Spectral Radius as a Measure of Variation in Node Degree for Complex Network Graphs

We present correlation analysis between the centrality values observed for nodes (a computationally lightweight metric) and the maximal clique size (a computationally hard metric) that each node is part of in complex real-world network graphs. We consider the four common centrality metrics: degree centrality (DegC), eigenvector centrality (EVC), closeness centrality (ClC) and betweenness centrality (BWC). We define the maximal clique size for a node as the size of the largest clique (in terms of the number of constituent nodes) the node is part of. The real-world network graphs studied range from regular random network graphs to scale-free network graphs. We observe that the correlation between the centrality value and the maximal clique size for a node increases with increase in the spectral radius ratio for node degree, which is a measure of the variation of the node degree in the network. We observe the degree-based centrality metrics (DegC and EVC) to be relatively better correlated with the maximal clique size compared to the shortest path-based centrality metrics (ClC and BWC).

The construction of complex networks from electroencephalography (EEG) proves to be an effective method for representing emotion patterns in affection computing as it offers rich spatiotemporal EEG features associated with brain emotions. In this paper, we propose a novel method for constructing complex networks from EEG signals for emotion recognition, which begins with phase space reconstruction to obtain ordinal patterns and subsequently forms a graph network representation from the sequence of ordinal patterns based on the visibility graph method, named ComNet-PSR-VG. For the proposed ComNet-PSR-VG, the initial step involves mapping EEG signals into a series of ordinal partitions using phase space reconstruction, generating a sequence of ordinal patterns. These ordinal patterns are then quantified to form a symbolized new sequence. Subsequently, the resulting symbolized sequence of ordinal patterns is transformed into a graph network using the visibility graph method. Two types of network node measures, average node degree (AND) and node degree entropy (NDE), are extracted from the graph networks as the inputs of machine learning for EEG emotion recognition. To evaluate the effectiveness of the proposed construction method of complex networks based on the visibility graph of ordinal patterns, comparative experiments are conducted using two types of simulated signals (random and Lorenz signals). Subsequently, EEG emotion recognition is performed on the SEED EEG emotion dataset. The experimental results show that, with AND as the feature, our proposed method is 4.88% higher than the existing visibility graph method and 12.23% higher than the phase space reconstruction method. These findings indicate that our proposed novel method for constructing complex networks from EEG signals not only achieves effective emotional EEG pattern recognition but also exhibits the potential for extension to other EEG pattern learning tasks, suggesting broad adaptability and application potential for our method.

This paper investigates topology control and seeks to find a distributed solution with low interference, high performance. The paper proposes a distributed topology control algorithm called CBDTG by using the method of adjusting transmission power and the achievements of computational geometry, especially Delaunay triangulation (DT). Each node in the network invokes the neighbor discovery process by using an initial power and adds the discovered neighbors into its neighbor set. Each node finds DT neighbors in order to generate a triangulation with good properties such as planarity and regularity. The experimental result shows that the average node degree is between that of basic CBTC and the condensed CBTC with several edges removed, and the resulting topology will not suffer from high interference due to high node degree, nor will it suffer from low connectivity because of extremely low node degree. CBDTG also outperforms the basic CBTC in throughput.

PurposeThe paper examines the impact of COVID-19 on the network structure of the Ghana Stock Exchange (GSE) market.Design/methodology/approachThe study utilises the minimum spanning tree (MST) and network measures, including normalised tree length, average shortest path length, node degree, and node strength, to analyse the GSE structure across three periods: pre-COVID-19 (January–December 2019), COVID-19 (January–December 2020) and post-COVID-19 (January–December 2021).FindingsThe study reveals a contraction of the MST, reduced normalised tree length and average shortest path length during COVID-19 compared to pre-COVID-19. The GSE market network structure underwent significant changes during COVID-19, with the MST expanding and normalised tree length increasing post-COVID-19. Node degree and strength analyses identified ICT stock as the central node pre-COVID-19, while financial stock assumed this role during and after COVID-19. These findings underscore the profound impact of the COVID-19 pandemic on the structure and dynamics of the GSE market network.Research limitations/implicationsThe study used correlation to measure the linear relationships between stocks without accounting for nonlinear associations. It focused exclusively on the GSE market. Despite this limitation, the empirical findings and robustness tests reveal the significant impact of COVID-19 on the GSE market’s network structure. These results highlight the dynamic nature of the GSE market’s network topology.Practical implicationsThis paper highlights the dynamic nature of the GSE market network structure, providing valuable insights for investors and market participants to navigate similar crises in the future. Additionally, GSE regulators can leverage these findings to identify stocks that may pose risks during extreme market events.Originality/valueThis study is the first in Ghana to examine the effects of extreme events, such as COVID-19, on the network structure of the GSE market.

In this work, we systematically analyzed the effects of various nodal definitions, as determined by a multi-granularity whole-brain segmentation scheme, upon the topological architecture of the human brain functional network using the resting-state functional magnetic resonance imaging data of 19 healthy, young subjects. A number of functional networks were created with their nodes defined according to two types of anatomical definitions (Type I and Type II) each of which consists of five granularity levels of whole brain segmentations with each level linked through ontology-based, hierarchical, structural relationships. Topological properties were computed for each network and then compared across levels within the same segmentation type as well as between Type I and Type II. Certain network architecture patterns were observed in our study: (1) As the granularity changes, the absolute values of each node's nodal degree and nodal betweenness change accordingly but the relative values within a single network do not change considerably; (2) The average nodal degree is generally affected by the sparsity level of the network whereas the other topological properties are more specifically affected by the nodal definitions; (3) Within the same ontology relationship type, as the granularity decreases, the network becomes more efficient at information propagation; (4) The small-worldness that we observe is an intrinsic property of the brain's resting-state functional network, independent of the ontology type and the granularity level. Furthermore, we validated the aforementioned conclusions and measured the reproducibility of this multi-granularity network analysis pipeline using another dataset of 49 healthy young subjects that had been scanned twice.

Phase changes in the evolution of the IPv4 and IPv6 AS-Level Internet topologies

Overlay networks are widely used to deploy functionality at edge nodes without changing network routers. Each node in an overlay network maintains connections with a number of peers, forming a graph upon which a distributed application or service is implemented. In an “Eclipse” attack, a set of malicious, colluding overlay nodes arranges for a correct node to peer only with members of the coalition. If successful, the attacker can mediate most or all communication to and from the victim. Furthermore, by supplying biased neighbor information during normal overlay maintenance, a modest number of malicious nodes can eclipse a large number of correct victim nodes. This paper studies the impact of Eclipse attacks on structured overlays and shows the limitations of known defenses. We then present the design, implementation, and evaluation of a new defense, in which nodes anonymously audit each other’s connectivity. The key observation is that a node that mounts an Eclipse attack must have a higher than average node degree. We show that enforcing a node degree limit by auditing is an effective defense against Eclipse attacks. Furthermore, unlike most existing defenses, our defense leaves flexibility in the selection of neighboring nodes, thus permitting important overlay optimizations like proximity neighbor selection (PNS).

We describe a class of new algorithms to construct bipartite networks that preserves a prescribed degree and joint-degree (degree-degree) distribution of the nodes. Bipartite networks are graphs that can represent real-world interactions between two disjoint sets, such as actor-movie networks, author-article networks, co-occurrence networks, and heterosexual partnership networks. Often there is a strong correlation between the degree of a node and the degrees of the neighbors of that node that must be preserved when generating a network that reflects the structure of the underling system. Our bipartite 2K (B2K) algorithms generate an ensemble of networks that preserve prescribed degree sequences for the two disjoint set of nodes in the bipartite network, and the joint-degree distribution that is the distribution of the degrees of all neighbors of nodes with the same degree. We illustrate the effectiveness of the algorithms on a romance network using the NetworkX software environment to compare other properties of a target network that are not directly enforced by the B2K algorithms. We observe that when average degree of nodes is low, as is the case for romance and heterosexual partnership networks, then the B2K networks tend to preserve additional properties, such as the cluster coefficients, than algorithms that do not preserve the joint-degree distribution of the original network.

As a branch of topology control, power control is of great importance to prolong the survival time of the network. A transmission power control algorithm is proposed under the minimum connected average node degree constraint in this paper. Using a Poisson point process in two dimensions, we derive an analytical expression that determines the required transmission range to achieve a specified connectivity probability for a given node density, and then the minimum connected average node degree is obtained. Based on the above results, an algorithm named MCAND is designed to achieve the given connected probability by adjusting the transmission power of each node dynamically, with the node degree equal to the minimum connected average node degree. Simulation results show that the performance of our algorithm is superior to that of the related algorithms based on node degree in terms of coverage and connectivity.

The study of robustness and connectivity properties are important in the analysis of complex networks. This paper reports on an effort to compare different network topologies according to their algebraic connectivity, network criticality, average node degree, and average node betweenness. We consider different network types and study the behavior of these various metrics as scale is increased. Based on extensive simulations, we suggest some guidelines for the design and simplification of networks. The main finding is that, algebraic connectivity, network criticality, average degree, and average node betweenness capture different properties of a graph. Depending on the nature of the problem at hand, one needs to select which one is appropriate to use as the main metric for network analysis.

Understanding the correlation between the underlie network structure and overlay cascade effects in the interdependent networks is one of major challenges in complex network studies. There are some existing metrics that can be used to measure the cascades. However, different metrics such as average node degree interpret different characteristic of network topological structure, especially less metrics have been identified to effectively measure the cascading performance in the interdependent networks. In this paper, we propose to use a combined Laplacian matrix to model the interdependent networks and their interconnectivity, and then use its effective resistance metric as an indicator to its cascading behavior. Moreover, we have conducted extensive comparative studies among different metrics such as average node degree, and the proposed effective resistance. We have found that the effective resistance metric can describe more accurate and finer characteristics of topological structure of the interdependent networks than average node degree which is widely adapted by the existing research studies for measuring the cascading performance in the interdependent networks.

Algebraic connectivity metric for spare capacity allocation problem in survivable networks

Radio communications are the most important sources of energy consumption in Wireless Sensor Networks (WSN). Optimizing the sensors transmitting power with keeping the required nodes average degree preserves significantly the sensors energy. In this paper, we analyze the optimal transmitting power needed by each sensor to reach a given average node degree in heterogeneous WSN. The problem that we aim solving in this paper is the following: given a set of heterogeneous sensors having various capabilities: weak nodes and strong nodes that are scattered randomly in a field according to a Poisson Point Process with densities λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and λ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> respectively. What is the optimal transmitting power needed by each sensor to ensure k-average degree? We investigate also a related problem: What has the most important effect on weak nodes transmitting power: The quantity of the strong nodes or their quality in terms of energy supply? Our analytical study has been validated with Monte-Carlo simulations.

The focus of research in this paper is to investigate whether a random network whose degree sequence of the vertices is the same as the degree sequence of the vertices in a real-world network would exhibit values for other analysis metrics similar to those of the real-world network. We use the well-known Configuration Model to generate a random network on the basis of the degree sequence of the vertices in a real-world network wherein the degree sequence need not be Poisson-style. The extent of similarity between the vertices of the random network and real-world network with respect to a particular metric is evaluated in the form of the correlation coefficient of the values of the vertices for the metric. We involve a total of 24 real-world networks in this study, with the spectral radius ratio for node degree (measure of variation in node degree) ranging from 1.04 to 3.0 (i.e., from random networks to scale-free networks). We consider a suite of seven node-level metrics and three networklevel metrics for our analysis and identify the metrics for which the degree sequence would be just sufficient to generate random networks that have a very strong correlation (correlation coefficient of 0.8 or above) with that of the vertices in the corresponding real-world networks

This paper investigates the relationship between spectral graph theory and graph properties, specifically focusing on the spectral radius, which is the largest eigenvalue of the adjacency matrix of a graph. Our problem is finding sharp upper bounds for the spectral radius of bipartite graphs with given bounde vertex sets. We first review existing inequalities, noting and discuss their limitations, noting particularly that some, like Hong's inequality, are not always sharp for all graph types. Our primary contribution is an elementary and direct approach to solving an optimization problem: finding a bipartite graph with a bounded number of vertices that maximizes its spectral radius. We prove that for any bipartite graph with bounded vertex sets of \(n_1\) and \(n_2\), the spectral radius \(\rho(G)\) is bounded by \(\sqrt{n_1n_2}\). We demonstrate that this inequality is sharp, with equality holding exclusively for the complete bipartite graph \(K_{n_1,n_2}\).