Core decomposition has proven to be a useful primitive for a wide range of graph analyses. One of its most appealing features is that, unlike other notions of dense subgraphs, it can be computed linearly in the size of the input graph.In this paper we provide an analogous tool for uncertain graphs, i.e., graphs whose edges are assigned a probability of existence. The fact that core decomposition can be computed efficiently in deterministic graphs does not guarantee efficiency in uncertain graphs, where even the simplest graph operations may become computationally intensive. Here we show that core decomposition of uncertain graphs can be carried out efficiently as well.We extensively evaluate our definitions and methods on a number of real-world datasets and applications, such as influence maximization and task-driven team formation.

The irruption of social media in the political sphere is generating repositories of “Big Data,” which can be mined to gain insights into communication dynamics. The research reported here relies on a large data set from Twitter to examine the activity, emotional content, and interactions of political parties and politicians during the campaign for the Spanish national elections in November 2011. The aim of this study is to investigate the adaptation of political parties to this new communication and organizational paradigm originating in the evolution of the Internet and online social networks. We analyze the reply and retweet networks of seven political parties with significant offline differences to assess their conversation and information diffusion patterns. We observe that political parties, and especially the major traditional parties, still tend to use Twitter just as a one‐way flow communication tool. Moreover, we find evidence of a balkanization trend in the Spanish online political sphere, as observed in previous research for other countries.

PageRank with personalization is used in Web search as an importance measure for Web documents. The goal of this paper is to characterize the tail behavior of the PageRank distribution in the Web and other complex networks characterized by power laws. To this end, we model the PageRank as a solution of a stochastic equationwhere the R i s are distributed as R. This equation is inspired by the original definition of the PageRank. In particular, N models the number of incoming links to a page, and B stays for the user preference. Assuming that N or B are heavy tailed, we employ the theory of regular variation to obtain the asymptotic behavior of R under quite general assumptions on the involved random variables. Our theoretical predictions show good agreement with experimental data.

The PageRank is a popularity measure designed by Google to rank Web pages. Experiments confirm that the PageRank obeys a 'power law' with the same exponent as the In-Degree. This paper presents a novel mathematical model that explains this phenomenon. The relation between the PageRank and In-Degree is modelled through a stochastic equation, which is inspired by the original definition of the PageRank, and is analogous to the well-known distributional identity for the busy period in the M/G/1 queue. Further, we employ the theory of regular variation and Tauberian theorems to analytically prove that the tail behavior of the PageRank and the In-Degree differ only by a multiplicative factor, for which we derive a closed-form expression. Our analytical results are in good agreement with experimental data.

We study the relation between PageRank and other parameters of information networks such as in-degree, out-degree, and the fraction of dangling nodes. We model this relation through a stochastic equation inspired by the original definition of PageRank. Further, we use the theory of regular variation to prove that PageRank and in-degree follow power laws with the same exponent. The difference between these two power laws is in a multiple coefficient, which depends mainly on the fraction of dangling nodes, average indegree, the power law exponent, and damping factor. The out-degree distribution has a minor effect, which we explicitly quantify. Our theoretical predictions show a good agreement with experimental data on three different samples of the Web.