Sequence Of Graphs Research Articles

A hierarchical structure of the quad-cube and the associated diameter, domination, and Hamiltonicity

The quad-cube is a special case of the metacube that itself is derivable from the hypercube. It is amenable to an application as a network topology, especially when the node size exceeds several million. This paper presents a sequence of graphs, leading up to the quad-cube, where the graphs in the sequence are important in their own right. For example, they exhibit hypercube-like low diameters and efficient domination parameters, and most of them admit a Hamiltonian cycle. The hierarchy is likely to be useful in the future.

A coarse-to-fine adaptive spatial–temporal graph convolution network with residuals for motor imagery decoding from the same limb

In the field of Brain Computer Interface (BCI) technology, Motor Imagery (MI) plays an important role as a paradigm. One of the primary focuses of this research area lies in exploring the MI of various upper limbs. Decoding MI signals from distinct joints within the same limb poses more intricate challenges in comparison to decoding MI signals originating from different upper limbs. In order to explore more efficient decoding methods, we propose a novel coarse-to-fine classification approach to investigate categorical decoding across three tasks, namely ‘rest’, ‘hand’, and ‘elbow’. This approach consists of two classification stages performed from coarse to fine. In the coarse classification stage, in order to capture the features of both resting state and the moving state in the temporal domain, the EEGNet network with temporal domain convolution is used to extract temporal domain features and classify the original samples into categories of ‘rest’ and ‘move’ (‘hand’, ‘elbow’). In the fine classification stage, the samples of ‘move’ category are segmented by time to form a graph sequence. Then, an adaptive spatial–temporal graph convolutional network with residuals is utilized to extract both temporal and spatial domains’ features from the graph sequence. The proposed algorithm has been validated experimentally on the MI-2 dataset and compared with contemporary methods. Its classification performance is quantified by the average accuracy which achieves a value of 72.21%. Extensive experimental results indicate that the novel coarse-to-fine classification approach is superior to the single classification approach.

Thermodynamic limit of spin systems on random graphs

We utilize the graphon—a continuous mathematical object which represents the limit of convergent sequences of dense graphs—to formulate a general, continuous description of quantum spin systems in thermal equilibrium when the average coordination number grows extensively in the system size. Specifically, we derive a closed set of coupled nonlinear Fredholm integral equations which governs the properties of the system. The graphon forms the kernel of these equations, and their solution yields exact expressions for the macroscopic observables in the system in the thermodynamic limit. We analyze these equations for both quantum and classical spin systems, recovering known results and providing analytical solutions for a range of more complex cases. We supplement this with controlled, finite-size numerical calculations using Monte Carlo and tensor network methods, showing their convergence towards our analytical results with increasing system size. Published by the American Physical Society 2024

DSANet: Dynamic and Structure-Aware GCN for Sparse and Incomplete Point Cloud Learning.

Learning 3-D structures from incomplete point clouds with extreme sparsity and random distributions is a challenge since it is difficult to infer topological connectivity and structural details from fragmentary representations. Missing large portions of informative structures further aggravates this problem. To overcome this, a novel graph convolutional network (GCN) called dynamic and structure-aware NETwork (DSANet) is presented in this article. This framework is formulated based on a pyramidic auto-encoder (AE) architecture to address accurate structure reconstruction on the sparse and incomplete point clouds. A PointNet-like neural network is applied as the encoder to efficiently aggregate the global representations of coarse point clouds. On the decoder side, we design a dynamic graph learning module with a structure-aware attention (SAA) to take advantage of the topology relationships maintained in the dynamic latent graph. Relying on gradually unfolding the extracted representation into a sequence of graphs, DSANet is able to reconstruct complicated point clouds with rich and descriptive details. To associate analogous structure awareness with semantic estimation, we further propose a mechanism, called structure similarity assessment (SSA). This method allows our model to surmise semantic homogeneity in an unsupervised manner. Finally, we optimize the proposed model by minimizing a new distortion-aware objective end-to-end. Extensive qualitative and quantitative experiments demonstrate the impressive performance of our model in reconstructing unbroken 3-D shapes from deficient point clouds and preserving semantic relationships among different regional structures.

Double dominating sequences in bipartite and co-bipartite graphs

Double dominating sequences in bipartite and co-bipartite graphs

Modeling Sparse Graph Sequences and Signals Using Generalized Graphons

Modeling Sparse Graph Sequences and Signals Using Generalized Graphons

Information cascade prediction of complex networks based on physics-informed graph convolutional network

Cascade prediction aims to estimate the popularity of information diffusion in complex networks, which is beneficial to many applications from identifying viral marketing to fake news propagation in social media, estimating the scientific impact (citations) of a new publication, and so on. How to effectively predict cascade growth size has become a significant problem. Most previous methods based on deep learning have achieved remarkable results, while concentrating on mining structural and temporal features from diffusion networks and propagation paths. Whereas, the ignorance of spread dynamic information restricts the improvement of prediction performance. In this paper, we propose a novel framework called Physics-informed graph convolutional network (PiGCN) for cascade prediction, which combines explicit features (structural and temporal features) and propagation dynamic status in learning diffusion ability of cascades. Specifically, PiGCN is an end-to-end predictor, firstly splitting a given cascade into sub-cascade graph sequence and learning local structures of each sub-cascade via graph convolutional network , then adopting multi-layer perceptron to predict the cascade growth size. Moreover, our dynamic neural network, combining PDE-like equations and a deep learning method, is designed to extract potential dynamics of cascade diffusion, which captures dynamic evolution rate both on structural and temporal changes. To evaluate the performance of our proposed PiGCN model, we have conducted extensive experiment on two well-known large-scale datasets from Sina Weibo and ArXIv subject listing HEP-PH to verify the effectiveness of our model. The results of our proposed model outperform the mainstream model, and show that dynamic features have great significance for cascade size prediction.

Pangenome comparison via ED strings.

An elastic-degenerate (ED) string is a sequence of sets of strings. It can also be seen as a directed acyclic graph whose edges are labeled by strings. The notion of ED strings was introduced as a simple alternative to variation and sequence graphs for representing a pangenome, that is, a collection of genomic sequences to be analyzed jointly or to be used as a reference. In this study, we define notions of matching statistics of two ED strings as similarity measures between pangenomes and, consequently infer a corresponding distance measure. We then show that both measures can be computed efficiently, in both theory and practice, by employing the intersection graph of two ED strings. We also implemented our methods as a software tool for pangenome comparison and evaluated their efficiency and effectiveness using both synthetic and real datasets. As for efficiency, we compare the runtime of the intersection graph method against the classic product automaton construction showing that the intersection graph is faster by up to one order of magnitude. For showing effectiveness, we used real SARS-CoV-2 datasets and our matching statistics similarity measure to reproduce a well-established clade classification of SARS-CoV-2, thus demonstrating that the classification obtained by our method is in accordance with the existing one.

Learning multi-satellite scheduling policy with heterogeneous graph neural network

In this paper, we investigate the multi-satellite scheduling (MSS) problem, a combinatorial optimization problem (COP) that aims to enhance satellite utilization. Numerous scholars have conducted in-depth studies on this matter. Handcrafted policies are commonly employed in various algorithms. Despite the use of graphical representations, designing policies manually becomes increasingly difficult, particularly as the number of satellites and tasks grows. Consequently, we propose a novel approach for automatically summarizing graph structures and learning policies for MSS problem. The proposed approach is developed based on graph neural network, Transformer, and sequence to sequence framework. First, MSS problem is modeled as a heterogeneous graph following various constraints. Next, the encoder, which is based on GNN and Transformer, is designed to summarize crucial information from the aforementioned graph. Finally, a two-stage decoder is proposed taking into account dynamic information in stepwise solution generation. Since obtaining optimal solutions in practical applications is challenging, we train the model using reinforcement learning. The results of the experiment demonstrate that our proposed method performs effectively in various scenarios. The quality of the solutions generated through this method transcends that of conventional construction methods. Additionally, the method generates solutions quickly, which are comparable in quality to those produced by meta-heuristic algorithms.

Greedy maximal independent sets via local limits

AbstractThe random greedy algorithm for finding a maximal independent set in a graph constructs a maximal independent set by inspecting the graph's vertices in a random order, adding the current vertex to the independent set if it is not adjacent to any previously added vertex. In this paper, we present a general framework for computing the asymptotic density of the random greedy independent set for sequences of (possibly random) graphs by employing a notion of local convergence. We use this framework to give straightforward proofs for results on previously studied families of graphs, like paths and binomial random graphs, and to study new ones, like random trees and sparse random planar graphs. We conclude by analysing the random greedy algorithm more closely when the base graph is a tree.

Optimization in graphical small cancellation theory

Gromov (2003) [5] constructed finitely generated groups whose Cayley graphs contain all graphs from a given infinite sequence of expander graphs of unbounded girth and bounded diameter-to-girth ratio. These so-called Gromov monster groups provide examples of finitely generated groups that do not coarsely embed into Hilbert space, among other interesting properties. If graphs in Gromov's construction admit graphical small cancellation labellings, then one gets similar examples of Cayley graphs containing all the graphs of the family as isometric subgraphs. Osajda (2020) [11] recently showed how to obtain such labellings using the probabilistic method. In this short note, we simplify Osajda's approach, decreasing the number of generators of the resulting group significantly.

On the Convergence Properties of a Distributed Projected Subgradient Algorithm

A weight-balanced network plays an important role in the exact convergence of distributed optimization algorithms, but is not always satisfied in practice. Different from most of existing works focusing on designing distributed algorithms, we analyze the convergence of a well-known distributed projected subgradient algorithm over time-varying general graph sequences, i.e., the weight matrices of the network are only required to be row stochastic instead of doubly stochastic. We first show that there may exist a graph sequence such that the algorithm is not convergent when the network switches freely within finitely many graphs. Then to guarantee its convergence under any uniformly jointly strongly connected graph sequence, we provide a necessary and sufficient condition on the cost functions, i.e., the intersection of optimal solution sets to all local optimization problems is not empty. Furthermore, we surprisingly find that the algorithm is convergent for any periodically switching graph sequence, but the converged solution minimizes a weighted sum of local cost functions, where the weights depend on the Perron vectors of some product matrices of the underlying switching graphs. Finally, we consider a slightly broader class of quasi-periodically switching graph sequences, and show that the algorithm is convergent for any quasi-periodic graph sequence if and only if the network switches between only two graphs. This work helps us understand impacts of communication networks on the convergence of distributed algorithms, and complements existing results from a different viewpoint.

Multi-level category-aware graph neural network for session-based recommendation

Graph neural network (GNN)-based models have achieved state-of-the-art performance in session-based recommendation (SBR). In our research, we observe that the subsequence-level intra-category item–item transition can reflect user preference under specific category, thus enhancing recommendation performance. This is overlooked by some existing category-aware SBR models. Moreover, most existing SBR models focus solely on item prediction. However, we find that item prediction and category prediction are closely related, which can respectively serve as the main task and the auxiliary task within a joint multi-task learning (MTL) framework to enhance the performance of the main task. In this paper, we propose an SBR model named Multi-level Category-aware Graph Neural Network (MCGNN) that captures multi-level information involved in a session including subsequence-level intra-category item–item transition, and performs both item prediction and category prediction in an MTL manner. To achieve this, we propose to construct an intra-category graph for each session to model subsequence-level intra-category item–item transition, sequence-level category–category transition and item–category relation. We also construct a sequence graph for each session to model sequence-level item–item transition. These two graphs capture the multi-level information involved in a session. Next, we apply GNNs to learn item representations and category representations through the two graph views. Finally, we perform item prediction and category prediction in a joint MTL framework. Our experimental results on three widely used datasets show the superiority of our proposed MCGNN.

Dynamic multi-fusion spatio-temporal graph neural network for multivariate time series forecasting

Multivariate time series data with multiple points is a kind of spatio-temporal data, and in recent years, spatio-temporal graph neural networks (STGNNs) have become one of the most effective methods for spatio-temporal data prediction. By leveraging graph neural networks and sequence models, STGNNs simultaneously capture the hidden temporal and spatial patterns in spatio-temporal data. However, there exist two key limitations: (1) Most existing STGNNs whether based on static graphs or dynamic graphs, fail to introduce time periodicity into graph structure learning and ignore the importance of multi-relational graph structure learning. (2) Most multivariate time series prediction models with multiple points ignore the relation between multiple attributes. To this end, we propose a novel framework called the Dynamic Multi-fusion Spatio-temporal Graph Neural Network (DMF-STNet), where multi-fusion embodies two meanings: spatio-temporal fusion and multi-attribute fusion, as well as multi-graph fusion and multi-layer fusion. The dynamic periodic graph is proposed to mine time periodicity that is hidden in graph structures, and we further offer multi-relational graph structure learning methods to capture static and dynamic relations. Moreover, we introduce a dual-stage feature aggregation method to aggregate hidden features from both primary and auxiliary attributes. Extensive experiments on three real-world datasets demonstrate the efficacy of our proposed DMF-STNet, and the visualization study proves the interpretability of learned weights.

Computation of Grundy dominating sequences in (co-)bipartite graphs

Computation of Grundy dominating sequences in (co-)bipartite graphs

Note on locality of volume growth rate of graphs

In this note, we discuss locality of volume growth rate gr(G) and obtain the following results: (i) The volume growth rate is upper semi-continuous with respect to the local convergence, thus the locality holds when target graph is of volume growth rate 1. (ii) For any Cayley graph sequence {Gn}n=1∞ converging locally to regular tree Td(d≥3),limn→∞gr(Gn)=gr(Td). (iii) Given a finitely generated infinite group Γ=〈S|R〉 satisfying that there is a∈S of infinite order, and ∀b∈S,∃c∈〈S∖{a}|R〉 such that ba=ac. Then for Cayley graph sequence {Gn}n=1∞ of groups 〈S|R,an〉, which converges locally to Cayley graph G of Γ,gr(Gn)=gr(G).

DECENTRALIZED CONDITIONAL GRADIENT METHOD ON TIME-VARIABLE GRAPHS

In this paper, we consider a generalization of the decentralized Frank-Wulff algorithm for network time variables, study the convergence properties of the algorithm, and carry out the corresponding numerical experiments. The changing network is modeled as a deterministic or stochastic sequence of graphs.

Honeywords Generation Mechanism Based on Zero-Divisor Graph Sequences

Honeywords Generation Mechanism Based on Zero-Divisor Graph Sequences

Finite-time error bounds for distributed linear stochastic approximation

This paper considers a novel multi-agent linear stochastic approximation algorithm driven by Markovian noise and general consensus-type interaction, in which each agent evolves according to its local stochastic approximation process which depends on the information from its neighbors. The interconnection structure among the agents is described by a time-varying directed graph. While the convergence of consensus-based stochastic approximation algorithms when the interconnection among the agents is described by doubly stochastic matrices (at least in expectation) has been studied, less is known about the case when the interconnection matrix is simply stochastic. For any uniformly strongly connected graph sequences whose associated interaction matrices are stochastic, the paper derives finite-time bounds on the mean-square error, defined as the deviation of the output of the algorithm from the unique equilibrium point of the associated ordinary differential equation. For the case of interconnection matrices being stochastic, the equilibrium point can be any unspecified convex combination of the local equilibria of all the agents in the absence of communication. Both the cases with constant and time-varying step-sizes are considered. In the case when the convex combination is required to be a straight average and interaction between any pair of neighboring agents may be uni-directional, so that doubly stochastic matrices cannot be implemented in a distributed manner, the paper proposes a push-sum-type distributed stochastic approximation algorithm and provides its finite-time bound for the time-varying step-size case by leveraging the analysis for the consensus-type algorithm with stochastic matrices and developing novel properties of the push-sum algorithm. Distributed temporal difference learning is discussed as an illustrative application.

Grundy Total Hop Dominating Sequences in Graphs

Let G = (V (G), E(G)) be an undirected graph with γ(C) ̸= 1 for each component C of G. Let S = (v1, v2, · · · , vk) be a sequence of distint vertices of a graph G, and let Sˆ ={v1, v2, . . . , vk}. Then S is a legal open hop neighborhood sequence if N2G(vi) \Si−1j=1 N2G(vj ) ̸= ∅for every i ∈ {2, . . . , k}. If, in addition, Sˆ is a total hop dominating set of G, then S is a Grundy total hop dominating sequence. The maximum length of a Grundy total hop dominating sequence in a graph G, denoted by γth gr(G), is the Grundy total hop domination number of G. In this paper, we show that the Grundy total hop domination number of a graph G is between the total hop domination number and twice the Grundy hop domination number of G. Moreover, determine values or bounds of the Grundy total hop domination number of some graphs.