KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation

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 $|E|^{\frac{1}{2}+o(1)}$ 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.Comment: Some typos correcte

Into the Square: On the Complexity of Some Quadratic-time Solvable Problems

International audienceWe analyze several quadratic-time solvable problems, and we show that these problems are not solvable in truly subquadratic time (that is, in time O(n2−ϵ) for some ϵ>0), unless the well known Strong Exponential Time Hypothesis (in short, SETH) is false. In particular, we start from an artificial quadratic-time solvable variation of the k-Sat problem (already introduced and used in the literature) and we will construct a web of Karp reductions, proving that a truly subquadratic-time algorithm for any of the problems in the web falsifies SETH. Some of these results were already known, while others are, as far as we know, new. The new problems considered are: computing the betweenness centrality of a vertex (the same result was proved independently by Abboud et al.), computing the minimum closeness centrality in a graph, computing the hyperbolicity of a graph, and computing the subset graph of a collection of sets. On the other hand, we will show that testing if a directed graph is transitive and testing if a graph is a comparability graph are subquadratic-time solvable (our algorithm is practical, since it is not based on intricate matrix multiplication algorithms)

KADABRA is an ADaptive Algorithm for Betweenness via Random Approximation

International audienceWe 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 Θ(|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

Hyperbolicity Measures "Democracy" in Real-World Networks

We analyze the hyperbolicity of real-world networks, a geometric quantity that measures if a space is negatively curved. In our interpretation, a network with small hyperbolicity is "aristocratic", because it contains a small set of vertices involved in many shortest paths, so that few elements "connect" the systems, while a network with large hyperbolicity has a more "democratic" structure with a larger number of crucial elements. We prove mathematically the soundness of this interpretation, and we derive its consequences by analyzing a large dataset of real-world networks. We confirm and improve previous results on hyperbolicity, and we analyze them in the light of our interpretation. Moreover, we study (for the first time in our knowledge) the hyperbolicity of the neighborhood of a given vertex. This allows to define an "influence area" for the vertices in the graph. We show that the influence area of the highest degree vertex is small in what we define "local" networks, like most social or peer-to-peer networks. On the other hand, if the network is built in order to reach a "global" goal, as in metabolic networks or autonomous system networks, the influence area is much larger, and it can contain up to half the vertices in the graph. In conclusion, our newly introduced approach allows to distinguish the topology and the structure of various complex networks

On Computing the Hyperbolicity of Real-World Graphs

International audienceThe (Gromov) hyperbolicity is a topological property of a graph, which has been recently applied in several different contexts, such as the design of routing schemes, network security, computational biology, the analysis of graph algorithms, and the classification of complex networks. Computing the hyperbolicity of a graph can be very time consuming: indeed, the best available algorithm has running-time O(n^{3.69}), which is clearly prohibitive for big graphs. In this paper, we provide a new and more efficient algorithm: although its worst-case complexity is O(n^4), in practice it is much faster, allowing, for the first time, the computation of the hyperbolicity of graphs with up to 200,000 nodes. We experimentally show that our new algorithm drastically outperforms the best previously available algorithms, by analyzing a big dataset of real-world networks. Finally, we apply the new algorithm to compute the hyperbolicity of random graphs generated with the Erdös-Renyi model, the Chung-Lu model, and the Configuration Model

Computing Top-k Closeness Centrality Faster in Unweighted Graphs

International audienceGiven a connected graph G = (V,E), the closeness centrality of a vertex v is defined as (n-1 / \Sigma_{w \in V} d(v,w). 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 analysed in the last decade. However, this problem is computationally not easy, especially for large networks: in the first part of the paper, we prove that it is not solvable in time O(|E|^{2-epsilon) on directed graphs, for any constant epsilon > 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 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

Algorithms for metric properties of large real-world networks from theory to practice and back

Motivated by complex networks analysis, we study algorithms that compute metric properties of real-world graphs. In the worst-case, we prove that, under reasonable assumptions, the trivial algorithms based on computing the distance between all pairs of nodes are almost optimal. Then, we try to overcome these bottlenecks by designing new algorithms that work surprisingly well in practice, even if they are not efficient in the worst-case. We propose new algorithms for the computation of the diameter, the radius, the closeness centrality, the betweenness centrality, and and the hyperbolicity: these algorithms are much faster than the textbook algorithms, when tested on real-world complex networks, and they also outperform similar approaches that were published in the literature. For example, to solve several problems, our algorithms are thousands, and even billions of times faster than the textbook algorithm, on standard inputs. However, the experimental results are not completely satisfactory from a theoretical point of view. In order to fill this gap, we develop an axiomatic framework where these algorithms can be evaluated and compared: we define some axioms, we show that real-world networks satisfy these axioms, and we prove that our algorithms are efficient if the input satisfies these axioms. This way, we obtain results that do not depend on the specific dataset used, and we highlight the main properties of the input that are exploited. A further confirmation of the validity of this approach is that the results obtained mirror very well the empirical results. Finally, we prove that the axioms are verified on realistic models of random graphs, such as the Configuration Model, the Chung-Lu model, and the Norros-Reittu model. This way, our axiomatic analyses can be turned into average-case analyses on these models, with no modification. This modular approach to average-case complexity has two advantages: we can prove results in several models with a single worst-case analysis, and we can validate the choice of the model by showing that the axioms used are verified in practic