Showing papers on "Connectivity published in 2019"

Pyramid Graph Networks With Connection Attentions for Region-Based One-Shot Semantic Segmentation

Abstract: One-shot image segmentation aims to undertake the segmentation task of a novel class with only one training image available. The difficulty lies in that image segmentation has structured data representations, which yields a many-to-many message passing problem. Previous methods often simplify it to a one-to-many problem by squeezing support data to a global descriptor. However, a mixed global representation drops the data structure and information of individual elements. In this paper, we propose to model structured segmentation data with graphs and apply attentive graph reasoning to propagate label information from support data to query data. The graph attention mechanism could establish the element-to-element correspondence across structured data by learning attention weights between connected graph nodes. To capture correspondence at different semantic levels, we further propose a pyramid-like structure that models different sizes of image regions as graph nodes and undertakes graph reasoning at different levels. Experiments on PASCAL VOC 2012 dataset demonstrate that our proposed network significantly outperforms the baseline method and leads to new state-of-the-art performance on 1-shot and 5-shot segmentation benchmarks.

Learning Trajectory Dependencies for Human Motion Prediction

Abstract: Human motion prediction, i.e., forecasting future body poses given observed pose sequence, has typically been tackled with recurrent neural networks (RNNs). However, as evidenced by prior work, the resulted RNN models suffer from prediction errors accumulation, leading to undesired discontinuities in motion prediction. In this paper, we propose a simple feed-forward deep network for motion prediction, which takes into account both temporal smoothness and spatial dependencies among human body joints. In this context, we then propose to encode temporal information by working in trajectory space, instead of the traditionally-used pose space. This alleviates us from manually defining the range of temporal dependencies (or temporal convolutional filter size, as done in previous work). Moreover, spatial dependency of human pose is encoded by treating a human pose as a generic graph (rather than a human skeletal kinematic tree) formed by links between every pair of body joints. Instead of using a pre-defined graph structure, we design a new graph convolutional network to learn graph connectivity automatically. This allows the network to capture long range dependencies beyond that of human kinematic tree. We evaluate our approach on several standard benchmark datasets for motion prediction, including Human3.6M, the CMU motion capture dataset and 3DPW. Our experiments clearly demonstrate that the proposed approach achieves state of the art performance, and is applicable to both angle-based and position-based pose representations. The code is available at https://github.com/wei-mao-2019/LearnTrajDep

Learning Trajectory Dependencies for Human Motion Prediction

Abstract: Human motion prediction, i.e., forecasting future body poses given observed pose sequence, has typically been tackled with recurrent neural networks (RNNs). However, as evidenced by prior work, the resulted RNN models suffer from prediction errors accumulation, leading to undesired discontinuities in motion prediction. In this paper, we propose a simple feed-forward deep network for motion prediction, which takes into account both temporal smoothness and spatial dependencies among human body joints. In this context, we then propose to encode temporal information by working in trajectory space, instead of the traditionally-used pose space. This alleviates us from manually defining the range of temporal dependencies (or temporal convolutional filter size, as done in previous work). Moreover, spatial dependency of human pose is encoded by treating a human pose as a generic graph (rather than a human skeletal kinematic tree) formed by links between every pair of body joints. Instead of using a pre-defined graph structure, we design a new graph convolutional network to learn graph connectivity automatically. This allows the network to capture long range dependencies beyond that of human kinematic tree. We evaluate our approach on several standard benchmark datasets for motion prediction, including Human3.6M, the CMU motion capture dataset and 3DPW. Our experiments clearly demonstrate that the proposed approach achieves state of the art performance, and is applicable to both angle-based and position-based pose representations. The code is available at this https URL

Densely Connected Graph Convolutional Networks for Graph-to-Sequence Learning

Abstract: We focus on graph-to-sequence learning, which can be framed as transducing graph structures to sequences for text generation. To capture structural information associated with graphs, we investigat...

EdMot: An Edge Enhancement Approach for Motif-aware Community Detection

Abstract: Network community detection is a hot research topic in network analysis. Although many methods have been proposed for community detection, most of them only take into consideration the lower-order structure of the network at the level of individual nodes and edges. Thus, they fail to capture the higher-order characteristics at the level of small dense subgraph patterns, e.g., motifs. Recently, some higher-order methods have been developed but they typically focus on the motif-based hypergraph which is assumed to be a connected graph. However, such assumption cannot be ensured in some real-world networks. In particular, the hypergraph may become fragmented. That is, it may consist of a large number of connected components and isolated nodes, despite the fact that the original network is a connected graph. Therefore, the existing higher-order methods would suffer seriously from the above fragmentation issue, since in these approaches, nodes without connection in hypergraph can't be grouped together even if they belong to the same community. To address the above fragmentation issue, we propose an Edge enhancement approach for Motif-aware community detection (EdMot ). The main idea is as follows. Firstly, a motif-based hypergraph is constructed and the top K largest connected components in the hypergraph are partitioned into modules. Afterwards, the connectivity structure within each module is strengthened by constructing an edge set to derive a clique from each module. Based on the new edge set, the original connectivity structure of the input network is enhanced to generate a rewired network, whereby the motif-based higher-order structure is leveraged and the hypergraph fragmentation issue is well addressed. Finally, the rewired network is partitioned to obtain the higher-order community structure. Extensive experiments have been conducted on eight real-world datasets and the results show the effectiveness of the proposed method in improving the community detection performance of state-of-the-art methods.

Second-Order Continuous-Time Algorithm for Optimal Resource Allocation in Power Systems

Abstract: In this paper, based on differential inclusions and the saddle point dynamics, a novel second-order continuous-time algorithm is proposed to solve the optimal resource allocation problem in power systems. The considered cost function is the sum of all local cost functions with a set of affine equality demand constraints and an inequality constraint on generating capacity of the generator. In virtue of nonsmooth analysis, geometric graph theory, and Lyapunov stability theory, all generators achieve consensus on the Lagrange multipliers associated with a set of affine equality constraints while the proposed algorithm converges exponentially to the optimal solution of the resource allocation problem starting from any initial states over an undirected and connected graph. Moreover, the obtained results can be further extended to the optimal resource allocation problem in case of switching communication topologies. Finally, two numerical examples involving a smart grid system composed of five generators and the IEEE 30-bus system demonstrate the effectiveness and the performance of the theoretical results.

Densely Connected Graph Convolutional Networks for Graph-to-Sequence Learning

Abstract: We focus on graph-to-sequence learning, which can be framed as transducing graph structures to sequences for text generation. To capture structural information associated with graphs, we investigate the problem of encoding graphs using graph convolutional networks (GCNs). Unlike various existing approaches where shallow architectures were used for capturing local structural information only, we introduce a dense connection strategy, proposing a novel Densely Connected Graph Convolutional Networks (DCGCNs). Such a deep architecture is able to integrate both local and non-local features to learn a better structural representation of a graph. Our model outperforms the state-of-the-art neural models significantly on AMRto-text generation and syntax-based neural machine translation.

Cooperative Control of Multiple Agents With Unknown High-Frequency Gain Signs Under Unbalanced and Switching Topologies

Abstract: Existing results on cooperative control of multiagent systems with unknown control directions require that the underlying topology is either fixed with a strongly connected graph or switching between different strongly connected graphs. Furthermore, in most cases the graph is assumed to be balanced. This paper proposes a new class of nonlinear proportional-integral (PI) based algorithms to relax these requirements and allow for unbalanced and switching topologies having a jointly strongly connected basis. This is made possible for single-integrator (SI) and double-integrator (DI) agents with nonidentical unknown control directions by a suitable selection of the distributed nonlinear PI functions. Moreover, as a special case, the proposed algorithms are applied to strongly connected and fixed graphs. Finally, simulation examples are given to show the validity of our theoretical results.

Some results about component factors in graphs

Abstract: For a set ℋ of connected graphs, a spanning subgraph H of a graph G is called an ℋ-factor of G if every component of H is isomorphic to a member ofℋ An H -factor is also referred as a component factor If each component of H is a star (resp path), H is called a star (resp path) factor By a P ≥ k -factor (k positive integer) we mean a path factor in which each component path has at least k vertices (ie it has length at least k − 1) A graph G is called a P ≥ k -factor covered graph, if for each edge e of G , there is a P ≥ k -factor covering e In this paper, we prove that (1) a graph G has a {K 1,1 ,K 1,2 , … ,K 1,k }-factor if and only if bind(G ) ≥ 1/k , where k ≥ 2 is an integer; (2) a connected graph G is a P ≥ 2 -factor covered graph if bind(G ) > 2/3; (3) a connected graph G is a P ≥ 3 -factor covered graph if bind(G ) ≥ 3/2 Furthermore, it is shown that the results in this paper are best possible in some sense

Computing First General Zagreb Index of Operations on Graphs

Abstract: The numerical coding of the molecular structures on the bases of topological indices plays an important role in the subject of Cheminformatics which is a combination of Computer, Chemistry, and Information Science. In 1972, it was shown that the total π-electron energy of a molecular graph depends upon its structure and it can be obtained by the sum of the square of degrees of the vertices of a molecular graph. Later on, this sum was named as the first Zagreb index. In 2005, for γeR - {0, 1}, the first general Zagreb index of a graph G is defined as M γ (G) = Σ veV(G) [d G (v)] γ , where d G (v) is degree of the vertex v in G. In this paper, for each γeR - {0, 1}, we study the first general Zagreb index of the cartesian product of two graphs such that one of the graphs is D-sum graph and the other is any connected graph, where D-sum graph is obtained by using certain D operations on a connected graph. The obtained results are general extensions of the results of Deng et al. [Applied Mathematics and Computation 275(2016): 422-431] and Akhter et al. [AKCE Int. J. Graphs Combin. 14(2017): 70-79] who proved only for γ = 2 and γ = 3, respectively.

A new algorithm for finding minimum spanning trees with undirected neutrosophic graphs

Abstract: In this paper, we discuss the minimum spanning tree (MST) problem of an undirected neutrosophic weighted connected graph in which a single-valued neutrosophic number, instead of a real number/fuzzy number, is assigned to each arc as its arc length. We define this type of MST as neutrosophic minimum spanning tree (NMST). We describe the utility of neutrosophic numbers as arc lengths and its application in different real world MST problems. Here, a new algorithm for designing the MST of a neutrosophic graph is introduced. In the proposed algorithm, we incorporate the uncertainty in Kruskal algorithm for designing MST using neutrosophic number as arc length. A score function is used to compare different NMSTs whose weights are computed using the addition operation of neutrosophic numbers. We compare this weight of the NMST with that of an equivalent classical MST with real numbers as arc lengths. Compared with the existing algorithms for NMST, the proposed algorithm is more efficient due to the fact that the addition operation and the ranking of neutrosophic number can be done in straightforward manners. The proposed algorithm is illustrated by numerical examples.

Improved spectral convergence rates for graph Laplacians on epsilon-graphs and k-NN graphs

Abstract: In this paper we improve the spectral convergence rates for graph-based approximations of Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of the graph connectivity $\varepsilon$, our results show that the eigenvalues and eigenvectors of the graph Laplacian converge to those of the Laplace-Beltrami operator at a rate of $O(n^{-1/(m+4)})$, up to log factors, where $m$ is the manifold dimension and $n$ is the number of vertices in the graph. Our approach is general and allows us to analyze a large variety of graph constructions that include $\varepsilon$-graphs and $k$-NN graphs.

A Unified Framework for Structured Graph Learning via Spectral Constraints

Abstract: Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying graphical models from data. Learning a graph with a specific structure is essential for interpretability and identification of the relationships among data. Useful structured graphs include the multi-component graph, bipartite graph, connected graph, sparse graph, and regular graph. In general, structured graph learning is an NP-hard combinatorial problem, therefore, designing a general tractable optimization method is extremely challenging. In this paper, we introduce a unified graph learning framework lying at the integration of Gaussian graphical models and spectral graph theory. To impose a particular structure on a graph, we first show how to formulate the combinatorial constraints as an analytical property of the graph matrix. Then we develop an optimization framework that leverages graph learning with specific structures via spectral constraints on graph matrices. The proposed algorithms are provably convergent, computationally efficient, and practically amenable for numerous graph-based tasks. Extensive numerical experiments with both synthetic and real data sets illustrate the effectiveness of the proposed algorithms. The code for all the simulations is made available as an open source repository.

Computing Metric Dimension of Certain Families of Toeplitz Graphs

Abstract: The position of a moving point in a connected graph can be identified by computing the distance from the point to a set of sonar stations which have been appropriately situated in the graph. Let Q = {q 1 , q 2 , ... , q k } be an ordered set of vertices of a graph G and a is any vertex in G, then the code/representation of a w.r.t Q is the k-tuple (r(a, q 1 ), r(a, q 2 ), ... , r(a, q k )), denoted by r(a|Q). If the different vertices of G have the different representations w.r.t Q, then Q is known as a resolving set/locating set. A resolving/locating set having the least number of vertices is the basis for G and the number of vertices in the basis is called metric dimension of G and it is represented as dim(G). In this paper, the metric dimension of Toeplitz graphs generated by two and three parameters denoted by T n 〈1, t〉 and T n 〈1, 2, t〉, respectively is discussed and proved that it is constant.

Quadratization in discrete optimization and quantum mechanics.

Abstract: A book about turning high-degree optimization problems into quadratic optimization problems that maintain the same global minimum (ground state). This book explores quadratizations for pseudo-Boolean optimization, perturbative gadgets used in QMA completeness theorems, and also non-perturbative k-local to 2-local transformations used for quantum mechanics, quantum annealing and universal adiabatic quantum computing. The book contains ~70 different Hamiltonian transformations, each of them on a separate page, where the cost (in number of auxiliary binary variables or auxiliary qubits, or number of sub-modular terms, or in graph connectivity, etc.), pros, cons, examples, and references are given. One can therefore look up a quadratization appropriate for the specific term(s) that need to be quadratized, much like using an integral table to look up the integral that needs to be done. This book is therefore useful for writing compilers to transform general optimization problems, into a form that quantum annealing or universal adiabatic quantum computing hardware requires; or for transforming quantum chemistry problems written in the Jordan-Wigner or Bravyi-Kitaev form, into a form where all multi-qubit interactions become 2-qubit pairwise interactions, without changing the desired ground state. Applications cited include computer vision problems (e.g. image de-noising, un-blurring, etc.), number theory (e.g. integer factoring), graph theory (e.g. Ramsey number determination), and quantum chemistry. The book is open source, and anyone can make modifications here: this https URL.

Computing top-k Closeness Centrality Faster in Unweighted Graphs

Abstract: Given a connected graph G=(V,E), where V denotes the set of nodes and E the set of edges of the graph, the length (that is, the number of edges) of the shortest path between two nodes v and w is denoted by d(v,w). The closeness centrality of a vertex v is then defined as n=1/Σw ∈ V d(v,w), where n=|V|. This measure is widely used in the analysis of real-world complex networks, and the problem of selecting the k most central vertices has been deeply analyzed in the last decade. However, this problem is computationally not easy, especially for large networks: in the first part of the article, we prove that it is not solvable in time O(|E|2=ϵ) on directed graphs, for any constant ϵ > 0, under reasonable complexity assumptions. Furthermore, we propose a new algorithm for selecting the k most central nodes in a graph: we experimentally show that this algorithm improves significantly both the textbook algorithm, which is based on computing the distance between all pairs of vertices, and the state of the art. For example, we are able to compute the top k nodes in few dozens of seconds in real-world networks with millions of nodes and edges. Finally, as a case study, we compute the 10 most central actors in the Internet Movie Database (IMDB) collaboration network, where two actors are linked if they played together in a movie, and in the Wikipedia citation network, which contains a directed edge from a page p to a page q if p contains a link to q.

Mincut pooling in Graph Neural Networks.

Abstract: The advance of node pooling operations in Graph Neural Networks (GNNs) has lagged behind the feverish design of new message-passing techniques, and pooling remains an important and challenging endeavor for the design of deep architectures. In this paper, we propose a pooling operation for GNNs that leverages a differentiable unsupervised loss based on the mincut optimization objective. For each node, our method learns a soft cluster assignment vector that depends on the node features, the target inference task (e.g., graph classification), and, thanks to the mincut objective, also on the graph connectivity. Graph pooling is obtained by applying the matrix of assignment vectors to the adjacency matrix and the node features. We validate the effectiveness of the proposed pooling method on a variety of supervised and unsupervised tasks.

Near-Optimal Massively Parallel Graph Connectivity

Abstract: Identifying the connected components of a graph, apart from being a fundamental problem with countless applications, is a key primitive for many other algorithms. In this paper, we consider this problem in parallel settings. Particularly, we focus on the Massively Parallel Computations (MPC) model, which is the standard theoretical model for modern parallel frameworks such as MapReduce, Hadoop, or Spark. We consider the truly sublinear regime of MPC for graph problems where the space per machine is n^δ for some desirably small constant δ ∊ (0, 1). We present an algorithm that for graphs with diameter D in the wide range [log^e n, n], takes O(log D) rounds to identify the connected components and takes O(log log n) rounds for all other graphs. The algorithm is randomized, succeeds with high probability, does not require prior knowledge of D, and uses an optimal total space of O(m). We complement this by showing a conditional lower-bound based on the widely believed TwoCycle conjecture that Ω(log D) rounds are indeed necessary in this setting. Studying parallel connectivity algorithms received a resurgence of interest after the pioneering work of Andoni etal [FOCS 2018] who presented an algorithm with O(log D log log n) round-complexity. Our algorithm improves this result for the whole range of values of D and almost settles the problem due to the conditional lower-bound. Additionally, we show that with minimal adjustments, our algorithm can also be implemented in a variant of (CRCW) PRAM in asymptotically the same number of rounds.

Parallel Batch-Dynamic Graph Connectivity

Abstract: In this paper, we study batch parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in the sequential setting. The best sequential algorithm for dynamic connectivity is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves O(log2 n) amortized time per edge insertion or deletion, and O(log n) time per query. We design a parallel batch-dynamic connectivity algorithm that is work-efficient with respect to the HDT algorithm for small batch sizes, and is asymptotically faster when the average batch size is sufficiently large. Given a sequence of batched updates, where Δ is the average batch size of all deletions, our algorithm achieves O(log n log(1+n / Δ)) expected amortized work per edge insertion and deletion and O(log3 n) depth w.h.p. Our algorithm answers a batch of k connectivity queries in O(k log(1 + n/k)) expected work and O(log n) depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.

Parallel Batch-Dynamic Graph Connectivity

Abstract: In this paper, we study batch parallel algorithms for the dynamic connectivity problem, a fundamental problem that has received considerable attention in the sequential setting. The most well known sequential algorithm for dynamic connectivity is the elegant level-set algorithm of Holm, de Lichtenberg and Thorup (HDT), which achieves $O(\log^2 n)$ amortized time per edge insertion or deletion, and $O(\log n / \log\log n)$ time per query. We design a parallel batch-dynamic connectivity algorithm that is work-efficient with respect to the HDT algorithm for small batch sizes, and is asymptotically faster when the average batch size is sufficiently large. Given a sequence of batched updates, where $\Delta$ is the average batch size of all deletions, our algorithm achieves $O(\log n \log(1 + n / \Delta))$ expected amortized work per edge insertion and deletion and $O(\log^3 n)$ depth w.h.p. Our algorithm answers a batch of $k$ connectivity queries in $O(k \log(1 + n/k))$ expected work and $O(\log n)$ depth w.h.p. To the best of our knowledge, our algorithm is the first parallel batch-dynamic algorithm for connectivity.

Complexity and Geometry of Sampling Connected Graph Partitions.

Abstract: In this paper, we prove intractability results about sampling from the set of partitions of a planar graph into connected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, we use gadgets inspired by their technique to provide families of graphs where the "flip walk" Markov chain used in practice for this sampling task exhibits exponentially slow mixing. Supporting our theoretical results we present some empirical evidence demonstrating the slow mixing of the flip walk on grid graphs and on real data. Inspired by connections to the statistical physics of self-avoiding walks, we investigate the sensitivity of certain popular sampling algorithms to the graph topology. Finally, we discuss a few cases where the sampling problem is tractable. Applications to political redistricting have recently brought increased attention to this problem, and we articulate open questions about this application that are highlighted by our results.

Algorithms for generating all possible spanning trees of a simple undirected connected graph: an extensive review

Abstract: Generation of all possible spanning trees of a graph is a major area of research in graph theory as the number of spanning trees of a graph increases exponentially with graph size. Several algorithms of varying efficiency have been developed since early 1960s by researchers around the globe. This article is an exhaustive literature survey on these algorithms, assuming the input to be a simple undirected connected graph of finite order, and contains detailed analysis and comparisons in both theoretical and experimental behavior of these algorithms.

Graph Filtration Learning

Abstract: We propose an approach to learning with graph-structured data in the problem domain of graph classification. In particular, we present a novel type of readout operation to aggregate node features into a graph-level representation. To this end, we leverage persistent homology computed via a real-valued, learnable, filter function. We establish the theoretical foundation for differentiating through the persistent homology computation. Empirically, we show that this type of readout operation compares favorably to previous techniques, especially when the graph connectivity structure is informative for the learning problem.

A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Subgraph Problem

Abstract: We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge connected spanning subgraph of a given undirected multigraph. The algorithm is based upon a reduction to a restricted class of graphs. In these graphs, the approximation algorithm constructs a 2-edge connected spanning subgraph by modifying the smallest 2-edge cover.

Hypergraph-based connectivity measures for signaling pathway topologies.

Abstract: Characterizing cellular responses to different extrinsic signals is an active area of research, and curated pathway databases describe these complex signaling reactions. Here, we revisit a fundamental question in signaling pathway analysis: are two molecules "connected" in a network? This question is the first step towards understanding the potential influence of molecules in a pathway, and the answer depends on the choice of modeling framework. We examined the connectivity of Reactome signaling pathways using four different pathway representations. We find that Reactome is very well connected as a graph, moderately well connected as a compound graph or bipartite graph, and poorly connected as a hypergraph (which captures many-to-many relationships in reaction networks). We present a novel relaxation of hypergraph connectivity that iteratively increases connectivity from a node while preserving the hypergraph topology. This measure, B-relaxation distance, provides a parameterized transition between hypergraph connectivity and graph connectivity. B-relaxation distance is sensitive to the presence of small molecules that participate in many functionally unrelated reactions in the network. We also define a score that quantifies one pathway's downstream influence on another, which can be calculated as B-relaxation distance gradually relaxes the connectivity constraint in hypergraphs. Computing this score across all pairs of 34 Reactome pathways reveals pairs of pathways with statistically significant influence. We present two such case studies, and we describe the specific reactions that contribute to the large influence score. Finally, we investigate the ability for connectivity measures to capture functional relationships among proteins, and use the evidence channels in the STRING database as a benchmark dataset. STRING interactions whose proteins are B-connected in Reactome have statistically significantly higher scores than interactions connected in the bipartite graph representation. Our method lays the groundwork for other generalizations of graph-theoretic concepts to hypergraphs in order to facilitate signaling pathway analysis.