General Random Graphs Research Articles

A graph residual generation network for node classification based on multi-information aggregation

The key to improving the performance of graph convolutional networks (GCN) is to fully explore the correlation between neighboring and distant information. Aiming at the over-smoothing problem of GCN, in order to make full use of the relationship among features, graphs and labels, a graph residual generation network based on multi-information aggregation (MIA-GRGN) is proposed. Firstly, aiming at the defects of GCN, we design a deep initial residual graph convolution network (DIRGCN), which connects the initial input through residuals, so that each layer node retains part of the information of the initial features, ensuring the localization of the graph structure and effectively alleviating the problem of over-smoothing. Secondly, we propose a random graph generation method (RGGM) by utilizing graph edge sampling and negative edge sampling, and optimize the supervision loss function of DIRGCN in the form of generation framework. Finally, applying RGGM and DIRGCN as inference modules for modeling hypotheses and obtaining approximate posterior distributions of unknown labels, an optimized loss function is obtained, we construct a multi-information aggregation MIA-GRGN that combines graph structure, node characteristics and label joint distribution. Experiments on benchmark graph classification datasets show that MIA-GRGN achieves better classification results compared with the benchmark models and mainstream models, especially for datasets with less dense edge relationships between nodes.

The Ordered Median Tree Location Problem

In this paper, we propose the Ordered Median Tree Location Problem (OMT). The OMT is a single-allocation facility location problem where p facilities must be placed on a network connected by a non-directed tree. The objective is to minimize the sum of the ordered weighted averaged allocation costs plus the sum of the costs of connecting the facilities in the tree. We present different MILP formulations for the OMT based on properties of the minimum spanning tree problem and the ordered median optimization. Given that ordered median location problems are rather difficult to solve we have improved the OMT solution performance by introducing covering variables in a valid reformulation plus developing two pre-processing phases to reduce the size of this formulations. In addition, we propose a Benders decomposition algorithm to approach the OMT. We establish an empirical comparison between these new formulations and we also provide enhancements that together with a proper formulation allow to solve medium size instances on general random graphs.

Structify-Net: Random Graph generation with controlled size and customized structure

Structify-Net: Random Graph generation with controlled size and customized structure

Quantum walk-based protocol for secure communication between any two directly connected nodes on a network

The utilization of quantum entanglement as a cryptographic resource has superseded conventional approaches to secure communication. Security and fidelity of intranetwork communication between quantum devices is the backbone of a quantum network. This work presents an protocol that generates entanglement between any two directly connected nodes of a quantum network to be used as a resource to enable quantum communication across that pair in the network. The protocol is based on a directed discrete-time quantum walk and paves the way for private inter-node quantum communication channels in the network. We also present the simulation results of this protocol on random networks generated from various models. We show that after implementation, the probability of the walker being at all nodes other than the source and target is negligible and this holds independent of the random graph generation model. This constitutes a viable method for the practical realisation of secure communication over any random network topology.

Entropy of the ensemble of acyclic graphs

The structural phase transition in complex network models is known to yield knowledge relevant for several problems of practical application. Examples include resilience of artificial networks, dynamics of epidemic spreading, among others. The model of random graphs is probably the simplest model of complex networks, and the solution of this model falls into the universality class mean field percolation. In this work, we concentrate on a similar problem, namely, the structural phase transition in the ensemble of random acyclic graphs. It should be noted that several approaches to the problem of random graphs, such as generating functions or the Molloy–Reed criterion, rely on the fact that before the critical point, cycles should not be present in random graphs. In this way, up to the critical point our solution should produce results that are equivalent to these other methods. Our approach takes advantage of the fact that acyclic graphs allow for an exact combinatorial enumeration of the whole ensemble, what leads to an exact expression for the entropy of this system. With this definition of entropy we can determine the onset of the critical transition as well as the critical exponents associated with the transition. Our results are illustrated with Monte-Carlo results and are discussed within the context of general random graphs, as well as in comparison with another model of acyclic graphs.

The Support of MISRA C++ Analyzer for Reliability of Embedded Systems

Cyber-Physical Systems (CPS) are increasingly used in many complex applications, such as autonomous delivery drones, the automotive CPS design, power grid control systems, and medical robotics. However, existing programming languages lack certain design patterns for CPS designs, including temporal semantics and concurrency models. Future research directions may involve programming language extensions to support CPS designs. On the other hand, JSF++, MISRA, and MISRA C++ are providing specifications intended to increase the reliability of safety-critical systems. This article also describes the development of rule checkers based on the MISRA C++ specification using the Clang open-source tool, which allows for the annotation of code and the easy extension of the MISRA C++ specification to other programming languages and systems. This is potentially useful for future CPS language research extensions to work with reliability software specifications using the Clang tool. Experiments were performed using key C++ benchmarks to validate our method in comparison with the well-known Coverity commercial tool. We illustrate key rules related to class, inheritance, template, overloading, and exception handling. Open-source benchmarks that violate the rules detected by our checkers are also illustrated. A random graph generator is further used to generate diamond case with multiple inheritance testdata for our software validations. The experimental results demonstrate that our method can provide information that is more detailed than that obtained using Coverity for nine open-source C++ benchmarks. Since the Clang tool is widely used, it will further allow developers to annotate their own extensions.

Optimization on large interconnected graphs and networks using adiabatic quantum computation

In this paper, we demonstrate that it is possible to create an adiabatic quantum computing algorithm that solves the shortest path between any two vertices on an undirected graph with at most [Formula: see text] qubits, where [Formula: see text] is the number of vertices of the graph. We do so without relying on any classical algorithms, aside from creating an ([Formula: see text]) adjacency matrix. The objective of this paper is to demonstrate the fact that it is possible to model large graphs on an adiabatic quantum computer using the maximum number of qubits available and random graph generators such as the Barabási–Albert and the Erdős–Rényi methods which can scale based on a power law.

Graph Generation and Diffusion using Random Walks

Simulations are often used in the study of metro network systems and the interactions of passengers with such systems in the real life. Graph theory is used to represent such metro systems. Simple random graphs are generated using a random graph generation algorithm revolving around random walks. The goal is to use such graphs to analyze the effects of the topologies of the graph parallel to the events which happen in real metro systems. This is done through random walks on the graphs by Monte Carlo Simulation of those random walkers. The simulations showed that the degree of a node in the graph has a near linear relationship with the number of times a specific node has been visited.

Random graph generator for leader and community detection in networks

AbstractIn complex network analyses, mainly in social networks, the detection of communities is an important source of information for revealing an internal organization of nodes. On the other hand, if the network reveals a leadership structure, it is possible to understand the mechanisms of information dissemination on it. The detection of leaders and communities is a big challenge depending on the complexity level of the network. In the literature, there are some metaheuristic algorithms for detecting leaders and communities based on pattern recognition on the graph associated with the network. In this paper, we developed a random graph system to generate a synthetic network instance with leader and community structures that define a ground truth. We compare this ground truth to determine the performance of the algorithms LCDA 1 and LCDA 2 for detecting leaders and communities. The results corroborate that the benchmarking system would help in selecting useful configurations for practical applications.

Lyapunov Optimization Based Mobile Edge Computing for Internet of Vehicles Systems

Mobile-Edge Computing (MEC) is an emerging paradigm in the Internet of Vehicles (IoV) to meet the ever-increasing computation demands of smart applications. To provide satisfactory computation performance, it is of significant importance to conduct computation offloading in IoV. In this paper, we investigate a multi-vehicle IoV system assisted by MECs with limited computation resources, where vehicles with complex applications can offload their subtasks to MEC servers. Applications are modeled as interdependent subtasks with general random task graphs, different from existing works with independent ones. To maximize the average logarithmic data processing rate (LDPR), the computation offloading problem is formulated as a time-average optimization with long-term constraints, which results from variable vehicle number, various applications and time-varying communication channels. To reduce the cooperation overhead, we propose a multi-agent Proximal Policy Optimization algorithm (Ly-MAPPO) which requires local observation only to solve the subproblems achieved by Lyapunov optimization technique in real time. In addition, to improve the performance of the Ly-MAPPO algorithm, Graph Convolutional Neural Network (GCN) is introduced to extract inter-dependencies between subtasks. Extensive simulations show that the GCN embedded Ly-MAPPO outperforms other baseline algorithms, e.g., greedy algorithm and gene algorithm, etc., for different traffic loads and computation resources in MEC servers.

Random processes with high variance produce scale free networks

Real-world networks tend to be scale free, having heavy-tailed degree distributions with more hubs than predicted by classical random graph generation methods. Preferential attachment and growth are the most commonly accepted mechanisms leading to these networks and are incorporated in the Barabási–Albert (BA) model (Barabási, 2009 [1]). We provide an alternative model using a randomly stopped linking process inspired by a generalized Central Limit Theorem (CLT) for geometric distributions with widely varying parameters. The common characteristic of both the BA model and our randomly stopped linking model is the mixture of widely varying geometric distributions, suggesting the critical characteristic of scale free networks is high variance, not growth or preferential attachment. The limitation of classical random graph models is low variance in parameters, while scale free networks are the natural, expected result of real-world variance.

Numerical Simulation of Mechanical Properties of Irregular Columnar Jointed Rock Mass

Columnar jointed rock masses (CJRM) are characterized by significant anisotropy. The mechanical properties of CJRM are complicated by the presence of columnar joints. In this paper, based on the Voronoi random graph generation algorithm, the irregular CJRM numerical model is established, in which columns are simulated by solid elements while joints are simulated by contact surface elements. Uniaxial compression tests on irregular CJRM are then conducted by using FLAC3D to investigate the deformation and strength characteristics, and failure modes. Results indicate that irregular CJRM exhibit strong anisotropy under uniaxial compression, and the compressive strength varies with joint angle in a typical U-shape curve. Failure modes can be classified as three categories: splitting failure along loading direction, shear-sliding failure along columnar joint, and combined failure of column fracturing and joint splitting. The numerical results are then compared with the experimental results. It demonstrates the effectiveness and reliability of applying numerical simulation to investigations on mechanical properties and failure mechanism of irregular CJRM. The results of this study provide helpful information for the design and construction of the project involving CJRM.

Generation of random chordal graphs using subtrees of a tree

Chordal graphs form one of the most studied graph classes. Several graph problems that are NP-hard in general become solvable in polynomial time on chordal graphs, whereas many others remain NP-hard. For a large group of problems among the latter, approximation algorithms, parameterized algorithms, and algorithms with moderately exponential or sub-exponential running time have been designed. Chordal graphs have also gained increasing interest during the recent years in the area of enumeration algorithms. Being able to test these algorithms on instances of chordal graphs is crucial for understanding the concepts of tractability of hard problems on graph classes. Unfortunately, only few published papers give algorithms for generating chordal graphs. Even in these papers, only very few methods aim for generating a large variety of chordal graphs. Surprisingly, none of these methods is directly based on the “intersection of subtrees of a tree” characterization of chordal graphs. In this paper, we give an algorithm for generating chordal graphs, based on the characterization that a graph is chordal if and only if it is the intersection graph of subtrees of a tree. Upon generating a random host tree, we give and test various methods that generate subtrees of the host tree. We compare our methods to one another and to existing ones for generating chordal graphs. Our experiments show that one of our methods is able to generate the largest variety of chordal graphs in terms of maximal clique sizes. Moreover, two of our subtree generation methods result in an overall complexity of our generation algorithm that is the best possible time complexity for a method generating the entire node set of subtrees in an “intersection of subtrees of a tree” representation. The instances corresponding to the results presented in this paper, and also a set of relatively small-sized instances are made available online.

The Random Plots Graph Generation Model for Studying Systems with Unknown Connection Structures.

We consider the problem of modeling complex systems where little or nothing is known about the structure of the connections between the elements. In particular, when such systems are to be modeled by graphs, it is unclear what vertex degree distributions these graphs should have. We propose that, instead of attempting to guess the appropriate degree distribution for a poorly understood system, one should model the system via a set of sample graphs whose degree distributions cover a representative range of possibilities and account for a variety of possible connection structures. To construct such a representative set of graphs, we propose a new random graph generator, Random Plots, in which we (1) generate a diversified set of vertex degree distributions and (2) target a graph generator at each of the constructed distributions, one-by-one, to obtain the ensemble of graphs. To assess the diversity of the resulting ensembles, we (1) substantialize the vague notion of diversity in a graph ensemble as the diversity of the numeral characteristics of the graphs within this ensemble and (2) compare such formalized diversity for the proposed model with that of three other common models (Erdős–Rényi–Gilbert (ERG), scale-free, and small-world). Computational experiments show that, in most cases, our approach produces more diverse sets of graphs compared with the three other models, including the entropy-maximizing ERG. The corresponding Python code is available at GitHub.

Assessing the impact of the density and sparsity of the network on community detection using a Gaussian mixture random partition graph generator.

Identification of sub-networks within a network is essential to understand the functionality of a network. This process is called as ’Community detection’. There are various existing community detection algorithms, and the performance of these algorithms can be varied based on the network structure. In this paper, we introduce a novel random graph generator using a mixture of Gaussian distributions. The community sizes of the generated network depend on the given Gaussian distributions. We then develop simulation studies to understand the impact of density and sparsity of the network on community detection. We use Infomap, Label propagation, Spinglass, and Louvain algorithms to detect communities. The similarity between true communities and detected communities is evaluated using Adjusted Rand Index, Adjusted Mutual Information, and Normalized Mutual Information similarity scores. We also develop a method to generate heatmaps to compare those similarity score values. The results indicate that the Louvain algorithm has the highest capacity to detect perfect communities while Label Propagation has the lowest capacity

The Impact of Edge Correlations in Random Networks

Random graphs are frequently used models of real-life random networks. The classical Erdös–Rényi random graph model is very well explored and has numerous nontrivial properties. In particular, a good number of important graph parameters that are hard to compute in the deterministic case often become much easier in random graphs. However, a fundamental restriction in the Erdös–Rényi random graph is that the edges are required to be probabilistically independent. This is a severe restriction, which does not hold in most real-life networks. We consider more general random graphs in which the edges may be dependent. Specifically, two models are analyzed. The first one is called a p-robust random graph. It is defined by the requirement that each edge exist with probability at least p, no matter how we condition on the presence/absence of other edges. It is significantly more general than assuming independent edges existing with probability p, as exemplified via several special cases. The second model considers the case when the edges are positively correlated, which means that the edge probability is at least p for each edge, no matter how we condition on the presence of other edges (but absence is not considered). We prove some interesting, nontrivial properties about both models.

Sublinear Random Access Generators for Preferential Attachment Graphs

We consider the problem of sampling from a distribution on graphs, specifically when the distribution is defined by an evolving graph model, and consider the time, space, and randomness complexities of such samplers. In the standard approach, the whole graph is chosen randomly according to the randomized evolving process, stored in full, and then queries on the sampled graph are answered by simply accessing the stored graph. This may require prohibitive amounts of time, space, and random bits, especially when only a small number of queries are actually issued. Instead, we propose a setting where one generates parts of the sampled graph on-the-fly, in response to queries, and therefore requires amounts of time, space, and random bits that are a function of the actual number of queries. Yet, the responses to the queries correspond to a graph sampled from the distribution in question. Within this framework, we focus on two random graph models: the Barabási-Albert Preferential Attachment model (BA-graphs) ( Science , 286 (5439):509–512) (for the special case of out-degree 1) and the random recursive tree model ( Theory of Probability and Mathematical Statistics , (51):1–28). We give on-the-fly generation algorithms for both models. With probability 1-1/poly( n ), each and every query is answered in polylog( n ) time, and the increase in space and the number of random bits consumed by any single query are both polylog( n ), where n denotes the number of vertices in the graph. Our work thus proposes a new approach for the access to huge graphs sampled from a given distribution, and our results show that, although the BA random graph model is defined by a sequential process, efficient random access to the graph’s nodes is possible. In addition to the conceptual contribution, efficient on-the-fly generation of random graphs can serve as a tool for the efficient simulation of sublinear algorithms over large BA-graphs, and the efficient estimation of their on such graphs.

Geometry of the minimal spanning tree of a random 3-regular graph

The global structure of the minimal spanning tree (MST) is expected to be universal for a large class of underlying random discrete structures. However, very little is known about the intrinsic geometry of MSTs of most standard models, and so far the scaling limit of the MST viewed as a metric measure space has only been identified in the case of the complete graph (Addario-Berry et al. in Ann Probab 45(5):3075–3144, 2017). In this work, we show that the MST constructed by assigning i.i.d. continuous edge weights to either the random (simple) 3-regular graph or the 3-regular configuration model on n vertices, endowed with the tree distance scaled by \(n^{-1/3}\) and the uniform probability measure on the vertices, converges in distribution with respect to Gromov–Hausdorff–Prokhorov topology to a random compact metric measure space. Further, this limiting space has the same law as the scaling limit of the MST of the complete graph identified in Addario-Berry et al. (2017) up to a scaling factor of \(6^{1/3}\). Our proof relies on a novel argument that proceeds via a comparison between a 3-regular configuration model and the largest component in the critical Erdős–Rényi random graph. The techniques of this paper can be used to establish the scaling limit of the MST in the setting of general random graphs with given degree sequences provided two additional technical conditions are verified.

Fast uniform generation of random graphs with given degree sequences

AbstractIn this paper we provide an algorithm that generates a graph with given degree sequence uniformly at random. Provided that , where is the maximal degree and m is the number of edges, the algorithm runs in expected time O(m). Our algorithm significantly improves the previously most efficient uniform sampler, which runs in expected time for the same family of degree sequences. Our method uses a novel ingredient which progressively relaxes restrictions on an object being generated uniformly at random, and we use this to give fast algorithms for uniform sampling of graphs with other degree sequences as well. Using the same method, we also obtain algorithms with expected run time which is (i) linear for power‐law degree sequences in cases where the previous best was O(n4.081), and (ii) O(nd + d4) for d‐regular graphs when , where the previous best was O(nd3).

The complexity of subtree intersection representation of chordal graphs and linear time chordal graph generation

It is known that any chordal graph on n vertices can be represented as the intersection of n subtrees in a tree on n nodes (Gavril in J Comb Theory 16:47–56, 1974). This characterization has been recently used to generate random chordal graphs on n vertices by generating n subtrees of a tree on n nodes. The space (and thus time) complexity of an algorithm generating n subtrees of a tree on n nodes is at least the sum of the sizes of the generated subtrees. The determination of this sum was left as an open question in Şeker et al. (Generation of random chordal graphs using subtrees of a tree. arXiv preprint arXiv:1810.13326, 2018). In this paper, we show that the sum of the sizes of n subtrees in a tree on n nodes is \(\varTheta (m\sqrt{n})\). We also show that we can confine ourselves to contraction-minimal subtree intersection representations because they are sufficient to generate every chordal graph with strictly positive probability. Moreover, the sum of the sizes of the subtrees in a contraction-minimal representation is at most \(2m+n\). We use this result to derive the first linear-time random chordal graph generator. Based on contraction-minimal representations, we also derive connectivity-related structural properties of chordal graphs. Besides these theoretical results, we also conduct experiments to study the quality of the chordal graphs generated by our algorithm and compare them to those generated by existing methods from the literature. Our algorithm does not generate chordal graphs uniformly at random, which is a quite challenging open question, irrespective of the time complexity of the generator. However, our experimental study suggests that the generated graphs have a fairly varied structure as indicated by the sizes of maximal cliques. Furthermore, our algorithm is simple to implement and produces graphs with 10,000 vertices and \(4 \times 10^7\) edges in less than one second on a laptop computer.