Eigenvector centrality is an established measure of global connectivity, from which the importance and influence of nodes can be inferred. We introduce local eigenvector centrality (LEC), a scale-adaptive extension that integrates local and global connectivity into a single centrality measure. LEC identifies a topologically relevant mesoscopic scale for community-level centrality assessment, informed by prominent eigengaps, and combines information from the associated eigenvectors, via the Euclidean norm, to yield a centrality that reflects community-driven node influence. Networks with known community structures are employed to demonstrate that LEC closely reproduces eigenvector centrality for each community in isolation. Discrepancies between these measures are shown to be attributable to the global connectivity of affected nodes. LEC therefore provides a principled integration of community-level and global notions of node importance. Comparisons with PageRank and Katz centrality demonstrate that damping-based approaches do not recover community-scale centrality as effectively as LEC.
The computational demands of community detection algorithms such as Louvain and spectral optimization can be prohibitive for large networks. Eigenvector centrality and Katz centrality are two network statistics commonly used to describe the relative importance of nodes; and their calculation can be closely approximated on large networks by scalable iterative methods. In this paper, we present and leverage a surprising relationship between Katz centrality and eigenvector centrality to detect communities. Beyond the computational gains, we demonstrate that our approach identifies communities tha…
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…
The modern age of digital music access has increased the availability of data about music consumption and creation, facilitating the large-scale analysis of the complex networks that connect music together. Data about user streaming behaviour, and the musical collaboration networks are particularly important with new data-driven recommendation systems. Without thorough analysis, such collaboration graphs can lead to false or misleading conclusions. Here we present a new collaboration network of artists from the online music streaming service Spotify, and demonstrate a critical change in the ei…
Eigenvector centrality is one of the outstanding measures of central tendency in graph theory. In this paper we consider the problem of calculating eigenvector centrality of graph partitioned into components and how this partitioning can be used. Two cases are considered; first where the a single component in the graph has the dominant eigenvalue, secondly when there are at least two components that share the dominant eigenvalue for the graph. In the first case we implement and compare the method to the usual approach (power method) for calculating eigenvector centrality while in the second ca…
The purpose of the research is to find a centrality measure that can be used in place of PageRank and to find out the conditions where we can use it in place of PageRank. After analysis and comparison of graphs with a large number of nodes using Spearman's Rank Coefficient Correlation, the conclusion is evident that Eigenvector can be safely used in place of PageRank in directed networks to improve the performance in terms of the time complexity.
This paper characterizes the difficulty of estimating a network's eigenvector centrality only from data on the nodes, i.e., with no information about the topology of the network. We model this nodal data as graph signals generated by passing white noise through generic (not necessarily low-pass) graph filters. Leveraging the spectral properties of graph filters, we estimate the eigenvectors of the adjacency matrix of the underlying network. To this end, a simple selection algorithm is proposed, which chooses the correct eigenvector of the signal covariance matrix with minimal assumptions on th…
We consider the problem of estimating a network's eigenvector centrality only from data on the nodes, with no information about network topology. Leveraging the versatility of graph filters to model network processes, data supported on the nodes is modeled as a graph signal obtained via the output of a graph filter applied to white noise. We seek to simplify the downstream task of centrality ranking by bypassing network topology inference methods and, instead, inferring the centrality structure of the graph directly from the graph signals. To this end, we propose two simple algorithms for rank…
Eigenvector-based centrality measures are among the most popular centrality measures in network science. The underlying idea is intuitive and the mathematical description is extremely simple in the framework of standard, mono-layer networks. Moreover, several efficient computational tools are available for their computation. Moving up in dimensionality, several efforts have been made in the past to describe an eigenvector-based centrality measure that generalizes Bonacich index to the case of multiplex networks. In this work, we propose a new definition of eigenvector centrality that relies on…
There are different measures to classify a network's data set that, depending on the problem, have different success. For example, the resistance distance and eigenvector centrality measures have been successful in revealing ecological pathways and differentiating between biomedical images of patients with Alzheimer's disease, respectively. The resistance distance measures the effective distance between any two nodes of a network taking into account all possible shortest paths between them and the eigenvector centrality measures the relative importance of each node in the network. However, bot…
Numerous centrality measures have been developed to quantify the importances of nodes in time-independent networks, and many of them can be expressed as the leading eigenvector of some matrix. With the increasing availability of network data that changes in time, it is important to extend such eigenvector-based centrality measures to time-dependent networks. In this paper, we introduce a principled generalization of network centrality measures that is valid for any eigenvector-based centrality. We consider a temporal network with N nodes as a sequence of T layers that describe the network duri…
We develop a new sampling method to estimate eigenvector centrality on incomplete networks. Our goal is to estimate this global centrality measure having at disposal a limited amount of data. This is the case in many real-world scenarios where data collection is expensive, the network is too big for data storage capacity or only partial information is available. The sampling algorithm is theoretically grounded by results derived from spectral approximation theory. We studied the problem on both synthetic and real data and tested the performance comparing with traditional methods, such as rando…
A two-steps tensor eigenvector centrality for nodes and hyperedges in hypergraphs
School of Mathematical Sciences, Harbin Engineering University（哈尔滨工程大学数学科学学院）；Academy for Advanced Interdisciplinary Studies, Northeast Normal University（东北师范大学前沿交叉科学研究院）
Eigenvector centrality is a common measure of the importance of nodes in a network. Here we show that under common conditions the eigenvector centrality displays a localization transition that causes most of the weight of the centrality to concentrate on a small number of nodes in the network. In this regime the measure is no longer useful for distinguishing among the remaining nodes and its efficacy as a network metric is impaired. As a remedy, we propose an alternative centrality measure based on the nonbacktracking matrix, which gives results closely similar to the standard eigenvector cent…
We study the lobby index (l-index for short) as a local node centrality measure for complex networks. The l-inde is compared with degree (a local measure), betweenness and Eigenvector centralities (two global measures) in the case of biological network (Yeast interaction protein-protein network) and a linguistic network (Moby Thesaurus II). In both networks, the l-index has poor correlation with betweenness but correlates with degree and Eigenvector. Being a local measure, one can take advantage by using the l-index because it carries more information about its neighbors when compared with deg…
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 scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency matrix. Recent work indicates that there are further benefits from accounting directly for higher order interactions, notably through a hypergraph representation where an edge may involve multiple nodes. Building on these ideas, we motivate, define and analyze a class of spectral centrality measures for identifying important nodes and hyperedges in hypergraphs, g…
Generalizing Perron--Frobenius theory and eigenvector-based centralities to networks with complex edge weights
Center for Systems Biology Dresden（德累斯顿系统生物学中心）；Max Planck Institute for the Physics of Complex Systems（马克斯·普朗克复杂系统物理研究所）；Max-Planck Institute of Molecular Cell Biology and Genetics（马克斯·普朗克分子细胞生物学与遗传学研究所）；TU Dresden（德累斯顿工业大学）；University of California, Los Angeles（加州大学洛杉矶分校）；Sante Fe Institute（圣塔菲研究所）；Frankfurt School of Finance and Management（法兰克福金融管理学院）；University of Florida（佛罗里达大学）
The existing methods to calculate the Eigenvector Centrality(EC) tend to not be robust enough for determination of EC in low time complexity or not well-scalable for large networks, hence rendering them practically unreliable/ computationally expensive. So, it is of the essence to develop a method that is scalable in low computational time. Hence, we propose a deep learning model for the identification of nodes with high Eigenvector Centrality. There have been a few previous works in identifying the high ranked nodes with supervised learning methods, but in real-world cases, the graphs are not…
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…
Centrality is widely recognized as one of the most critical measures to provide insight in the structure and function of complex networks. While various centrality measures have been proposed for single-layer networks, a general framework for studying centrality in multilayer networks (i.e., multicentrality) is still lacking. In this study, a tensor-based framework is introduced to study eigenvector multicentrality, which enables the quantification of the impact of interlayer influence on multicentrality, providing a systematic way to describe how multicentrality propagates across different la…
Graph Isomorphism is one of the classical problems of graph theory for which no deterministic polynomial-time algorithm is currently known, but has been neither proven to be NP-complete. Several heuristic algorithms have been proposed to determine whether or not two graphs are isomorphic (i.e., structurally the same). In this research, we propose to use the sequence (either the non-decreasing or nonincreasing order) of eigenvector centrality (EVC) values of the vertices of two graphs as a precursor step to decide whether or not to further conduct tests for graph isomorphism. The eigenvector ce…
The graph invariant examined in this paper is the largest eigenvalue of the adjacency matrix of a graph. Previous work demonstrates the tight relationship between this invariant, the birth and death rate of a contagion spreading on the graph, and the trajectory of the contagion over time. We begin by conducting a simulation confirming this and explore bounds on the birth and death rate in terms of well-known graph invariants. As a result, the change in the largest eigenvalue resulting from removal of a vertex in the network is the best measure of effectiveness of interventions that slow the sp…
Edge centrality measures are functions that evaluate the importance of edges in a network. They can be used to assess the role of a backlink for the popularity of a website as well as the importance of a flight in virus spreading. Various node centralities have been translated to apply for edges, including Edge Betweenness, Eigenedge (edge version of Eigenvector centrality) and Edge PageRank. With this paper, we initiate the discussion on the axiomatic properties of edge centrality measures. We do it by proposing an axiomatic characterization of Edge PageRank. Our characterization is the first…
Finding the source of a disturbance or fault in complex systems such as industrial chemical processing plants can be a difficult task and consume a significant number of engineering hours. In many cases, a systematic elimination procedure is considered to be the only feasible approach but can cause undesired process upsets. Practitioners desire robust alternative approaches. This paper presents an unsupervised, data-driven method for ranking process elements according to the magnitude and novelty of their influence. Partial bivariate transfer entropy estimation is used to infer a weighted dire…
We study the blind centrality ranking problem, where our goal is to infer the eigenvector centrality ranking of nodes solely from nodal observations, i.e., without information about the topology of the network. We formalize these nodal observations as graph signals and model them as the outputs of a network process on the underlying (unobserved) network. A simple spectral algorithm is proposed to estimate the leading eigenvector of the associated adjacency matrix, thus serving as a proxy for the centrality ranking. A finite rate performance analysis of the algorithm is provided, where we find …
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…
Eigenvector centrality is a standard network analysis tool for determining the importance of (or ranking of) entities in a connected system that is represented by a graph. However, many complex systems and datasets have natural multi-way interactions that are more faithfully modeled by a hypergraph. Here we extend the notion of graph eigenvector centrality to uniform hypergraphs. Traditional graph eigenvector centralities are given by a positive eigenvector of the adjacency matrix, which is guaranteed to exist by the Perron-Frobenius theorem under some mild conditions. The natural representati…
Several expressions for the $j$-th component $\left( x_{k}\right)_{j}$ of the $k$-th eigenvector $x_{k}$ of a symmetric matrix $A$ belonging to eigenvalue $λ_{k}$ and normalized as $x_{k}^{T}x_{k}=1$ are presented. In particular, the expression \[ \left( x_{k}\right)_{j}^{2}=-\frac{1}{c_{A}^{\prime}\left( λ_{k}\right) }\det\left( A_{\backslash\left\{ j\right\} }-λ_{k}I\right) \] where $c_{A}\left( λ\right) =\det\left( A-λI\right) $ is the characteristic polynomial of $A$, $c_{A}^{\prime}\left( λ\right) =\frac{dc_{A}\left( λ\right) }{dλ}$ and $A_{\backslash\left\{ j\right\} }$ is obtained from …
We perform an extensive analysis of how sampling impacts the estimate of several relevant network measures. In particular, we focus on how a sampling strategy optimized to recover a particular spectral centrality measure impacts other topological quantities. Our goal is on one hand to extend the analysis of the behavior of TCEC [Ruggeri2019], a theoretically-grounded sampling method for eigenvector centrality estimation. On the other hand, to demonstrate more broadly how sampling can impact the estimation of relevant network properties like centrality measures different than the one aimed at o…
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.
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…
With great theoretical and practical significance, locating influential nodes of complex networks is a promising issues. In this paper, we propose a dynamics-sensitive (DS) centrality that integrates topological features and dynamical properties. The DS centrality can be directly applied in locating influential spreaders. According to the empirical results on four real networks for both susceptible-infected-recovered (SIR) and susceptible-infected (SI) spreading models, the DS centrality is much more accurate than degree, $k$-shell index and eigenvector centrality.
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…
In this paper analogies between different (dis)similarity matrices are derived. These matrices, which are connected to path enumeration and random walks, are used in community detection methods or in computation of centrality measures for complex networks. The focus is on a number of known centrality measures, which inherit the connections established for similarity matrices. These measures are based on the principal eigenvector of the adjacency matrix, path enumeration, as well as on the stationary state, stochastic matrix or mean first-passage times of a random walk. Particular attention is …
Epidemic modeling in complex networks has become one of the latest topics in recent times. The Susceptible-Infectious-Recovered (SIR) model and its variants are often used for epidemic modeling. One important issue in epidemic modeling is the determination of the spreading ability of the nodes in the network. Thus, for example, can be detected in the early stages. In this study, we developed a centrality measure called Lexical Sorting Centrality (LSC) that distinguishes the spreading ability of the nodes. Using other centrality measures calculated for the nodes, LSC sorts the nodes in a way si…
The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding of nodes centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution $P(q) \sim q^{-γ}$, localization occurs on the largest hub if $γ>5/2$; for $γ<5/2$ a new type of localization arises on a meso…
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…
Centrality measures are used in network science to identify the most important vertices for transmission of information and dynamics on a graph. One of these measures, introduced by Estrada and collaborators, is the $β$-subgraph centrality, which is based on the exponential of the matrix $βA$, where $A$ is the adjacency matrix of the graph and $β$ is a real parameter ("inverse temperature"). We prove that for algebraic $β$, two vertices with equal $β$-subgraph centrality are necessarily cospectral. We further show that two such vertices must have the same degree and eigenvector centralities. O…
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…
We use the susceptible-infected-recovered (SIR) model for disease spread over a network, and empirically study how well various centrality measures perform at identifying which nodes in a network will be the best spreaders of disease on 10 real-world networks. We find that the relative performance of degree, shell number and other centrality measures can be sensitive to B, the probability that an infected node will transmit the disease to a susceptible node. We also find that eigenvector centrality performs very well in general for values of B above the epidemic threshold.
Typing Yesterday into the search-bar of your browser provides a long list of websites with, in top places, a link to a video by The Beatles. The order your browser shows its search results is a notable example of the use of network centrality. Centrality is a measure of the importance of the nodes in a network and it plays a crucial role in a huge number of fields, ranging from sociology to engineering, and from biology to economics. Many metrics are available to evaluate centrality. However, centrality measures are generally based on ad hoc assumptions, and there is no commonly accepted way t…
Networks are versatile representations of the interactions between entities in complex systems. Cycles on such networks represent feedback processes which play a central role in system dynamics. In this work, we introduce a measure of the importance of any individual cycle, as the fraction of the total information flow of the network passing through the cycle. This measure is computationally cheap, numerically well-conditioned, induces a centrality measure on arbitrary subgraphs and reduces to the eigenvector centrality on vertices. We demonstrate that this measure accurately reflects the impa…
Measures of complex network analysis, such as vertex centrality, have the potential to unveil existing network patterns and behaviors. They contribute to the understanding of networks and their components by analyzing their structural properties, which makes them useful in several computer science domains and applications. Unfortunately, there is a large number of distinct centrality measures and little is known about their common characteristics in practice. By means of an empirical analysis, we aim at a clear understanding of the main centrality measures available, unveiling their similariti…
We develop a complete theory for the combinatorics of walk-counting on a directed graph in the case where each backtracking step is downweighted by a given factor. By deriving expressions for the associated generating functions, we also obtain linear systems for computing centrality measures in this setting. In particular, we show that backtrack-downweighted Katz-style network centrality can be computed at the same cost as standard Katz. Studying the limit of this centrality measure at its radius of convergence also leads to a new expression for backtrack-downweighted eigenvector centrality th…
Information flow, opinion, and epidemics spread over structured networks. When using individual node centrality indicators to predict which nodes will be among the top influencers or spreaders in a large network, no single centrality has consistently good ranking power. We show that statistical classifiers using two or more centralities as input are instead consistently predictive over many diverse, static real-world topologies. Certain pairs of centralities cooperate particularly well in statistically drawing the boundary between the top spreaders and the rest: local centralities measuring th…
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…
Social networks are discrete systems with a large amount of heterogeneity among nodes (individuals). Measures of centrality aim at a quantification of nodes' importance for structure and function. Here we ask to which extent the most central nodes can be found by purely local search. We find that many networks have close-to-optimal searchability under eigenvector centrality, outperforming searches for degree and betweenness. Searchability of the strongest spreaders in epidemic dynamics tends to be substantially larger for supercritical than for subcritical spreading.
We perform social network analysis on 53 students split over three semesters and 13 groups, using conventional measures like eigenvector centrality, betweeness centrality, and degree centrality, as well as defining a variant of the Abelian Sandpile Model (ASM) with the intention of modeling stress propagation in the college classroom. We correlate the results of these analyses with group project grades received; due to a small or poorly collected dataset, we are unable to conclude that any of these network measures relates to those grades. However, we are successful in using this dataset to de…
