Structural Properties Of Graphs Research Articles

Extremal Bicyclic Graphs with Respect to Permanental Sums and Hosoya Indices

Graph polynomials is one of the important research directions in mathematical chemistry. The coefficients of some graph polynomials, such as matching polynomial and permanental polynomial, are related to structural properties of graphs. The Hosoya index of a graph is the sum of the absolute value of all coefficients for the matching polynomial. And the permanental sum of a graph is the sum of the absolute value of all coefficients of the permanental polynomial. In this paper, we characterize the second to sixth minimal Hosoya indices of all bicyclic graphs. Furthermore, using the results, the second to sixth minimal permanental sums of all bicyclic graphs are also characterized.

Graphs with at most two moplexes

AbstractA moplex is a natural graph structure that arises when lifting Dirac's classical theorem from chordal graphs to general graphs. The notion is known to be closely related to lexicographic searches in graphs as well as to asteroidal triples, and has been applied in several algorithms related to graph classes, such as interval graphs, claw‐free, and diamond‐free graphs. However, while every noncomplete graph has at least two moplexes, little is known about structural properties of graphs with a bounded number of moplexes. The study of these graphs is, in part, motivated by the parallel between moplexes in general graphs and simplicial modules in chordal graphs: unlike in the moplex setting, properties of chordal graphs with a bounded number of simplicial modules are well understood. For instance, chordal graphs having at most two simplicial modules are intervals. In this work, we initiate an investigation of ‐moplex graphs, which are defined as graphs containing at most moplexes. Of particular interest is the smallest nontrivial case , which forms a counterpart to the class of interval graphs. As our main structural result, we show that, when restricted to connected graphs, the class of 2‐moplex graphs is sandwiched between the classes of proper interval graphs and cocomparability graphs; moreover, both inclusions are tight for hereditary classes. From a complexity‐theoretic viewpoint, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. We develop new reductions that answer this question negatively for two prominent problems fitting this profile, namely, Graph Isomorphism and Max‐Cut. On the other hand, we prove that every connected 2‐moplex graph contains a Hamiltonian path, generalising the same property of connected proper interval graphs. Furthermore, for graphs with a higher number of moplexes, we lift the previously known result that graphs without asteroidal triples have at most two moplexes to the more general setting of larger asteroidal sets.

Localization transition in non-Hermitian systems depending on reciprocity and hopping asymmetry.

We studied the single-particle Anderson localization problem for non-Hermitian systems on directed graphs. Random regular graph and various undirected standard random graph models were modified by controlling reciprocity and hopping asymmetry parameters. We found the emergence of left, biorthogonal, and right localized states depending on both parameters and graph structure properties such as node degree d. For directed random graphs, the occurrence of biorthogonal localization near exceptional points is described analytically and numerically. The clustering of localized states near the center of the spectrum and the corresponding mobility edge for left and right states are shown numerically. Structural features responsible for localization, such as topologically invariant nodes or drains and sources, were also described. Considering the diagonal disorder, we observed the disappearance of localization dependence on reciprocity around W∼20 for a random regular graph d=4. With a small diagonal disorder, the average biorthogonal fractal dimension drastically reduces. Around W∼5, localization scars occur within the spectrum, alternating as vertical bands of clustering of left and right localized states.

GRAPH FRACTALS WITH THE VARIABILITY OF THE FORMATION PROCESS

Fractals, which are characterized by their self-similarity at different scales, are complex geometric entities created using recursive algorithms. They are widely used in computer graphics to create complex visual effects and to model natural phenomena such as river networks and mountain landscapes. Graph fractals combine the properties of fractals and graph structures and can be used for research in fields such as computer networks or medicine. This work presents the approach of constructive-production modeling, based on formal grammars, for the generation of graph fractals with the variability of the formation process.

Hub‐aware random walk graph embedding methods for classification

AbstractIn the last two decades, we are witnessing a huge increase of valuable big data structured in the form of graphs or networks. To apply traditional machine learning and data analytic techniques to such data it is necessary to transform graphs into vector‐based representations that preserve the most essential structural properties of graphs. For this purpose, a large number of graph embedding methods have been proposed in the literature. Most of them produce general‐purpose embeddings suitable for a variety of applications such as node clustering, node classification, graph visualization and link prediction. In this article, we propose two novel graph embedding algorithms based on random walks that are specifically designed for the node classification problem. Random walk sampling strategies of the proposed algorithms have been designed to pay special attention to hubs–high‐degree nodes that have the most critical role for the overall connectedness in large‐scale graphs. The proposed methods are experimentally evaluated by analyzing the classification performance of three classification algorithms trained on embeddings of real‐world networks. The obtained results indicate that our methods considerably improve the predictive power of examined classifiers compared with currently the most popular random walk method for generating general‐purpose graph embeddings (node2vec).

Recent advances on mechanisms of network generation: Community, exchangeability, and scale‐free properties

AbstractThe mechanisms of network generation have undergone extensive analysis and found broad applications in various real‐world scenarios. Among the fruitful literature on network models, numerous studies seek to explore and interpret fundamental graph structure properties, including the clustering effect, exchangeability, and scale‐free properties. In this paper, we present a comprehensive review of the statistical modeling methods for the mechanisms of network generation. We specifically focus on three representative classes of models, namely the stochastic block models, the exchangeable network models, and the preferential attachment models. For each model type, our approach begins by reviewing existing methods and model setups, followed by an exploration of the core modeling principles behind them. We also summarize relevant statistical inference techniques and provide a unified understanding of theoretical analyses. Furthermore, we emphasize several challenges and open problems that could shed light on future research. We conclude this review with the identification of some possible directions for future study.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms Algorithms and Computational Methods > Networks and Security

On Lanzhou Index of Graphs

Let G = ( V , E ) be a simple graph with vertex set V ( G ) and edge set E ( G ) . The Lanzhou index of a graph G is defined by L z ( G ) = ∑ u ∈ V ( G ) d 2 u ¯¯¯ d u , where d u ( ¯¯¯ d u resp.) denotes the degree of the vertex u in G ( ¯¯¯¯ G , the complement graph of G resp.). It has predictive powers to provide insights of chemical relevant properties of chemical graph structures. In this paper we discuss some properties of Lanzhou index. Several inequalities having lower and upper bound for the Lanzhou index in terms of first, second and third Zagreb indices, radius of graph, eccentric connectivity index, Schultz index, inverse sum indeg index and symmetric division deg index, are discussed. At the end the Lanzhou index of corona and join of graphs have been derived.

Improving Augmentation Consistency for Graph Contrastive Learning

Graph contrastive learning (GCL) enhances unsupervised graph representation by generating different contrastive views, in which properties of augmented nodes are required to be aligned with their anchors. However, we find that in some existing GCL methods, it is hard to inherit semantic and structural properties of graphs from anchor views due to inconsistent augmentation schemes, which may hurt node consistency in augmented views. In this paper, we present ConGCL to improve node consistency and enhance node classification. Specifically, we first consider context entailment, which integrates the semantic and structural properties to better mine the underlying consistency relationships of nodes. Beneficial from this, we then design a novel consistency improvement loss to maintain augmentation consistency agreement among positive node pairs under stochastic augmentation schemes. To investigate the effectiveness of ConGCL on improving augmentation consistency and enhancing node classification, we conduct empirical study and extensive experiments on benchmark datasets. The code is available at: https://github.com/brysonwx/ConGCL.

Towards Adaptive Information Fusion in Graph Convolutional Networks

Graph Convolutional Networks (GCNs) have gained great popularity in tackling various analytic tasks on graph and network data. However, some recent studies raise concerns about whether GCNs can optimally integrate node features and topological structures in a complex graph. In this paper, we first present an experimental investigation. Surprisingly, our experimental results clearly show that the capability of the state-of-the-art GCNs in fusing node features and topological structures is distant from optimal or even satisfactory. The weakness may severely hinder the capability of GCNs in some classification tasks, since GCNs may not be able to adaptively learn some deep correlation information between topological structures and node features. Can we remedy the weakness and design a new type of GCNs that can retain the advantages of the state-of-the-art GCNs and, at the same time, enhance the capability of fusing topological structures and node features substantially? We tackle the challenge and propose an <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A</b> daptive <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</b> ulti-channel <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</b> raph <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</b> onvolutional <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</b> etwork for semi-supervised classification ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">AM-GCN</b> ). The central idea is that we extract the specific and common embeddings from node features, topological structures, and their combinations simultaneously, and use the attention mechanism to learn adaptive importance weights of the embeddings. However, considering that the input topology and feature structure in AM-GCN are still predefined and fixed, once the properties of graph structures are not consistent with tasks, the fusion performance of AM-GCN will be hindered from the beginning. Therefore, we need to adjust the structure and further propose the <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</b> abel <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</b> ropagation guided <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</b> ulti-channel <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</b> raph <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</b> onvolutional <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</b> etwork ( <bold xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LPM-GCN</b> ). LPM-GCN introduces edge weights learning on both topology and feature spaces to improve structural homophily, which can better promote the fusion process of graph convolutional networks. Our extensive experiments on benchmark data sets clearly show that our proposed models extract the most correlated information from both node features and topological structures substantially, and improves the classification accuracy with a clear margin.

Revealing Social Group Long-Term Survival for Smart Cities Based on Behavior Graph Structures Using Virtual Game

With the transformation of human life into virtual worlds based on the reality, current researches of smart cities and sustainability should integrate the factors of virtual social behaviors. Virtual games are an ideal domain and tools for exploring social science. Therefore, our work innovatively using Nova Empire as a case study to reveal the social group long-term survival via complexity analysis of behavior graph structural properties for building smart cities. Specifically, behavioral data of 101,424 players and 5,324 alliances from September 2021 to February 2022 are used. Our observations show that the phenomenon of “gap of wealth” in real society also exists in the virtual game. Meanwhile, the player online time is significantly positive related to alliance survival time, which is the foundation of our work. Then, correlation and regression analysis are performed to understand the significance of different structural properties on alliance survival time. Our original findings demonstrate that larger alliance, more sub-groups, balanced player distribution and frequent behavioral interactions will promote long-term survival of the alliance. Small sub-group and a relaxed social environment can improve player online time. Furthermore, we transfer the conclusions from virtual game to real society based on the mapping principle. Finally, new virtual tools and solutions for policymakers to improve smart cities and sustainable society ecosystem, and operation strategies for game designers to improve players retention rate are proposed.

Disparate Vulnerability in Link Inference Attacks against Graph Neural Networks

Graph Neural Networks (GNNs) have been widely used in various graph-based applications. Recent studies have shown that GNNs are vulnerable to link-level membership inference attacks (LMIA) which can infer whether a given link was included in the training graph of a GNN model. While most of the studies focus on the privacy vulnerability of the links in the entire graph, none have inspected the privacy risk of specific subgroups of links (e.g., links between LGBT users). In this paper, we present the first study of disparity in subgroup vulnerability (DSV) of GNNs against LMIA. First, with extensive empirical evaluation, we demonstrate the existence of non-negligible DSV under various settings of GNN models and input graphs. Second, by both statistical and causal analysis, we identify the difference between three specific graph structural properties of subgroups as one of the underlying reasons for DSV. Among the three properties, the difference between subgroup density has the largest causal effect on DSV. Third, inspired by the causal analysis, we design a new defense mechanism named FairDefense to mitigate DSV while providing protection against LMIA. At a high level, at each iteration of target model training, FairDefense randomizes the membership of edges in the training graph with a given probability, aiming to reduce the gap between the density of different subgroups for DSV mitigation. Our empirical results demonstrate that FairDefense outperforms the existing defense methods in the trade-off between defense and target model accuracy. More importantly, it offers better DSV mitigation.

GT-CHES: Graph transformation for classification in human evolutionary systems

While increasingly complex algorithms are being developed for graph classification in highly-structured domains, such as image processing and climate forecasting, they often lead to over-fitting and inefficiency when applied to human interaction networks where the confluence of cooperation, conflict, and evolutionary pressures produces chaotic environments. We propose a graph transformation approach for efficient classification in chaotic human systems that is based on game theoretic, network theoretic, and chaos theoretic principles. Graph structural properties are compiled into time-series that are then transposed into the frequency domain to offer a dynamic view of the system for classification. We propose a set of benchmark data sets and show through experiments that the approach is efficient and appropriate for many dynamic networks in which agents both compete and cooperate, such as social media networks, stock markets, political campaigns, legislation, and geopolitical events.

Some New Bounds for α-Adjacency Energy of Graphs

Let G be a graph with the adjacency matrix A(G), and let D(G) be the diagonal matrix of the degrees of G. Nikiforov first defined the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G), 0≤α≤1, which shed new light on A(G) and Q(G)=D(G)+A(G), and yielded some surprises. The α−adjacency energy EAα(G) of G is a new invariant that is calculated from the eigenvalues of Aα(G). In this work, by combining matrix theory and the graph structure properties, we provide some upper and lower bounds for EAα(G) in terms of graph parameters (the order n, the edge size m, etc.) and characterize the corresponding extremal graphs. In addition, we obtain some relations between EAα(G) and other energies such as the energy E(G). Some results can be applied to appropriately estimate the α-adjacency energy using some given graph parameters rather than by performing some tedious calculations.

Physio-chemical properties of HCP ( U ) and HEX ( U ) crystal structure lattice via eccentricity dependent chemical invariants

A crystal lattice is the three-dimensional structural arrangement of atoms or molecules inside a crystalline solid as points. Each crystal lattice consists of repeating unit cells in different directions. From earlier scientific research, it was recognised that the physio-chemical properties of any crystal structure lattice are highly reliant on the molecular structure of the network. This study led us to investigate complex structures through a mathematical perspective using a chemical invariant (topological index) by a chemical graph theory that correlates a chemical graph structure and physio-chemical properties, where atoms represent the nodes and a bond between atoms is considered as an edge. In this research, we have given explicit expressions for eccentricity-based chemical invariants, namely, eccentric connectivity (EC), total eccentricity (TE), eccentricity-based Zagreb (EZ) and connective eccentric (CE) for inorganic symmetrical Hexagonal Close-Packed and Hexagonal ( crystal structure lattices.

Improving gene regulatory network inference and assessment: The importance of using network structure.

Gene regulatory networks are graph models representing cellular transcription events. Networks are far from complete due to time and resource consumption for experimental validation and curation of the interactions. Previous assessments have shown the modest performance of the available network inference methods based on gene expression data. Here, we study several caveats on the inference of regulatory networks and methods assessment through the quality of the input data and gold standard, and the assessment approach with a focus on the global structure of the network. We used synthetic and biological data for the predictions and experimentally-validated biological networks as the gold standard (ground truth). Standard performance metrics and graph structural properties suggest that methods inferring co-expression networks should no longer be assessed equally with those inferring regulatory interactions. While methods inferring regulatory interactions perform better in global regulatory network inference than co-expression-based methods, the latter is better suited to infer function-specific regulons and co-regulation networks. When merging expression data, the size increase should outweigh the noise inclusion and graph structure should be considered when integrating the inferences. We conclude with guidelines to take advantage of inference methods and their assessment based on the applications and available expression datasets.

Exploring robust architectures for deep artificial neural networks

The architectures of deep artificial neural networks (DANNs) are routinely studied to improve their predictive performance. However, the relationship between the architecture of a DANN and its robustness to noise and adversarial attacks is less explored, especially in computer vision applications. Here we investigate the relationship between the robustness of DANNs in a vision task and their underlying graph architectures or structures. First we explored the design space of architectures of DANNs using graph-theoretic robustness measures and transformed the graphs to DANN architectures using various image classification tasks. Then we explored the relationship between the robustness of trained DANNs against noise and adversarial attacks and their underlying architectures. We show that robustness performance of DANNs can be quantified before training using graph structural properties such as topological entropy and Olivier-Ricci curvature, with the greatest reliability for complex tasks and large DANNs. Our results can also be applied for tasks other than computer vision such as natural language processing and recommender systems.

An automatic hypothesis generation for plausible linkage between xanthium and diabetes

There has been a significant increase in text mining implementation for biomedical literature in recent years. Previous studies introduced the implementation of text mining and literature-based discovery to generate hypotheses of potential candidates for drug development. By conducting a hypothesis-generation step and using evidence from published journal articles or proceedings, previous studies have managed to reduce experimental time and costs. First, we applied the closed discovery approach from Swanson’s ABC model to collect publications related to 36 Xanthium compounds or diabetes. Second, we extracted biomedical entities and relations using a knowledge extraction engine, the Public Knowledge Discovery Engine for Java or PKDE4J. Third, we built a knowledge graph using the obtained bio entities and relations and then generated paths with Xanthium compounds as source nodes and diabetes as the target node. Lastly, we employed graph embeddings to rank each path and evaluated the results based on domain experts’ opinions and literature. Among 36 Xanthium compounds, 35 had direct paths to five diabetes-related nodes. We ranked 2,740,314 paths in total between 35 Xanthium compounds and three diabetes-related phrases: type 1 diabetes, type 2 diabetes, and diabetes mellitus. Based on the top five percentile paths, we concluded that adenosine, choline, beta-sitosterol, rhamnose, and scopoletin were potential candidates for diabetes drug development using natural products. Our framework for hypothesis generation employs a closed discovery from Swanson’s ABC model that has proven very helpful in discovering biological linkages between bio entities. The PKDE4J tools we used to capture bio entities from our document collection could label entities into five categories: genes, compounds, phenotypes, biological processes, and molecular functions. Using the BioPREP model, we managed to interpret the semantic relatedness between two nodes and provided paths containing valuable hypotheses. Lastly, using a graph-embedding algorithm in our path-ranking analysis, we exploited the semantic relatedness while preserving the graph structure properties.

Faster Graph Coloring in Polynomial Space

We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time O(1.1389^n) for graphs with maximum degree 3 and in time O(1.2356^n) for general graphs, where n is the number of vertices in the input graph. Together with the inclusion-exclusion approach of Björklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time O(2.2356^n) as well as an exponential-space O(1.2330^n) time algorithm for counting independent sets. Our main algorithm counts independent sets in graphs with maximum degree at most 3 and no vertex with three neighbors of degree 3. This polynomial-space algorithm is designed and analyzed using the recently introduced Separate, Measure and Conquer approach [Gaspers & Sorkin, ICALP 2015]. Using Wahlström’s compound measure approach, this improvement in running time for small degree graphs is then bootstrapped to larger degrees, giving the improvement for general graphs. Combining both approaches leads to some inflexibility in choosing vertices to branch on for the small-degree cases, which we counter by structural graph properties.

Spectra of closeness Laplacian and closeness signless Laplacian of graphs

For a graph G with vertex set V(G) and u, v ∈ V(G), the distance between vertices u and v in G, denoted by dG(u,v), is the length of a shortest path connecting them and it is ∞ if there is no such a path, and the closeness of vertex u in G is cG(u) = ∑w∈V(G)2-dG(u,w). Given a graph G that is not necessarily connected, for u, v∈V(G), the closeness matrix of G is the matrix whose (u,v)-entry is equal to 2-dG(u,v) if u≠v and 0 otherwise, the closeness Laplacian is the matrix whose (u,v)-entry is equal to $$ \left\{\begin{array}{c}-{2}^{-{d}_G(u,v)}\hspace{1em}\mathrm{if}\enspace u\ne v,\enspace \\ \enspace {c}_G(u)\hspace{1em}\hspace{1em}\mathrm{otherwise}\hspace{0.5em}\end{array}\right.\hspace{0.5em} $$ and the closeness signless Laplacian is the matrix whose (u,v)-entry is equal to $$ \left\{\begin{array}{c}{2}^{-{d}_G(u,v)}\hspace{1em}\hspace{1em}\&amp;\mathrm{if}\enspace \mathrm{u}\ne \mathrm{v},\\ {c}_G(u)\hspace{1em}\hspace{1em}\mathrm{otherwise}.\end{array}\right. $$ We establish relations connecting the spectral properties of closeness Laplacian and closeness signless Laplacian and the structural properties of graphs. We give tight upper bounds for all nontrivial closeness Laplacian eigenvalues and characterize the extremal graphs, and determine all trees and unicyclic graphs that maximize the second smallest closeness Laplacian eigenvalue. Also, we give tight upper bounds for the closeness signless Laplacian eigenvalues and determine the trees whose largest closeness signless Laplacian eigenvalues achieve the first two largest values.

Encoding edge type information in graphlets.

Graph embedding approaches have been attracting increasing attention in recent years mainly due to their universal applicability. They convert network data into a vector space in which the graph structural information and properties are maximumly preserved. Most existing approaches, however, ignore the rich information about interactions between nodes, i.e., edge attribute or edge type. Moreover, the learned embeddings suffer from a lack of explainability, and cannot be used to study the effects of typed structures in edge-attributed networks. In this paper, we introduce a framework to embed edge type information in graphlets and generate a Typed-Edge Graphlets Degree Vector (TyE-GDV). Additionally, we extend two combinatorial approaches, i.e., the colored graphlets and heterogeneous graphlets approaches to edge-attributed networks. Through applying the proposed method to a case study of chronic pain patients, we find that not only the network structure of a patient could indicate his/her perceived pain grade, but also certain social ties, such as those with friends, colleagues, and healthcare professionals, are more crucial in understanding the impact of chronic pain. Further, we demonstrate that in a node classification task, the edge-type encoded graphlets approaches outperform the traditional graphlet degree vector approach by a significant margin, and that TyE-GDV could achieve a competitive performance of the combinatorial approaches while being far more efficient in space requirements.