Tail Index Estimation of PageRanks in Evolving Random Graphs

Abstract

Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal part of random graphs is considered. We used the extreme value theory regarding sums and maxima of non-stationary random length sequences to evaluate the tail index of the PageRanks and max-linear models of superstar nodes in the evolving graphs where existing nodes or edges can be deleted or not. The evolution is provided by a linear preferential attachment. Our approach is based on the analysis of maxima and sums of the node PageRanks over communities (block maxima and block sums), which can be independent or weakly dependent random variables. By an empirical study, it was found that tail indices of the block maxima and block sums are close to the minimum tail index of representative series extracted from the communities. The tail indices are estimated by data of simulated graphs.

Extreme Value Statistics for Evolving Random Networks

“…Thus, the mentioned pair-wise dependency is not required. The results of the empirical study in [19] are in agreement with theoretical results in [14,15].…”

“…Using a simulation study, it was found in [19] that the tail indices of the sums and maxima over communities are close to the minimum tail index of the representative series extracted from the communities. The representative series can be formed by taking one of the nodes of a community as a representative.…”

“…Although node indices within the communities are not i.i.d., the Hill estimation still works in practice; see [19,30]. Its consistency for undirected graphs is proven in [60], but the result regarding directed graphs remains, to the best of our knowledge, an unresolved problem.…”

“…From another side, network communities may be interpreted as rows of A (0) ; see, Figure 1a in [19]. Such an approach has the advantage that sums and maxima over communities may be considered independent or weakly dependent r.v.s.…”

“…This creates an obstacle to defining the stationarity in the graphs. In [19], it is determined that "a graph is stationary if, for all finite sets of vertices with the same adjacency matrices, the joint distributions of their in-and out-degrees are the same." Tail and extremal indices of sets of injected nodes considered in time depend both on the choice of the seed network and the attachment policy.…”