Node Centrality Measures Research Articles

Representing Spatial Data with Graph Contrastive Learning

Large-scale geospatial data pave the way for geospatial machine learning algorithms, and a good representation is related to whether the machine learning model is effective. Hence, it is a critical task to learn effective feature representation for geospatial data. In this paper, we construct a spatial graph from the locations and propose a geospatial graph contrastive learning method to learn the location representations. Firstly, we propose a skeleton graph in order to preserve the primary structure of the geospatial graph to solve the positioning bias problem of remote sensing. Then, we define a novel mixed node centrality measure and propose four data augmentation methods based on the measure. Finally, we propose a heterogeneous graph attention network to aggregate information from both the structural neighborhood and semantic neighborhood separately. Extensive experiments on both geospatial datasets and non-geospatial datasets are conducted to illustrate that the proposed method outperforms state-of-the-art baselines.

Random Walks on Networks with Centrality-Based Stochastic Resetting

We introduce a refined way to diffusely explore complex networks with stochastic resetting where the resetting site is derived from node centrality measures. This approach differs from previous ones, since it not only allows the random walker with a certain probability to jump from the current node to a deliberately chosen resetting node, rather it enables the walker to jump to the node that can reach all other nodes faster. Following this strategy, we consider the resetting site to be the geometric center, the node that minimizes the average travel time to all the other nodes. Using the established Markov chain theory, we calculate the Global Mean First Passage Time (GMFPT) to determine the search performance of the random walk with resetting for different resetting node candidates individually. Furthermore, we compare which nodes are better resetting node sites by comparing the GMFPT for each node. We study this approach for different topologies of generic and real-life networks. We show that, for directed networks extracted for real-life relationships, this centrality focused resetting can improve the search to a greater extent than for the generated undirected networks. This resetting to the center advocated here can minimize the average travel time to all other nodes in real networks as well. We also present a relationship between the longest shortest path (the diameter), the average node degree and the GMFPT when the starting node is the center. We show that, for undirected scale-free networks, stochastic resetting is effective only for networks that are extremely sparse with tree-like structures as they have larger diameters and smaller average node degrees. For directed networks, the resetting is beneficial even for networks that have loops. The numerical results are confirmed by analytic solutions. Our study demonstrates that the proposed random walk approach with resetting based on centrality measures reduces the memoryless search time for targets in the examined network topologies.

New results and open problems on subgraph centrality

Subgraph centrality, introduced by Estrada and Rodr\'iguez-Vel\'azquez in [12], has become a widely used centrality measure in the analysis of networks, with applications in biology, neuroscience, economics and many other fields. It is also worthy of study from a strictly mathematical point of view, in view of its connections to topics in spectral graph theory, number theory, analytic matrix functions, and combinatorics. In this paper we present some new results and a list of open questions about subgraph centrality and other node centrality measures based on graph walks.

Flowthrough Centrality: A Stable Node Centrality Measure

This paper introduces flowthrough centrality, a node centrality measure determined from the hierarchical maximum concurrent flow problem (HMCFP). Based upon the extent to which a node is acting as a hub within a network, this centrality measure is defined to be the fraction of the flow passing through the node to the total flow capacity of the node. Flowthrough centrality is compared to the commonly-used centralities of closeness centrality, betweenness centrality, and flow betweenness centrality, as well as to stable betweenness centrality to measure the stability (i.e., accuracy) of the centralities when knowledge of the network topology is incomplete or in transition. Perturbations do not alter the flowthrough centrality values of nodes that are based upon flow as much as they do other types of centrality values that are based upon geodesics. The flowthrough centrality measure overcomes the problem of overstating or understating the roles that significant actors play in social networks. The flowthrough centrality is canonical in that it is determined from a natural, realized flow universally applicable to all networks.

An influential node identification method considering multi-attribute decision fusion and dependency

It is essential to study the robustness and centrality of interdependent networks for building reliable interdependent systems. Here, we consider a nonlinear load-capacity cascading failure model on interdependent networks, where the initial load distribution is not random, as usually assumed, but determined by the influence of each node in the interdependent network. The node influence is measured by an automated entropy-weighted multi-attribute algorithm that takes into account both different centrality measures of nodes and the interdependence of node pairs, then averaging for not only the node itself but also its nearest neighbors and next-nearest neighbors. The resilience of interdependent networks under such a more practical and accurate setting is thoroughly investigated for various network parameters, as well as how nodes from different layers are coupled and the corresponding coupling strength. The results thereby can help better monitoring interdependent systems.

Linking the network centrality measures closeness and degree

Measuring the importance of nodes in a network with a centrality measure is an core task in any network application. There many measures available and it is speculated that many encode similar information. We give an explicit non-linear relationship between two of the most popular measures of node centrality: degree and closeness. Based on a shortest-path tree approximation, we give an analytic derivation that shows the inverse of closeness is linearly dependent on the logarithm of degree. We show that our hypothesis works well for a range of networks produced from stochastic network models and for networks derived from 130 real-world data sets. We connect our results with previous results for other network distance scales such as average distance. Our results imply that measuring closeness is broadly redundant unless our relationship is used to remove the dependence on degree from closeness. The success of our relationship suggests that most networks can be approximated by shortest-path spanning trees which are all statistically similar two or more steps away from their root nodes.

Principal centrality measures: a comprehensive approach to the Spanish stocks market

In social network analysis, for determining the relevance or significance of a node in the network, several node centrality measures are often used such as degree centrality, betwenness centrality, closeness centrality, eigenvector, subgraph and page rank centrality. In this paper we apply a principal components analysis over the traditional centrality measures for obtaining an overall single metric that combines the best attributes of the traditional centrality measures and permits to detect relevant nodes in the network. Concretely, a detailed study of the Spanish stocks market will be used for demonstrating the advantages of this approach.

An Experimental Study on the Scalability of Recent Node Centrality Metrics in Sparse Complex Networks.

Node centrality measures are among the most commonly used analytical techniques for networks. They have long helped analysts to identify “important” nodes that hold power in a social context, where damages could have dire consequences for transportation applications, or who should be a focus for prevention in epidemiology. Given the ubiquity of network data, new measures have been proposed, occasionally motivated by emerging applications or by the ability to interpolate existing measures. Before analysts use these measures and interpret results, the fundamental question is: are these measures likely to complete within the time window allotted to the analysis? In this paper, we comprehensively examine how the time necessary to run 18 new measures (introduced from 2005 to 2020) scales as a function of the number of nodes in the network. Our focus is on giving analysts a simple and practical estimate for sparse networks. As the time consumption depends on the properties in the network, we nuance our analysis by considering whether the network is scale-free, small-world, or random. Our results identify that several metrics run in the order of O(nlogn) and could scale to large networks, whereas others can require O(n2) or O(n3) and may become prime targets in future works for approximation algorithms or distributed implementations.

Semi-Local Integration Measure of Node Importance

Numerous centrality measures have been introduced as tools to determine the importance of nodes in complex networks, reflecting various network properties, including connectivity, survivability, and robustness. In this paper, we introduce Semi-Local Integration (SLI), a node centrality measure for undirected and weighted graphs that takes into account the coherence of the locally connected subnetwork and evaluates the integration of nodes within their neighbourhood. We illustrate SLI node importance differentiation among nodes in lexical networks and demonstrate its potential in natural language processing (NLP). In the NLP task of sense identification and sense structure analysis, the SLI centrality measure evaluates node integration and provides the necessary local resolution by differentiating the importance of nodes to a greater extent than standard centrality measures. This provides the relevant topological information about different subnetworks based on relatively local information, revealing the more complex sense structure. In addition, we show how the SLI measure can improve the results of sentiment analysis. The SLI measure has the potential to be used in various types of complex networks in different research areas.

TOPOLOGICAL AND DYNAMICAL PROPERTIES OF THE NETWORK OF SHAREHOLDERS IN S&P 500 COMPANIES BASED ON GRAPH DATABASES

The interesting properties of scale-free and small-world networks recently observed have triggered the attention of the research community to the study of real growing complex networks. In scale-free networks, most vertices are sparsely connected, while a few vertices are intensively connected to many others, indicating a “preferential linking” during growing. In small-world networks, the average length of the shortest path between two randomly chosen nodes is small. In this paper, we study the topological and dynamical properties of the network of shareholders (NOS) in 11593 different companies. Based on Graph Databases, we calculate all the well-known in the literature topological and dynamical properties of a network along with centrality measures of nodes of NOS, which quantify the role that a node plays in the overall structure of NOS. We prove that NOS is both a scale-free and smallworld network. An understanding of NOS helps in predicting the emergence of important new phenomena affecting portfolio management in general. Also, this work reveals the fact that graph databases could serve as an efficient tool for analyzing such network models for stock markets. To the best of the authors’ knowledge, this is the first study calculating all the well-known in the literature topological and dynamical properties for Market Investments Networks, that is based on graph databases.

The robustness of the photosynthetic system I energy transfer complex network to targeted node attack and random node failure

Abstract In this article, we implement and compare 10 node removal (attack) strategies from the literature over the photosystem I (PSI) complex network of the common pea plant (Pisum sativum), representing the FRET energy transfer among its nodes/chromophores. We measure the network robustness (functioning) with four indicators. The node attack strategies and the network robustness indicators consider both the binary-topological and the weighted structure of the network. First, we find that the well-known node betweenness centrality attack, which has proven highly effective in dismantling most real-world networks’ topological connectivity, is ineffective over the PSI network. Second, the degeneracy of the node properties caused by the PSI’s higher network connectivity level induces a random-like node removal even when nodes are removed according to a specific node centrality measure. This phenomenon triggers a very low decrease of the PSI network functioning even when subjected to node attack. Such an outcome would indicate that the node attack strategies based on classic node properties, such as the degree or the betweenness centrality, may show low efficacy in dismantling real-world networks with very high connectivity levels. Last, the PSI network can be built by tuning a cut-off distance (CD) that defines the viable energy transfers among nodes/chromophores and progressively discards the lower energy transfer links among distant nodes/chromophores. This represents a ‘weight thresholding’ procedure allowing us to investigate the efficacy of the node attack strategies when links of lower weight are progressively pruned from the PSI network. We find that the best node attack strategies change by decreasing the CD, showing that the weight thresholding procedure affects the network response to node removal. This last outcome outlines the importance of investigating the stability of the system response for real-world weighted complex networks subjected to the weight thresholding procedure.

Efficient parallel algorithms for dynamic closeness‐ and betweenness centrality

AbstractFinding the centrality measures of nodes in a graph is a problem of fundamental importance due to various applications from social networks, biological networks, and transportation networks. Given the large size of such graphs, it is natural to use parallelism as a recourse. Several studies show how to compute the various centrality measures of nodes in a graph on parallel architectures, including multi‐core systems and GPUs. However, as these graphs evolve and change, it is pertinent to study how to update the centrality measures on changes to the underlying graph. In this article, we show novel parallel algorithms for updating the betweenness‐ and closeness‐centrality values of nodes in a dynamic graph. Our algorithms process a batch of updates in parallel by extending the approach of handling a single update for betweenness‐ and closeness‐centrality. For the latter, we also introduce techniques based on traversals of the block‐cut tree of a graph. Besides, our algorithms incorporate mechanisms to exploit the structural properties of graphs for enhanced performance. We implement our algorithms on two parallel architectures: an Intel 24‐core CPU and an Nvidia Tesla V100 GPU. To the best of our knowledge, we are the first to show GPU algorithms for the above two problems. In addition, we conduct detailed experiments to study the impact of various parameters associated with our algorithms and their implementation. Our results on a collection of real‐world graphs indicate that our algorithms achieve a significant speedup over corresponding state‐of‐the‐art algorithms.

Exploring the representational power of graph autoencoder

While representation learning has yielded a great success on many graph learning tasks, there is little understanding behind the structures that are being captured by these embeddings. For example, we wonder if the topological features, such as the Triangle Count, the Degree of the node and other centrality measures are concretely encoded in the embeddings. Furthermore, we ask if the presence of these structures in the embeddings is necessary for a better performance on the downstream tasks, such as clustering and classification. To address these questions, we conduct an extensive empirical study over three classes of unsupervised graph embedding models and seven different variants of Graph Autoencoders.Our results show that five topological features: the Degree, the Local Clustering Score, the Betweenness Centrality, the Eigenvector Centrality, and Triangle Count are concretely preserved in the first layer of the graph autoencoder that employs the SUM aggregation rule, under the condition that the model preserves the second-order proximity. We supplement further evidence for the presence of these features by revealing a hierarchy in the distribution of the topological features in the embeddings of the aforementioned model. We also show that a model with such properties can outperform other models on certain downstream tasks, especially when the preserved features are relevant to the task at hand. Finally, we evaluate the suitability of our findings through a test case study related to social influence prediction.

A Matrix-Variate t Model for Networks.

Networks represent a useful tool to describe relationships among financial firms and network analysis has been extensively used in recent years to study financial connectedness. An aspect, which is often neglected, is that network observations come with errors from different sources, such as estimation and measurement errors, thus a proper statistical treatment of the data is needed before network analysis can be performed. We show that node centrality measures can be heavily affected by random errors and propose a flexible model based on the matrix-variate t distribution and a Bayesian inference procedure to de-noise the data. We provide an application to a network among European financial institutions.

Discriminating Power of Centrality Measures in Complex Networks.

Centrality metrics are one of the most fundamental tools in social network analysis and network science, and various measures for evaluating node importance metrics have been devised. However, the crucial issue of testing the discriminating power of different centrality measures is still open. In this article, we propose to assess the discriminating power of node centrality measures by using the notion of automorphism and orbit: nodes in the same orbit have identical metric scores, while nodes in different orbits should have different centrality values. Under this assumption, we present a benchmark for the discriminating power of node centrality measures. Moreover, we propose an efficient approach to evaluate centrality measures in terms of the discriminating power, which is devoid of finding orbits. Extensive experiments on real and model networks are executed to compare seven commonly used node centrality metrics.

A measure of reliability for scientific co-authorship networks using fuzzy logic

Studies on the reliability of scientific co-authorship networks identify whether they are reliable, according to their researchers’ participation and how strong the co-authorship relationships are. Co-authorship among members of a research group can usually be represented by a graph in which each node represents one of the researchers belonging to this group, and each edge represents a connection (co-authorship relationship) between two researchers. The aim of this investigation is to propose a mathematical analysis using fuzzy logic to estimate the reliability of scientific co-authorship networks, based on node centrality measures and the existence of uncertainties in estimating the reliability of the individual components (researchers). To develop the proposed methodology, a research group from Sao Paulo State University–UNESP registered with the National Council for Scientific and Technological Development (CNPq) in Brazil was analysed. The results show the simplicity of implementation and the viability of mathematical modelling to estimate the reliability of scientific co-authorship networks.

Node Centrality Measures Identify Relevant Structural MRI Features of Subjects with Autism.

Autism spectrum disorders (ASDs) are a heterogeneous group of neurodevelopmental conditions characterized by impairments in social interaction and communication and restricted patterns of behavior, interests, and activities. Although the etiopathogenesis of idiopathic ASD has not been fully elucidated, compelling evidence suggests an interaction between genetic liability and environmental factors in producing early alterations of structural and functional brain development that are detectable by magnetic resonance imaging (MRI) at the group level. This work shows the results of a network-based approach to characterize not only variations in the values of the extracted features but also in their mutual relationships that might reflect underlying brain structural differences between autistic subjects and healthy controls. We applied a network-based analysis on sMRI data from the Autism Brain Imaging Data Exchange I (ABIDE-I) database, containing 419 features extracted with FreeSurfer software. Two networks were generated: one from subjects with autistic disorder (AUT) (DSM-IV-TR), and one from typically developing controls (TD), adopting a subsampling strategy to overcome class imbalance (235 AUT, 418 TD). We compared the distribution of several node centrality measures and observed significant inter-class differences in averaged centralities. Moreover, a single-node analysis allowed us to identify the most relevant features that distinguished the groups.

New models for multi-class networks

Many complex phenomena can be modeled by networks, that is, by a set of nodes connected by edges. Networks are represented by graphs, and several algebraic and analytical methods have been developed for their study. However, in order to obtain a more useful representation of a system, it is often appropriate to include more information about the nodes and/or edges, and those additions make it necessary to adapt or modify such methods of study.Multi-class networks, in which the set of nodes and/or the set of edges are partitioned in two or more classes, are useful when different nodes and edges can play fundamentally distinct roles in the system. In this article we introduce new models and methods for multi-class networks, based on how the adjacency matrix is formed.We apply this approach to obtain measures of node importance or centrality, in particular using the Perron eigenvector. Perturbation results shed light on how the relative importance of a node changes by the addition of a single edge, and experiments with both synthetic and real data sets illustrate features of the methods discussed.

Event-related network changes unfold the dynamics of cortical integration during face processing.

Face perception arises from a collective activation of brain regions in the occipital, parietal and temporal cortices. Despite the wide acknowledgment that these regions act in an intertwined network, the network behavior itself is poorly understood. Here we present a study in which time-varying connectivity estimated from EEG activity elicited by facial expressions presentation was characterized using graph-theoretical measures of node centrality and global network topology. Results revealed that face perception results from a dynamic reshaping of the network architecture, characterized by the emergence of hubs located in the occipital and temporal regions of the scalp. The importance of these nodes can be observed from the early stages of visual processing and reaches a climax in the same time-window in which the face-sensitive N170 is observed. Furthermore, using Granger causality, we found that the time-evolving centrality of these nodes is associated with ERP amplitude, providing a direct link between the network state and local neural response. Additionally, investigating global network topology by means of small-worldness and modularity, we found that face processing requires a functional network with a strong small-world organization that maximizes integration, at the cost of segregated subdivisions. Interestingly, we found that this architecture is not static, but instead, it is implemented by the network from stimulus onset to ~200 ms. Altogether, this study reveals the event-related changes underlying face processing at the network level, suggesting that a distributed processing mechanism operates through dynamically weighting the contribution of the cortical regions involved.

Comparative Analysis of Centrality Measures of Network Nodes based on Principal Component Analysis

. The analysis of networks of a diverse nature, which are citation networks, social networks or information and communication networks, includes the study of topological properties that allow one to assess the relationships between network nodes and evaluate various characteristics, such as the density and diameter of the network, related subgroups of nodes, etc. For this, the network is represented as a graph – a set of vertices and edges between them. One of the most important tasks of network analysis is to estimate the significance of a node (or in terms of graph theory – a vertex). For this, various measures of centrality have been developed, which make it possible to assess the degree of significance of the nodes of the network graph in the structure of the network under consideration.
 The existing variety of measures of centrality gives rise to the problem of choosing the one that most fully describes the significance and centrality of the node.
 The relevance of the work is due to the need to analyze the centrality measures to determine the significance of vertices, which is one of the main tasks of studying networks (graphs) in practical applications.
 The study made it possible, using the principal component method, to identify collinear measures of centrality, which can be further excluded both to reduce the computational complexity of calculations, which is especially important for networks that include a large number of nodes, and to increase the reliability of the interpretation of the results obtained when evaluating the significance node within the analyzed network in solving practical problems.
 In the course of the study, the patterns of representation of various measures of centrality in the space of principal components were revealed, which allow them to be classified in terms of the proximity of the images of network nodes formed in the space determined by the measures of centrality used.