Metric tree‐like structures in real‐world networks: an empirical study

Abstract

Based on solid theoretical foundations, we present strong evidence that a number of real-world networks, taken from different domains (such as Internet measurements, biological data, web graphs, and social and collaboration networks) exhibit tree-like structures from a metric point of view. We investigate a few graph parameters, namely, the tree-distortion and the tree-stretch, the tree-length and the tree-breadth, Gromov's hyperbolicity, the cluster-diameter and the cluster-radius in a layering partition of a graph; such parameters capture and quantify this phenomenon of being metrically close to a tree. By bringing all those parameters together, we provide efficient means for detecting such metric tree-like structures in large-scale networks. We also show how such structures can be used. For example, they are helpful in efficient and compact encoding of approximate distance and almost shortest path information and in quick and accurate estimation of diameters and radii of those networks. Estimating the diameter and estimating the radius of a graph (or distances between arbitrary vertices) are fundamental primitives in many network and graph mining algorithms.

Parameterized Approximation Algorithms for Some Location Problems in Graphs

“…1 Create a layering partition T of G. 2 For each cluster C of T , set r(C) := minv∈C r(v). 3 Compute a minimum r-dominating subtree Tr for T (see [12]).…”

“…To compare these results with the results of [7,13], notice that, for any graph G, its hyperbolicity δ is at most ∆ [2] and at most two times its tree-breadth ρ [6], and the inequalities are sharp.…”

“…The Domination problem is notorious also in the theory of fixed-parameter tractability (see, e. g., [11,18] for an introduction to parameterized complexity). It was the first problem to be shown W [2]-complete [11], and it is hence unlikely to be FPT, i. e., unlikely to have an algorithm with runtime f (k)n c for f a computable function, k the size of an optimal solution, c a constant, and n the number of vertices of the input graph. Similar results are known also for the connected domination problem [17].…”

“…Empirical analysis [2], however, have shown that ∆ is usually very small. Therefore, we use a so-called one-sided binary search.…”

Gromov Hyperbolicity in Mycielskian Graphs

“…The concept of hyperbolicity appears also in discrete mathematics, algorithms and networking [12]. For example, it has been shown empirically in [13] that the Internet topology embeds with better accuracy into a hyperbolic space than into a Euclidean space of comparable dimension (formal proofs that the distortion is related to the hyperbolicity can be found in [14]); furthermore, it is evidenced that many real networks are hyperbolic (see, e.g., [15][16][17][18][19]). Recently, among the practical network applications, hyperbolic spaces were used to study secure transmission of information on the Internet or the way viruses are spread through the network (see [20,21]); also to traffic flow and effective resistance of networks [22][23][24].…”