The sensitivity of networks to the removal of vertices has been studied extensively over the last 15 years. A common approach to measuring this sensitivity is (i) successively removing vertices following a specific removal strategy and (ii) comparing the original and the modified network using a specific comparison method. In this paper we apply a wide range of removal strategies and comparison methods in order to study the sensitivity of medium-sized networks from the real world and randomly generated networks. In the first part of our study we observe that social networks and web graphs differ in sensitivity. When removing vertices, social networks are robust, web graphs are not. This effect is consistent with the work of who analyzed very large social networks and web graphs. For randomly generated networks we find that their sensitivity depends significantly on the comparison method. The choice of removal strategy has surprisingly marginal impact on the sensitivity for removal strategies derived from common centrality measures. However, the removal strategy has a strong impact when removing the vertices in random order.
Modifying graphs and comparing the outcome with the respective source graph is a common approach to tackle a variety of questions. In the field of social network analysis, networks are modified in order to simulate measurement errors and to examine the robustness of centrality measures. An empirical network was altered by Bolland @cite_5 and Pearson correlation was used to measure the robustness. Random samples have been taken from empirical networks by Costenbader and Valente @cite_19 to investigate the stability of various centrality measures. Four error types have been applied to Erd o s-R 'enyi graphs by @cite_25 to measure centrality robustness regarding different types of accuracy measures. Six types of measurement errors have been applied to real-world and generated networks by @cite_14 in order to examine the robustness of node-level network measures by means of Spearman's rho.
@cite_19 @cite_5 @cite_14 @cite_25
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An analysis is conducted on the robustness of measures of centrality in the face of random error in the network data. We use random networks of varying sizes and densities and subject them (separately) to four kinds of random error in varying amounts. The types of error are edge deletion, node deletion, edge addition, and node addition. The results show that the accuracy of centrality measures declines smoothly and predictably with the amount of error. This suggests that, for random networks and random error, we shall be able to construct confidence intervals around centrality scores. In addition, centrality measures were highly similar in their response to error. Dense networks were the most robust in the face of all kinds of error except edge deletion. For edge deletion, sparse networks were more accurately measured. Abstract Research on measurement error in network data has typically focused on missing data. We embed missing data, which we term false negative nodes and edges, in a broader classification of error scenarios. This includes false positive nodes and edges and falsely aggregated and disaggregated nodes. We simulate these six measurement errors using an online social network and a publication citation network, reporting their effects on four node-level measures – degree centrality, clustering coefficient, network constraint, and eigenvector centrality. Our results suggest that in networks with more positively-skewed degree distributions and higher average clustering, these measures tend to be less resistant to most forms of measurement error. In addition, we argue that the sensitivity of a given measure to an error scenario depends on the idiosyncracies of the measure's calculation, thus revising the general claim from past research that the more ‘global’ a measure, the less resistant it is to measurement error. Finally, we anchor our discussion to commonly-used networks in past research that suffer from these different forms of measurement error and make recommendations for correction strategies. 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. 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.