Small Graphs Research Articles

A Graph Convolution Network Based on Improved Density Clustering for Recommendation System

Recommendation systems have been widely used in various applications to solve information overload andimprove user experience. Traditional recommendation algorithms mainly used Euclidean data for calculationand abandoned the graph structure features in user and item data. Aiming at the problems in the current recommendation algorithms, this paper proposes an improved user density clustering method and extracts userfeatures through optimized graph neural network. Firstly, the improved density clustering method is used toform the clustering subgraph of users based on the influence value of users. Secondly, the user data and itemdata features of cluster subgraph are extracted by graph convolution network. Finally, the features of clustersubgraphs are processed by global graph convolution network and the recommendation results are generatedaccording to the global graph features. This model not only improves the efficiency of decomposing large graphinto small graph through the improved user density clustering algorithm, but also extracts the features of usergroups through graph convolution neural network to improve the recommendation effect. The experiment alsoproves the validity of this model.

On 2-Factors Splitting an Embedded Graph into Two Plane Graphs

We investigate 2-planarizing 2-factors, i.e. 2-factors of embedded graphs so that cutting along the cycles of the 2-factor we get two plane graphs where the cycles of the 2-factors are a spanning set of face boundaries in each of the graphs. We will give necessary criteria for an abstract graph to have an embedding with a 2-planarizing 2-factor as well as necessary criteria for embedded graphs to have such a 2-factor. Along the way, we discuss to which degree classical results from planar hamiltonicity theory can be extended in our framework. In addition we present computational results on how common 2-planarizing 2-factors are in small cubic graphs.

A general model of hierarchical fractal scale-free networks.

We propose a general model of unweighted and undirected networks having the scale-free property and fractal nature. Unlike the existing models of fractal scale-free networks (FSFNs), the present model can systematically and widely change the network structure. In this model, an FSFN is iteratively formed by replacing each edge in the previous generation network with a small graph called a generator. The choice of generators enables us to control the scale-free property, fractality, and other structural properties of hierarchical FSFNs. We calculate theoretically various characteristic quantities of networks, such as the exponent of the power-law degree distribution, fractal dimension, average clustering coefficient, global clustering coefficient, and joint probability describing the nearest-neighbor degree correlation. As an example of analyses of phenomena occurring on FSFNs, we also present the critical point and critical exponents of the bond-percolation transition on infinite FSFNs, which is related to the robustness of networks against edge removal. By comparing the percolation critical points of FSFNs whose structural properties are the same as each other except for the clustering nature, we clarify the effect of the clustering on the robustness of FSFNs. As demonstrated by this example, the present model makes it possible to elucidate how a specific structural property influences a phenomenon occurring on FSFNs by varying systematically the structures of FSFNs. Finally, we extend our model for deterministic FSFNs to a model of non-deterministic ones by introducing asymmetric generators and reexamine all characteristic quantities and the percolation problem for such non-deterministic FSFNs.

Multicolor star-critical Ramsey numbers and Ramsey-good graphs

This paper seeks to develop the multicolor version of star-critical Ramsey numbers, which serve as a measure of the strength of the corresponding Ramsey numbers. We offer several general theorems, some of which focus on Ramsey-good cases (i.e., cases in which the corresponding Ramsey number is equal to a general lower bound). We also prove some specific cases for small graphs, and conclude with a table of known multicolor star-critical Ramsey numbers.

Magnetically Induced Current Densities in Zinc Porphyrin Nanoshells.

The molecular structures of porphyrinoid cages were obtained by constructing small polyhedral graphs whose vertices have degree-4. The initial structures were then fully optimized at the density functional theory (DFT) level using the generalized gradient approximation. Some of polyhedral vertices were replaced with Zn–porphyrin units and their edges were replaced with ethyne or butadiyne bridges or connected by fusing two neighboring Zn–porphyrin units. Molecule 1 is an ethyne-bridge porphyrinoid nanotube, whose ends are sealed with a Zn–porphyrin. Molecule 2 is the corresponding open porphyrinoid nanotube. Molecule 3 is a clam-like porphyrinoid cage, whose shells consist of fused Zn–porphyrins, and the two halves are connected via butadiyne bridges. Molecule 4 is a cross-belt of fused Zn–porphyrins, and molecule 5 is a cross-belt of Zn–porphyrins connected with butadiyne bridges. The magnetically induced current density of the optimized porphyrinoid cages was calculated for determining the aromatic character, the degree of aromaticity and the current-density pathways. The current-density calculations were performed at the DFT level with the gauge—including magnetically induced currents (GIMIC) method using the B3LYP hybrid functional and def2-SVP basis sets. Calculations of the current densities show that molecule 2 sustains a paratropic ring current around the nanotube, whereas sealing the ends as in molecule 1 leads to an almost nonaromatic nanotube. Fusing porphyrinoids as in molecules 3 and 4 results in complicated current-density pathways that differ from the ones usually appearing in porphyrinoids. The aromatic character of molecules 4 and 5 changes upon oxidation. The neutral molecule 4 is antiaromatic, whereas the dication is nonaromatic. Molecule 5 is nonaromatic, and its dication is aromatic.

Neuronal network inference and membrane potential model using multivariate Hawkes processes

BackgroundIn this work, we propose to catch the complexity of the membrane potential’s dynamic of a motoneuron between its spikes, taking into account the spikes from other neurons around. Our approach relies on two types of data: extracellular recordings of multiple spikes trains and intracellular recordings of the membrane potential of a central neuron. New methodWe provide a unified framework and a complete pipeline to analyze neuronal activity from data extraction to statistical inference. To the best of our knowledge, this is the first time that a Hawkes-diffusion model is investigated on such complex data. The first step of the proposed procedure is to select a subnetwork of neurons impacting the central neuron using a multivariate Hawkes process. Then we infer a jump-diffusion dynamic in which jumps are driven from a Hawkes process, the occurrences of which correspond to the spike trains of the aforementioned subset of neurons that interact with the central neuron. ResultsFrom the Hawkes estimation step we recover a small connectivity graph which contains the central neuron, and we show that taking into account this information improves the inference of membrane potential through the proposed jump-diffusion model. A goodness of fit test is applied to validate the relevance of the Hawkes model in such context. Comparison with existing methodsWe compare an empirical inference method and two sparse estimation procedures based on the Hawkes assumption for the reconstruction of the connectivity graph using the spike-trains. Then, the Hawkes-diffusion model is competed with the simple diffusion in terms of best fit to describe the behavior of the membrane potential of a central neuron surrounded by a network. ConclusionsThe present method takes advantage of both spike trains and membrane potential to understand the behavior of a fixed neuron. The entire code has been developed and is freely available on GitHub.

Discovering association rules from big graphs

This paper tackles two challenges to discovery of graph rules. Existing discovery methods often (a) return an excessive number of rules, and (b) do not scale with large graphs given the intractability of the discovery problem. We propose an application-driven strategy to cut back rules and data that are irrelevant to users' interests, by training a machine learning (ML) model to identify data pertaining to a given application. Moreover, we introduce a sampling method to reduce a big graph G to a set H of small sample graphs. Given expected support and recall bounds, the method is able to deduce samples in H and mine rules from H to satisfy the bounds in the entire G . As proof of concept, we develop an algorithm to discover Graph Association Rules (GARs), which are a combination of graph patterns and attribute dependencies, and may embed ML classifiers as predicates. We show that the algorithm is parallelly scalable, i.e. , it guarantees to reduce runtime when more machines are used. We experimentally verify that the method is able to discover rules with recall above 91% when using sample ratio 10%, with speedup of 61 times.

Corona product of signed graphs and its application to modeling signed networks

The notion of corona of two graphs was introduced by Frucht and Harary in 1970. In this paper, we generalize their definition of corona product of two graphs and introduce corona product of two signed graphs by utilizing the framework of marked graphs, which was introduced by Beineke and Harary in 1978. We study structural and spectral properties of corona product of signed graphs. Further, we define signed corona graphs by considering corona product of a fixed small signed graph with itself iteratively, and we call the small graph as the seed graph for the corresponding corona product graphs. Signed corona graphs can be employed as a signed network generative model for large growing signed networks. We study structural properties of corona graphs that include statistics of signed links, all types of signed triangles and degree distribution. Besides we analyze algebraic conflict of signed corona graphs generated by specially structured seed graphs. Finally, we show that a suitable choice of a seed graph can produce corona graphs which preserve properties of real signed networks.

Making graphs compact by lossless contraction

This paper proposes a scheme to reduce big graphs to small graphs. It contracts obsolete parts and regular structures into supernodes. The supernodes carry a synopsis S_mathcal {Q} for each query class mathcal {Q} in use, to abstract key features of the contracted parts for answering queries of mathcal {Q}. Moreover, for various types of graphs, we identify regular structures to contract. The contraction scheme provides a compact graph representation and prioritizes up-to-date data. Better still, it is generic and lossless. We show that the same contracted graph is able to support multiple query classes at the same time, no matter whether their queries are label based or not, local or non-local. Moreover, existing algorithms for these queries can be readily adapted to compute exact answers by using the synopses when possible and decontracting the supernodes only when necessary. As a proof of concept, we show how to adapt existing algorithms for subgraph isomorphism, triangle counting, shortest distance, connected component and clique decision to contracted graphs. We also provide a bounded incremental contraction algorithm in response to updates, such that its cost is determined by the size of areas affected by the updates alone, not by the entire graphs. We experimentally verify that on average, the contraction scheme reduces graphs by 71.9% and improves the evaluation of these queries by 1.69, 1.44, 1.47, 2.24 and 1.37 times, respectively.

An analysis of the parameterized complexity of periodic timetabling

Public transportation networks are typically operated with a periodic timetable. The periodic event scheduling problem (PESP) is the standard mathematical modeling tool for periodic timetabling. PESP is a computationally very challenging problem: For example, solving the instances of the benchmarking library PESPlib to optimality seems out of reach. Since PESP can be solved in linear time on trees, and the treewidth is a rather small graph parameter in the networks of the PESPlib, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth, or even better, fixed-parameter tractable algorithms. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. Moreover, we show W[1]-hardness of the general feasibility problem with respect to treewidth, which means that we can most likely only accomplish pseudo-polynomial-time algorithms on input networks with bounded tree- or branchwidth. We present two such algorithms based on dynamic programming. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special—but standard—structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib.

Bringing Your Own View: Graph Contrastive Learning without Prefabricated Data Augmentations.

Self-supervision is recently surging at its new frontier of graph learning. It facilitates graph representations beneficial to downstream tasks; but its success could hinge on domain knowledge for handcraft or the often expensive trials and errors. Even its state-of-the-art representative, graph contrastive learning (GraphCL), is not completely free of those needs as GraphCL uses a prefabricated prior reflected by the ad-hoc manual selection of graph data augmentations. Our work aims at advancing GraphCL by answering the following questions: How to represent the space of graph augmented views? What principle can be relied upon to learn a prior in that space? And what framework can be constructed to learn the prior in tandem with contrastive learning? Accordingly, we have extended the prefabricated discrete prior in the augmentation set, to a learnable continuous prior in the parameter space of graph generators, assuming that graph priors per se, similar to the concept of image manifolds, can be learned by data generation. Furthermore, to form contrastive views without collapsing to trivial solutions due to the prior learnability, we have leveraged both principles of information minimization (InfoMin) and information bottleneck (InfoBN) to regularize the learned priors. Eventually, contrastive learning, InfoMin, and InfoBN are incorporated organically into one framework of bi-level optimization. Our principled and automated approach has proven to be competitive against the state-of-the-art graph self-supervision methods, including GraphCL, on benchmarks of small graphs; and shown even better generalizability on large-scale graphs, without resorting to human expertise or downstream validation. Our code is publicly released at https://github.com/Shen-Lab/GraphCL_Automated.

Wasserstein-Based Graph Alignment

A novel method for comparing non-aligned graphs of various sizes is proposed, based on the Wasserstein distance between graph signal distributions induced by the respective graph Laplacian matrices. Specifically, a new formulation for the one-to-many graph alignment problem is casted, which aims at matching a node in the smaller graph with one or more nodes in the larger graph. By incorporating optimal transport into our graph comparison framework, a structurally-meaningful graph distance, and a signal transportation plan that models the structure of graph data are generated. The resulting alignment problem is solved with stochastic gradient descent, where a novel Dykstra operator is used to ensure that the solution is a one-to-many (soft) assignment matrix. The performance of our novel framework is demonstrated on graph alignment, graph classification and graph signal transportation. Our method is shown to lead to significant improvements with respect to the state-of-the-art algorithms on each ofthese tasks.

OblivGM: Oblivious Attributed Subgraph Matching as a Cloud Service

In recent years there has been growing popularity of leveraging cloud computing for storing and querying attributed graphs, which have been widely used to model complex structured data in various applications. Such trend of outsourced graph analytics, however, is accompanied with critical privacy concerns regarding the information-rich and proprietary attributed graph data. In light of this, we design, implement, and evaluate OblivGM, a new system aimed at oblivious graph analytics services outsourced to the cloud. OblivGM focuses on the support for attributed subgraph matching, one popular and fundamental graph query functionality aiming to retrieve from a large attributed graph subgraphs isomorphic to a small query graph. Built from a delicate synergy of insights from attributed graph modelling and advanced lightweight cryptography, OblivGM protects the confidentiality of data content associated with attributed graphs and queries, conceals the connections among vertices in attributed graphs, and hides search access patterns. Meanwhile, OblivGM flexibly supports oblivious evaluation of varying subgraph queries, which may contain equality and/or range predicates. Extensive experiments over a real-world attributed graph dataset demonstrate that while providing strong security guarantees, OblivGM achieves practically affordable performance (with query latency on the order of a few seconds).

Properties of Large 2-Crossing-Critical Graphs

A $c$-crossing-critical graph is one that has crossing number at least $c$ but each of its proper subgraphs has crossing number less than $c$. Recently, a set of explicit construction rules was identified by Bokal, Oporowski, Richter, and Salazar to generate all large $2$-crossing-critical graphs (i.e., all apart from a finite set of small sporadic graphs). They share the property of containing a generalized Wagner graph $V_{10}$ as a subdivision. In this paper, we study these graphs and establish their order, simple crossing number, edge cover number, clique number, maximum degree, chromatic number, chromatic index, and treewidth. We also show that the graphs are linear-time recognizable and that all our proofs lead to efficient algorithms for the above measures. Keywords. Crossing number, crossing-critical graph, chromatic number, chromatic index, treewidth.

Strategic robustness in bi-level system-of-systems design

Abstract Robust designs protect system utility in the presence of uncertainty in technical and operational outcomes. Systems-of-systems, which lack centralized managerial control, are vulnerable to strategic uncertainty from coordination failures between partially or completely independent system actors. This work assesses the suitability of a game-theoretic equilibrium selection criterion to measure system robustness to strategic uncertainty and investigates the effect of strategically robust designs on collaborative behavior. The work models interactions between agents in a thematic representation of a mobile computing technology transition using an evolutionary game theory framework. Strategic robustness and collaborative solutions are assessed over a range of conditions by varying agent payoffs. Models are constructed on small world, preferential attachment and random graph topologies and executed in batch simulations. Results demonstrate that systems designed to reduce the impacts of coordination failure stemming from strategic uncertainty also increase the stability of the collaborative strategy by increasing the probability of collaboration by partners; a form of robustness by environment shaping that has not been previously investigated in design literature. The work also demonstrates that strategy selection follows the risk dominance equilibrium selection criterion and that changes in robustness to coordination failure can be measured with this criterion.

New Graphical Software Tool for Creating Cause-Effect Graph Specifications

Cause-effect graphing is a commonly used black-box technique with many applications in practice. It is important to be able to create accurate cause-effect graph specifications from system requirements before converting them to test case tables used for black-box testing. In this paper, a new graphical software tool for creating cause-effect graph specifications is presented. The tool uses standardized graphical notation for describing different types of nodes, logical relations and constraints, resulting in a visual representation of the desired cause-effect graph which can be exported for later usage and imported in the tool. The purpose of this work is to make the cause-effect graph specification process easier for users in order to solve some of the problems which arise due to the insufficient amount of understanding of cause-effect graph elements. The proposed tool was successfully used for creating cause-effect graph specifications for small, medium and large graphs. It was also successfully used for performing different types of tasks by users without any prior knowledge of the functionalities of the tool, indicating that the tool is easy to use, helpful and intuitive. The results indicate that the usage of standardized notation is easier to understand than non-standardized approaches from other tools.

Practical Near-Data-Processing Architecture for Large-Scale Distributed Graph Neural Network

Graph Neural Networks have drawn tremendous attention in the past few years due to their convincing performance and high interpretability in various graph-based tasks like link prediction and node classification. With the ever-growing graph size in the real world, especially for industrial graphs at a billion-level, the storage of graphs can easily consume Terabytes so that the process of GNNs has to be processed in a distributed manner. As a result, the execution could be inefficient due to the expensive cross-node communication and irregular memory access. Various GNN accelerators have been proposed for efficient GNN processing. They, however, mainly focused on small and medium-size graphs, which is not applicable to large-scale distributed graphs. In this paper, we present a practical Near-Data-Processing architecture based on a memory-pool system for large-scale distributed GNNs. We propose a customized memory fabric interface to construct the memory pool for low-latency and high throughput cross-node communication, which can provide flexible memory allocation and strong scalability. A practical Near-Data-Processing design is proposed for efficient work offloading and bandwidth utilization improvement. Moreover, we also introduce a partition and scheduling scheme to further improve performance and achieve workload balance. Comprehensive evaluations demonstrate that the proposed architecture can achieve up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$27\times $ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$8\times $ </tex-math></inline-formula> higher training speed compared to two state-of-the-art distributed GNN frameworks: Deep Graph Library and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P^{3}$ </tex-math></inline-formula> , respectively.

Multicriteria Scalable Graph Drawing via Stochastic Gradient Descent, (SGD)2.

Readability criteria, such as distance or neighborhood preservation, are often used to optimize node-link representations of graphs to enable the comprehension of the underlying data. With few exceptions, graph drawing algorithms typically optimize one such criterion, usually at the expense of others. We propose a layout approach, Multicriteria Scalable Graph Drawing via Stochastic Gradient Descent, (SGD)2, that can handle multiple readability criteria. (SGD)2 can optimize any criterion that can be described by a differentiable function. Our approach is flexible and can be used to optimize several criteria that have already been considered earlier (e.g., obtaining ideal edge lengths, stress, neighborhood preservation) as well as other criteria which have not yet been explicitly optimized in such fashion (e.g., node resolution, angular resolution, aspect ratio). The approach is scalable and can handle large graphs. A variation of the underlying approach can also be used to optimize many desirable properties in planar graphs, while maintaining planarity. Finally, we provide quantitative and qualitative evidence of the effectiveness of (SGD)2: we analyze the interactions between criteria, measure the quality of layouts generated from (SGD)2 as well as the runtime behavior, and analyze the impact of sample sizes. The source code is available on github and we also provide an interactive demo for small graphs.

Computing some role assignments of Cartesian product of graphs

In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. When a smaller graph models the social roles in a network, this gives rise to the decision problem called r-Role Assignment whether it exists such an assignment of r distinct roles to the vertices of the graph. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is one of the most studied operation on graphs and has numerous applications in diverse areas, such as Mathematics, Computer Science, Chemistry and Biology. In this paper, we determine the computational complexity of r-Role Assignment restricted to Cartesian product of graphs, for r = 2, 3. In fact, we show that the Cartesian product of graphs is always 2-role assignable, however the problem of 3-Role Assignment is still NP-complete for this class.

Phase transition in random intersection graphs with communities

AbstractThe “random intersection graph with communities” (RIGC) models networks with communities, assuming an underlying bipartite structure of groups and individuals. Each group has its own internal structure described by a (small) graph, while groups may overlap. The group memberships are generated by a bipartite configuration model. The model generalizes the classical random intersection graph model, a special case where each community is a complete graph. The RIGC model is analytically tractable. We prove a phase transition in the size of the largest connected component in terms of the model parameters. We prove that percolation on RIGC produces a graph within the RIGC family, also undergoing a phase transition with respect to size of the largest component. Our proofs rely on the connection to the bipartite configuration model. Our related results on the bipartite configuration model are of independent interest, since they shed light on interesting differences from the unipartite case.