We study the betweenness centrality of fractal and non-fractal scale-free network models as well as real networks. We show that the correlation between degree and betweenness centrality $C$ of nodes is much weaker in fractal network models compared to non-fractal models. We also show that nodes of both fractal and non-fractal scale-free networks have power law betweenness centrality distribution $P(C)\sim C^{-δ}$. We find that for non-fractal scale-free networks $δ= 2$, and for fractal scale-free networks $δ= 2-1/d_{B}$, where $d_{B}$ is the dimension of the fractal network. We support these results by explicit calculations on four real networks: pharmaceutical firms (N=6776), yeast (N=1458), WWW (N=2526), and a sample of Internet network at AS level (N=20566), where $N$ is the number of nodes in the largest connected component of a network. We also study the crossover phenomenon from fractal to non-fractal networks upon adding random edges to a fractal network. We show that the crossover length $\ell^{*}$, separating fractal and non-fractal regimes, scales with dimension $d_{B}$ of the network as $p^{-1/d_{B}}$, where $p$ is the density of random edges added to the network. We find that the correlation between degree and betweenness centrality increases with $p$.
In this paper, we empirically investigate correlations among four centrality measures, originated from the social science, of various complex networks. For each network, we compute the centrality measures, from which the partial correlation as well as the correlation coefficient among measures is estimated. We uncover that the degree and the betweenness centrality are highly correlated; furthermore, the betweenness follows a power-law distribution irrespective of the type of networks. This characteristic is further examined in terms of the conditional probability distribution of the betweennes…
A class of centrality measures called betweenness centralities reflects degree of participation of edges or nodes in communication between different parts of the network. The original shortest-path betweenness centrality is based on counting shortest paths which go through a node or an edge. One of shortcomings of the shortest-path betweenness centrality is that it ignores the paths that might be one or two steps longer than the shortest paths, while the edges on such paths can be important for communication processes in the network. To rectify this shortcoming a current flow betweenness centr…
In addition to science citation indicators of journals like impact and immediacy, social network analysis provides a set of centrality measures like degree, betweenness, and closeness centrality. These measures are first analyzed for the entire set of 7,379 journals included in the Journal Citation Reports of the Science Citation Index and the Social Sciences Citation Index 2004, and then also in relation to local citation environments which can be considered as proxies of specialties and disciplines. Betweenness centrality is shown to be an indicator of the interdisciplinarity of journals, bu…
Betweenness centrality of a vertex in a graph measures the fraction of shortest paths going through the vertex. This is a basic notion for determining the importance of a vertex in a network. The k-betweenness centrality of a vertex is defined similarly, but only considers shortest paths of length at most k. The sequence of k-betweenness centralities for all possible values of k forms the betweenness centrality profile of a vertex. We study properties of betweenness centrality profiles in trees. We show that for scale-free random trees, for fixed k, the expectation of k-betweenness centrality …
By considering the eigenratio of the Laplacian of the connection graph as synchronizability measure, we propose a procedure for weighting dynamical networks to enhance theirsynchronizability. The method is based on node and edge betweenness centrality measures and is tested on artificially const ructed scale-free, Watts-Strogatz and random networks as well as on some real-world graphs. It is also numerically shown that the same procedure could be used to enhance the phase synchronizability of networks of nonidentical oscillators.
We analyze whether preferential attachment in scientific coauthorship networks is different for authors with different forms of centrality. Using a complete database for the scientific specialty of research about "steel structures," we show that betweenness centrality of an existing node is a significantly better predictor of preferential attachment by new entrants than degree or closeness centrality. During the growth of a network, preferential attachment shifts from (local) degree centrality to betweenness centrality as a global measure. An interpretation is that supervisors of PhD projects …
Betweenness centrality quantifies the importance of a vertex for the information flow in a network. We propose a flexible definition of betweenness for temporal multiplexes, where geodesics are determined accounting for the topological and temporal structure and the duration of paths. We propose an algorithm to compute the new metric via a mapping to a static graph. We show the importance of considering the temporal multiplex structure and an appropriate distance metric comparing the results with those obtained with static or single-layer metrics on a dataset of $\sim 20$k European flights.
Betweenness centrality is a metric that seeks to quantify a sense of the importance of a vertex in a network graph in terms of its "control" on the distribution of information along geodesic paths throughout that network. This quantity however does not capture how different vertices participate together in such control. In order to allow for the uncovering of finer details in this regard, we introduce here an extension of betweenness centrality to pairs of vertices, which we term co-betweenness, that provides the basis for quantifying various analogous pairwise notions of importance and contro…
The Betweenness Centrality index is a very important centrality measure in the analysis of a large number of networks. Despite its significance in a lot of interdisciplinary applications, its computation is very expensive. The fastest known algorithm presently is by Brandes which takes O(|V || E|) time for computation. In real life scenarios, it happens very frequently that a single vertex or a set of vertices is sequentially removed from a network. The recomputation of Betweenness Centrality on removing a single vertex becomes expensive when the Brandes algorithm is repeated. It is to be unde…
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…
Betweenness centrality ranks the importance of nodes by their participation in all shortest paths of the network. Therefore computing exact betweenness values is impractical in large networks. For static networks, approximation based on randomly sampled paths has been shown to be significantly faster in practice. However, for dynamic networks, no approximation algorithm for betweenness centrality is known that improves on static recomputation. We address this deficit by proposing two incremental approximation algorithms (for weighted and unweighted connected graphs) which provide a provable gu…
Centrality measures, erstwhile popular amongst the sociologists and psychologists, have seen broad and increasing applications across several disciplines of late. Amongst a plethora of application specific definitions available in the literature to rank the vertices, closeness centrality, betweenness centrality and eigenvector centrality (page-rank) have been the most important and widely applied ones. Networks where information, signal or commodities are flowing on the edges, surrounds us. Betweenness centrality comes as a handy tool to analyze such systems, but betweenness computation is a d…
Urban morphology and socioeconomic aspects of cities have been explored by analysing urban street network. To analyse the network, several variations of the centrality indices are often used. However, its nature has not yet been widely studied, thus leading to an absence of robust visualisation method of urban road network characteristics. To fill this gap, we propose to use a set of local betweenness centrality and a new simple and robust visualisation method. By analysing 30 European cities, we found that our method illustrates common structures of the cities: road segments important for lon…
This paper introduces two new closely related betweenness centrality measures based on the Randomized Shortest Paths (RSP) framework, which fill a gap between traditional network centrality measures based on shortest paths and more recent methods considering random walks or current flows. The framework defines Boltzmann probability distributions over paths of the network which focus on the shortest paths, but also take into account longer paths depending on an inverse temperature parameter. RSP's have previously proven to be useful in defining distance measures on networks. In this work we stu…
This paper expands the degree-based consideration of the preferential attachment growth process and applies five different connectivity criteria (node degree, clustering coefficient, betweenness centrality, closeness centrality, and eigenvector centrality) to define the development of new links in the networks. Based on statistical inference, the analysis shows that all the available control attributes are capable generating SF networks, that the proposed generalized preferential attachment growth process produces networks of statistically different topologies, under different control-attribut…
Betweenness centrality (BC) was proposed as an indicator of the extent of an individual's influence in a social network. It is measured by counting how many times a vertex (i.e., an individual) appears on all the shortest paths between pairs of vertices. A question naturally arises as to how the influence of a team or group in a social network can be measured. Here, we propose a method of measuring this influence on a bipartite graph comprising vertices (individuals) and hyperedges (teams). When the hyperedge size varies, the number of shortest paths between two vertices in a hypergraph can be…
Random geometric networks consist of 1) a set of nodes embedded randomly in a bounded domain $\mathcal{V} \subseteq \mathbb{R}^d$ and 2) links formed probabilistically according to a function of mutual Euclidean separation. We quantify how often all paths in the network characterisable as topologically `shortest' contain a given node (betweenness centrality), deriving an expression in terms of a known integral whenever 1) the network boundary is the perimeter of a disk and 2) the network is extremely dense. Our method shows how similar formulas can be obtained for any convex geometry. Numerica…
In this paper, we propose a general framework for optimal control measures, which follows the evolution of COVID-19 infection counts collected by Surveillance Units on a country level. We employ an autoregressive model that allows to decompose the mean number of infections into three components that describe: intra-locality infections, inter-locality infections, and infections from other sources such as travelers arriving to a country from abroad. We identify the inter-locality term as a time-evolving network and when it drives the dynamics of the disease we focus on its properties. Tools from…
Recent decades have witnessed the tremendous development of network science, which indeed brings a new and insightful language to model real systems of different domains. Betweenness, a widely employed centrality in network science, is a decent proxy in investigating network loads and rankings. However, the extremely high computational cost greatly prevents its applying on large networks. Though several parallel algorithms have been presented to reduce its calculation cost on unweighted networks, a fast solution for weighted networks, which are in fact more ubiquitous than unweighted ones in r…
We devise methods for finding approximations of the generalized inverse of the graph Laplacian matrix, which arises in many graph-theoretic applications. Finding this matrix in its entirety involves solving a matrix inversion problem, which is resource demanding in terms of consumed time and memory and hence impractical whenever the graph is relatively large. Our approximations use only few eigenpairs of the Laplacian matrix and are parametric with respect to this number, so that the user can compromise between effectiveness and efficiency of the approximated solution. We apply the devised app…
Betweenness centrality lies at the core of both transport and structural vulnerability properties of complex networks, however, it is computationally costly, and its measurement for networks with millions of nodes is near impossible. By introducing a multiscale decomposition of shortest paths, we show that the contributions to betweenness coming from geodesics not longer than L obey a characteristic scaling vs L, which can be used to predict the distribution of the full centralities. The method is also illustrated on a real-world social network of 5.5*10^6 nodes and 2.7*10^7 links.
Who is more important in a network? Who controls the flow between the nodes or whose contribution is significant for connections? Centrality metrics play an important role while answering these questions. The betweenness metric is useful for network analysis and implemented in various tools. Since it is one of the most computationally expensive kernels in graph mining, several techniques have been proposed for fast computation of betweenness centrality. In this work, we propose and investigate techniques which compress a network and shatter it into pieces so that the rest of the computation ca…
Centrality, which quantifies the "importance" of individual nodes, is among the most essential concepts in modern network theory. As there are many ways in which a node can be important, many different centrality measures are in use. Here, we concentrate on versions of the common betweenness and it closeness centralities. The former measures the fraction of paths between pairs of nodes that go through a given node, while the latter measures an average inverse distance between a particular node and all other nodes. Both centralities only consider shortest paths (i.e., geodesics) between pairs o…
Nowadays, identification and detection community structures in complex networks is an important factor in extracting useful information from networks. Label propagation algorithm with near linear-time complexity is one of the most popular methods for detecting community structures, yet its uncertainty and randomness is a defective factor. Merging LPA with other community detection metrics would improve its accuracy and reduce instability of LPA. Considering this point, in this paper we tried to use edge betweenness centrality to improve LPA performance. On the other hand, calculating edge betw…
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…
A citation-based indicator for interdisciplinarity has been missing hitherto among the set of available journal indicators. In this study, we investigate network indicators (betweenness centrality), journal indicators (Shannon entropy, the Gini coefficient), and more recently proposed Rao-Stirling measures for "interdisciplinarity." The latter index combines the statistics of both citation distributions of journals (vector-based) and distances in citation networks among journals (matrix-based). The effects of various normalizations are specified and measured using the matrix of 8,207 journals …
Estimating influential nodes in large scale networks including but not limited to social networks, biological networks, communication networks, emerging smart grids etc. is a topic of fundamental interest. To understand influences of nodes in a network, a classical metric is centrality within which there are multiple specific instances including degree centrality, closeness centrality, betweenness centrality and more. As of today, existing algorithms to identify nodes with high centrality measures operate upon the entire (or rather global) network, resulting in high computational complexity. I…
We demonstrate that the distribution of betweenness centrality (BC), a global structural metric based on network flow, is an invariant quantity in most planar graphs. We confirm this invariance through an empirical analysis of street networks from 97 of the most populous cities worldwide, at scales significantly larger than previous studies. We also find that the BC distribution is robust to major alterations in the network, including significant changes to its topology and edge weight structure, indicating that the only relevant factors shaping the distribution are the number of nodes and edg…
The study of network robustness focuses on the way the overall functionality of a network is affected as some of its constituent parts fail. Failures can occur at random or be part of an intentional attack and, in general, networks behave differently against different removal strategies. Although much effort has been put on this topic, there is no unified framework to study the problem. While random failures have been mostly studied under percolation theory, targeted attacks have been recently restated in terms of network dismantling. In this work, we link these two approaches by performing a …
The control of epidemic spreading is essential to avoid potential fatal consequences and also, to lessen unforeseen socio-economic impact. The need for effective control is exemplified during the severe acute respiratory syndrome (SARS) in 2003, which has inflicted near to a thousand deaths as well as bankruptcies of airlines and related businesses. In this article, we examine the efficacy of control strategies on the propagation of infectious diseases based on removing connections within real world airline network with the associated economic and social costs taken into account through defini…
We study the h Hirsch index as a local node centrality measure for complex networks in general. The h index is compared with the Degree centrality (a local measure), the Betweenness and Eigenvector centralities (two non-local measures) in the case of a biological network (Yeast interaction protein-protein network) and a linguistic network (Moby Thesaurus II) as test environments. In both networks, the Hirsch index has poor correlation with Betweenness centrality but correlates well with Eigenvector centrality, specially for the more important nodes that are relevant for ranking purposes, say i…
Recent research on finding important intermediate nodes in a network suspected to contain criminal activity is highly dependent on network centrality values. Betweenness centrality, for example, is widely used to rank the nodes that act as brokers in the shortest paths connecting all source and all the end nodes in a network. However both the shortest path node betweenness and the linearly scaled betweenness can only show rankings for all the nodes in a network. In this paper we explore the mathematical concept of pair-dependency on intermediate nodes, adapting the concept to criminal relation…
Betweenness Centrality (BC) is an important measure used widely in complex network analysis, such as social network, web page search, etc. Computing the exact BC values is highly time consuming. Currently the fastest exact BC determining algorithm is given by Brandes, taking $O(nm)$ time for unweighted graphs and $O(nm+n^2\log n)$ time for weighted graphs, where $n$ is the number of vertices and $m$ is the number of edges in the graph. Due to the extreme difficulty of reducing the time complexity of exact BC determining problem, many researchers have considered the possibility of any satisfact…
The betweenness centrality of graphs using random walk paths instead of geodesics is studied. A scaling collapse with no adjustable parameters is obtained as the graph size $N$ is varied; the scaling curve depends on the graph model. A normalized random betweenness, that counts each walk passing through a node only once, is also defined. It is argued to be more useful and seen to have simpler scaling behavior. In particular, the probability for a random walk on a preferential attachment graph to pass through the root node is found to tend to unity as $N\rightarrow\infty.$
Existing centrality measures for social network analysis suggest the im-portance of an actor and give consideration to actor's given structural position in a network. These existing measures suggest specific attribute of an actor (i.e., popularity, accessibility, and brokerage behavior). In this study, we propose new hybrid centrality measures (i.e., Degree-Degree, Degree-Closeness and Degree-Betweenness), by combining existing measures (i.e., degree, closeness and betweenness) with a proposition to better understand the importance of actors in a given network. Generalized set of measures are …
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…
Robustness estimation is critical for the design and maintenance of resilient networks, one of the global challenges of the 21st century. Existing studies exploit network metrics to generate attack strategies, which simulate intentional attacks in a network, and compute a metric-induced robustness estimation. While some metrics are easy to compute, e.g. degree centrality, other, more accurate, metrics require considerable computation efforts, e.g. betweennes centrality. We propose a new algorithm for estimating the robustness of a network in sub-quadratic time, i.e., significantly faster than …
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.
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…
In this comment, we investigate a common used algorithm proposed by Newman [M. E. J. Newman, Phys. Rev. E {\bf 64}, 016132(2001)] to calculate the betweenness centrality for all vertices. The inaccurateness of Newman's algorithm is pointed out and a corrected algorithm, also with O($MN$) time complexity, is given. In addition, the comparison of calculating results for these two algorithm aiming the protein interaction network of Yeast is shown.
In this work we investigate the betweenness centrality in geographical networks and its relationship with network communities. We show that vertices with large betweenness define what we call characteristic betweenness paths in both modeled and real-world geographical networks. We define a geographical network model that possess a simple topology while still being able to present such betweenness paths. Using this model, we show that such paths represent pathways between entry and exit points of highly connected regions, or communities, of geographical networks. By defining a new network, cont…
The temporal component of social networks is often neglected in their analysis, and statistical measures are typically performed on a "static" representation of the network. As a result, measures of importance (like betweenness centrality) cannot reveal any temporal role of the entities involved. Our goal is to start filling this limitation by proposing a form of temporal betweenness measure, and by using it to analyse a knowledge mobilization network. We show that this measure, which takes time explicitly into account, allows us to detect centrality roles that were completely hidden in the cl…
In complex networks a common task is to identify the most important or "central" nodes. There are several definitions, often called centrality measures, which often lead to different results. Here we study extensively correlations between four local and global measures namely the degree, the shortest-path-betweenness, the random-walk betweenness and the subgraph centrality on different random-network models like Erdos-Renyi, Small-World and Barabasi-Albert as well as on different real networks like metabolic pathways, social collaborations and computer networks. Correlations are quite differen…
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…
The solution of today's complex problems requires the grouping of task forces whose members are usually connected remotely over long physical distances and different time zones. Hence, understanding the effects of imposed communication patterns (i.e., who can communicate with whom) on group performance is important. Here, we use an agent-based model to explore the influence of the betweenness centrality of the nodes on the time the group requires to find the global maxima of NK-fitness landscapes. The agents cooperate by broadcasting messages, informing on their fitness to their neighbors, and…
Much of the past work in network analysis has focused on analyzing discrete graphs, where binary edges represent the "presence" or "absence" of a relationship. Since traditional network measures (e.g., betweenness centrality) utilize a discrete link structure, complex systems must be transformed to this representation in order to investigate network properties. However, in many domains there may be uncertainty about the relationship structure and any uncertainty information would be lost in translation to a discrete representation. Uncertainty may arise in domains where there is moderating lin…
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…
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…
This study looks into employees' communication, offering novel metrics which can help to predict a company's stock price. We studied the intranet forum of a large Italian company, exploring the interactions and the use of language of about 8,000 employees. We built a network linking words included in the general discourse. In this network, we focused on the position of the node representing the company brand. We found that a lower sentiment, a higher betweenness centrality of the company brand, a denser word co-occurrence network and more equally distributed centrality scores of employees (low…
The identification of nodes occupying important positions in a network structure is crucial for the understanding of the associated real-world system. Usually, betweenness centrality is used to evaluate a node capacity to connect different graph regions. However, we argue here that this measure is not adapted for that task, as it gives equal weight to "local" centers (i.e. nodes of high degree central to a single region) and to "global" bridges, which connect different communities. This distinction is important as the roles of such nodes are different in terms of the local and global organisat…
This study develops the epidemic hitting time (EHT) metric on graphs measuring the expected time an epidemic starting at node $a$ in a fully susceptible network takes to propagate and reach node $b$. An associated EHT centrality measure is then compared to degree, betweenness, spectral, and effective resistance centrality measures through exhaustive numerical simulations on several real-world network data-sets. We find two surprising observations: first, EHT centrality is highly correlated with effective resistance centrality; second, the EHT centrality measure is much more delocalized compare…
In this study we propose a method based on e-mail social network analysis to compare the communication behavior of managers who voluntarily quit their job and managers who decide to stay. Collecting 18 months of e-mail, we analyzed the communication behavior of 866 managers, out of which 111 left a large global service company. We compared differences in communication patterns by computing social network metrics, such as betweenness and closeness centrality, and content analysis indicators, such as emotionality and complexity of the language used. To study the emergence of managers' disengagem…
In various applications involving complex networks, network measures are employed to assess the relative importance of network nodes. However, the robustness of such measures in the presence of link inaccuracies has not been well characterized. Here we present two simple stochastic models of false and missing links and study the effect of link errors on three commonly used node centrality measures: degree centrality, betweenness centrality, and dynamical importance. We perform numerical simulations to assess robustness of these three centrality measures. We also develop an analytical theory, w…
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 …
