Centrality Measures Based on Matrix Functions

ABSTRACTPurposeThis study examined how social network strength (i.e., the intensity and frequency of interactions within one's personal network, assessed using centrality measures such as degree, closeness, betweenness, and eigenvector centrality) affects mental health.MethodsThe study was conducted with 108 first‐year university students. Data were collected using a structured questionnaire, and NetMiner 4.0 was employed to analyze degree centrality, closeness centrality, betweenness centrality, and eigenvector centrality as indicators of social network centrality. Additionally, Quadratic Assignment Procedure (QAP) correlation analysis and QAP regression analysis were conducted to assess the relationship between social network strength and mental health.ResultsThe study found that stronger social networks were linked to better mental health in late adolescence. Specifically, higher social network centrality was associated with lower levels of hopelessness and depression. QAP correlation analysis revealed that hopelessness had a significant negative correlation with in‐closeness centrality (which reflects how easily one can be reached by others within the network) (r = −0.252, p < 0.05), while depression was negatively correlated with in‐degree centrality (the number of direct incoming connections from peers) (r = −0.233, p < 0.05), in‐closeness centrality (r = −0.256, p < 0.05), and eigenvector centrality (a measure of how well a person is connected to popular or influential members of the network) (r = −0.291, p < 0.01). QAP regression analysis further confirmed that weaker social ties, indicated by higher out‐closeness centrality (how easily one can reach others, especially those at a distance), were associated with higher depressive symptoms (adj. R2 = 0.143, F = 2.423, p < 0.05). These results suggest that building stronger and more integrated social networks may help reduce psychological distress and support mental well‐being in late adolescence.ConclusionStronger social networks in late adolescence are associated with better mental health, highlighting the importance of social connections. While correlational findings suggest general links between centrality and emotional well‐being, the regression results underscore specific predictors with practical implications. In‐degree, in‐closeness, and eigenvector centrality were associated with reduced depressive symptoms, indicating that being well‐integrated within a social network may offer protective benefits. In contrast, high out‐closeness centrality predicted increased depression, suggesting that the effort required to maintain outward‐reaching connections may impose psychological burdens. These findings suggest that targeted interventions should focus on strengthening inward social integration while addressing the stress related to maintaining extensive outward connections, providing direction for more effective youth mental health strategies.

Feedback centralities are one of the key classes of centrality measures. They assess the importance of a vertex recursively, based on the importance of its neighbours. Feedback centralities includes the Eigenvector Centrality, as well as its variants, such as the Katz Centrality or the PageRank, and are used in various AI applications, such as ranking the importance of websites on the Internet and most influential users in the Twitter social network. In this paper, we study the theoretical underpinning of the feedback centralities. Specifically, we propose a novel axiomatization of the Eigenvector Centrality and the Katz Centrality based on six simple requirements. Our approach highlights the similarities and differences between both centralities which may help in choosing the right centrality for a specific application.

Human Brain can be studied and analysed using neuroimages viz. MRI(Magnetic Resonance Imaging) / fMRI(Functional Magnetic Resonance Imaging) / PET(Positron Emission Tomography) / EEG(Electroencephalography) / MEG(Magnetoencephalography), which brings out the hidden information from it. Connectomes are graph like structure which represents complex brain network connectivity which are of two type's mainly structural connectivity and functional connectivity. Connectivity can be anatomical and functional properties of the brain. Nodes are voxels or Regions of Interest whereas edges are fibre bundles, temporal correlation between regions, in structural connectome and functional connectome respectively. This work focuses on functional connectivity analysis of brain network obtained through RS- fMRI images for identification of important regions in the human brain using image processing and graph theoretical approaches. The RS-fMRI images are obtained from 1000 Functional connectomes project and preprocessed using image processing techniques. Then the image is parcellated using AAL(Automated Anatomical Labeling) atlas and binary matrix is obtained. The graph is constructed from the derived matrix that exhibits functional connectivity between ROIs(Region of Interest). The various centrality measures (degree centrality, eigenvector centrality, betweenness centrality and closeness centrality) are used to identify the ROIs that act as provincial and/or connector hubs. The prominent provincial hubs are Rolandic Operculum, Thalamus, Insula, Hippocampus, Olfactory and connector hubs are Insula, Putamen, Occipital superior gyrus, Parietal Superior gyrus and Supramarginal gyrus. This work highlights the key regions in human brain which is involved in massive communication and information flow within the network.

Food addiction (FA) is a condition in which ultra-processed foods (UPFs) activate the brain's reward pathways, leading to binge eating, loss of control, and continued consumption despite negative consequences. It can appear early in childhood and is linked to behavioral, sociodemographic, and metabolic factors. This study assessed the contribution of FA, its structure, and connectivity in relation to sociodemographic, nutritional status, and metabolic variables in network analysis. A cross-sectional study was conducted with 93 children (7-11years old) living in Vitória de Santo Antão, Brazil. FA was assessed using the Yale Food Addiction Scale for Children, which was translated and validated for the Brazilian child population. Sociodemographic (age, sex, race, socioeconomic class), anthropometric (body weight, height, waist circumference, BMI, BMI-for-age, body fat percentage, lean mass, and fat mass), and metabolic (blood pressure, total cholesterol, triglycerides, HDL, LDL, and fasting glucose) factors were analyzed. For network analysis, the degree centrality (DC), closeness centrality (CC), betweenness centrality (BC), and eigenvector centrality (EC) were evaluated. FA exhibited moderate centrality in sociodemographic and metabolic networks, acting as a connector between key variables such as age and socioeconomic class (BC = 0.071-0.500; EC = 0.301-0.500; CC = 0.636-0.667). These metrics indicate that FA, while not dominant, maintains access to influential nodes and participates in relevant information pathways. In contrast, within the anthropometric network, FA showed a peripheral role, with fewer direct links (DC = 0.222-0.285) and limited intermediation (BC = 0.111). Variation in centrality across domains underscores the selective integration of FA, suggesting that its impact is context dependent.

Essential proteins are indispensable for cellular life. It is of great significance to identify essential proteins that can help us understand the minimal requirements for cellular life and is also very important for drug design. However, identification of essential proteins based on experimental approaches are always time-consuming and expensive. With the development of high-throughput technology in the post-genomic era, more and more protein-protein interaction data can be obtained, which make us study essential proteins from the network level become possible. There have been a series of computational approaches proposed for predicting essential proteins based on network topologies. Most of these topology based essential protein discovery methods were to use network centrality. In this paper, we investigate the essential proteins' topological characters from a completely new perspective. To our knowledge it is the first time that topology potential is used to identify essential proteins from protein-protein interaction network. The basic idea is that each protein in the network can be viewed as a material particle which creates a potential field around itself and the interaction of all proteins forms a topological field over the network. By defining and computing the value of each protein's topology potential, we can obtain a more precise ranking which reflects the importance of proteins from the protein-protein interaction network. The experiment results show that topology potential outperforms traditional topology measures: Degree Centrality (DC), Betweenness Centrality (BC), Closeness Centrality (CC), Subgraph Centrality(SC), Eigenvector Centrality(EC), Information Centrality(IC), and Sum of ECC (NC) for predicting essential proteins. In addition, these centrality measures are improved on their performance for identifying essential proteins in biological network when controlled by topology potential.

The Economic Effect of Interlocking Directorates in Italy: New Evidence Using Centrality Measures

We investigate the use of centrality measures to determine connected dominating sets (CDSs) for mobile ad hoc networks (MANETs) whose topology changes dynamically with time. A CDS is typically considered the graph theory equivalent for the backbone of a network and has been commonly used as the underlying topology for network-wide broadcasts with minimal retransmissions (accomplished by including fewer nodes in the CDS). In the case of MANETs, the degree centrality has been so far the most commonly used measure to determine a CDS with minimum node size (number of constituent nodes in the CDS). But, degree centrality-based CDS has been observed to be quite unstable in the presence of node mobility. In the rapidly emerging area of Network Science, centrality measures such as Eigenvector centrality, Betweenness centrality and Closeness centrality are used for complex network analysis. In this paper, we explore the use of these three centrality measures as the underlying criterion for inclusion of nodes in a CDS for MANETs and evaluate the lifetime and node size of such CDSs in comparison to that incurred for the degree centrality-based CDS and the maximum stable CDS determined using a benchmarking algorithm. We observe the Eigenvector centrality-based CDS to be the most stable (but the CDS node size is also the largest); the Betweenness centrality-based CDS is the least stable (but incurs the smallest CDS node size). The Betweenness centrality-based CDS incurs the lowest values for the CDS Node Size / CDS Lifetime tradeoff ratios when most of the nodes, if not all of them, are mobile.

Cerebral microinfarcts (CMIs) might cause measurable disruption to brain connections and are associated with cognitive decline, but the association between CMIs and motor impairment is still unclear. To assess the CMIs effect on motor function in vivo and explore the potential neuropathological mechanism based on graph-based network method. We identified 133 non-demented middle-aged and elderly participants who underwent MRI scanning, cognitive, and motor assessment. The short physical performance battery (SPPB) assessed motor function, including balance, walking speed, and chair stand. We grouped participants into 34 incident CMIs carriers and 99 non-CMIs carriers as controls, depending on diffusion-weighted imaging. Then we assessed the independent CMIs effects on motor function and explored neural mechanisms of CMIs on motor impairment via mapping of degree centrality (DC) and eigenvector centrality (EC). CMIs carriers had worse motor function than non-carriers. Linear regression analyses showed that CMIs independently contributed to motor function. CMIs carriers had decreased EC in the precuneus, while increased DC and EC in the middle temporal gyrus and increased DC in the inferior frontal gyrus compared to controls (p < 0.05, corrected). Correlation analyses showed that EC of precuneus was related to SPPB (r = 0.25) and balance (r = 0.27); however, DC (r = -0.25) and EC (r = -0.25) of middle temporal gyrus was related with SPPB in all participants (p < 0.05, corrected). CMIs represent an independent risk factor for motor dysfunction. The relationship between CMIs and motor function may be attributed to suppression of functional hub region and compensatory activation of motor-related regions.

Network theory concepts form the core of algorithms that are designed to uncover valuable insights from various datasets. Especially, network centrality measures such as Eigenvector centrality, Katz centrality, PageRank centrality etc., are used in retrieving top-K viral information propagators in social networks,while web page ranking in efficient information retrieval, etc. In this paper, we propose a novel method for identifying top-K viral information propagators from a reduced search space. Our algorithm computes the Katz centrality and Local average centrality values of each node and tests the values against two threshold (constraints) values. Only those nodes, which satisfy these constraints, form the search space for top-K propagators. Our proposed algorithm is tested against four datasets and the results show that the proposed algorithm is capable of reducing the number of nodes in search space at least by 70%. We also considered the parameter (alpha and beta) dependency of Katz centrality values in our experiments and established a relationship between the alpha values, number of nodes in search space and network characteristics. Later, we compare the top-K results of our approach against the top-K results of degree centrality.

As relational datasets modeled as graphs keep increasing in size and their data-acquisition is permeated by uncertainty, graph-based analysis techniques can become computationally and conceptually challenging. In particular, node centrality measures rely on the assumption that the graph is perfectly known -- a premise not necessarily fulfilled for large, uncertain networks. Accordingly, centrality measures may fail to faithfully extract the importance of nodes in the presence of uncertainty. To mitigate these problems, we suggest a statistical approach based on graphon theory: we introduce formal definitions of centrality measures for graphons and establish their connections to classical graph centrality measures. A key advantage of this approach is that centrality measures defined at the modeling level of graphons are inherently robust to stochastic variations of specific graph realizations. Using the theory of linear integral operators, we define degree, eigenvector, Katz and PageRank centrality functions for graphons and establish concentration inequalities demonstrating that graphon centrality functions arise naturally as limits of their counterparts defined on sequences of graphs of increasing size. The same concentration inequalities also provide high-probability bounds between the graphon centrality functions and the centrality measures on any sampled graph, thereby establishing a measure of uncertainty of the measured centrality score. The same concentration inequalities also provide high-probability bounds between the graphon centrality functions and the centrality measures on any sampled graph, thereby establishing a measure of uncertainty of the measured centrality score.

&lt;p&gt;The article presents a comparative analysis of basic centrality measures (degree centrality, closeness centrality, betweenness centrality, eigenvector centrality, PageRank, and Katz centrality) to identify influential agents in the network of regional marketing communities. The study is based on data from marketing communities in Arkhangelsk, collected through the API of the &amp;laquo;VKontakte&amp;raquo; platform. The analysis revealed that the network has a pronounced cluster structure and contains hub nodes that ensure its connectivity. Based on the centrality distributions, an assessment of the resilience of the marketing community network was conducted, identifying groups of key nodes that play a significant role in information dissemination.&lt;/p&gt;

This chapter provides an overview of social network analysis (SNA), a robust tool for studying the complex network of relationships inside social systems, encompassing individuals, groups, and organizations. The chapter provides insights into theoretical frameworks that help understand network structures. For example, the chapter examines social influence theory, which investigates the impact of network members on one other’s beliefs, attitudes, behaviors, and views. Social network analysis helps identify influential individuals in a network using metrics like degree centrality, betweenness centrality, and eigenvector centrality. Centrality measures are discussed in detail, as these are essential for understanding the network’s information flow and unity. Collectively, these principles and instruments provide a thorough comprehension of the dynamics and importance of social networks. The chapter also explains the rationale for network maps that depict persons, groups, or organizations as nodes, which are connected by edges or links.

This paper is concerned with distributed computation of several commonly used centrality measures in complex networks. In particular, we propose deterministic algorithms, which converge in finite time, for the distributed computation of the degree, closeness and betweenness centrality measures in directed graphs. Regarding eigenvector centrality, we consider the PageRank problem as its typical variant, and design distributed randomized algorithms to compute PageRank for both fixed and time-varying graphs. A key feature of the proposed algorithms is that they do not require to know the network size, which can be simultaneously estimated at every node, and that they are clock-free. To address the PageRank problem of time-varying graphs, we introduce the novel concept of persistent graph, which eliminates the effect of spamming nodes. Moreover, we prove that these algorithms converge almost surely and in the sense of $L^p$. Finally, the effectiveness of the proposed algorithms is illustrated via extensive simulations using a classical benchmark.

Despite the huge importance that the centrality metrics have in understanding the topology of a network, too little is known about the effects that small alterations in the topology of the input graph induce in the norm of the vector that stores the node centralities. If so, then it could be possible to avoid re-calculating the vector of centrality metrics if some minimal changes occur in the network topology, which would allow for significant computational savings. Hence, after formalising the notion of centrality, three of the most basic metrics were herein considered (i.e., Degree, Eigenvector, and Katz centrality). To perform the simulations, two probabilistic failure models were used to describe alterations in network topology: Uniform (i.e., all nodes can be independently deleted from the network with a fixed probability) and Best Connected (i.e., the probability a node is removed depends on its degree). Our analysis suggests that, in the case of degree, small variations in the topology of the input graph determine small variations in Degree centrality, independently of the topological features of the input graph; conversely, both Eigenvector and Katz centralities can be extremely sensitive to changes in the topology of the input graph. In other words, if the input graph has some specific features, even small changes in the topology of the input graph can have catastrophic effects on the Eigenvector or Katz centrality.

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.