Discriminative Distance-Based Network Indices and the Tiny-World Property

Distance-based indices, including closeness centrality, average path length, eccentricity and average eccentricity, are important tools for network analysis. In these indices, the distance between two vertices is measured by the size of shortest paths between them. However, this measure has shortcomings. A well-studied shortcoming is that extending it to disconnected graphs (and also directed graphs) is controversial. The second shortcoming is that when this measure is used in real-world networks, a huge number of vertices may have exactly the same closeness/eccentricity scores. The third shortcoming is that in many applications, the distance between two vertices not only depends on the size of shortest paths, but also on the number of shortest paths between them. In this paper, we develop a new distance measure between vertices of a graph that yields discriminative distance-based centrality indices. This measure is proportional to the size of shortest paths and inversely proportional to the number of shortest paths. We present algorithms for exact computation of the proposed discriminative indices. We then 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. Finally, we preform extensive experiments over several real-world networks from different domains and show that compared to the traditional indices, discriminative indices have usually much more discriminability. Our experiments reveal that real-world networks have usually a tiny average discriminative path length, bounded by a constant (e.g., 2). We refer to this property as the tiny-world property.