Many popular measures used in social network analysis, including centrality, are based on the random walk. The random walk is a model of a stochastic process where a node interacts with one other node at a time. However, the random walk may not be appropriate for modeling social phenomena, including epidemics and information diffusion, in which one node may interact with many others at the same time, for example, by broadcasting the virus or information to its neighbors. To produce meaningful results, social network analysis algorithms have to take into account the nature of interactions between the nodes. In this paper we classify dynamical processes as conservative and non-conservative and relate them to well-known measures of centrality used in network analysis: PageRank and Alpha-Centrality. We demonstrate, by ranking users in online social networks used for broadcasting information, that non-conservative Alpha-Centrality generally leads to a better agreement with an empirical ranking scheme than the conservative PageRank.
@cite_28 studied measures similar to Alpha-Centrality and personalized PageRank (with attenuation factor 1) which they call communicability. They linked the communicability functions to dynamics by showing their relationship to the thermal Green's function of oscillators. They used communicability to identify important actors in small social networks, demonstrating that different communicability functions led to different judgements of centrality, but did not justify the choice of the particular communicability function in terms of the interactions taking place between actors. Although we study a similar function, the goal of our work is to contrast conservative and non-conservative dynamics and explain how these differences should guide the choice of centrality measure for a given social network.
@cite_28
2950028823 2041757699 2116844604 2127405101 2042914057 2768318902 2092598586 2620696915 1988183975 2161283706 2127387319 2046868922 2171137858 1605385825 2096646197 2241780750 2123726161 2134784378 2131492273 2053507997
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 position in network hierarchy as well as from the potential to mobilize resources through intra-community (bonding) and inter-community (bridging) ties. Inspired by this idea, we propose a novel centrality measure SC (Social Centrality) for actors in social networks. Our measure accounts for - i) an individual's propensity to socialize, and ii) his connections within and outside the community. These two factors are suitably aggregated to produce social centrality score. Comparative analysis of SC measure with classical and recent centrality measures using large public networks shows that it consistently produces a more realistic ranking of nodes. The inference is based on the available ground truth for each tested networks. Extensive analysis of rankings delivered by SC measure and mapping with known facts in well-studied networks justifies its effectiveness in diverse social networks. Scalability evaluation of SC measure justifies its efficacy for real-world large networks. A complex network can be modeled as a graph representing the ''who knows who'' relationship. In the context of graph theory for social networks, the notion of centrality is used to assess the relative importance of nodes in a given network topology. For example, in a network composed of large dense clusters connected through only a few links, the nodes involved in those links are particularly critical as far as the network survivability is concerned. This may also impact any application running on top of it. Such information can be exploited for various topological maintenance issues to prevent congestion and disruption. This can also be used offline to identify the most important actors in large social interaction graphs. Several forms of centrality have been proposed so far. Yet, they suffer from imperfections: initially designed for small social graphs, they are either of limited use (degree centrality), either incompatible in a distributed setting (e.g. random walk betweenness centrality). In this paper we introduce a novel form of centrality: the second order centrality which can be computed in a distributed manner. This provides locally each node with a value reflecting its relative criticity and relies on a random walk visiting the network in an unbiased fashion. To this end, each node records the time elapsed between visits of that random walk (called return time in the sequel) and computes the standard deviation (or second order moment) of such return times. The key point is that central nodes see regularly the random walk compared to other topology nodes. Both through theoretical analysis and simulation, we show that the standard deviation can be used to accurately identify critical nodes as well as to globally characterize graphs topology in a distributed way. We finally compare our proposal to well-known centralities to assess its competitivity. In recent work, Ozgur, Leveque, and Tse (2007) obtained a complete scaling characterization of throughput scaling for random extended wireless networks (i.e., n nodes are placed uniformly at random in a square region of area n). They showed that for small path-loss exponents alpha isin (2,3], cooperative communication is order optimal, and for large path-loss exponents alpha > 3, multihop communication is order optimal. However, their results (both the communication scheme and the proof technique) are strongly dependent on the regularity induced with high probability by the random node placement. In this paper, we consider the problem of characterizing the throughput scaling in extended wireless networks with arbitrary node placement. As a main result, we propose a more general novel cooperative communication scheme that works for arbitrarily placed nodes. For small path-loss exponents alpha isin (2,3], we show that our scheme is order optimal for all node placements, and achieves exactly the same throughput scaling as in Ozgur. This shows that the regularity of the node placement does not affect the scaling of the achievable rates for alpha isin (2,3]. The situation is, however, markedly different for large path-loss exponents alpha > 3. We show that in this regime the scaling of the achievable per-node rates depends crucially on the regularity of the node placement. We then present a family of schemes that smoothly ldquointerpolaterdquo between multihop and cooperative communication, depending upon the level of regularity in the node placement. We establish order optimality of these schemes under adversarial node placement for alpha > 3. The concept of centrality is often invoked in social network analysis, and diverse indices have been proposed to measure it. This paper develops a unified framework for the measurement of centrality. All measures of centrality assess a node's involvement in the walk structure of a network. Measures vary along four key dimensions: type of nodal involvement assessed, type of walk considered, property of walk assessed, and choice of summary measure. If we cross-classify measures by type of nodal involvement (radial versus medial) and property of walk assessed (volume versus length), we obtain a four-fold polychotomization with one cell empty which mirrors Freeman's 1979 categorization. At a more substantive level, measures of centrality summarize a node's involvement in or contribution to the cohesiveness of the network. Radial measures in particular are reductions of pair-wise proximities cohesion to attributes of nodes or actors. The usefulness and interpretability of radial measures depend on the fit of the cohesion matrix to the one-dimensional model. In network terms, a network that is fit by a one-dimensional model has a core-periphery structure in which all nodes revolve more or less closely around a single core. This in turn implies that the network does not contain distinct cohesive subgroups. Thus, centrality is shown to be intimately connected with the cohesive subgroup structure of a network. Centrality has revealed crucial for understanding the structural properties of complex relational networks. Centrality is also relevant for various spatial factors affecting human life and behaviors in cities. Here, we present a comprehensive study of centrality distributions over geographic networks of urban streets. Five different measures of centrality, namely degree, closeness, betweenness, straightness and information, are compared over 18 1-square-mile samples of different world cities. Samples are represented by primal geographic graphs, i.e., valued graphs defined by metric rather than topologic distance where intersections are turned into nodes and streets into edges. The spatial behavior of centrality indices over the networks is investigated graphically by means of color-coded maps. The results indicate that a spatial analysis, that we term multiple centrality assessment, grounded not on a single but on a set of different centrality indices, allows an extended comprehension of the city structur... Abstract As network analysis methods prevail, more metrics are applied to co-word networks to reveal hot topics in a field. However, few studies have examined the relationships among these metrics. To bridge this gap, this study explores the relationships among different ranking metrics, including one frequency-based and six network-based metrics, in order to understand the impact of network structural features on ranking themes on co-word networks. We collected bibliographic data from three disciplines from Web of Science (WoS), and generated 40 simulation networks following the preferential attachment assumption. Correlation analysis on the empirical and simulated networks shows strong relationships among the metrics. Their relationships are consistent across disciplines. The metrics can be categorized into three groups according to the strength of their correlations, where Degree Centrality, H-index, and Coreness are in one group, Betweenness Centrality, Clustering Coefficient, and frequency in another, and Weighted PageRank by itself. Regression analysis on the simulation networks reveals that network topology properties, such as connectivity, sparsity, and aggregation, influence the relationships among selected metrics. In addition, when comparing the top keywords ranked by the metrics in the three disciplines, we found the metrics exhibit different discriminative capacity. Coreness and H-index may be better suited for categorizing keywords rather than ranking keywords. Findings from this study contribute to a better understanding of the relationships among different metrics and provide guidance for using them effectively in different contexts. Abstract A fundamental problem in the study of complex networks is to provide quantitative measures of correlation and information flow between different parts of a system. To this end, several notions of communicability have been introduced and applied to a wide variety of real-world networks in recent years. Several such communicability functions are reviewed in this paper. It is emphasized that communication and correlation in networks can take place through many more routes than the shortest paths, a fact that may not have been sufficiently appreciated in previously proposed correlation measures. In contrast to these, the communicability measures reviewed in this paper are defined by taking into account all possible routes between two nodes, assigning smaller weights to longer ones. This point of view naturally leads to the definition of communicability in terms of matrix functions, such as the exponential, resolvent, and hyperbolic functions, in which the matrix argument is either the adjacency matrix or the graph Laplacian associated with the network. Considerable insight on communicability can be gained by modeling a network as a system of oscillators and deriving physical interpretations, both classical and quantum-mechanical, of various communicability functions. Applications of communicability measures to the analysis of complex systems are illustrated on a variety of biological, physical and social networks. The last part of the paper is devoted to a review of the notion of locality in complex networks and to computational aspects that by exploiting sparsity can greatly reduce the computational efforts for the calculation of communicability functions for large networks. Centrality measures are widely used to identify leaders of covert networks. We study how a group of such leaders can avoid being detected by such measures. More concretely, we study the hardness of choosing a set of edges that can be added to the network in order to decrease the leaders' ranking according to two fundamental centrality measures, namely degree, and closeness. We prove that this problem is NP-complete for each measure. We then study how the leaders can construct a network from scratch, designed specifically for them to hide in disguise. We identify a network structure that not only guarantees to hide the leaders to a certain extent, but also allows them to spread their influence across the network. The ability to measure centrality in social networks has been a particularly useful development in social network analysis. For researchers trying to decide which centrality measure is most meaningful and valid for their research purposes, various papers have explored the conceptual foundations of centrality measures. Less well documented is the empirical performance of centrality measures under different research scenarios or constraints. This study uses bootstrap sampling procedures to determine how sampling affects the stability of 11 different network centrality measures. Results indicate that some measures are more stable than others, and that stability is also a function of network and study properties. Simulation studies using susceptible-infectious-recovered models were conducted to estimate individuals’ risk of infection and time to infection in small-world and randomly mixing networks. Infection transmitted more rapidly but ultimately resulted in fewer infected individuals in the small-world, compared with the random, network. The ability of measures of network centrality to identify high-risk individuals was also assessed. ‘‘Centrality’’ describes an individual’s position in a population; numerous parameters are available to assess this attribute. Here, the authors use the centrality measures degree (number of contacts), random-walk betweenness (a measure of the proportion of times an individual lies on the path between other individuals), shortest-path betweenness (the proportion of times an individual lies on the shortest path between other individuals), and farness (the sum of the number of steps between an individual and all other individuals). Each was associated with time to infection and risk of infection in the simulated outbreaks. In the networks examined, degree (which is the most readily measured) was at least as good as other network parameters in predicting risk of infection. Identification of more central individuals in populations may be used to inform surveillance and infection control strategies. Centrality measures, or at least popular interpretations of these measures, make implicit assumptions about the manner in which traffic flows through a network. For example, some measures count only geodesic paths, apparently assuming that whatever flows through the network only moves along the shortest possible paths. This paper lays out a typology of network flows based on two dimensions of variation, namely the kinds of trajectories that traffic may follow (geodesics, paths, trails, or walks) and the method of spread (broadcast, serial replication, or transfer). Measures of centrality are then matched to the kinds of flows that they are appropriate for. Simulations are used to examine the relationship between type of flow and the differential importance of nodes with respect to key measurements such as speed of reception of traffic and frequency of receiving traffic. It is shown that the off-the-shelf formulas for centrality measures are fully applicable only for the specific flow processes they are designed for, and that when they are applied to other flow processes they get the “wrong” answer. It is noted that the most commonly used centrality measures are not appropriate for most of the flows we are routinely interested in. A key claim made in this paper is that centrality measures can be regarded as generating expected values for certain kinds of node outcomes (such as speed and frequency of reception) given implicit models of how traffic flows, and that this provides a new and useful way of thinking about centrality. © 2004 Elsevier B.V. All rights reserved. Abstract Egocentric centrality measures (for data on a node’s first-order zone) parallel to Freeman’s [Social Networks 1 (1979) 215] centrality measures for complete (sociocentric) network data are considered. Degree-based centrality is in principle identical for egocentric and sociocentric network data. A closeness measure is uninformative for egocentric data, since all geodesic distances from ego to other nodes in the first-order zone are 1 by definition. The extent to which egocentric and sociocentric versions of Freeman’s betweenness centrality measure correspond is explored empirically. Across seventeen diverse networks, that correspondence is found to be relatively close—though variations in egocentric network composition do lead to some notable differences in egocentric and sociocentric betweennness. The findings suggest that research design has a relatively modest impact on assessing the relative betweenness of nodes, and that a betweenness measure based on egocentric network data could be a reliable substitute for Freeman’s betweenness measure when it is not practical to collect complete network data. However, differences in the research methods used in sociocentric and egocentric studies could lead to additional differences in the respective betweenness centrality measures. Centrality measures are used in network science to rank the relative importance of nodes and edges of a graph. Here we define new measures of centrality for power grid structure that are based on its functionality. We show that the relative importance analysis based on centrality in graph theory can be generalized to power grid network with its electrical parameters taken into account. In the paper we experiment with the proposed electrical centrality measures on the NYISO-2935 system and the IEEE 300-bus system. We analyze the centrality distribution in order to identify important nodes or branches in the system which are of essential importance in terms of system vulnerability. We also present and discuss a number of interesting discoveries regarding the importance rank of power grid nodes and branches. Analyzing networks requires complex algorithms to extract meaningful information. Centrality metrics have shown to be correlated with the importance and loads of the nodes in network traffic. Here, we are interested in the problem of centrality-based network management. The problem has many applications such as verifying the robustness of the networks and controlling or improving the entity dissemination. It can be defined as finding a small set of topological network modifications which yield a desired closeness centrality configuration. As a fundamental building block to tackle that problem, we propose incremental algorithms which efficiently update the closeness centrality values upon changes in network topology, i.e., edge insertions and deletions. Our algorithms are proven to be efficient on many real-life networks, especially on small-world networks, which have a small diameter and a spike-shaped shortest distance distribution. In addition to closeness centrality, they can also be a great arsenal for the shortest-path-based management and analysis of the networks. We experimentally validate the efficiency of our algorithms on large networks and show that they update the closeness centrality values of the temporal DBLP-coauthorship network of 1.2 million users 460 times faster than it would take to compute them from scratch. To the best of our knowledge, this is the first work which can yield practical large-scale network management based on closeness centrality values. We consider variations of two well-known centrality measures, betweenness and closeness, with a different model of information spread. Rather than along shortest paths only, it is assumed that information spreads efficiently like an electrical current. We prove that the current-flow variant of closeness centrality is identical with another known measure, information centrality, and give improved algorithms for computing both measures exactly. Since running times and space requirements are prohibitive for large networks, we also present a randomized approximation scheme for current-flow betweenness. Measuring the relative importance of each vertex in a network is one of the most fundamental building blocks in network analysis. Among several importance measures, betweenness centrality, in particular, plays key roles in many real applications. Considerable effort has been made for developing algorithms for static settings. However, real networks today are highly dynamic and are evolving rapidly, and scalable dynamic methods that can instantly reflect graph changes into centrality values are required. In this paper, we present the first fully dynamic method for managing betweenness centrality of all vertices in a large dynamic network. Its main data structure is the weighted hyperedge representation of shortest paths called hypergraph sketch. We carefully design dynamic update procedure with theoretical accuracy guarantee. To accelerate updates, we further propose two auxiliary data structures called two-ball index and special-purpose reachability index. Experimental results using real networks demonstrate its high scalability and efficiency. In particular, it can reflect a graph change in less than a millisecond on average for a large-scale web graph with 106M vertices and 3.7B edges, which is several orders of magnitude larger than the limits of previous dynamic methods. We empirically analyze five online communities: Friendster, Livejournal, Facebook, Orkut, and Myspace, to study how social networks decline. We define social resilience as the ability of a community to withstand changes. We do not argue about the cause of such changes, but concentrate on their impact. Changes may cause users to leave, which may trigger further leaves of others who lost connection to their friends. This may lead to cascades of users leaving. A social network is said to be resilient if the size of such cascades can be limited. To quantify resilience, we use the k-core analysis, to identify subsets of the network in which all users have at least k friends. These connections generate benefits (b) for each user, which have to outweigh the costs (c) of being a member of the network. If this difference is not positive, users leave. After all cascades, the remaining network is the k-core of the original network determined by the cost-to-benefit (c b) ratio. By analysing the cumulative distribution of k-cores we are able to calculate the number of users remaining in each community. This allows us to infer the impact of the c b ratio on the resilience of these online communities. We find that the different online communities have different k-core distributions. Consequently, similar changes in the c b ratio have a different impact on the amount of active users. Further, our resilience analysis shows that the topology of a social network alone cannot explain its success of failure. As a case study, we focus on the evolution of Friendster. We identify time periods when new users entering the network observed an insufficient c b ratio. This measure can be seen as a precursor of the later collapse of the community. Our analysis can be applied to estimate the impact of changes in the user interface, which may temporarily increase the c b ratio, thus posing a threat for the community to shrink, or even to collapse. AbstractGiven 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 study, 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 to do. Our axioms suggest some simple, basic properties that a centrality measure should exhibit.Surprisingly, only a new simple measure based on distances, harmonic centrality, turns out to satisfy all axioms; essentially, harmonic centrality is a correction to Bavelas’s classic closeness centrality [Bavelas 50] designed to take unreachable nodes into account in a natural way.As a sanity check, we examine in turn each measure under the lens of information... We perform sensitivity analyses to assess the impact of missing data on the structural properties of social networks. The social network is conceived of as being generated by a bipartite graph, in which actors are linked together via multiple interaction contexts or affiliations. We discuss three principal missing data mechanisms: network boundary specification (non-inclusion of actors or affiliations), survey non-response, and censoring by vertex degree (fixed choice design), examining their impact on the scientific collaboration network from the Los Alamos E-print Archive as well as random bipartite graphs. The simulation results show that network boundary specification and fixed choice designs can dramatically alter estimates of network-level statistics. The observed clustering and assortativity coefficients are overestimated via omission of affiliations or fixed choice thereof, and underestimated via actor non-response, which results in inflated measurement error. We also find that social networks with multiple interaction contexts may have certain interesting properties due to the presence of overlapping cliques. In particular, assortativity by degree does not necessarily improve network robustness to random omission of nodes as predicted by current theory. Three measures of actors' network centrality are derived from an elementary rocess model of social influence. The measures are closely related to, and cast new light on, widely used measures of actors' centrality; for example, the essential social organization of status that has been assumed by Hubbell, Bonacich, Coleman, and Burt appears as a deducible outcome of this social influence process. Unlike previous measures, which have been viewed as competing alternatives, the present measures are complementary and, in their juxtaposition, provide for a rich description of social structure. The complementary indicates a degree of theoretical unification in the work on network centrality that was heretofore unsuspected.