Abstract: This paper addresses the hyperlink prediction problem in hypernetworks. Different from the traditional link prediction problem where only pairwise relations are considered as links, our task here is to predict the linkage of multiple nodes, i.e., hyperlink. Each hyperlink is a set of an arbitrary number of nodes which together form a multiway relationship. Hyperlink prediction is challenging – since the cardinality of a hyperlink is variable, existing classifiers based on a fixed number of input features become infeasible. Heuristic methods, such as the common neighbors and Katz index, do not work for hyperlink prediction, since they are restricted to pairwise similarities. In this paper, we formally define the hyperlink prediction problem, and propose a new algorithm called Coordinated Matrix Minimization (CMM), which alternately performs nonnegative matrix factorization and least square matching in the vertex adjacency space of the hypernetwork, in order to infer a subset of candidate hyperlinks that are most suitable to fill the training hypernetwork. We evaluate CMM on two novel tasks: predicting recipes of Chinese food, and finding missing reactions of metabolic networks. Experimental results demonstrate the superior performance of our method over many seemingly promising baselines.

Abstract: Echo state network is a novel kind of recurrent neural networks, with a trainable linear readout layer and a large fixed recurrent connected hidden layer, which can be used to map the rich dynamics of complex real-world data sets. It has been extensively studied in time series prediction. However, there may be an ill-posed problem caused by the number of real-world training samples less than the size of the hidden layer. In this brief, a Laplacian echo state network (LAESN), is proposed to overcome the ill-posed problem and obtain low-dimensional output weights. First, an echo state network is used to map the multivariate time series into a large reservoir. Then, assuming that an unknown underlying manifold is inside the reservoir, we employ the Laplacian eigenmaps to estimate the manifold by constructing an adjacency graph associated with the reservoir states. Finally, the output weights are calculated by the low-dimensional manifold. In addition, some criteria of transient stability, local controllability, and local observability are given. Experimental results based on two real-world data sets substantiate the effectiveness and characteristics of the proposed LAESN model.

Abstract: By considering the cubic nature of hyperspectral image (HSI) to address the issue of the curse of dimensionality, we have introduced a tensor locality preserving projection (TLPP) algorithm for HSI dimensionality reduction and classification. The TLPP algorithm reveals the local structure of the original data through constructing an adjacency graph. However, the hyperspectral data are often susceptible to noise, which may lead to inaccurate graph construction. To resolve this issue, we propose a modified TLPP (MTLPP) via building an adjacency graph on a dual feature space rather than the original space. To this end, the region covariance descriptor is exploited to characterize a region of interest around each hyperspectral pixel. The resulting covariances are the symmetric positive definite matrices lying on a Riemannian manifold such that the Log-Euclidean metric is utilized as the similarity measure for the search of the nearest neighbors. Since the defined covariance feature is more robust against noise, the constructed graph can preserve the intrinsic geometric structure of data and enhance the discriminative ability of features in the low-dimensional space. The experimental results on two real HSI data sets validate the effectiveness of our proposed MTLPP method.

Abstract: We explore further the notion of cluster adjacency, focussing on non-MHV amplitudes. We extend the notion of adjacency to the BCFW decomposition of tree-level amplitudes. Adjacency controls the appearance of poles, both physical and spurious, in individual BCFW terms. We then discuss how this notion of adjacency is connected to the adjacency already observed at the level of symbols of scattering amplitudes which controls the appearance of branch cut singularities. Poles and symbols become intertwined by cluster adjacency and we discuss the relation of this property to the $\bar{Q}$-equation which imposes constraints on the derivatives of the transcendental functions appearing in loop amplitudes.

Abstract: We exploit the recently described property of cluster adjacency for scattering amplitudes in planar $\mathcal{N}=4$ super Yang-Mills theory to construct the symbol of the four-loop NMHV heptagon amplitude. We use a manifestly cluster adjacent ansatz and describe how the parameters of this ansatz are determined using simple physical consistency requirements. We then specialise our answer for the amplitude to the multi-Regge limit, finding agreement with previously available results up to the next-to-leading logarithm, and obtaining new predictions up to (next-to)$^3$-leading-logarithmic accuracy.

Abstract: We consider spectral clustering algorithms for community detection under a general bipartite stochastic block model (SBM). A modern spectral clustering algorithm consists of three steps: (1) regularization of an appropriate adjacency or Laplacian matrix (2) a form of spectral truncation and (3) a k-means type algorithm in the reduced spectral domain. We focus on the adjacency-based spectral clustering and for the first step, propose a new data-driven regularization that can restore the concentration of the adjacency matrix even for the sparse networks. This result is based on recent work on regularization of random binary matrices, but avoids using unknown population level parameters, and instead estimates the necessary quantities from the data. We also propose and study a novel variation of the spectral truncation step and show how this variation changes the nature of the misclassification rate in a general SBM. We then show how the consistency results can be extended to models beyond SBMs, such as inhomogeneous random graph models with approximate clusters, including a graphon clustering problem, as well as general sub-Gaussian biclustering. A theme of the paper is providing a better understanding of the analysis of spectral methods for community detection and establishing consistency results, under fairly general clustering models and for a wide regime of degree growths, including sparse cases where the average expected degree grows arbitrarily slowly.

Abstract: In this paper, we present a fully-dynamic graph data structure for the Graphics Processing Unit (GPU). It delivers high update rates while keeping a low memory footprint using autonomous memory management directly on the GPU. The data structure is fully-dynamic, allowing not only for edge but also vertex updates. Performing the memory management on the GPU allows for fast initialization times and efficient update procedures without additional intervention or reallocation procedures from the host. Our optimized approach performs initialization completely in parallel; up to 300x faster compared to previous work. It achieves up to 200 million edge updates per second for sorted and unsorted update batches; up to 30x faster than previous work. Furthermore, it can perform more than 300 million adjacency queries and millions of vertex updates per second. On account of efficient memory management techniques like a queuing approach, currently unused memory is reused later on by the framework, permitting the storage of tens of millions of vertices and hundreds of millions of edges in GPU memory. We evaluate algorithmic performance using a PageRank and a Static Triangle Counting (STC) implementation, demonstrating the suitability of the framework even for memory access intensive algorithms.

Abstract: This article has two main objectives: one is to describe some extensions of an adaptive Algebraic Multigrid (AMG) method of the form previously proposed by the first and third authors, and a second one is to present a new software framework, named BootCMatch, which implements all the components needed to build and apply the described adaptive AMG both as a stand-alone solver and as a preconditioner in a Krylov method. The adaptive AMG presented is meant to handle general symmetric and positive definite (SPD) sparse linear systems, without assuming any a priori information of the problem and its origin; the goal of adaptivity is to achieve a method with a prescribed convergence rate. The presented method exploits a general coarsening process based on aggregation of unknowns, obtained by a maximum weight matching in the adjacency graph of the system matrix. More specifically, a maximum product matching is employed to define an effective smoother subspace (complementary to the coarse space), a process referred to as compatible relaxation, at every level of the recursive two-level hierarchical AMG process.Results on a large variety of test cases and comparisons with related work demonstrate the reliability and efficiency of the method and of the software.

Abstract: Given a directed graph, two vertices v and w are 2-vertex-connected if there are two internally vertex-disjoint paths from v to w and two internally vertex-disjoint paths from w to v. In this paper, we show how to compute this relation in \(O(m+n)\) time, where n is the number of vertices and m is the number of edges of the graph. As a side result, we show how to build in linear time an O(n)-space data structure, which can answer in constant time queries on whether any two vertices are 2-vertex-connected. Additionally, when two query vertices v and w are not 2-vertex-connected, our data structure can produce in constant time a “witness” of this property, by exhibiting a vertex or an edge that is contained in all paths from v to w or in all paths from w to v. We are also able to compute in linear time a sparse certificate for 2-vertex connectivity, i.e., a subgraph of the input graph that has O(n) edges and maintains the same 2-vertex connectivity properties as the input graph.

Abstract: The graph similarity problem, also known as approximate graph isomorphism or graph matching problem, has been extensively studied in the machine learning community, but has not received much attention in the algorithms community: Given two graphs G,H of the same order n with adjacency matrices A_G,A_H, a well-studied measure of similarity is the Frobenius distance dist(G,H):=min_{pi}|A_G^{pi}-A_H|_F, where pi ranges over all permutations of the vertex set of G, where A_G^pi denotes the matrix obtained from A_G by permuting rows and columns according to pi, and where |M |_F is the Frobenius norm of a matrix M. The (weighted) graph similarity problem, denoted by GSim (WSim), is the problem of computing this distance for two graphs of same order. This problem is closely related to the notoriously hard quadratic assignment problem (QAP), which is known to be NP-hard even for severely restricted cases. It is known that GSim (WSim) is NP-hard; we strengthen this hardness result by showing that the problem remains NP-hard even for the class of trees. Identifying the boundary of tractability for WSim is best done in the framework of linear algebra. We show that WSim is NP-hard as long as one of the matrices has unbounded rank or negative eigenvalues: hence, the realm of tractability is restricted to positive semi-definite matrices of bounded rank. Our main result is a polynomial time algorithm for the special case where the associated (weighted) adjacency graph for one of the matrices has a bounded number of twin equivalence classes. The key parameter underlying our algorithm is the clustering number of a graph; this parameter arises in context of the spectral graph drawing machinery.

Abstract: We propose a new methodology for the analysis of spatial fields of object data distributed over complex domains. Our approach enables to jointly handle both data and domain complexities, through a divide et impera approach. As a key element of innovation, we propose to use a random domain decomposition, whose realizations define sets of homogeneous sub-regions where to perform simple, independent, weak local analyses (divide), eventually aggregated into a final strong one (impera). In this broad framework, the complexity of the domain (e.g., strong concavities, holes or barriers) can be accounted for by defining its partitions on the basis of a suitable metric, which allows to properly represent the adjacency relationships among the complex data (such as scalar, functional or constrained data) over the domain. As an illustration of the potential of the methodology, we consider the analysis and spatial prediction (Kriging) of the probability density function of dissolved oxygen in the Chesapeake Bay.

Abstract: In this paper, the cooperative caching problem in fog radio access networks (F-RAN) is investigated. To maximize the incremental offloaded traffic, we formulate the clustering optimization problem with the consideration of cooperative caching and local content popularity, which falls into the scope of combinatorial programming. We then propose an effective graph-based approach to solve this challenging problem. Firstly, a node graph is constructed with its vertex set representing the considered fog access points (F-APs) and its edge set reflecting the potential cooperations among the F-APs. Then, by exploiting the adjacency table of each vertex of the node graph, we propose to get the complete subgraphs through indirect searching for the maximal complete subgraphs for the sake of a reduced searching complexity. Furthermore, by using the complete subgraphs so obtained, a weighted graph is constructed. By setting the weights of the vertices of the weighted graph to be the incremental offloaded traffics of their corresponding complete subgraphs, the original clustering optimization problem can be transformed into an equivalent 0–1 integer programming problem. The max-weight independent subset of the vertex set of the weighted graph, which is equivalent to the objective cluster sets, can then be readily obtained by solving the above optimization problem through the greedy algorithm that we propose. Our proposed graph-based approach has an apparently low complexity in comparison with the brute force approach which has an exponential complexity. Simulation results show the remarkable improvements in terms of offloading gain by using our proposed approach.

Abstract: We propose an automatic multiorgan segmentation method for 3-D radiological images of different anatomical contents and modalities. The approach is based on a simultaneous multilabel graph cut optimization of location, appearance, and spatial configuration criteria of target structures. Organ location is defined by target-specific probabilistic atlases (PA) constructed from a training dataset using a fast (2+1)D SURF-based multiscale registration method involving a simple four-parameter transformation. PAs are also used to derive target-specific organ appearance models represented as intensity histograms. The spatial configuration prior is derived from shortest-path constraints defined on the adjacency graph of structures. Thorough evaluations on Visceral project benchmarks and training dataset, as well as comparisons with the state-of-the-art confirm that our approach is comparable to and often outperforms similar approaches in multiorgan segmentation, thus proving that the combination of multiple suboptimal but complementary information sources can yield very good performance.

Abstract: Statistical inference on graphs often proceeds via spectral methods involving low-dimensional embeddings of matrix-valued graph representations, such as the graph Laplacian or adjacency matrix. In this paper, we analyze the asymptotic information-theoretic relative performance of Laplacian spectral embedding and adjacency spectral embedding for block assignment recovery in stochastic block model graphs by way of Chernoff information. We investigate the relationship between spectral embedding performance and underlying network structure (e.g.~homogeneity, affinity, core-periphery, (un)balancedness) via a comprehensive treatment of the two-block stochastic block model and the class of $K$-block models exhibiting homogeneous balanced affinity structure. Our findings support the claim that, for a particular notion of sparsity, loosely speaking, "Laplacian spectral embedding favors relatively sparse graphs, whereas adjacency spectral embedding favors not-too-sparse graphs." We also provide evidence in support of the claim that "adjacency spectral embedding favors core-periphery network structure."

Abstract: Read-only memory (ROM) model is a classical model of computation to study time-space tradeoffs of algorithms A classical result on the ROM model is that any algorithm to sort n numbers using O(s) words of extra space requires Omega (n^2/s) comparisons for lg n <= s <= n/lg n and the bound has also been recently matched by an algorithm However, if we relax the model, we do have sorting algorithms (say Heapsort) that can sort using O(n lg n) comparisons using O(lg n) bits of extra space, even keeping a permutation of the given input sequence at anytime during the algorithm We address similar relaxations for graph algorithms We show that a simple natural relaxation of ROM model allows us to implement fundamental graph search methods like BFS and DFS more space efficiently than in ROM By simply allowing elements in the adjacency list of a vertex to be permuted, we show that, on an undirected or directed connected graph G having n vertices and m edges, the vertices of G can be output in a DFS or BFS order using O(lg n) bits of extra space and O(n^3 lg n) time Thus we obtain similar bounds for reachability and shortest path distance (both for undirected and directed graphs) With a little more (but still polynomial) time, we can also output vertices in the lex-DFS order As reachability in directed graphs (even in DAGs) and shortest path distance (even in undirected graphs) are NL-complete, and lex-DFS is P-complete, our results show that our model is more powerful than ROM if L != P En route, we also introduce and develop algorithms for another relaxation of ROM where the adjacency lists of the vertices are circular lists and we can modify only the heads of the lists Here we first show a linear time DFS implementation using n + O(lg n) bits of extra space Improving the extra space exponentially to only O(lg n) bits, we also obtain BFS and DFS albeit with a slightly slower running time Both the models we propose maintain the graph structure throughout the algorithm, only the order of vertices in the adjacency list changes In sharp contrast, for BFS and DFS, to the best of our knowledge, there are no algorithms in ROM that use even O(n^{1-epsilon}) bits of extra space; in fact, implementing DFS using cn bits for c<1 has been mentioned as an open problem Furthermore, DFS (BFS, respectively) algorithms using n+o(n) (o(n), respectively) bits of extra use Reingold's [JACM, 2008] or Barnes et al's reachability algorithm [SICOMP, 1998] and hence have high runtime Our results can be contrasted with the recent result of Buhrman et al [STOC, 2014] which gives an algorithm for directed st-reachability on catalytic Turing machines using O(lg n) bits with catalytic space O(n^2 lg n) and time O(n^9)

Abstract: We consider the skeleton of the polytope of pyramidal tours. A Hamiltonian tour is called pyramidal if the salesperson starts in city 1, then visits some cities in increasing order of their numbers, reaches city n, and returns to city 1 visiting the remaining cities in decreasing order. The polytope PYR(n) is defined as the convex hull of the characteristic vectors of all pyramidal tours in the complete graph K n . The skeleton of PYR(n) is the graph whose vertex set is the vertex set of PYR(n) and the edge set is the set of geometric edges or one-dimensional faces of PYR(n). We describe the necessary and sufficient condition for the adjacency of vertices of the polytope PYR(n). On this basis we developed an algorithm to check the vertex adjacency with linear complexity. We establish that the diameter of the skeleton of PYR(n) equals 2, and the asymptotically exact estimate of the clique number of the skeleton of PYR(n) is Θ(n2). It is known that this value characterizes the time complexity in a broad class of algorithms based on linear comparisons.

Abstract: In studying the mechanisms that create portmanteaux, it is often difficult to distinguish structural constituency from linear adjacency. This paper examines the Irish verbal complex, the particular intricacies of which allow us to disentangle linear adjacency from morphosyntactic constituency. We observe that portmanteaux are formed between linearly adjacent nodes, without reference to structural constituency. A theory is developed in which Vocabulary Insertion operates over linearized structures, allowing Insertion of linearly adjacent nodes. This mechanism is termed ‘Stretching.’ In this way the characteristic slogan of Distributed Morphology (Halle and Marantz 1993) “syntax all the way down” is maintained, but the relevant sense of syntax is one in which post-syntactic processing creates complex structures which imperfectly mirror the narrow syntax. Importantly, it is linear adjacency within these post-syntactic structures which regulates patterns of Vocabulary Insertion, not structural adjacency.

Abstract: FP-Growth algorithm is an association rule mining algorithm based on frequent pattern tree (FP-Tree), which doesn’t need to generate a large number of candidate sets. However, constructing FP-Tree requires two scansof the original transaction database and the recursive mining of FP-Tree to generate frequent itemsets. In addition, the algorithm can’t work effectively when the dataset is dense. To solve the problems of large memory usage and low time-effectiveness of data mining in this algorithm, this paper proposes an improved algorithm based on adjacency table using a hash table to store adjacency table, which considerably saves the finding time. The experimental results show that the improved algorithm has good performance especially for mining frequent itemsets in dense data sets.

Abstract: We study space and time efficient quantum algorithms for two graph problems -- deciding whether an $n$-vertex graph is a forest, and whether it is bipartite. Via a reduction to the s-t connectivity problem, we describe quantum algorithms for deciding both properties in $\tilde{O}(n^{3/2})$ time and using $O(\log n)$ classical and quantum bits of storage in the adjacency matrix model. We then present quantum algorithms for deciding the two properties in the adjacency array model, which run in time $\tilde{O}(n\sqrt{d_m})$ and also require $O(\log n)$ space, where $d_m$ is the maximum degree of any vertex in the input graph.

Abstract: It is shown how LCD codes with a particularly useful feature can be found from row spans over finite fields of adjacency matrices of graphs by considering these together with the codes from the associated reflexive graphs and complementary graphs. Application is made to some particular classes, including uniform subset graphs and strongly regular graphs where, if a p-ary code from a graph has this special LCD feature, the dimension can be found from the multiplicities modulo p of the eigenvalues of an adjacency matrix and, bounds on the minimum weight of the code and the dual code follow from the valency of the graph.

Abstract: This study analyzed the performance of distance-based objective variables as an alternative to adjacency-based variables in spatial optimization when the aim is to aggregate small forest segments i...