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

Computing Top-k Closeness Centrality Faster in Unweighted Graphs. (Technical Report)

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

Scaling up Group Closeness Maximization

Closeness is a widely-used centrality measure in social network analysis. For a node it indicates the inverse average shortest-path distance to the other nodes of the network. While the identification of the k nodes with highest closeness received significant attention, many applications are actually interested in finding a group of nodes that is central as a whole. For this problem, only recently a greedy algorithm with approximation ratio (1−1/e) has been proposed [Chen et al., ADC 2016]. Since this algorithm’s running time is still expensive for large networks, a heuristic without approximation guarantee has also been proposed in the same paper. In the present paper we develop new techniques to speed up the greedy algorithm without losing its theoretical guarantee. Compared to a straightforward implementation, our approach is orders of magnitude faster and, compared to the heuristic proposed by Chen et al., we always find a solution with better quality in a comparable running time in our experiments. Our method Greedy++ allows us to approximate the group with maximum closeness on networks with up to hundreds of millions of edges in minutes or at most a few hours. To have the same theoretical guarantee, the greedy approach by [Chen et al., ADC 2016] would take several days already on networks with hundreds of thousands of edges. In a comparison with the optimum, our experiments show that the solution found by Greedy++ is actually much better than the theoretical guarantee. Over all tested networks, the empirical approximation ratio is never lower than 0.97. Finally, we study for the first time the correlation between the top-k nodes with highest closeness and an approximation of the most central group in large complex networks and show that the overlap between the two is relatively small

A Faster Method to Estimate Closeness Centrality Ranking

Closeness centrality is one way of measuring how central a node is in the given network. The closeness centrality measure assigns a centrality value to each node based on its accessibility to the whole network. In real life applications, we are mainly interested in ranking nodes based on their centrality values. The classical method to compute the rank of a node first computes the closeness centrality of all nodes and then compares them to get its rank. Its time complexity is $O(n \cdot m + n)$, where $n$ represents total number of nodes, and $m$ represents total number of edges in the network. In the present work, we propose a heuristic method to fast estimate the closeness rank of a node in $O(\alpha \cdot m)$ time complexity, where $\alpha = 3$. We also propose an extended improved method using uniform sampling technique. This method better estimates the rank and it has the time complexity $O(\alpha \cdot m)$, where $\alpha \approx 10-100$. This is an excellent improvement over the classical centrality ranking method. The efficiency of the proposed methods is verified on real world scale-free social networks using absolute and weighted error functions

Discriminative Distance-Based Network Indices with Application to Link Prediction

In large networks, using the length of shortest paths as the distance measure has shortcomings. A well-studied shortcoming is that extending it to disconnected graphs and directed graphs is controversial. The second shortcoming is that a huge number of vertices may have exactly the same score. The third shortcoming is that in many applications, the distance between two vertices not only depends on the length of shortest paths, but also on the number of shortest paths. In this paper, first we develop a new distance measure between vertices of a graph that yields discriminative distance-based centrality indices. This measure is proportional to the length of shortest paths and inversely proportional to the number of shortest paths. We present algorithms for exact computation of the proposed discriminative indices. Second, we develop randomized algorithms that precisely estimate average discriminative path length and average discriminative eccentricity and show that they give $(\epsilon,\delta)$-approximations of these indices. Third, we perform extensive experiments over several real-world networks from different domains. In our experiments, we first show that compared to the traditional indices, discriminative indices have usually much more discriminability. Then, we show that our randomized algorithms can very precisely estimate average discriminative path length and average discriminative eccentricity, using only few samples. Then, we show that real-world networks have usually a tiny average discriminative path length, bounded by a constant (e.g., 2). Fourth, in order to better motivate the usefulness of our proposed distance measure, we present a novel link prediction method, that uses discriminative distance to decide which vertices are more likely to form a link in future, and show its superior performance compared to the well-known existing measures

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