In complex networks, each node has some unique characteristics that define the importance of the node based on the given application-specific context. These characteristics can be identified using various centrality metrics defined in the literature. Some of these centrality measures can be computed using local information of the node, such as degree centrality and semi-local centrality measure. Others use global information of the network like closeness centrality, betweenness centrality, eigenvector centrality, Katz centrality, PageRank, and so on. In this survey, we discuss these centrality measures and the state of the art literature that includes the extension of centrality measures to different types of networks, methods to update centrality values in dynamic networks, methods to identify top-k nodes, approximation algorithms, open research problems related to the domain, and so on. The paper is concluded with a discussion on application specific centrality measures that will help to choose a centrality measure based on the network type and application requirements.
Centrality metrics have been used in various networks, such as communication, social, biological, geographic, or contact networks. In particular, they have been used in order to study and analyze targeted attack behaviors and investigated their effect on network resilience. Although a rich volume of centrality metrics has been developed for decades, a limited set of centrality metrics have been commonly in use. This paper aims to introduce various existing centrality metrics and discuss their applicabilities and performance based on the results obtained from extensive simulation experiments to…
Centrality measures are crucial in quantifying the influence of the members of a social network. Although there has been a great deal of work dealing with this issue, the vast majority of classical centrality measures are agnostic of the community structure characterizing many social networks. Recent works have developed community-aware centrality measures that exploit features of the community structure information encountered in most real-world complex networks. In this paper, we investigate the interactions between 5 popular classical centrality measures and 5 community-aware centrality mea…
We perform the first axiomatic analysis of medial centrality measures. These measures, also called betweenness-like centralities, assess the role of a node in connecting others in the network. We focus on a setting with one target node and several source nodes. We consider three classic medial centrality measures adapted to this setting: Betweenness Centrality, Stress Centrality and Random Walk Betweenness Centrality. While Betweenness and Stress Centralities assume that the information in the network follows shortest paths, Random Walk Betweenness Centrality assumes it moves randomly along th…
The importance of a node in a social network is identified through a set of measures called centrality. Degree centrality, closeness centrality, betweenness centrality and clustering coefficient are the most frequently used metrics to compute node centrality. Their computational complexity in some cases makes unfeasible, when not practically impossible, their computations. For this reason we focused on two alternative measures, WERW-Kpath and Game of Thieves, which are at the same time highly descriptive and computationally affordable. Our experiments show that a strong correlation exists betw…
Centrality, in some sense, captures the extent to which a vertex controls the flow of information in a network. Here, we propose Local Detour Centrality as a novel centrality-based betweenness measure that captures the extent to which a vertex shortens paths between neighboring vertices as compared to alternative paths. After presenting our measure, we demonstrate empirically that it differs from other leading central measures, such as betweenness, degree, closeness, and the number of triangles. Through an empirical case study, we provide a possible interpretation for Local Detour Centrality a…
Identifying key nodes is crucial for accelerating or impeding dynamic spreading in a network. Community-aware centrality measures tackle this problem by exploiting the community structure of a network. Although there is a growing trend to design new community-aware centrality measures, there is no systematic investigation of the proposed measures' effectiveness. This study performs an extensive comparative evaluation of prominent community-aware centrality measures using the Susceptible-Infected-Recovered (SIR) model on real-world online social networks. Overall, results show that K-shell with…
Experts from several disciplines have been widely using centrality measures for analyzing large as well as complex networks. These measures rank nodes/edges in networks by quantifying a notion of the importance of nodes/edges. Ranking aids in identifying important and crucial actors in networks. In this chapter, we summarize some of the centrality measures that are extensively applied for mining social network data. We also discuss various directions of research related to these measures.
Hierarchy and centrality are two popular notions used to characterize the importance of entities in complex systems. Indeed, many complex systems exhibit a natural hierarchical structure, and centrality is a fundamental characteristic allowing to identify key constituents. Several measures based on various aspects of network topology have been proposed in order to quantify these concepts. While numerous studies have investigated whether centrality measures convey redundant information, how centrality and hierarchy measures are related is still an open issue. In this paper, we investigate the i…
Unlike classical centrality measures, recently developed community-aware centrality measures use a network's community structure to identify influential nodes in complex networks. This paper investigates their relationship on a set of fifty real-world networks originating from various domains. Results show that classical and community-aware centrality measures generally exhibit low to medium correlation values. These results are consistent across networks. Transitivity and efficiency are the most influential macroscopic network features driving the correlation variation between classical and c…
The roles of different nodes within a network are often understood through centrality analysis, which aims to quantify the capacity of a node to influence, or be influenced by, other nodes via its connection topology. Many different centrality measures have been proposed, but the degree to which they offer unique information, and such whether it is advantageous to use multiple centrality measures to define node roles, is unclear. Here we calculate correlations between 17 different centrality measures across 212 diverse real-world networks, examine how these correlations relate to variations in…
A Survey on Centrality and Importance Measures in Hypergraphs: Categorization and Empirical Insights
Kim Jaechul Graduate School of AI, KAIST（KAIST 金在澈人工智能研究生院）；School of Electrical Engineering, KAIST（KAIST 电气工程学院）；Intesa Sanpaolo（圣保罗银行）；Intesa Sanpaolo AI Research（圣保罗银行AI研究院）
The position value, introduced by Meessen (1988), is a solution concept for cooperative games in which the value assigned to a player depends on the value of the connections or links he has with other players. This concept has been studied by Borm et al. (1992) and characterised by Slikker (2005). In this paper, we analyse the position value from the point of view of the typical properties of a measure of centrality in a social network. We extend the analysis already developed in Gomez et al. (2003) for the Myerson centrality measure, where the symmetric effect on the centralities of the end n…
Targeting influential nodes in complex networks allows fastening or hindering rumors, epidemics, and electric blackouts. Since communities are prevalent in real-world networks, community-aware centrality measures exploit this information to target influential nodes. Researches show that they compare favorably with classical measures that are agnostic about the community structure. Although the diffusion process is of prime importance, previous studies consider mainly the famous Susceptible-Infected-Recovered (SIR) epidemic propagation model. This work investigates the consistency of previous a…
Social Network Analysis (SNA) is used to study the exchange of resources among individuals, groups, or organizations. The role of individuals or connections in a network is described by a set of centrality metrics which represent one of the most important results of SNA. Degree, closeness, betweenness and clustering coefficient are the most used centrality measures. Their use is, however, severely hampered by their computation cost. This issue can be overcome by an algorithm called Game of Thieves (GoT). Thanks to this new algorithm, we can compute the importance of all elements in a network (…
Graph centrality measures use the structure of a network to quantify central or "important" nodes, with applications in web search, social media analysis, and graphical data mining generally. Traditional centrality measures such as the well known PageRank interpret a directed edge as a vote in favor of the importance of the linked node. We study the case where nodes may belong to diverse communities or interests and investigate centrality measures that can identify nodes that are simultaneously important to many such diverse communities. We propose a family of diverse centrality measures forme…
Several centrality measures have been formulated to quantify the notion of 'importance' of actors in social networks. Current measures scrutinize either local or global connectivity of the nodes and have been found to be inadequate for social networks. Ignoring hierarchy and community structure, which are inherent in all human social networks, is the primary cause of this inadequacy. Positional hierarchy and embeddedness of an actor in the community are intuitively crucial determinants of his importance. The theory of social capital asserts that an actor's importance is derived from his positi…
This paper provides the first analysis of the differentially private computation of three centrality measures, namely eigenvector, Laplacian and closeness centralities, on arbitrary weighted graphs, using the smooth sensitivity approach. We do so by finding lower bounds on the amounts of noise that a randomised algorithm needs to add in order to make the output of each measure differentially private. Our results indicate that these computations are either infeasible, in the sense that there are large families of graphs for which smooth sensitivity is unbounded; or impractical, in the sense tha…
The discriminant power of centrality indices for the degree, eigenvector, closeness, betweenness and subgraph centrality is analyzed. It is defined by the number of graphs for which the standard deviation of the centrality of its nodes is zero. On the basis of empirical analysis it is concluded that the subgraph centrality displays better discriminant power than the rest of centralities. We also propose some new conjectures about the types of graphs for which the subgraph centrality does not discriminate among nonequivalent nodes.
In recent years, the problem of identifying the spreading ability and ranking social network users according to their influence has attracted a lot of attention; different approaches have been proposed for this purpose. Most of these approaches rely on the topological location of nodes and their neighbours in the graph to provide a measure that estimates the spreading ability of users. One of the most well-known measures is k-shell; additional measures have been proposed based on it. However, as the same k-shell index may be assigned to nodes with different degrees, this measure suffers from l…
Group centrality is an extension of the classical notion of centrality for individuals, to make it applicable to sets of them. We perform a SWOT (strengths, weaknesses, opportunities and threats) analysis of the use of group centrality in semantic networks, for different centrality notions: degree, closeness, betweenness, giving prominence to random walks. Among our main results stand out the relevance and NP-hardness of the problem of finding the most central set in a semantic network for an specific centrality measure.
Centrality measures are used in network science to evaluate the centrality of vertices or the position they occupy in a network. There are a large number of centrality measures according to some criterion. However, the generalizations of the most well-known centrality measures for weighted networks, degree centrality, closeness centrality, and betweenness centrality have solely assumed the edge weights to be constants. This paper proposes a methodology to generalize degree, closeness and betweenness centralities taking into account the variability of edge weights in the form of closed interval…
Centrality is an important notion in complex networks; it could be used to characterize how influential a node or an edge is in the network. It plays an important role in several other network analysis tools including community detection. Even though there are a small number of axiomatic frameworks associated with this notion, the existing formalizations are not generic in nature. In this paper we propose a generic axiomatic framework to capture all the intrinsic properties of a centrality measure (a.k.a. centrality index). We analyze popular centrality measures along with other novel measures…
Centrality measures quantify the importance of a node in a network based on different geometric or diffusive properties, and focus on different scales. Here, we adopt a geometrical viewpoint to define a multi-scale centrality in networks. Given a metric distance between the nodes, we measure the centrality of a node by its tendency to be close to geodesics between nodes in its neighborhood, via the concept of triangle inequality excess. Depending on the size of the neighborhood, the resulting Gromov centrality defines the importance of a node at different scales in the graph, and recovers as l…
This paper presents a formal definition of stability for node centrality measures in weighted graphs. It is shown that the commonly used measures of degree, closeness and eigenvector centrality are stable whereas betweenness centrality is not. An alternative definition of the latter that preserves the same centrality notion while satisfying the stability criteria is introduced. Continuity is presented as a less stringent alternative to stability. Betweenness centrality is shown to be not only unstable but discontinuous. Numerical experiments in synthetic random networks and real-world data sho…
Among the most fundamental tools for social network analysis are centrality measures, which quantify the importance of every node in the network. This centrality analysis typically disregards the possibility that the network may have been deliberately manipulated to mislead the analysis. To solve this problem, a recent study attempted to understand how a member of a social network could rewire the connections therein to avoid being identified as a leader of that network. However, the study was based on the assumption that the network analyzer - the seeker - is oblivious to any evasion attempts…
Mobile Social Networks (MSNs) have been evolving and enabling various fields in recent years. Recent advances in mobile edge computing, caching, and device-to-device communications, can have significant impacts on 5G systems. In those settings, identifying central users is crucial. It can provide important insights into designing and deploying diverse services and applications. However, it is challenging to evaluate the centrality of nodes in MSNs with dynamic environments. In this paper, we propose a Social-Relation based Centrality (SoReC) measure, in which social network information is used…
Identifying influential nodes in a network is a major issue due to the great deal of applications concerned, such as disease spreading and rumor dynamics. That is why, a plethora of centrality measures has emerged over the years in order to rank nodes according to their topological importance in the network. Local metrics such as degree centrality make use of a very limited information and are easy to compute. Global metrics such as betweenness centrality exploit the information of the whole network structure at the cost of a very high computational complexity. Recent works have shown that com…
We study network centrality based on dynamic influence propagation models in social networks. To illustrate our integrated mathematical-algorithmic approach for understanding the fundamental interplay between dynamic influence processes and static network structures, we focus on two basic centrality measures: (a) Single Node Influence (SNI) centrality, which measures each node's significance by its influence spread; and (b) Shapley Centrality, which uses the Shapley value of the influence spread function --- formulated based on a fundamental cooperative-game-theoretical concept --- to measure …
Today, there exist many centrality measures for assessing the importance of nodes in a network as a function of their position and the underlying topology. One class of such measures builds on eigenvector centrality, where the importance of a node is derived from the importance of its neighboring nodes. For directed and weighted complex networks, where the nodes can carry some intrinsic property value, there have been centrality measures proposed that are variants of eigenvector centrality. However, these expressions all suffer from shortcomings. Here, an extension of such centrality measures …
Network data is usually not error-free, and the absence of some nodes is a very common type of measurement error. Studies have shown that the reliability of centrality measures is severely affected by missing nodes. This paper investigates the reliability of centrality measures when missing nodes are likely to belong to the same community. We study the behavior of five commonly used centrality measures in uniform and scale-free networks in various error scenarios. We find that centrality measures are generally more reliable when missing nodes are likely to belong to the same community than in …
This work deals with the issue of assessing the influence of a node in the entire network and in the subnetwork to which it belongs as well, adapting the classical idea of vertex centrality. We provide a general definition of relative vertex centrality measure with respect to the classical one, referred to the whole network. Specifically, we give a decomposition of the relative centrality measure by including also the relative influence of the single node with respect to a given subgraph containing it. The proposed measure of relative centrality is tested in the empirical networks generated by…
Influence Maximization (IM) aims at finding the most influential users in a social network, i. e., users who maximize the spread of an opinion within a certain propagation model. Previous work investigated the correlation between influence spread and nodal centrality measures to bypass more expensive IM simulations. The results were promising but incomplete, since these studies investigated the performance (i. e., the ability to identify influential users) of centrality measures only in restricted settings, e. g., in undirected/unweighted networks and/or within a propagation model less common …
To measure node importance, network scientists employ centrality scores that typically take a microscopic or macroscopic perspective, relying on node features or global network structure. However, traditional centrality measures such as degree centrality, betweenness centrality, or PageRank neglect the community structure found in real-world networks. To study node importance based on network flows from a mesoscopic perspective, we analytically derive a community-aware information-theoretic centrality score based on network flow and the coding principles behind the map equation: map equation c…
Given a social network, which of its nodes are more central? This question has been asked many times in sociology, psychology and computer science, and a whole plethora of centrality measures (a.k.a. centrality indices, or rankings) were proposed to account for the importance of the nodes of a network. In this paper, we try to provide a mathematically sound survey of the most important classic centrality measures known from the literature and propose an axiomatic approach to establish whether they are actually doing what they have been designed for. Our axioms suggest some simple, basic proper…
The calculation of centrality measures is common practice in the study of networks, as they attempt to quantify the importance of individual vertices, edges, or other components. Different centralities attempt to measure importance in different ways. In this paper, we examine a conjecture posed by E. Estrada regarding the ability of several measures to distinguish the vertices of networks. Estrada conjectured that if all vertices of a graph have the same subgraph centrality, then all vertices must also have the same degree, eigenvector, closeness, and betweenness centralities. We provide a cou…
Most previous work of centralities focuses on metrics of vertex importance and methods for identifying powerful vertices, while related work for edges is much lesser, especially for weighted networks, due to the computational challenge. In this paper, we propose to use the well-known Kirchhoff index as the measure of edge centrality in weighted networks, called $θ$-Kirchhoff edge centrality. The Kirchhoff index of a network is defined as the sum of effective resistances over all vertex pairs. The centrality of an edge $e$ is reflected in the increase of Kirchhoff index of the network when the …
Node centrality is one of the integral measures in network analysis with wide range of applications from socio-economic to personalized recommendation. We argue that an effective centrality measure should undertake stability even under information loss or noise introduced in the network. With six local information based centrality metric, we investigate the effect of varying assortativity while keeping degree distribution unchanged, using networks with scale free and exponential degree distribution. This model provides a novel scope to analyze stability of centrality metric which can further f…
Given a directed network $ G $, we are interested in studying the qualitative features of $ G $ which govern how perturbations propagate across $ G $. Various classical centrality measures have been already developed and proven useful to capture qualitative features and behaviors for undirected networks. In this paper, we use topological data analysis (TDA) to adapt measures of centrality to capture both directedness and non-local propagating behaviors in networks. We introduce a new metric for computing centrality in directed weighted networks, namely the quasi-centrality measure. We compute …
We show that prominent centrality measures in network analysis are all based on additively separable and linear treatments of statistics that capture a node's position in the network. This enables us to provide a taxonomy of centrality measures that distills them to varying on two dimensions: (i) which information they make use of about nodes' positions, and (ii) how that information is weighted as a function of distance from the node in question. The three sorts of information about nodes' positions that are usually used -- which we refer to as "nodal statistics" -- are the paths from a given…
Identifying vital nodes in networks exhibiting a community structure is a fundamental issue. Indeed, community structure is one of the main properties of real-world networks. Recent works have shown that community-aware centrality measures compare favorably with classical measures agnostic about this ubiquitous property. Nonetheless, there is no clear consensus about how they relate and in which situation it is better to use a classical or a community-aware centrality measure. To this end, in this paper, we perform an extensive investigation to get a better understanding of the relationship be…
Classic measures of graph centrality capture distinct aspects of node importance, from the local (e.g., degree) to the global (e.g., closeness). Here we exploit the connection between diffusion and geometry to introduce a multiscale centrality measure. A node is defined to be central if it breaks the metricity of the diffusion as a consequence of the effective boundaries and inhomogeneities in the graph. Our measure is naturally multiscale, as it is computed relative to graph neighbourhoods within the varying time horizon of the diffusion. We find that the centrality of nodes can differ widely…
Influential nodes play a critical role in boosting or curbing spreading phenomena in complex networks. Numerous centrality measures have been proposed for identifying and ranking the nodes according to their importance. Classical centrality measures rely on various local or global properties of the nodes. They do not take into account the network community structure. Recently, a growing number of researches have shifted to community-aware centrality measures. Indeed, it is a ubiquitous feature in a vast majority of real-world networks. In the literature, the focus is on designing community-awa…
We extend the concept of eigenvector centrality to multiplex networks, and introduce several alternative parameters that quantify the importance of nodes in a multi-layered networked system, including the definition of vectorial-type centralities. In addition, we rigorously show that, under reasonable conditions, such centrality measures exist and are unique. Computer experiments and simulations demonstrate that the proposed measures provide substantially different results when applied to the same multiplex structure, and highlight the non-trivial relationships between the different measures o…
Graph matching is the process of computing the similarity between two graphs. Depending on the requirement, it can be exact or inexact. Exact graph matching requires a strict correspondence between nodes of two graphs, whereas inexact matching allows some flexibility or tolerance during the graph matching. In this chapter, we describe an approximate inexact graph matching by reducing the size of the graphs using different centrality measures. Experimental evaluation shows that it can reduce running time for inexact graph matching.
