An algorithm for embedding Turán graphs into incomplete hypercubes with minimum wirelength

An effective routing algorithm in incomplete hypercubes

An effective approach to the enhancement of incomplete hypercube computers

Incomplete hypercubes make the hypercubes more flexible on task allocation in large cubes, cost of manufacturing hardware, and hypercubes with faulty nodes. The authors devise and analyze broadcasting algorithms based on the hierarchical binomial spanning trees, hierarchical edge-disjoint spanning trees and edge-disjoint spanning trees in an incomplete hypercube of 2/sup n/+2/sup k/ nodes, where 0<or=k<n. The hierarchical binomial spanning trees have the shortcoming of unbalanced load on parallel paths. In the one-port communication model, in which each node can send message along only one link at a time, the hierarchical edge-disjoint spanning tree algorithms are optimal within a factor of two. The edge-disjoint spanning trees algorithms are optimal with a factor of /sup n+1///sub k+1/ For all-port communication, in which a node can send messages to any number of links adjacent to it, the edge-disjoint spanning trees algorithms are strictly optimal.<<ETX>>

Incomplete hypercubes are analyzed. The elementary properties of complete hypercubes and a routing algorithm for incomplete hypercubes are briefly reviewed. Structural properties, including diameter, mean message traversal, and traffic density, of incomplete hypercube computers with size 2/sup n/+2/sup k/, 0 >

Incomplete hypercubes make the hypercubes more flexible on task allocation in large cubes, cost of manufacturing hardware, and hypercubes with faulty nodes. The authors devise and analyze a broadcasting algorithm based on edge-disjoint spanning trees in an incomplete hypercube of 2/sup n/+2/sup k/ nodes, where 0 >

The 3-ary n-cube, denoted as $$ {Q}_n^3 $$ , is an important interconnection network topology proposed for parallel computers, owing to its many desirable properties such as regular and symmetrical structure, and strong scalability, among others. In this paper, we first obtain an exact formula for the minimum wirelength to embed $$ {Q}_n^3 $$ into grids. We then propose a load balancing algorithm for embedding $$ {Q}_n^3 $$ into a square grid with minimum dilation and congestion. Finally, we derive an O(N2) algorithm for embedding $$ {Q}_n^3 $$ into a gird with balanced communication, where N is the number of nodes in $$ {Q}_n^3 $$ . Simulation experiments are performed to verify the total wirelength and evaluate the network cost of our proposed embedding algorithm.

The deferred-merge embedding (DME) algorithm is presented. In linear time, it embeds any given connection topology into the Manhattan plane to create a clock tree with zero skew while minimizing total wirelength. Experimental results show that the algorithm yields exact zero skew trees with 9% to 16% wirelength reduction over previous constructions. The DME algorithm may be applied to either the Elmore or the linear delay model and yields optimal total wirelength for linear delay.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The incrementally extensible hypercube (IEH) graph is a generalization of binary hypercube networks in that the number of nodes can be arbitrary in contrast to a strict power of 2. In this paper, the authors present an efficient model to fulfill the embedding of a full binary tree into a full IEH graph. As the model the authors proposed, an algorithm is developed for the embedding with expansion 1, load 1, congestion 2, and dilation 2. Moreover, the embedding algorithm can be also applied into hypercubes with one faulty node.

Deriving an effective VLSI layout for interconnected network is important, since it increases the cost-effectiveness of parallel architectures. Graph embedding is the key to solving the problems of parallel structure simulation and layout design of VLSI. Wirelength is a criterion measuring the quality for graph embedding. And it is extensively used for VLSI design. Owing to the limitation of the chip area, the total wirelength of embedded network becomes a key issue affecting the network-on-chip communication performance. \(AQ_{n}\), the n-dimensional augmented cube, is an important interconnection network topology proposed for parallel computers. In this paper, we first study the minimum wirelength of embedding augmented cube into a linear array based on the maximum induced subgraph problem. Furthermore, we obtain the exact wirelength of embedding augmented cubes into grids and propose a linear embedding algorithm to prepare for further study of efficient layout areas.

Efficient routing of messages is a key to the performance of multicomputers. Multicast communication refers to the delivery of the same message from a source node to an arbitrary number of destination nodes. While multicast communication is highly demanded in many applications, most of the existing multicomputers do not directly support this service; rather it is indirectly supported by multiple one-to-one or broadcast communications, which result in more network traffic and a waste of system resources. In this paper, we study routing evaluation criteria for multicast communication under different switching technologies. Multicast communication in multicomputers is formulated as a graph theoretical problem. Depending on the evaluation criteria and switching technologies, we study three optimal multicast communication problems, which are equivalent to the finding of the following three subgraphs: optimal multicast path, optimal multicast cycle, and minimal Steiner tree, where the interconnection of a multicomputer defines a host graph. We will show that all these optimization problems are NP-complete for the popular 2D-mesh and hypercube host graphs. Heuristic multicast algorithms for these routing problems are proposed. © 1993 IEEE

Graph embedding algorithms embed a graph into a vector space where the structure and the inherent properties of the graph are preserved. The existing graph embedding methods cannot preserve the asymmetric transitivity well, which is a critical property of directed graphs. Asymmetric transitivity depicts the correlation among directed edges, that is, if there is a directed path from u to v, then there is likely a directed edge from u to v. Asymmetric transitivity can help in capturing structures of graphs and recovering from partially observed graphs. To tackle this challenge, we propose the idea of preserving asymmetric transitivity by approximating high-order proximity which are based on asymmetric transitivity. In particular, we develop a novel graph embedding algorithm, High-Order Proximity preserved Embedding (HOPE for short), which is scalable to preserve high-order proximities of large scale graphs and capable of capturing the asymmetric transitivity. More specifically, we first derive a general formulation that cover multiple popular high-order proximity measurements, then propose a scalable embedding algorithm to approximate the high-order proximity measurements based on their general formulation. Moreover, we provide a theoretical upper bound on the RMSE (Root Mean Squared Error) of the approximation. Our empirical experiments on a synthetic dataset and three real-world datasets demonstrate that HOPE can approximate the high-order proximities significantly better than the state-of-art algorithms and outperform the state-of-art algorithms in tasks of reconstruction, link prediction and vertex recommendation.

Entity Resolution is the process of identifying whether or not various entities from different sources are referring to the same real-world entity. Entity Resolution hasn't been extensively researched in graph databases, whereas it has been for relational databases. This paper focuses on providing comparisons of experiments on various datasets to determine the most appropriate method used in the Entity Resolution process from among literature's similarity algorithms, graph embedding techniques, and graph embedding algorithms combined to link prediction. Moreover, if the embedding algorithm employed has an impact on the given results. The results show that the Entity Resolution process performed better when graph embedding techniques were paired with link prediction, and the chosen graph embedding algorithm also has an impact on the results.

Graph embeddings play a significant role in the design and analysis of parallel algorithms. It is a mapping of the topological structure of a guest graph G into a host graph H, which is represented as a one-to-one mapping from the vertex set of the guest graph to the vertex set of the host graph. In multiprocessing systems, the interconnection networks enhance the efficient communication between the components in the system. Obtaining minimum wirelength in embedding problems is significant in the designing of networks and simulating one architecture by another. In this paper, we determine the wirelength of embedding 3-ary n-cubes into cylinders and certain trees.

Traditionally, the study of complex networks has employed mathematical toolsfrom graph theory and statistical physics. More recently, topological data analysis has introduced the application of algebraic topology to network science, opening up new possibilities. This dissertation takes a similar direction by drawing on topological and geometrical aspects of graph theory to develop data mining algorithms that provide new scientific insights into complex networks. I consider the geometric structure of networks at many levels. First, I study the intrinsic metric structure of a graph using the theory of length spectrum and its relation to the eigenvalues of the non-backtracking matrix of the graph. I use this relationship to develop a principled algorithm for measuring graph distance. Second, I study the geometric structure of graph embeddings. Graph embedding techniques place a graph in a surrounding metric space. I use the geometry of this space to develop two embedding algorithms that enable the performance of such tasks such as link prediction and anomaly detection in an efficient and interpretable manner. Third, because considering distances between graphs depends on the assumption that graphs with similar structure exhibit similar behavior, I study how certain dynamical properties of graphs change under small perturbations of their structure. In particular, I focus on how small perturbations to the structure of the graph affect the epidemic threshold of certain dynamics. I introduce two algorithms that effectively manipulate the epidemic threshold by exploiting the relationship between the graph structure and its spectral properties. I conclude with a reflection on the current trend in network science to rethink the study of complex systems in terms of polyadic relations. My collaborators and I find that a necessary step in this direction is to focus on the properties of real-world systems that determine which relationships affect the existence of other relationships. I discuss this property, which I call dependency, in the context of graphs, simplicial complexes, and hypergraphs.--Author's abstract

Network analysis has been widely applied in many real-world tasks, such as gene analysis and targeted marketing. To extract effective features for these analysis tasks, network embedding automatically learns a low-dimensional vector representation for each node, such that the meaningful topological proximity is well preserved. While the embedding algorithms on pure topological structure have attracted considerable attention, in practice, nodes are often abundantly accompanied with other types of meaningful information, such as node attributes, second-order proximity, and link directionality. A general framework for incorporating the heterogeneous information into network embedding could be potentially helpful in learning better vector representations. However, it remains a challenging task to jointly embed the geometrical structure and a distinct type of information due to the heterogeneity. In addition, the real-world networks often contain a large number of nodes, which put demands on the scalability of the embedding algorithms. To bridge the gap, in this article, we propose a general embedding framework named Heterogeneous Information Learning in Large-scale networks (HILL) to accelerate the joint learning. It enables the simultaneous node proximity assessing process to be done in a distributed manner by decomposing the complex modeling and optimization into many simple and independent sub-problems. We validate the significant correlation between the heterogeneous information and topological structure, and illustrate the generalizability of HILL by applying it to perform attributed network embedding and second-order proximity learning. A variation is proposed for link directionality modeling. Experimental results on real-world networks demonstrate the effectiveness and efficiency of HILL.