good expanders, our result implies, in particular, that there exists a positive constantcsuch that the random graphG(n,cn−1/3log2n) is asymptotically almost surely (a.a.s.) universal for(n,O(1)). Moreover, a corresponding result holds for the random regular graph of degreecn2/3log2n. Another interesting consequence is the existence of locally sparsen-vertex(n,Δ)-universal graphs. For example, we show that one can (randomly) constructn-vertex(n,O(1))-universal graphs with clique number at most five. This complements the construction of Bhatt, Chung, Leighton and Rosenberg (1989), whose(n,Δ)-universal graphs with merelyO(n)edges contain large cliques of size Ω(Δ). Finally, we show that random graphs are robustly(n,Δ)-universal in the context of the Maker–Breaker tree-universality game.

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Mikheenko, Alla, and Mikhail Kolmogorov. "Assembly Graph Browser: interactive visualization of assembly graphs." Bioinformatics 35, no. 18 (February 4, 2019): 3476–78. http://dx.doi.org/10.1093/bioinformatics/btz072.

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Abstract Summary Currently, most genome assembly projects focus on contigs and scaffolds rather than assembly graphs that provide a more comprehensive representation of an assembly. Since interactive visualization of large assembly graphs remains an open problem, we developed an Assembly Graph Browser (AGB) tool that visualizes large assembly graphs, extending the functionality of previously developed visualization approaches. Assembly Graph Browser includes a number of novel functions including repeat analysis, construction of the contracted assembly graphs (i.e. the graphs obtained by collapsing a selected set of edges) and a new approach to visualizing large assembly graphs. Availability and implementation http://www.github.com/almiheenko/AGB. Supplementary information Supplementary data are available at Bioinformatics online.

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Journal articles on the topic 'Large graph'

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