Pairwise Edges Research Articles

Dual-view hypergraph attention network for news recommendation

News Recommendation (NR) helps users to quickly find their mostly interested information. Recently, some NR systems based on graph neural networks has achieved significant performance improvement in that they use a graph structure to pair-wisely link two nodes (e.g., user, news, topic node) for representation learning. However, such kind of pairwise edges may be not enough to describe multifaceted relations involving more than two nodes; While a hypergraph structure with one hyperedge connecting two and more nodes might be a better choice. Considering this, we propose a dual-view hypergraph attention network for news recommendations (Hyper4NR) in this paper. In particular, we design a dual-view hypergraph structure to model users’ click history, containing both topic-view hyperedges and semantic-view hyperedges. On the constructed hypergraph, we use hyperedge-specific attention network (HSAN) to pass messages in between hyperedges and nodes to encode their representations based on a self-supervised learning approach. Furthermore, we construct another kind of candidate hypergraph on which we apply the HyperGAT to obtain enhanced candidate news encoding. Extensive experiments on the widely used MIND news dataset and Adressa dataset show that our Hyper4NR outperforms these state-of-the-art NR methods.

A critical probability for biclique partition of Gn,p

The biclique partition number of a graph G=(V,E), denoted bp(G), is the minimum number of pairwise edge disjoint complete bipartite subgraphs of G so that each edge of G belongs to exactly one of them. It is easy to see that bp(G)≤n−α(G), where α(G) is the maximum size of an independent set of G. Erdős conjectured in the 80's that for almost every graph G equality holds; i.e., if G=Gn,1/2 then bp(G)=n−α(G) with high probability. Alon showed that this is false. We show that the conjecture of Erdős is true if we instead take G=Gn,p, where p is constant and less than a certain threshold value p0≈0.312. This verifies a conjecture of Chung and Peng for these values of p. We also show that if p0<p<1/2 then bp(Gn,p)=n−(1+Θ(1))α(Gn,p) with high probability.

Network Differential in Gaussian Graphical Models from Multimodal Neuroimaging Data.

Multimodal brain network analysis has the potential to provide insights into the mechanisms of brain disorders. Most previous studies have analyzed either unimodal brain graphs or focused on local/global graphic metrics with little consideration of details of disrupted paths in the patient group. As we show, the combination of multimodal brain graphs and disrupted path-based analysis can be highly illuminating to recognize path-based disease biomarkers. In this study, we first propose a way to estimate multimodal brain graphs using static functional network connectivity (sFNC) and gray matter features using a Gaussian graphical model of schizophrenia versus controls. Next, applying the graph theory approach we identify disconnectors or connectors in the patient group graph that create additional paths or cause absent paths compared to the control graph. Results showed several edges in the schizophrenia group graph that trigger missing or additional paths. Identified edges associated with these disrupted paths were identified both within and between dFNC and gray matter which highlights the importance of considering multimodal studies and moving beyond pairwise edges to provide a more comprehensive understanding of brain disorders.Clinical Relevance- We identified a path-based biomarker in schizophrenia, by imitating the structure of paths in a multimodal (sMIR+fMRI) brain graph of the control group. Identified cross-modal edges associated with disrupted paths were related to the middle temporal gyrus and cerebellar regions.

Inserting Multiple Edges into a Planar Graph

Let $G$ be a connected planar (but not yet embedded) graph and $F$ a set of edges with ends in $V(G)$ and not belonging to $E(G)$. The multiple edge insertion problem (MEI) asks for a drawing of $G+F$ with the minimum number of pairwise edge crossings, such that the subdrawing of $G$ is plane. A solution to this problem is known to approximate the crossing number of the graph $G+F$, but unfortunately, finding an exact solution to MEI is NP-hard for general $F$. The MEI problem is linear-time solvable for the special case of $|F|=1$ (SODA 01 and Algorithmica), and there is a polynomial-time solvable extension in which all edges of $F$ are incident to a common vertex which is newly introduced into $G$ (SODA 09). The complexity for general $F$ but with constant $k=|F|$ was open, but algorithms both with relative and absolute approximation guarantees have been presented (SODA 11, ICALP 11 and JoCO). We present a fixed-parameter algorithm for the MEI problem in the case that $G$ is biconnected, which is extended to also cover the case of connected $G$ with cut vertices of bounded degree. These are the first exact algorithms for the general MEI problem, and they run in time $O(|V(G)|)$ for any constant $k$.

Vital node identification in hypergraphs via gravity model.

Hypergraphs that can depict interactions beyond pairwise edges have emerged as an appropriate representation for modeling polyadic relations in complex systems. With the recent surge of interest in researching hypergraphs, the centrality problem has attracted much attention due to the challenge of how to utilize higher-order structure for the definition of centrality metrics. In this paper, we propose a new centrality method (HGC) on the basis of the gravity model as well as a semi-local HGC, which can achieve a balance between accuracy and computational complexity. Meanwhile, two comprehensive evaluation metrics, i.e., a complex contagion model in hypergraphs, which mimics the group influence during the spreading process and network s-efficiency based on the higher-order distance between nodes, are first proposed to evaluate the effectiveness of our methods. The results show that our methods can filter out nodes that have fast spreading ability and are vital in terms of hypergraph connectivity.

A Bipartite Graph Neural Network Approach for Scalable Beamforming Optimization

Deep learning (DL) techniques have been intensively studied for the optimization of multi-user multiple-input single-output (MU-MISO) downlink systems owing to the capability of handling nonconvex formulations. However, the fixed computation structure of existing deep neural networks (DNNs) lacks flexibility with respect to the system size, i.e., the number of antennas or users. This paper develops a bipartite graph neural network (BGNN) framework, a scalable DL solution designed for multi-antenna beamforming optimization. The MU-MISO system is first characterized by a bipartite graph where two disjoint vertex sets, each of which consists of transmit antennas and users, are connected via pairwise edges. These vertex interconnection states are modeled by channel fading coefficients. Thus, a generic beamforming optimization process is interpreted as a computation task over a weighted bipartite graph. This approach partitions the beamforming optimization procedure into multiple suboperations dedicated to individual antenna vertices and user vertices. Separated vertex operations lead to scalable beamforming calculations that are invariant to the system size. The vertex operations are realized by a group of DNN modules that collectively form the BGNN architecture. Identical DNNs are reused at all antennas and users so that the resultant learning structure becomes flexible to the network size. Component DNNs of the BGNN are trained jointly over numerous MU-MISO configurations with randomly varying network sizes. As a result, the trained BGNN can be universally applied to arbitrary MU-MISO systems. Numerical results validate the advantages of the BGNN framework over conventional methods.

Vertex and edge metric dimensions of cacti

In a graph G , a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S . The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G . In Sedlar and Škrekovski (0000) we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of these configurations is present in the graph. In this paper we extend the result to cactus graphs i.e. graphs in which all cycles are pairwise edge disjoint. We do so by defining a unicyclic subgraph of G for every cycle of G and applying the already introduced approach for unicyclic graphs which involves the configurations. The obtained results enable us to prove the cycle rank conjecture for cacti. They also yield a simple upper bound on metric dimensions of cactus graphs and we conclude the paper by conjecturing that the same upper bound holds in general.

Dynamic dense CRF inference for video segmentation and semantic SLAM

The dense conditional random field (dense CRF) is an effective post-processing tool for image/video segmentation and semantic SLAM. In this paper, we extend the traditional dense CRF inference algorithm to incremental sensor data modelling. The algorithm efficiently infers the maximum a posteriori probability (MAP) solution for a dynamically changing dense CRF model that is applied to incremental multi-class video segmentation and semantic SLAM. The computational cost is roughly proportional to the total change in the Gaussian pairwise edges of the dense CRF. In our system, with an increase in the number of frames of the sensor data, MAP calculations take approximately the same time to compute the overall three-dimensional dense CRF modelled for the entire video. Compared with the traditional dense CRF for video segmentation, this method is more suitable for incremental (in-line) video segmentation and robot semantic SLAM. The results of experiments show that if part of a pairwise edge is altered, our dynamic algorithm is significantly faster than the widely known standard dense CRF algorithm. In addition, the accuracy of its inference does not change. Several multi-class video segmentation tests confirmed the efficiency of inference of the algorithm. In another application, we used the dynamic dense CRF to incrementally integrate robot SLAM and video segmentation. The results show that an accurate SLAM can improve the accuracy of video segmentation, and the computational cost of the dense CRF MAP can be constrained over a constant range. The application of our algorithm is not limited to video segmentation: It is generic, and can be used to yield similar improvements in many optimization solutions for MAP in dynamically changing models.

Vertex Nomination Between Graphs via Spectral Embedding and Quadratic Programming

Given a network and a subset of interesting vertices whose identities are only partially known, the vertex nomination problem seeks to rank the remaining vertices in such a way that the interesting vertices are ranked at the top of the list. An important variant of this problem is vertex nomination in the multiple graphs setting. Given two graphs G 1, G 2 with common vertices and a vertex of interest , we wish to rank the vertices of G 2 such that the vertices most similar to x are ranked at the top of the list. The current article addresses this problem and proposes a method that first applies adjacency spectral graph embedding to embed the graphs into a common Euclidean space, and then solves a penalized linear assignment problem to obtain the nomination lists. Since the spectral embedding of the graphs are only unique up to orthogonal transformations, we present two approaches to eliminate this potential nonidentifiability. One approach is based on orthogonal Procrustes and is applicable when there are enough vertices with known correspondence between the two graphs. Another approach uses adaptive point set registration and is applicable when there are few or no vertices with known correspondence. We show that our nomination scheme leads to accurate nomination under a generative model for pairs of random graphs that are approximately low-rank and possibly with pairwise edge correlations. We illustrate our algorithm’s performance through simulation studies on synthetic data as well as analysis of a high-school friendship network and analysis of transition rates between web pages on the Bing search engine. Supplementary materials for this article are available and include R code, data, and an appendix with detailed proofs.

The Sufficient Conditions of (δ(G) − 2)-(|F|-)Fault-Tolerant Maximal Local-(Edge-)Connectivity of Connected Graphs

An interconnection network is usually modeled by a connected graph in which vertices represent processors and edges represent links between processors. The connectivity is an important parameter to evaluate the fault tolerance of interconnection networks. A connected graph [Formula: see text] is maximally local-(edge-)connected if each pair vertices [Formula: see text] of [Formula: see text] is connected by min[Formula: see text] pairwise (edge-)disjoint paths between [Formula: see text] and [Formula: see text] in [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant maximally local-(edge-)connected if [Formula: see text] is maximally local-(edge-)connected for any [Formula: see text] ([Formula: see text]) with [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-fault-tolerant maximally local-(edge-)connected of order [Formula: see text] if [Formula: see text] is maximally local-(edge-)connected for any [Formula: see text] with [Formula: see text], where [Formula: see text] is a conditional faulty vertex (edge) set of order [Formula: see text]. In this paper, we obtain the sufficient condition of connected graphs to be [Formula: see text]-edge-fault-tolerant maximally local-edge-connected. Moreover, we consider the sufficient condition of connected graphs to be [Formula: see text]-fault-tolerant maximally local-(edge-)connected of order [Formula: see text]. Some previous results in [Theor. Comput. Sci. 731 (2018) 50–67] and [Theor. Comput. Sci. 847 (2020) 39–48] are extended.

Local linear graphon estimation using covariates

Summary We consider local linear estimation of the graphon function, which determines probabilities of pairwise edges between nodes in an unlabelled network. Real-world networks are typically characterized by node heterogeneity, with different nodes exhibiting different degrees of interaction. Existing approaches to graphon estimation are limited to local constant approximations, and are not designed to estimate heterogeneity across the full network. In this paper, we show how continuous node covariates can be employed to estimate heterogeneity in the network via a local linear graphon estimator. We derive the bias and variance of an oracle-based local linear graphon estimator, and thus obtain the mean integrated squared error optimal bandwidth rule. We also provide a plug-in bandwidth selection procedure that makes local linear estimation for unlabelled networks practically feasible. The finite-sample performance of our approach is investigated in a simulation study, and the method is applied to a school friendship network and an email network to illustrate its advantages over existing methods.

Body Field: Structured Mean Field with Human Body Skeleton Model and Shifted Gaussian Edge Potentials

An efficient method for simultaneous human body part segmentation and pose estimation is introduced. A conditional random field with a fully-connected graphical model is used. Possible node (image pixel) labels comprise of the human body parts and the background. In the human body skeleton model, the spatial dependencies among body parts are encoded in the definition of pairwise energy functions according to the conditional random fields. Proper pairwise edge potentials between image pixels are defined according to the presence or absence of human body parts that are near to each other. Various Gaussian kernels in position, color, and histogram of oriented gradients spaces are used for defining the pairwise energy terms. Shifted Gaussian kernels are defined between each two body parts that are connected to each other according to the human body skeleton model. As shifted Gaussian kernels impose a high computational cost to the inference, an efficient inference process is proposed by a mean field approximation method that uses high dimensional shifted Gaussian filtering. The experimental results evaluated on the challenging KTH Football, Leeds Sports Pose, HumanEva, and Penn-Fudan datasets show that the proposed method increases the per-pixel accuracy measure for human body part segmentation and also improves the probability of correct parts metric of human body joint locations.

Tetra-Sodium Nitrilo-Tris(methylenephosphonato)-Manganate(II) Na4[MnII{N(CH2PO3)3}] · 13H2O: Synthesis and Some Structural Features

A sodium salt of the complex of nitrilo-tris(methylenephosphonic) (NTP) acid with Mn(II) (Na4[MnII{N(CH2PO3)3}] · 13H2O, sp. gr. P$$\bar {1}$$, Z = 2, a = 11.2146(4) A, b = 11.2687(4) A, c = 12.4625(4) A, α = 108.619(3)°, β = 97.149(3)°, γ = 116.991(4)°) has been synthesized and investigated. This complex is binuclear with a chelate structure; the Mn coordination polyhedron is a distorted trigonal bipyramid with three O atoms of different PO3 groups of the NTP molecule in the base, with the nitrogen atom of the NTP molecule in one vertex, and the O atom of the neighboring NTP molecule in the opposite vertex. The crystal packing is of island type; Mn polyhedra share (pairwise) edges and are surrounded by a spatial binding of Na polyhedra and water molecules.

Random walks on hypergraphs.

In the past 20 years network science has proven its strength in modeling many real-world interacting systems as generic agents, the nodes, connected by pairwise edges. Nevertheless, in many relevant cases, interactions are not pairwise but involve larger sets of nodes at a time. These systems are thus better described in the framework of hypergraphs, whose hyperedges effectively account for multibody interactions. Here we propose and study a class of random walks defined on such higher-order structures and grounded on a microscopic physical model where multibody proximity is associated with highly probable exchanges among agents belonging to the same hyperedge. We provide an analytical characterization of the process, deriving a general solution for the stationary distribution of the walkers. The dynamics is ultimately driven by a generalized random-walk Laplace operator that reduces to the standard random-walk Laplacian when all the hyperedges have size 2 and are thus meant to describe pairwise couplings. We illustrate our results on synthetic models for which we have full control of the high-order structures and on real-world networks where higher-order interactions are at play. As the first application of the method, we compare the behavior of random walkers on hypergraphs to that of traditional random walkers on the corresponding projected networks, drawing interesting conclusions on node rankings in collaboration networks. As the second application, we show how information derived from the random walk on hypergraphs can be successfully used for classification tasks involving objects with several features, each one represented by a hyperedge. Taken together, our work contributes to unraveling the effect of higher-order interactions on diffusive processes in higher-order networks, shedding light on mechanisms at the heart of biased information spreading in complex networked systems.

Crossing Number for Graphs with Bounded Pathwidth

The crossing number is the smallest number of pairwise edge crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth. In this paper, we show that the crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth 3. The technique also shows that the crossing number and the rectilinear (a.k.a. straight-line) crossing number are identical for this graph class, and that we require only an $$O(n)\times O(n)$$-grid to achieve such a drawing. Our techniques can further be extended to devise a 2-approximation for general graphs with pathwidth 3. One crucial ingredient here is that the crossing number of a graph with a separation pair can be lower-bounded using the crossing numbers of its cut-components, a result that may be interesting in its own right. Finally, we give a $$4{\mathbf{w}}^3$$-approximation of the crossing number for maximal graphs of pathwidth $${\mathbf{w}}$$. This is a constant approximation for bounded pathwidth. We complement this with an NP-hardness proof of the weighted crossing number already for pathwidth 3 graphs and bicliques $$K_{3,n}$$.

FCN-Based DenseNet Framework for Automated Detection and Classification of Skin Lesions in Dermoscopy Images

Skin Lesion detection and classification are very critical in diagnosing skin malignancy. Existing Deep learning-based Computer-aided diagnosis (CAD) methods still perform poorly on challenging skin lesions with complex features such as fuzzy boundaries, artifacts presence, low contrast with the background and, limited training datasets. They also rely heavily on a suitable turning of millions of parameters which often leads to over-fitting, poor generalization, and heavy consumption of computing resources. This study proposes a new framework that performs both segmentation and classification of skin lesions for automated detection of skin cancer. The proposed framework consists of two stages: the first stage leverages on an encoder-decoder Fully Convolutional Network (FCN) to learn the complex and inhomogeneous skin lesion features with the encoder stage learning the coarse appearance and the decoder learning the lesion borders details. Our FCN is designed with the sub-networks connected through a series of skip pathways that incorporate long skip and short-cut connections unlike, the only long skip connections commonly used in the traditional FCN, for residual learning strategy and effective training. The network also integrates the Conditional Random Field (CRF) module which employs a linear combination of Gaussian kernels for its pairwise edge potentials for contour refinement and lesion boundaries localization. The second stage proposes a novel FCN-based DenseNet framework that is composed of dense blocks that are merged and connected via the concatenation strategy and transition layer. The system also employs hyper-parameters optimization techniques to reduce network complexity and improve computing efficiency. This approach encourages feature reuse and thus requires a small number of parameters and effective with limited data. The proposed model was evaluated on publicly available HAM10000 dataset of over 10000 images consisting of 7 different categories of diseases with 98% accuracy, 98.5% recall, and 99% of AUC score respectively.

Strong-Structural Convolution Neural Network for Semantic Segmentation

We present a combinatorial deep convolutional neural network architecture, termed strong convolution neural network (SSN), for semantic segmentation task. The structure of SSN consists of two components: Increment feature convolution neural network and post-process Conditional Random Fields unit (CRFs). The increment feature CNN unit has three parts: I-Block, Deconvolution layer and Transition Block. I-Block employs increment convolution to efficiently maintain feature information. Before passing through pooling layer, we put the feature map into activate layer ReLU, and batch normalization layer. In Decoding stage, we use skip-connects to keep the pooling index information. To enforce the correlation of same semantic labels, we define the strong semantic label (SSL) stage to intensify the pairwise potential energy. To achieve high computation performance, we make further improvement on SSL by employing the adaptive soft semantic sections label method. We proposed the adaptive strong semantic label selection algorithm to generate the SSL. Through the CRFs unit, with unitary energy and pairwise edge energy, the semantic segmentation initial labels transform semantic segmentation labels. Experimental evaluation reveals the training time versus accuracy trade-off involved in achieving good segmentation performance.

Book embedding of 3-crossing-critical graphs with rational average degree between 3.5 and 4

We consider the graphs P(m, n) with average degree between 3.5 and 4, made up of m identical pieces together with n identical pieces glued together in a circular fashion, such that in any drawing of P(m, n) on the plane, there are exactly three pairwise edges crossings, and when deleting any edge of the graph, the number of crossings of the remaining graph decreases. We then embed P(m, n) into a book such that the vertices are put on a line called the spine and the edges are put on half-planes called the pages, which have the spine as their common boundary, without crossings. In this paper, we show that the minimal number of pages needed to embed P(m, n) into a book is three.

Decomposition of complete graphs into connected unicyclic graphs with eight edges and pentagon

&lt;p&gt;A &lt;span class="math"&gt;&lt;em&gt;G&lt;/em&gt;&lt;/span&gt;-decomposition of the complete graph &lt;span class="math"&gt;&lt;em&gt;K&lt;/em&gt;&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt;&lt;/span&gt; is a family of pairwise edge disjoint subgraphs of &lt;span class="math"&gt;&lt;em&gt;K&lt;/em&gt;&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt;&lt;/span&gt;, all isomorphic to &lt;span class="math"&gt;&lt;em&gt;G&lt;/em&gt;&lt;/span&gt;, such that every edge of &lt;span class="math"&gt;&lt;em&gt;K&lt;/em&gt;&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt;&lt;/span&gt; belongs to exactly one copy of &lt;span class="math"&gt;&lt;em&gt;G&lt;/em&gt;&lt;/span&gt;. Using standard decomposition techniques based on &lt;span class="math"&gt;&lt;em&gt;ρ&lt;/em&gt;&lt;/span&gt;-labelings, introduced by Rosa in 1967, and their modifications we show that each of the ten non-isomorphic connected unicyclic graphs with eight edges containing the pentagon decomposes the complete graph &lt;span class="math"&gt;&lt;em&gt;K&lt;/em&gt;&lt;sub&gt;&lt;em&gt;n&lt;/em&gt;&lt;/sub&gt;&lt;/span&gt; whenever the necessary conditions are satisfied.&lt;/p&gt;

The undirected two disjoint shortest paths problem

The k disjoint shortest paths problem (k-DSPP) on a graph with k source–sink pairs (si,ti) asks if there exist k pairwise edge- or vertex-disjoint shortest si–ti-paths. It is known to be NP-complete if k is part of the input. Restricting to 2-DSPP with strictly positive lengths, it becomes solvable in polynomial time. We extend this result by allowing zero edge lengths and give a polynomial-time algorithm based on dynamic programming for 2-DSPP on undirected graphs with non-negative edge lengths.