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Artykuły w czasopismach na temat „Hierarchical graphs”

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Autor: Grafiati

Data publikacji: 4 czerwca 2021

Data aktualizacji: 27 kwietnia 2022

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1

EADES, PETER, XUEMIN LIN i ROBERTO TAMASSIA. "AN ALGORITHM FOR DRAWING A HIERARCHICAL GRAPH". International Journal of Computational Geometry & Applications 06, nr 02 (czerwiec 1996): 145–55. http://dx.doi.org/10.1142/s0218195996000101.

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Hierarchical graphs appear in several graph drawing applications, where nodes are assigned layers for semantic reasons. More importantly, general methods for drawing directed graphs usually begin by transforming the input digraph into a hierarchical graph, then applying a hierarchical graph drawing algorithm. This paper introduces the Degree Weighted Barycentre (DWB) algorithm for drawing hierarchical graphs. We show that drawings output by DWB satisfy several important aesthetic criteria: under certain connectivity conditions, they are planar, convex, and symmetric whenever such drawings are possible. The algorithm can be implemented as a simple Gauss — Seidel iteration.

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BUSATTO, GIORGIO, HANS-JÖRG KREOWSKI i SABINE KUSKE. "Abstract hierarchical graph transformation". Mathematical Structures in Computer Science 15, nr 4 (15.07.2005): 773–819. http://dx.doi.org/10.1017/s0960129505004846.

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In this paper we introduce a new hierarchical graph model to structure large graphs into small components by distributing the nodes (and, likewise, edges) into a hierarchy of packages. In contrast to other known approaches, we do not fix the type of underlying graphs. Moreover, our model is equipped with a rule-based transformation concept such that hierarchical graphs are not restricted to being used only for the static representation of complex system states, but can also be used to describe dynamic system behaviour.

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3

Kasyanov, V. N. "Methods and Tools for Visualization of Graphs and Graph Algorithms". International Journal of Applied Mathematics and Informatics 15 (16.11.2021): 78–84. http://dx.doi.org/10.46300/91014.2021.15.13.

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Graphs are the most common abstract structure encountered in computer science and are widely used for structural information visualization. In the paper, we consider practical and general graph formalism of so called hierarchical graphs and present the Higres and ALVIS systems aimed at supporting of structural information visualization on the base of hierarchical graph models.

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Faran, Rachel, i Orna Kupferman. "A Parametrized Analysis of Algorithms on Hierarchical Graphs". International Journal of Foundations of Computer Science 30, nr 06n07 (wrzesień 2019): 979–1003. http://dx.doi.org/10.1142/s0129054119400252.

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Hierarchical graphs are used in order to describe systems with a sequential composition of sub-systems. A hierarchical graph consists of a vector of subgraphs. Vertices in a subgraph may “call” other subgraphs. The reuse of subgraphs, possibly in a nested way, causes hierarchical graphs to be exponentially more succinct than equivalent flat graphs. Early research on hierarchical graphs and the computational price of their succinctness suggests that there is no strong correlation between the complexity of problems when applied to flat graphs and their complexity in the hierarchical setting. That is, the complexity in the hierarchical setting is higher, but all “jumps” in complexity up to an exponential one are exhibited, including no jumps in some problems. We continue the study of the complexity of algorithms for hierarchical graphs, with the following contributions: (1) In many applications, the subgraphs have a small, often a constant, number of exit vertices, namely vertices from which control returns to the calling subgraph. We offer a parameterized analysis of the complexity and point to problems where the complexity becomes lower when the number of exit vertices is bounded by a constant. (2) We describe a general methodology for algorithms on hierarchical graphs. The methodology is based on an iterative compression of subgraphs in a way that maintains the solution to the problems and results in subgraphs whose size depends only on the number of exit vertices, and (3) we handle labeled hierarchical graphs, where edges are labeled by letters from some alphabet, and the problems refer to the languages of the graphs.

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5

Lin, Zhe, Fan Zhang, Xuemin Lin, Wenjie Zhang i Zhihong Tian. "Hierarchical core maintenance on large dynamic graphs". Proceedings of the VLDB Endowment 14, nr 5 (styczeń 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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6

SHARMA, ROHAN, BIBHAS ADHIKARI i TYLL KRUEGER. "SELF-ORGANIZED CORONA GRAPHS: A DETERMINISTIC COMPLEX NETWORK MODEL WITH HIERARCHICAL STRUCTURE". Advances in Complex Systems 22, nr 06 (wrzesień 2019): 1950019. http://dx.doi.org/10.1142/s021952591950019x.

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In this paper, we propose a self-organization mechanism for newly appeared nodes during the formation of corona graphs that define a hierarchical pattern in the resulting corona graphs and we call it self-organized corona graphs (SoCG). We show that the degree distribution of SoCG follows power-law in its tail with power-law exponent approximately 2. We also show that the diameter is less equal to 4 for SoCG defined by any seed graph and for certain seed graphs, the diameter remains constant during its formation. We derive lower bounds of clustering coefficients of SoCG defined by certain seed graphs. Thus, the proposed SoCG can be considered as a growing network generative model which is defined by using the corona graphs and a self-organization process such that the resulting graphs are scale-free small-world highly clustered growing networks. The SoCG defined by a seed graph can also be considered as a network with a desired motif which is the seed graph itself.

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7

Engels, Gregor, i Andy Schürr. "Encapsulated Hierarchical Graphs, Graph Types, and Meta Types". Electronic Notes in Theoretical Computer Science 2 (1995): 101–9. http://dx.doi.org/10.1016/s1571-0661(05)80186-0.

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8

WANG, XIAOQIAN, HUILING XU i CHANGBING MA. "CONSENSUS PROBLEMS IN WEIGHTED HIERARCHICAL GRAPHS". Fractals 27, nr 06 (wrzesień 2019): 1950086. http://dx.doi.org/10.1142/s0218348x19500865.

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We present the hierarchical graph for the growth of weighted networks in which the structural growth is coupled with the edges’ weight dynamical evolution. We investigate consensus problems of the graph from weighted Laplacian spectra perspective, focusing on three important quantities of consensus problems, convergence speed, delay robustness, and first-order coherence, which are determined by the second smallest eigenvalue, largest eigenvalue, and sum of reciprocals of each nonzero eigenvalue of weighted Laplacian matrix, respectively. Unlike previous enquiries, we want to emphasize the importance of weight factor in the study of coherence problems. In what follows, we attempt to study that the weighted Laplacian eigenvalues of the weighted hierarchical graphs, which are determined through analytic recursive equations. We find in our study that the value of convergence speed and delay robustness in weighted hierarchical graphs increases as weight factor increases and the value of first-order coherence decreases as weight factor increases. Moreover, as is expected, weight factor affects the performance of consensus behavior and can be regarded as a leverage in the problem of consensus problems. This paper puts forward the proposal and the countermeasure for stability optimization of networks from the perspective of weight factor for future researchers.

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9

Oggier, Frédérique, i Anwitaman Datta. "Renyi entropy driven hierarchical graph clustering". PeerJ Computer Science 7 (25.02.2021): e366. http://dx.doi.org/10.7717/peerj-cs.366.

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This article explores a graph clustering method that is derived from an information theoretic method that clusters points in ${{\mathbb{R}}^{n}}$ relying on Renyi entropy, which involves computing the usual Euclidean distance between these points. Two view points are adopted: (1) the graph to be clustered is first embedded into ${\mathbb{R}}^{d}$ for some dimension d so as to minimize the distortion of the embedding, then the resulting points are clustered, and (2) the graph is clustered directly, using as distance the shortest path distance for undirected graphs, and a variation of the Jaccard distance for directed graphs. In both cases, a hierarchical approach is adopted, where both the initial clustering and the agglomeration steps are computed using Renyi entropy derived evaluation functions. Numerical examples are provided to support the study, showing the consistency of both approaches (evaluated in terms of F-scores).

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10

Krinkin, Kirill, Alexander Ivanovich Vodyaho, Igor Kulikov i Nataly Zhukova. "Deductive Synthesis of Networks Hierarchical Knowledge Graphs". International Journal of Embedded and Real-Time Communication Systems 12, nr 3 (lipiec 2021): 32–48. http://dx.doi.org/10.4018/ijertcs.2021070103.

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The article focuses on developing of a deductive synthesis method for building telecommunications networks (TN) hierarchical knowledge graphs (KG). Synthesized KGs can be used to solve search, analytical, and recommendation (forecast) problems. TNs are complex heterogeneous objects. The synthesis of knowledge graphs of such objects requires much computational resources. The proposed method provides a low complexity of the synthesis of KG of TN by taking into account their hierarchical structure. The authors propose to do synthesis by direct downward multilevel inference and reverse multilevel inference. The article analyses existing graph models of TNs and methods for their building. Detailed description of the proposed method of networks hierarchical KGs synthesis is given. In order to evaluate the deductive synthesis method, a prototype of the system is developed. The provided real-world example shows how telecommunications networks hierarchical knowledge graphs are synthesized and used in practice. Finally, conclusions are formulated, and the areas of further research are identified.

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11

Kasyanov, V. N., A. M. Merculov i T. A. Zolotuhin. "A circular layout algorithm for attributed hierarchical graphs with ports". Journal of Physics: Conference Series 2099, nr 1 (1.11.2021): 012051. http://dx.doi.org/10.1088/1742-6596/2099/1/012051.

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Abstract Information visualization based on graph models is a key component of support tools for many applications in science and engineering. The Visual Graph system is intended for visualization of big amounts of complex information on the basis of attributed hierarchical graph models. In this paper, a circular layout algorithm for attributed hierarchical graphs with ports and its effective implementation in the Visual Graph system are presented.

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12

Gu, Mei-Mei, Jou-Ming Chang i Rong-Xia Hao. "On Component Connectivity of Hierarchical Star Networks". International Journal of Foundations of Computer Science 31, nr 03 (kwiecień 2020): 313–26. http://dx.doi.org/10.1142/s0129054120500100.

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For an integer [Formula: see text], the [Formula: see text]-component connectivity of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of vertices whose removal from [Formula: see text] results in a disconnected graph with at least [Formula: see text] components or a graph with fewer than [Formula: see text] vertices. This naturally generalizes the classical connectivity of graphs defined in term of the minimum vertex-cut. This kind of connectivity can help us to measure the robustness of the graph corresponding to a network. The hierarchical star networks [Formula: see text], proposed by Shi and Srimani, is a new level interconnection network topology, and uses the star graphs as building blocks. In this paper, by exploring the combinatorial properties and fault-tolerance of [Formula: see text], we study the [Formula: see text]-component connectivity of hierarchical star networks [Formula: see text]. We obtain the results: [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text].

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13

Béthune, Louis, Yacouba Kaloga, Pierre Borgnat, Aurélien Garivier i Amaury Habrard. "Hierarchical and Unsupervised Graph Representation Learning with Loukas’s Coarsening". Algorithms 13, nr 9 (21.08.2020): 206. http://dx.doi.org/10.3390/a13090206.

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We propose a novel algorithm for unsupervised graph representation learning with attributed graphs. It combines three advantages addressing some current limitations of the literature: (i) The model is inductive: it can embed new graphs without re-training in the presence of new data; (ii) The method takes into account both micro-structures and macro-structures by looking at the attributed graphs at different scales; (iii) The model is end-to-end differentiable: it is a building block that can be plugged into deep learning pipelines and allows for back-propagation. We show that combining a coarsening method having strong theoretical guarantees with mutual information maximization suffices to produce high quality embeddings. We evaluate them on classification tasks with common benchmarks of the literature. We show that our algorithm is competitive with state of the art among unsupervised graph representation learning methods.

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14

Zhang, H., J. J. Zhou i R. Li. "Enhanced Unsupervised Graph Embedding via Hierarchical Graph Convolution Network". Mathematical Problems in Engineering 2020 (26.07.2020): 1–9. http://dx.doi.org/10.1155/2020/5702519.

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Graph embedding aims to learn the low-dimensional representation of nodes in the network, which has been paid more and more attention in many graph-based tasks recently. Graph Convolution Network (GCN) is a typical deep semisupervised graph embedding model, which can acquire node representation from the complex network. However, GCN usually needs to use a lot of labeled data and additional expressive features in the graph embedding learning process, so the model cannot be effectively applied to undirected graphs with only network structure information. In this paper, we propose a novel unsupervised graph embedding method via hierarchical graph convolution network (HGCN). Firstly, HGCN builds the initial node embedding and pseudo-labels for the undirected graphs, and then further uses GCNs to learn the node embedding and update labels, finally combines HGCN output representation with the initial embedding to get the graph embedding. Furthermore, we improve the model to match the different undirected networks according to the number of network node label types. Comprehensive experiments demonstrate that our proposed HGCN and HGCN∗ can significantly enhance the performance of the node classification task.

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15

Maulik, Ujjwal. "Hierarchical Pattern Discovery in Graphs". IEEE Transactions on Systems, Man, and Cybernetics, Part C (Applications and Reviews) 38, nr 6 (listopad 2008): 867–72. http://dx.doi.org/10.1109/tsmcc.2008.2001719.

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16

Nakahira, Kenji, i Atsushi Miyamoto. "Parseval wavelets on hierarchical graphs". Applied and Computational Harmonic Analysis 44, nr 2 (marzec 2018): 414–45. http://dx.doi.org/10.1016/j.acha.2016.05.004.

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17

Barrière, L., F. Comellas, C. Dalfó i M. A. Fiol. "The hierarchical product of graphs". Discrete Applied Mathematics 157, nr 1 (styczeń 2009): 36–48. http://dx.doi.org/10.1016/j.dam.2008.04.018.

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18

Landmark, K., i E. Messel. "HIERARCHICAL PATH PLANNING FOR WALKING (ALMOST) ANYWHERE". ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-4/W8 (11.07.2018): 109–16. http://dx.doi.org/10.5194/isprs-archives-xlii-4-w8-109-2018.

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<p><strong>Abstract.</strong> Computerized path planning, not constrained to transportation networks, may be useful in a range of settings, from search and rescue to archaeology. This paper develops a method for general path planning intended to work across arbitrary distances and at the level of terrain detail afforded by aerial LiDAR scanning. Relevant information about terrain, trails, roads, and other infrastructure is encoded in a large directed graph. This basal graph is partitioned into strongly connected subgraphs such that the generalized diameter of each subgraphs is constrained by a set value, and with nominally as few subgraphs as possible. This is accomplished using the k-center algorithm adapted with heuristics suitable for large spatial graphs. A simplified graph results, with reduced (but known) position accuracy and complexity. Using a hierarchy of simplified graphs adapted to different length scales, and with careful selection of levels in the hierarchy based on geodesic distance, a shortest path search can be restricted to a small subset of the basal graph. The method is formulated using matrix-graph duality, suitable for linear algebra-oriented software. Extensive use is also made of public data, including LiDAR, as well as free and open software for geospatial data processing.</p>

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Lam, Ho-Ching, i Ivo D. Dinov. "Hyperbolic Wheel: A Novel Hyperbolic Space Graph Viewer for Hierarchical Information Content". ISRN Computer Graphics 2012 (31.10.2012): 1–10. http://dx.doi.org/10.5402/2012/609234.

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Tree and graph structures have been widely used to present hierarchical and linked data. Hyperbolic trees are special types of graphs composed of nodes (points or vertices) and edges (connecting lines), which are visualized on a non-Euclidean space. In traditional Euclidean space graph visualization, distances between nodes are measured by straight lines. Displays of large graphs in Euclidean spaces may not utilize efficiently the available space and may impose limitations on the number of graph nodes. The special hyperbolic space rendering of tree-graphs enables adaptive and efficient use of the available space and facilitates the display of large hierarchical structures. In this paper we report on a newly developed advanced hyperbolic graph viewer, Hyperbolic Wheel, which enables the navigation, traversal, discovery and interactive manipulation of information stored in large hierarchical structures. Examples of such structures include personnel records, disc directory structures, ontological constructs, web-pages and other nested partitions. The Hyperbolic Wheel framework provides an intuitive and dynamic graphical interface to explore and retrieve information about individual nodes (data objects) and their relationships (data associations). The Hyperbolic Wheel is freely available online for educational and research purposes.

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Leon, Leissi M. C., Krzysztof C. Ciesielski i Paulo A. V. Miranda. "Efficient Hierarchical Multi-Object Segmentation in Layered Graphs". Mathematical Morphology - Theory and Applications 5, nr 1 (1.01.2021): 21–42. http://dx.doi.org/10.1515/mathm-2020-0108.

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Abstract We propose a novel efficient seed-based method for the multi-object segmentation of images based on graphs, named Hierarchical Layered Oriented Image Foresting Transform (HLOIFT). It uses a tree of the relations between the image objects, with each node in the tree representing an object. Each tree node may contain different individual high-level priors of its corresponding object and defines a weighted digraph, named as layer. The layer graphs are then integrated into a hierarchical graph, considering the hierarchical relations of inclusion and exclusion. A single energy optimization is performed in the hierarchical layered weighted digraph leading to globally optimal results satisfying all the high-level priors. The experimental evaluations of HLOIFT, on medical, natural, and synthetic images, indicate promising results comparable to the related baseline methods that include structural information, but with lower computational complexity. Compared to the hierarchical segmentation by the min-cut/max-flow algorithm, our approach is less restrictive, leading to globally optimal results in more general scenarios, and has a better running time.

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Archambault, Daniel, i Helen C. Purchase. "On the effective visualisation of dynamic attribute cascades". Information Visualization 15, nr 1 (6.04.2015): 51–63. http://dx.doi.org/10.1177/1473871615576758.

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Cascades appear in many applications, including biological graphs and social media analysis. In a cascade, a dynamic attribute propagates through a graph, following its edges. We present the results of a formal user study that tests the effectiveness of different types of cascade visualisations on node-link diagrams for the task of judging cascade spread. Overall, we found that a small multiples presentation was significantly faster than animation with no significant difference in terms of error rate. Participants generally preferred animation over small multiples and a hierarchical layout to a force-directed layout. Considering each presentation method separately, when comparing force-directed layouts to hierarchical layouts, hierarchical layouts were found to be significantly faster for both presentation methods and significantly more accurate for animation. Representing the history of the cascade had no significant effect. Thus, for our task, this experiment supports the use of a small multiples interface with hierarchically drawn graphs for the visualisation of cascades. This work is important because without these empirical results, designers of dynamic multivariate visualisations (in many applications) would base their design decisions on intuition with little empirical support as to whether these decisions enhance usability.

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De, Nilanjan. "Application of generalised hierarchical product of graphs for computing F-index of four operations on graphs". Indonesian Journal of Combinatorics 2, nr 2 (21.12.2018): 97. http://dx.doi.org/10.19184/ijc.2018.2.2.5.

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The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph which was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced. In this paper we study the F-index of four operations on graphs which were introduced by Eliasi and Taeri, and hence using the derived results we find F-index of some particular and chemically interesting graphs.

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Tavakoli, M., F. Rahbarnia i A. R. Ashrafi. "Distribution of some graph invariants over hierarchical product of graphs". Applied Mathematics and Computation 220 (wrzesień 2013): 405–13. http://dx.doi.org/10.1016/j.amc.2013.06.009.

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DICKINSON, P. J., M. KRAETZL, H. BUNKE, M. NEUHAUS i A. DADEJ. "SIMILARITY MEASURES FOR HIERARCHICAL REPRESENTATIONS OF GRAPHS WITH UNIQUE NODE LABELS". International Journal of Pattern Recognition and Artificial Intelligence 18, nr 03 (maj 2004): 425–42. http://dx.doi.org/10.1142/s021800140400323x.

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A hierarchical abstraction scheme based on node contraction and two related similarity measures for graphs with unique node labels are proposed in this paper. The contraction scheme reduces the number of nodes in a graph and leads to a speed-up in the computation of graph similarity. Theoretical properties of the new graph similarity measures are derived and experimentally verified. A potential application of the proposed graph abstraction scheme in the domain of computer network monitoring is discussed.

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Li, Zitong, Xiang Cheng, Lixiao Sun, Ji Zhang i Bing Chen. "A Hierarchical Approach for Advanced Persistent Threat Detection with Attention-Based Graph Neural Networks". Security and Communication Networks 2021 (4.05.2021): 1–14. http://dx.doi.org/10.1155/2021/9961342.

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Advanced Persistent Threats (APTs) are the most sophisticated attacks for modern information systems. Currently, more and more researchers begin to focus on graph-based anomaly detection methods that leverage graph data to model normal behaviors and detect outliers for defending against APTs. However, previous studies of provenance graphs mainly concentrate on system calls, leading to difficulties in modeling network behaviors. Coarse-grained correlation graphs depend on handcrafted graph construction rules and, thus, cannot adequately explore log node attributes. Besides, the traditional Graph Neural Networks (GNNs) fail to consider meaningful edge features and are difficult to perform heterogeneous graphs embedding. To overcome the limitations of the existing approaches, we present a hierarchical approach for APT detection with novel attention-based GNNs. We propose a metapath aggregated GNN for provenance graph embedding and an edge enhanced GNN for host interactive graph embedding; thus, APT behaviors can be captured at both the system and network levels. A novel enhancement mechanism is also introduced to dynamically update the detection model in the hierarchical detection framework. Evaluations show that the proposed method outperforms the state-of-the-art baselines in APT detection.

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Qi, Yi, Yuze Dong, Zhongzhi Zhang i Zhang Zhang. "Hitting Times for Random Walks on Sierpiński Graphs and Hierarchical Graphs". Computer Journal 63, nr 9 (17.04.2020): 1385–96. http://dx.doi.org/10.1093/comjnl/bxz080.

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Abstract The Sierpiński graphs and hierarchical graphs are two much studied self-similar networks, both of which are iteratively constructed and have the same number of vertices and edges at any iteration, but display entirely different topological properties. Both graphs have a large variety of applications: Sierpiński graphs have a close connection with WK-recursive networks that are employed extensively in the design and implementation of local area networks and parallel processing architectures, while hierarchical graphs can be used to model complex networks. In this paper, we study hitting times for several absorbing random walks in Sierpiński graphs and hierarchical graphs. For all considered random walks, we determine exact solutions to hitting times for both graphs. The obtained explicit expressions indicate that the hitting times in both graphs behave quite differently. We show that the structural difference of the graphs is responsible for the disparate behaviors of their hitting times.

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Dutta, Anjan, Pau Riba, Josep Lladós i Alicia Fornés. "Hierarchical stochastic graphlet embedding for graph-based pattern recognition". Neural Computing and Applications 32, nr 15 (6.12.2019): 11579–96. http://dx.doi.org/10.1007/s00521-019-04642-7.

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AbstractDespite being very successful within the pattern recognition and machine learning community, graph-based methods are often unusable because of the lack of mathematical operations defined in graph domain. Graph embedding, which maps graphs to a vectorial space, has been proposed as a way to tackle these difficulties enabling the use of standard machine learning techniques. However, it is well known that graph embedding functions usually suffer from the loss of structural information. In this paper, we consider the hierarchical structure of a graph as a way to mitigate this loss of information. The hierarchical structure is constructed by topologically clustering the graph nodes and considering each cluster as a node in the upper hierarchical level. Once this hierarchical structure is constructed, we consider several configurations to define the mapping into a vector space given a classical graph embedding, in particular, we propose to make use of the stochastic graphlet embedding (SGE). Broadly speaking, SGE produces a distribution of uniformly sampled low-to-high-order graphlets as a way to embed graphs into the vector space. In what follows, the coarse-to-fine structure of a graph hierarchy and the statistics fetched by the SGE complements each other and includes important structural information with varied contexts. Altogether, these two techniques substantially cope with the usual information loss involved in graph embedding techniques, obtaining a more robust graph representation. This fact has been corroborated through a detailed experimental evaluation on various benchmark graph datasets, where we outperform the state-of-the-art methods.

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Ranjan, Ekagra, Soumya Sanyal i Partha Talukdar. "ASAP: Adaptive Structure Aware Pooling for Learning Hierarchical Graph Representations". Proceedings of the AAAI Conference on Artificial Intelligence 34, nr 04 (3.04.2020): 5470–77. http://dx.doi.org/10.1609/aaai.v34i04.5997.

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Graph Neural Networks (GNN) have been shown to work effectively for modeling graph structured data to solve tasks such as node classification, link prediction and graph classification. There has been some recent progress in defining the notion of pooling in graphs whereby the model tries to generate a graph level representation by downsampling and summarizing the information present in the nodes. Existing pooling methods either fail to effectively capture the graph substructure or do not easily scale to large graphs. In this work, we propose ASAP (Adaptive Structure Aware Pooling), a sparse and differentiable pooling method that addresses the limitations of previous graph pooling architectures. ASAP utilizes a novel self-attention network along with a modified GNN formulation to capture the importance of each node in a given graph. It also learns a sparse soft cluster assignment for nodes at each layer to effectively pool the subgraphs to form the pooled graph. Through extensive experiments on multiple datasets and theoretical analysis, we motivate our choice of the components used in ASAP. Our experimental results show that combining existing GNN architectures with ASAP leads to state-of-the-art results on multiple graph classification benchmarks. ASAP has an average improvement of 4%, compared to current sparse hierarchical state-of-the-art method. We make the source code of ASAP available to encourage reproducible research 1.

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Hong, Seok-Hee, i Hiroshi Nagamochi. "Convex drawings of hierarchical planar graphs and clustered planar graphs". Journal of Discrete Algorithms 8, nr 3 (wrzesień 2010): 282–95. http://dx.doi.org/10.1016/j.jda.2009.05.003.

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Eades, Peter, Qingwen Feng, Xuemin Lin i Hiroshi Nagamochi. "Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs". Algorithmica 44, nr 1 (21.02.2005): 1–32. http://dx.doi.org/10.1007/s00453-004-1144-8.

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31

MIYAMOTO, Sadaaki. "On Hierarchical Clustering by Fuzzy Graphs". Journal of Japan Society for Fuzzy Theory and Systems 5, nr 6 (1993): 1354–71. http://dx.doi.org/10.3156/jfuzzy.5.6_1354.

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32

Ene, Nneka Chinelo, Maribel Fernández i Bruno Pinaud. "Attributed Hierarchical Port Graphs and Applications". Electronic Proceedings in Theoretical Computer Science 265 (16.02.2018): 2–19. http://dx.doi.org/10.4204/eptcs.265.2.

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33

Andjelković, Miroslav, Bosiljka Tadić, Slobodan Maletić i Milan Rajković. "Hierarchical sequencing of online social graphs". Physica A: Statistical Mechanics and its Applications 436 (październik 2015): 582–95. http://dx.doi.org/10.1016/j.physa.2015.05.075.

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34

Lopez-Rubio, E., i E. J. Palomo. "Growing Hierarchical Probabilistic Self-Organizing Graphs". IEEE Transactions on Neural Networks 22, nr 7 (lipiec 2011): 997–1008. http://dx.doi.org/10.1109/tnn.2011.2138159.

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35

Pelillo, M., K. Siddiqi i S. W. Zucker. "Matching hierarchical structures using association graphs". IEEE Transactions on Pattern Analysis and Machine Intelligence 21, nr 11 (1999): 1105–20. http://dx.doi.org/10.1109/34.809105.

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36

Balmas, F. "Displaying dependence graphs: a hierarchical approach". Journal of Software Maintenance and Evolution: Research and Practice 16, nr 3 (maj 2004): 151–85. http://dx.doi.org/10.1002/smr.291.

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Nakahira, Kenji, i Atsushi Miyamoto. "Multi-link wavelets on hierarchical graphs". Applied and Computational Harmonic Analysis 37, nr 1 (lipiec 2014): 1–11. http://dx.doi.org/10.1016/j.acha.2013.08.007.

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38

WOLSKI, RICH. "STATIC SCHEDULING OF HIERARCHICAL PROGRAM GRAPHS". Parallel Processing Letters 05, nr 04 (grudzień 1995): 611–22. http://dx.doi.org/10.1142/s0129626495000540.

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Many parallel compilation systems represent programs internally as Directed Acyclic Graphs (DAGs). However, the storage of these DAGs becomes prohibitive when the program being compiled is large. In this paper we describe a compile-time scheduling methodology for hierarchical DAG programs represented in the IFX intermediate form. The method we present is itself hierarchical reducing the storage that would otherwise be required by a single flat DAG representation. We describe the scheduling model and demonstrate the method using the Optimizing Sisal Compiler and two scientific applications.

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39

Barrière, L., C. Dalfó, M. A. Fiol i M. Mitjana. "The generalized hierarchical product of graphs". Discrete Mathematics 309, nr 12 (czerwiec 2009): 3871–81. http://dx.doi.org/10.1016/j.disc.2008.10.028.

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Pentus, A. E., i M. R. Pentus. "Object-oriented representation of hierarchical graphs". Journal of Mathematical Sciences 140, nr 2 (styczeń 2007): 286–94. http://dx.doi.org/10.1007/s10958-007-0423-8.

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41

Guadagnino, Tiziano, Luca Di Giammarino i Giorgio Grisetti. "HiPE: Hierarchical Initialization for Pose Graphs". IEEE Robotics and Automation Letters 7, nr 1 (styczeń 2022): 287–94. http://dx.doi.org/10.1109/lra.2021.3125046.

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42

Yoshikawa, Tomohiro, Yuki Uchida, Takeshi Furuhashi, Eiji Hirao i Hiroto Iguchi. "Extraction of Evaluation Keywords for Analyzing Product Evaluation in User-Reviews Using Hierarchical Keyword Graph". Journal of Advanced Computational Intelligence and Intelligent Informatics 13, nr 4 (20.07.2009): 457–62. http://dx.doi.org/10.20965/jaciii.2009.p0457.

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Recently, the number of sites on the Internet which give users the opportunity to write their ideas and opinions for the public to read have been increasing. In addition, the number of people who want to know the opinions of others about interesting products has also been increasing. However, it is very difficult for people to read complete reviews on the Internet. This study tries to develop a new system for the analysis of reviews, a system which shows evaluation information about products using graphs of evaluation keywords. This paper focuses on the extraction of evaluation keywords from reviews on the Internet. This paper proposes a method for extracting evaluation keywords and displays its results as graphs. It employs HK Graph (Hierarchical Keyword Graph), which can visualize the relationship among words in a hierarchical network structure based on the co-occurrence information for the keyword graph.

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43

Luo, Zhaoyang. "Applications on Hyper-Zagreb Index of Generalized Hierarchical Product Graphs". Journal of Computational and Theoretical Nanoscience 13, nr 10 (1.10.2016): 7355–61. http://dx.doi.org/10.1166/jctn.2016.5726.

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Let G be a connected graph. The Hyper-Zagreb index of a connected graph G is defined as HM(G) = Σuv∈EG [dG(u)+dG(v)]2, where dG(v) is the degree of the vertex v in G. In this paper, the Hyper-Zagreb Gindex of the generalized hierarchical, Cartesian, cluster, corona products and four new sums of graphs according to some invariants of the factors are computed, respectively. As applications, we present explicit formulas for the HM index of the linear phenylene Fn, the C4 nanotorus Cm□Cn, the C4 nanotubes Pm□Cn, the l-dimensional hypercubes Ql , the zig-zag polyhex nanotube TUHC6[2n, 2], the hexagonal chain ln, the regular dicentric dendrimer DDp,r and so forth.

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44

Gu, Mei-Mei, Rong-Xia Hao i Eddie Cheng. "Note on Applications of Linearly Many Faults". Computer Journal 63, nr 9 (15.11.2019): 1406–16. http://dx.doi.org/10.1093/comjnl/bxz088.

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Abstract Most graphs have this property: after removing a linear number of vertices from a graph, the surviving graph is either connected or consists of a large connected component and small components containing a small number of vertices. This property can be applied to derive fault-tolerance related network parameters: extra edge connectivity and component edge connectivity. Using this general property, we obtained the $h$-extra edge connectivity and $(h+2)$-component edge connectivity of augmented cubes, Cayley graphs generated by transposition trees, complete cubic networks (including hierarchical cubic networks), generalized exchanged hypercubes (including exchanged hypercubes) and dual-cube-like graphs (including dual cubes).

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45

Bruni, Roberto, Ugo Montanari, Gordon Plotkin i Daniele Terreni. "On Hierarchical Graphs: Reconciling Bigraphs, Gs-monoidal Theories and Gs-graphs". Fundamenta Informaticae 134, nr 3-4 (2014): 287–317. http://dx.doi.org/10.3233/fi-2014-1103.

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Cunningham, Steve, i Michael J. Bailey. "Lessons from scene graphs: using scene graphs to teach hierarchical modeling". Computers & Graphics 25, nr 4 (sierpień 2001): 703–11. http://dx.doi.org/10.1016/s0097-8493(01)00099-1.

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RAFE, VAHID, i ADEL T. RAHMANI. "A NOVEL APPROACH TO VERIFY GRAPH SCHEMA-BASED SOFTWARE SYSTEMS". International Journal of Software Engineering and Knowledge Engineering 19, nr 06 (wrzesień 2009): 857–70. http://dx.doi.org/10.1142/s0218194009004398.

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Graph Grammars have recently become more and more popular as a general formal modeling language. Behavioral modeling of dynamic systems and model to model transformations are a few well-known examples in which graphs have proven their usefulness in software engineering. A special type of graph transformation systems is layered graphs. Layered graphs are a suitable formalism for modeling hierarchical systems. However, most of the research so far concentrated on graph transformation systems as a modeling means, without considering the need for suitable analysis tools. In this paper we concentrate on how to analyze these models. We will describe our approach to show how one can verify the designed graph transformation systems. To verify graph transformation systems we use a novel approach: using Bogor model checker to verify graph transformation systems. The AGG-like graph transformation systems are translated to BIR — the input language of Bogor — and Bogor verifies that model against some properties defined by combining LTL and special purpose graph rules. Supporting schema-based and layered graphs characterize our approach among existing solutions for verification of graph transformation systems.

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48

Luo, Zhaoyang, i Jianliang Wu. "Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications". Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/241712.

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LetGbe a connected graph. The first and second Zagreb eccentricity indices ofGare defined asM1*(G)=∑v∈V(G)‍εG2(v)andM2*(G)=∑uv∈E(G)‍εG(u)εG(v), whereεG(v)is the eccentricity of the vertexvinGandεG2(v)=(εG(v))2. Suppose thatG(U)⊓H(∅≠U⊆V(G))is the generalized hierarchical product of two connected graphsGandH. In this paper, the Zagreb eccentricity indicesM1*andM2*ofG(U)⊓Hare computed. Moreover, we present explicit formulas for theM1*andM2*ofS-sum graph, Cartesian, cluster, and corona product graphs by means of some invariants of the factors.

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49

Hong, Seok-Hee, Nikola S. Nikolov i Alexandre Tarassov. "A 2.5D Hierarchical Drawing of Directed Graphs". Journal of Graph Algorithms and Applications 11, nr 2 (2007): 371–96. http://dx.doi.org/10.7155/jgaa.00151.

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Körner, Christof. "Sequential processing in comprehension of hierarchical graphs". Applied Cognitive Psychology 18, nr 4 (2004): 467–80. http://dx.doi.org/10.1002/acp.997.

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