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We present the first comprehensive characterization of the diffusion of ideas on Twitter, studying more than 5.96 million topics that include both popular and less popular topics. On a data set containing approximately 10 million users and a comprehensive scraping of 196 million tweets, we perform a rigorous temporal and spatial analysis, investigating the time-evolving properties of the subgraphs formed by the users discussing each topic. We focus on two different notions of the spatial: the network topology formed by follower-following links on Twitter, and the geospatial location of the users. We investigate the effect of initiators on the popularity of topics and find that users with a high number of followers have a strong impact on topic popularity. We deduce that topics become popular when disjoint clusters of users discussing them begin to merge and form one giant component that grows to cover a significant fraction of the network. Our geospatial analysis shows that highly popular topics are those that cross regional boundaries aggressively.

We design a variation of skip lists that performs well for generally biased access sequences. Given n items, each with a positive weight w i , 1 ≤ i ≤ n, the time to access item i is O(1 + log(W/w i )), where W = n i=1 w i ; the data structure is dynamic. We present two instantiations of biased skip lists, one of which achieves this bound in the worst case, the other in the expected case. The structures are nearly identical; the deterministic one simply ensures the balance condition that the randomized one achieves probabilistically. We use the same method to analyze both. Introduction.The primary goal of data structures research is to design data organization mechanisms that admit fast access and update operations. For a generic n-element ordered data set that is accessed and updated uniformly, this goal is typically satisfied by dictionaries that achieve O(log n)-time performance for searches and updates; for example, AVL-trees [2], red-black trees [12], and (a, b)-trees [13].Nevertheless, many dictionary applications involve sets of weighted data items that are searched and updated non-uniformly according to those weights; that is, they are biased. For example, most operating systems textbooks (e.g., see [23]) devote major coverage to methods for dealing with biasing in memory requests. Other recent examples of biased sets include client web server requests [11] and DNS lookups [6]. For such applications, a biased search structure is more appropriate-that is, a structure that achieves search times faster than log n for highly weighted items. Biased searching is also useful in auxiliary structures deployed inside other data structures [5], [10], [21].Formally, a biased dictionary is a data structure that maintains an ordered set X , each element i of which has a weight, w i ; without loss of generality, we assume w i ≥ 1. The operations are as follows:

In this paper we study the problem of how resilient networks are to node faults. Specifically, we investigate the question of how many faults a network can sustain so that it still contains a large (i.e. linear-sized) connected component that still has approximately the same expansion as the original fault-free network. For this we apply a pruning technique which culls away parts of the faulty network which have poor expansion. This technique can be applied to both adversarial faults and to random faults. For adversarial faults we prove that for every network with expansion alpha, a large connected component with basically the same expansion as the original network exists for up to a constant times alpha n faults. This result is tight in the sense that every graph G of size n and uniform expansion alpha(.), i.e. G has an expansion of alpha(n) and every subgraph G' of size m of G has an expansion of O(alpha(m)), can be broken into sublinear components with omega(alpha(n) n) faults. For random faults we observe that the situation is significantly different, because in this case the expansion of a graph only gives a very weak bound on its resilience to random faults. More specifically, there are networks of uniform expansion O(sqrt{n}) that are resilient against a constant fault probability but there are also networks of uniform expansion Omega(1/log n) that are not resilient against a O(1/log n) fault probability. Thus, a different parameter is needed. For this we introduce the span of a graph which allows us to determine the maximum fault probability in a much better way than the expansion can. We use the span to show the first known results for the effect of random faults on the expansion of d-dimensional meshes.Comment: 8 pages; to appear at SPAA 200

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