Size Of Graph Research Articles

Performance Evaluation of Parallel Graphs Algorithms Utilizing Graphcore IPU

Recent years have been characterized by increasing interest in graph computations. This trend can be related to the large number of potential application areas. Moreover, increasing computational capabilities of modern computers allowed turning theory of graph algorithms into explorations of best methods for their actual realization. These factors, in turn, brought about ideas like creation of a hardware component dedicated to graph computation; i.e., the Graphcore Intelligent Processor Unit (IPU). Interestingly, Graphcore systems are a hardware implementation of the Bulk Synchronous Parallel paradigm, which seemed to be a mostly theoretical concept from the end of last century. In this context, the question that has to be addressed experimentally is as follows: how good are Graphcore systems in comparison with standard systems that can be used to run graph algorithms, i.e., CPUs and GPUs. To provide a partial response to this broad question, in this contribution, PageRank, Single Source Shortest Path and Breadth-First Search algorithms are used to compare the performance of IPU-deployed algorithms to other parallel architectures. Obtained results clearly show that the Graphcore IPU outperforms other devices for the studied heterogeneous algorithms and, currently, provides best-in-class execution time results for a range of graph sizes and densities.

On the minimum size of graphs with given generalized connectivity

Let G be a connected graph and S⊆V(G) with |S|≥2. A tree T in G is called an S-tree if S⊆V(T). Two S-trees T1 and T2 are internally disjoint if E(T1)∩E(T2)=0̸ and V(T1)∩V(T2)=S. For an integer k≥2, the generalizedk-connectivity of a graph G, denoted by κk(G), is defined as κk(G)=min{κG(S):S⊆V(G) and |S|=k}, where κG(S) denotes the maximum number of pairwise internally disjoint S-trees in G. The generalized connectivity is a generalization of traditional connectivity. Let Gn be the class of connected graphs of order n and let f(n,k,t)=minG∈Gn{|E(G)|:κk(G)=t}. In this paper, we prove f(n,k,t)≥⌈t(t+2)2t+2n⌉ for 4≤k≤n and 1≤t≤n−⌈k2⌉. In particular, the lower bound is sharp when k=4 and t=2 (i.e., we explicitly provide a family of graphs fulfilling the bound in this case), improving the known results of Sun et al. (2021).

Omega Invariant of Complement Graphs and Nordhaus-Gaddum Type Results.

The study aimed to obtain relationships between the omega invariants of a graph and its complement. We used some graph parameters, including the cyclomatic numbers, number of components, maximum number of components, order, and size of both graphs G and G. Also, we used triangular numbers to obtain our results related to the cyclomatic numbers and omega invariants of G and G. Several bounds for the above graph parameters have been obtained by the direct application of the omega invariant. We used combinatorial and graph theoretical methods to study formulae, relations, and bounds on the omega invariant, the number of faces, and the number of compo-nents of all realizations of a given degree sequence. Especially so-called Nordhaus-Gaddum type resulted in our calculations. In these calculations, the triangular numbers less than a given number play an important role. Quadratic equations and inequalities are intensively used. Several relations between the size and order of the graph have been utilized in this study. In this paper, we have obtained relationships between the omega invariants of a graph and its complement in terms of several graph parameters, such as the cyclomatic numbers, number of components, maximum number of components, order, and size of G and G, and triangular numbers. Some relationships between the omega invariants of a graph and its complement have been obtained.

Improving Graph Compression for Efficient Resource-Constrained Graph Analytics

Recent studies have shown the promise of directly processing compressed graphs. However, its benefits have been limited by high peak-memory usage and unbearably long compression time. In this paper, we introduce Laconic, a novel rule-based graph processing solution that overcomes the challenges of restricted memory and impractical compression time faced by existing approaches. Laconic, for the first time, ensures minimal memory overhead during compression and significantly reduces graph sizes, thus reducing peak memory demand during computations. By employing an efficient parallel compression algorithm, Laconic achieves a remarkable reduction in compression time. In our experiments, we compare Laconic with state-of-the-art solutions. The results demonstrate that Laconic outperforms other methods, reducing peak memory consumption by an average of 70% during compression and 66% during computation. Additionally, Laconic reduces rule compression time by an average of 93% compared to traditional rule-based compression, achieving a 2.47× higher compression ratio, and providing a 2.12× performance speedup.

Iterative-Epoch Online Cycle Elimination for Context-Free Language Reachability

Context-free language reachability (CFL-reachability) is a fundamental framework for implementing various static analyses. CFL-reachability utilizes context-free grammar (CFG) to extend the expressiveness of ordinary graph reachability from an unlabeled graph to an edge-labeled graph. Solving CFL-reachability requires a (sub)cubic time complexity with respect to the graph size, which limits its scalability in practice. Thus, an approach that can effectively reduce the graph size while maintaining the reachability result is highly desirable. Most of the existing graph simplification techniques for CFL-reachability work during the preprocessing stage, i.e., before the dynamic CFL-reachability solving process. However, in real-world CFL-reachability analyses, there is a large number of reducible nodes and edges that can only be discovered during dynamic solving, leaving significant room for on-the-fly improvements. This paper aims to reduce the graph size of CFL-reachability dynamically via online cycle elimination. We propose a simple yet effective approach to detect collapsible cycles in the graph based on the input context-free grammar. Our key insight is that symbols with particular forms of production rules in the grammar are the essence of transitivity of reachability relations in the graph. Specifically, in the graph, a reachability relation to a node v_i can be "transited" to another node v_j if there is a transitive relation from v_i to v_j, and cycles formed by transitive relations are collapsible. In this paper, we present an approach to identify the transitive symbols in a context-free grammar and propose an iterative-epoch framework for online cycle elimination. From the perspective of non-parallelized CFL-reachability solving, our iterative-epoch framework is well compatible with both the standard (unordered) solver and the recent ordered solver, and can significantly improve their performance. Our experiment on context-sensitive value-flow analysis for C/C++ and field-sensitive alias analysis for Java demonstrates promising performance improvement by our iterative-epoch cycle elimination technique. By collapsing cycles online, our technique accelerates the standard solver by 17.17× and 13.94× for value-flow analysis and alias analysis, respectively, with memory reductions of 48.8% and 45.0%. Besides, our technique can also accelerate the ordered solver by 14.32× and 8.36× for value-flow analysis and alias analysis, respectively, with memory reductions of 55.2% and 57.8%.

Constructing Semantic Summaries Using Embeddings

The increase in the size and complexity of large knowledge graphs now available online has resulted in the emergence of many approaches focusing on enabling the quick exploration of the content of those data sources. Structural non-quotient semantic summaries have been proposed in this direction that involve first selecting the most important nodes and then linking them, trying to extract the most useful subgraph out of the original graph. However, the current state of the art systems use costly centrality measures for identifying the most important nodes, whereas even costlier procedures have been devised for linking the selected nodes. In this paper, we address both those deficiencies by first exploiting embeddings for node selection, and then by meticulously selecting approximate algorithms for node linking. Experiments performed over two real-world big KGs demonstrate that the summaries constructed using our method enjoy better quality. Specifically, the coverage scores obtained were 0.8, 0.81, and 0.81 for DBpedia v3.9 and 0.94 for Wikidata dump 2018, across 20%, 25%, and 30% summary sizes, respectively. Additionally, our method can compute orders of magnitude faster than the state of the art.

Maximum size of a triangle-free graph with bounded maximum degree and matching number

Maximum size of a triangle-free graph with bounded maximum degree and matching number

Hosoya polynomial and wiener index of Abid-Waheed graph (AW)a8 and (AW)a10

Molecular structures are characterised by the Hosoya polynomial and Wiener index, ideas from mathematical chemistry and graph theory. The graph representation of a chemical compound that has atoms as vertices and chemical bonds as edges is called a molecular graph, and the Hosoya polynomial is a polynomial related to this graph. As a graph attribute that remains unchanged under graph isomorphism, the Hosoya polynomial is known as a graph invariant. It offers details regarding the quantity of distinct non-empty subgraphs within a specified graph. A topological metric called the Wiener index is employed to measure the branching complexity and size of a molecular graph. For every pair of vertices in a molecular network, the Wiener index is the total of those distances. In this paper, discussed the Hosoya polynomial, Wiener index and Hyper-Wiener index of the Abid-Waheed graphs (AW)a8 and (AW)a10. This graph is similar to Jahangir’s graph. Further, we have extended the research work on the applications of the described graphs.

A comprehensive benchmark of graph-based genetic variant genotyping algorithms on plant genomes for creating an accurate ensemble pipeline.

Although sequencing technologies have boosted the measurement of the genomic diversity of plant crops, it remains challenging to accurately genotype millions of genetic variants, especially structural variations, with only short reads. In recent years, many graph-based variation genotyping methods have been developed to address this issue and tested for human genomes. However, their performance in plant genomes remains largely elusive. Furthermore, pipelines integrating the advantages of current genotyping methods might be required, considering the different complexity of plant genomes. Here we comprehensively evaluate eight such genotypers in different scenarios in terms of variant type and size, sequencing parameters, genomic context, and complexity, as well as graph size, using both simulated and real data sets from representative plant genomes. Our evaluation reveals that there are still great challenges to applying existing methods to plants, such as excessive repeats and variants or high resource consumption. Therefore, we propose a pipeline called Ensemble Variant Genotyper (EVG) that can achieve better genotyping performance in almost all experimental scenarios and comparably higher genotyping recall and precision even using 5× reads. Furthermore, we demonstrate that EVG is more robust with an increasing number of graphed genomes, especially for insertions and deletions. Our study will provide new insights into the development and application of graph-based genotyping algorithms. We conclude that EVG provides an accurate, unbiased, and cost-effective way for genotyping both small and large variations and will be potentially used in population-scale genotyping for large, repetitive, and heterozygous plant genomes.

Intersecting families of transformations

Consider [Formula: see text] and [Formula: see text] to represent the full transformation semigroup and the symmetric group on the set [Formula: see text], respectively. A subset [Formula: see text] of [Formula: see text] is called an intersecting family if any two transformations [Formula: see text] coincide at some point [Formula: see text]. An intersecting family [Formula: see text] is called maximum if no other intersecting family [Formula: see text] exists with [Formula: see text]. Frankl and Deza [On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22(3) (1977) 352–360] established that the cardinality of a maximum intersecting family of [Formula: see text] equals [Formula: see text]. Subsequent research demonstrated that a maximum intersecting family of [Formula: see text] forms a coset of a stabilizer of one point, as shown by Cameron and Ku [Intersecting families of permutations, European J. Combin. 24(7) (2003) 881–890] and independently by Larose and Malvenuto [Stable sets of maximal size in Kneser-type graphs, European J. Combin. 25(5) (2004) 657–673]. Define [Formula: see text], where [Formula: see text] denotes the image set of [Formula: see text]. This paper presents a formula for the cardinality of a maximum intersecting family of [Formula: see text]. It then characterizes the maximum intersecting families of [Formula: see text] and concludes by determining the number of maximum intersecting families of [Formula: see text].

Scene Graph Lossless Compression with Adaptive Prediction for Objects and Relations

The scene graph is a novel data structure describing objects and their pairwise relationship within image scenes. As the size of scene graphs in vision and multimedia applications increases, the need for lossless storage and transmission of such data becomes more critical. However, the compression of scene graphs is less studied because of the complicated data structures involved and complex distributions. Existing solutions usually involve general-purpose compressors or graph structure compression methods, which are weak at reducing the redundancy in scene graph data. This article introduces a novel lossless compression framework with adaptive predictors for the joint compression of objects and relations in scene graph data. The proposed framework comprises a unified prior extractor and specialized element predictors to adapt to different data elements. Furthermore, to exploit the context information within and between graph elements, Graph Context Convolution is proposed to support different graph context modeling schemes for different graph elements. Finally, an overarching framework incorporates the learned distribution model to predict numerical data under complicated conditional constraints. Experiments conducted on labeled or generated scene graphs demonstrate the effectiveness of the proposed framework for scene graph lossless compression.

Fully Binarized Graph Convolutional Network Accelerator Based on In‐Memory Computing with Resistive Random‐Access Memory

Artificial intelligence for graph‐structured data has achieved remarkable success in applications such as recommendation systems, social networks, drug discovery, and circuit annotation. Graph convolutional networks (GCNs) are an effective way to learn representations of various graphs. The increasing size and complexity of graphs call for in‐memory computing (IMC) accelerators for GCN to alleviate massive data transmission between off‐chip memory and processing units. However, GCN implementation with IMC is challenging because of the large memory consumption, irregular memory access, and device nonidealities. Herein, a fully binarized GCN (BGCN) accelerator based on computational resistive random‐access memory (RRAM) through software–hardware codesign is presented. The essential operations including aggregation and combination in GCN are implemented on the RRAM crossbar arrays with cooperation between multiply‐and‐accumulation and content‐addressable memory operations. By leveraging the model quantization and IMC on the RRAM, the BGCN accelerator demonstrates less RRAM usage, high robustness to the device variations, high energy efficiency, and comparable classification accuracy compared to the current state‐of‐the‐art GCN accelerators on both graph classification task using the MUTAG and PTC datasets and node classification task using the Cora and CiteSeer datasets. These results provide a promising approach for edge intelligent systems to efficiently process graph‐structured data.

DGCLUSTER: A Neural Framework for Attributed Graph Clustering via Modularity Maximization

Graph clustering is a fundamental and challenging task in the field of graph mining where the objective is to group the nodes into clusters taking into consideration the topology of the graph. It has several applications in diverse domains spanning social network analysis, recommender systems, computer vision, and bioinformatics. In this work, we propose a novel method, DGCluster, which primarily optimizes the modularity objective using graph neural networks and scales linearly with the graph size. Our method does not require the number of clusters to be specified as a part of the input and can also leverage the availability of auxiliary node level information. We extensively test DGCluster on several real-world datasets of varying sizes, across multiple popular cluster quality metrics. Our approach consistently outperforms the state-of-the-art methods, demonstrating significant performance gains in almost all settings.

ASWT-SGNN: Adaptive Spectral Wavelet Transform-Based Self-Supervised Graph Neural Network

Graph Comparative Learning (GCL) is a self-supervised method that combines the advantages of Graph Convolutional Networks (GCNs) and comparative learning, making it promising for learning node representations. However, the GCN encoders used in these methods rely on the Fourier transform to learn fixed graph representations, which is inherently limited by the uncertainty principle involving spatial and spectral localization trade-offs. To overcome the inflexibility of existing methods and the computationally expensive eigen-decomposition and dense matrix multiplication, this paper proposes an Adaptive Spectral Wavelet Transform-based Self-Supervised Graph Neural Network (ASWT-SGNN). The proposed method employs spectral adaptive polynomials to approximate the filter function and optimize the wavelet using contrast loss. This design enables the creation of local filters in both spectral and spatial domains, allowing flexible aggregation of neighborhood information at various scales and facilitating controlled transformation between local and global information. Compared to existing methods, the proposed approach reduces computational complexity and addresses the limitation of graph convolutional neural networks, which are constrained by graph size and lack flexible control over the neighborhood aspect. Extensive experiments on eight benchmark datasets demonstrate that ASWT-SGNN accurately approximates the filter function in high-density spectral regions, avoiding costly eigen-decomposition. Furthermore, ASWT-SGNN achieves comparable performance to state-of-the-art models in node classification tasks.

An Efficient Subgraph-Inferring Framework for Large-Scale Heterogeneous Graphs

Heterogeneous Graph Neural Networks (HGNNs) play a vital role in advancing the field of graph representation learning by addressing the complexities arising from diverse data types and interconnected relationships in real-world scenarios. However, traditional HGNNs face challenges when applied to large-scale graphs due to the necessity of training or inferring on the entire graph. As the size of the heterogeneous graphs increases, the time and memory overhead required by these models escalates rapidly, even reaching unacceptable levels. To address this issue, in this paper, we present a novel framework named (SubInfer), which conducts training and inferring on subgraphs instead of the entire graphs, hence efficiently handling large-scale heterogeneous graphs. The proposed framework comprises three main steps: 1) partitioning the heterogeneous graph from multiple perspectives to preserve various semantic information, 2) completing the subgraphs to improve the convergence speed of subgraph training and the performance of subgraph inference, and 3) training and inferring the HGNN model on distributed clusters to further reduce the time overhead. The framework is applicable to the vast majority of HGNN models. Experiments on five benchmark datasets demonstrate that SubInfer effectively optimizes the training and inference phase, delivering comparable performance to traditional HGNN models while significantly reducing time and memory overhead.

Improved Anonymous Multi-Agent Path Finding Algorithm

We consider an Anonymous Multi-Agent Path-Finding (AMAPF) problem where the set of agents is confined to a graph, a set of goal vertices is given and each of these vertices has to be reached by some agent. The problem is to find an assignment of the goals to the agents as well as the collision-free paths, and we are interested in finding the solution with the optimal makespan. A well-established approach to solve this problem is to reduce it to a special type of a graph search problem, i.e. to the problem of finding a maximum flow on an auxiliary graph induced by the input one. The size of the former graph may be very large and the search on it may become a bottleneck. To this end, we suggest a specific search algorithm that leverages the idea of exploring the search space not through considering separate search states but rather bulks of them simultaneously. That is, we implicitly compress, store and expand bulks of the search states as single states, which results in high reduction in runtime and memory. Empirically, the resultant AMAPF solver demonstrates superior performance compared to the state-of-the-art competitor and is able to solve all publicly available MAPF instances from the well-known MovingAI benchmark in less than 30 seconds.

Recurrent Graph Neural Networks and Their Connections to Bisimulation and Logic

The success of Graph Neural Networks (GNNs) in practice has motivated extensive research on their theoretical properties. This includes recent results that characterise node classifiers expressible by GNNs in terms of first order logic. Most of the analysis, however, has been focused on GNNs with fixed number of message-passing iterations (i.e., layers), which cannot realise many simple classifiers such as reachability of a node with a given label. In this paper, we start to fill this gap and study the foundations of GNNs that can perform more than a fixed number of message-passing iterations. We first formalise two generalisations of the basic GNNs: recurrent GNNs (RecGNNs), which repeatedly apply message-passing iterations until the node classifications become stable, and graph-size GNNs (GSGNNs), which exploit a built-in function of the input graph size to decide the number of message-passings. We then formally prove that GNN classifiers are strictly less expressive than RecGNN ones, and RecGNN classifiers are strictly less expressive than GSGNN ones. To get this result, we identify novel semantic characterisations of the three formalisms in terms of suitable variants of bisimulation, which we believe have their own value for our understanding of GNNs. Finally, we prove syntactic logical characterisations of RecGNNs and GSGNNs analogous to the logical characterisation of plain GNNs, where we connect the two formalisms to monadic monotone fixpoint logic---a generalisation of first-order logic that supports recursion.

Acceleration of Graph Neural Network-Based Prediction Models in Chemistry via Co-Design Optimization on Intelligence Processing Units.

Atomic structure prediction and associated property calculations are the bedrock of chemical physics. Since high-fidelity ab initio modeling techniques for computing the structure and properties can be prohibitively expensive, this motivates the development of machine-learning (ML) models that make these predictions more efficiently. Training graph neural networks over large atomistic databases introduces unique computational challenges, such as the need to process millions of small graphs with variable size and support communication patterns that are distinct from learning over large graphs, such as social networks. We demonstrate a novel hardware-software codesign approach to scale up the training of atomistic graph neural networks (GNN) for structure and property prediction. First, to eliminate redundant computation and memory associated with alternative padding techniques and to improve throughput via minimizing communication, we formulate the effective coalescing of the batches of variable-size atomistic graphs as the bin packing problem and introduce a hardware-agnostic algorithm to pack these batches. In addition, we propose hardware-specific optimizations, including a planner and vectorization for the gather-scatter operations targeted for Graphcore's Intelligence Processing Unit (IPU), as well as model-specific optimizations such as merged communication collectives and optimized softplus. Putting these all together, we demonstrate the effectiveness of the proposed codesign approach by providing an implementation of a well-established atomistic GNN on the Graphcore IPUs. We evaluate the training performance on multiple atomistic graph databases with varying degrees of graph counts, sizes, and sparsity. We demonstrate that such a codesign approach can reduce the training time of atomistic GNNs and can improve their performance by up to 1.5× compared to the baseline implementation of the model on the IPUs. Additionally, we compare our IPU implementation with a Nvidia GPU-based implementation and show that our atomistic GNN implementation on the IPUs can run 1.8× faster on average compared to the execution time on the GPUs.

Enhancing Energy Efficiency in IoT-NDN via Parameter Optimization

The IoT encompasses objects, sensors, and everyday items not typically considered computers. IoT devices are subject to severe energy, memory, and computation power constraints. Employing NDN for the IoT is a recent approach to accommodate these issues. To gain a deeper insight into how different network parameters affect energy consumption, analyzing a range of parameters using hyperparameter optimization seems reasonable. The experiments from this work’s ndnSIM-based hyperparameter setup indicate that the data packet size has the most significant impact on energy consumption, followed by the caching scheme, caching strategy, and finally, the forwarding strategy. The energy footprint of these parameters is orders of magnitude apart. Surprisingly, the packet request sequence influences the caching parameters’ energy footprint more than the graph size and topology. Regarding energy consumption, the results indicate that data compression may be more relevant than expected, and caching may be more significant than the forwarding strategy. The framework for ndnSIM developed in this work can be used to simulate NDN networks more efficiently. Furthermore, the work presents a valuable basis for further research on the effect of specific parameter combinations not examined before.

Autonomous Ride-Sharing Service Using Graph Embedding and Dial-a-Ride Problem: Application to the Last-Mile Transit in Lyon City

Autonomous vehicles are anticipated to revolutionize ride-sharing services and subsequently enhance the public transportation systems through a first–last-mile transit service. Within this context, a fleet of autonomous vehicles can be modeled as a Dial-a-Ride Problem with certain features. In this study, we propose a holistic solving approach to this problem, which combines the mixed-integer linear programming formulation with a novel graph dimension reduction method based on the graph embedding framework. This latter method is effective since accounting for heterogeneous travel demands of the covered territory tends to increase the size of the routing graph drastically, thus rendering the exact solving of small instances computationally infeasible. An application is provided for the real transport demand of the industrial district of “Vallée de la Chimie” in Lyon city, France. Instances involving more than 50 transport requests and 10 vehicles could be easily solved. Results suggest that this method generates routes of reduced nodes with lower vehicle kilometers traveled compared to the constrained K-means-based reduction. Reductions in terms of GHG emissions are estimated to be around 75% less than the private vehicle mode in our applied service. A sensitivity analysis is also provided.