The sliding window model of computation captures scenarios in which data is arriving continuously, but only the latest $w$ elements should be used for analysis. The goal is to design algorithms that update the solution efficiently with each arrival rather than recomputing it from scratch. In this work, we focus on $k$-clustering problems such as $k$-means and $k$-median. In this setting, we provide simple and practical algorithms that offer stronger performance guarantees than previous results. Empirically, we show that our methods store only a small fraction of the data, are orders of magnitude faster, and find solutions with costs only slightly higher than those returned by algorithms with access to the full dataset.

The streaming computation model is a standard model for large-scale data analysis: the input arrives one element at a time, and the goal is to maintain an approximately optimal solution using only a constant, or, at worst, polylogarithmic space.

Given a connected graph G =( V , E ), where V denotes the set of nodes and E the set of edges of the graph, the length (that is, the number of edges) of the shortest path between two nodes v and w is denoted by d ( v , w ). The closeness centrality of a vertex v is then defined as n =1/Σ w ∈ V d ( v , w ), where n =| V |. This measure is widely used in the analysis of real-world complex networks, and the problem of selecting the k most central vertices has been deeply analyzed in the last decade. However, this problem is computationally not easy, especially for large networks: in the first part of the article, we prove that it is not solvable in time O (| E | 2=ϵ ) on directed graphs, for any constant ϵ > 0, under reasonable complexity assumptions. Furthermore, we propose a new algorithm for selecting the k most central nodes in a graph: we experimentally show that this algorithm improves significantly both the textbook algorithm, which is based on computing the distance between all pairs of vertices, and the state of the art. For example, we are able to compute the top k nodes in few dozens of seconds in real-world networks with millions of nodes and edges. Finally, as a case study, we compute the 10 most central actors in the Internet Movie Database (IMDB) collaboration network, where two actors are linked if they played together in a movie, and in the Wikipedia citation network, which contains a directed edge from a page p to a page q if p contains a link to q .

In recent years, researchers proposed several algorithms that compute metric quantities of real-world complex networks, and that are very efficient in practice, although there is no worst-case guarantee. In this work, we propose an axiomatic framework to analyze the performances of these algorithms, by proving that they are efficient on the class of graphs satisfying certain properties. Furthermore, we prove that these properties are verified asymptotically almost surely by several probabilistic models that generate power law random graphs, such as the Configuration Model, the Chung-Lu model, and the Norros-Reittu model. Thus, our results imply average-case analyses in these models. For example, in our framework, existing algorithms can compute the diameter and the radius of a graph in subquadratic time, and sometimes even in time n1+o(1). Moreover, in some regimes, it is possible to compute the k most central vertices according to closeness centrality in subquadratic time, and to design a distance oracle with sublinear query time and subquadratic space occupancy. In the worst case, it is impossible to obtain comparable results for any of these problems, unless widely- believed conjectures are false.

Centrality indices are widely used analytic measures for the importance of nodes in a network. Closeness centrality is very popular among these measures. For a single node v, it takes the sum of the distances of v to all other nodes into account. The currently best algorithms in practical applications for computing the closeness for all nodes exactly in unweighted graphs are based on breadth-first search (BFS) from every node. Thus, even for sparse graphs, these algorithms require quadratic running time in the worst case, which is prohibitive for large networks. In many relevant applications, however, it is unnecessary to compute closeness values for all nodes. Instead, one requires only the k nodes with the highest closeness values in descending order. Thus, we present a new algorithm for computing this top-k ranking in unweighted graphs. Following the rationale of previous work, our algorithm significantly reduces the number of traversed edges. It does so by computing upper bounds on the closeness and stopping the current BFS search when k nodes already have higher closeness than the bounds computed for the other nodes. In our experiments with real-world and synthetic instances of various types, one of these new bounds is good for small-world graphs with low diameter (such as social networks), while the other one excels for graphs with high diameter (such as road networks). Combining them yields an algorithm that is faster than the state of the art for top-k computations for all test instances, by a wide margin for high-diameter graphs. Finally, we prove that the quadratic worst-case complexity cannot be improved on directed, disconnected graphs, under reasonable complexity assumptions.

We present KADABRA, a new algorithm to approximate betweenness centrality in directed and undirected graphs, which significantly outperforms all previous approaches on real-world complex networks. The efficiency of the new algorithm relies on two new theoretical contributions, of independent interest. The first contribution focuses on sampling shortest paths, a subroutine used by most algorithms that approximate betweenness centrality. We show that, on realistic random graph models, we can perform this task in time |E|^{1/2+o(1)} with high probability, obtaining a significant speedup with respect to the Theta(|E|) worst-case performance. We experimentally show that this new technique achieves similar speedups on real-world complex networks, as well. The second contribution is a new rigorous application of the adaptive sampling technique. This approach decreases the total number of shortest paths that need to be sampled to compute all betweenness centralities with a given absolute error, and it also handles more general problems, such as computing the k most central nodes. Furthermore, our analysis is general, and it might be extended to other settings, as well.

MOTIVATION: The increasing availability of metabolomics data enables to better understand the metabolic processes involved in the immediate response of an organism to environmental changes and stress. The data usually come in the form of a list of metabolites whose concentrations significantly changed under some conditions, and are thus not easy to interpret without being able to precisely visualize how such metabolites are interconnected. RESULTS: We present a method that enables to organize the data from any metabolomics experiment into metabolic stories. Each story corresponds to a possible scenario explaining the flow of matter between the metabolites of interest. These scenarios may then be ranked in different ways depending on which interpretation one wishes to emphasize for the causal link between two affected metabolites: enzyme activation, enzyme inhibition or domino effect on the concentration changes of substrates and products. Equally probable stories under any selected ranking scheme can be further grouped into a single anthology that summarizes, in a unique subnetwork, all equivalently plausible alternative stories. An anthology is simply a union of such stories. We detail an application of the method to the response of yeast to cadmium exposure. We use this system as a proof of concept for our method, and we show that we are able to find a story that reproduces very well the current knowledge about the yeast response to cadmium. We further show that this response is mostly based on enzyme activation. We also provide a framework for exploring the alternative pathways or side effects this local response is expected to have in the rest of the network. We discuss several interpretations for the changes we see, and we suggest hypotheses that could in principle be experimentally tested. Noticeably, our method requires simple input data and could be used in a wide variety of applications. AVAILABILITY AND IMPLEMENTATION: The code for the method presented in this article is available at http://gobbolino.gforge.inria.fr.