Tree decompositions and social graphs

Abstract

Recent work has established that large informatics graphs such as social and information networks have non-trivial tree-like structure when viewed at moderate size scales. Here, we present results from the first detailed empirical evaluation of the use of tree decomposition (TD) heuristics for structure identification and extraction in social graphs. Although TDs have historically been used in structural graph theory and scientific computing, we show that-even with existing TD heuristics developed for those very different areas-TD methods can identify interesting structure in a wide range of realistic informatics graphs. Our main contributions are the following: we show that TD methods can identify structures that correlate strongly with the core-periphery structure of realistic networks, even when using simple greedy heuristics; we show that the peripheral bags of these TDs correlate well with low-conductance communities (when they exist) found using local spectral computations; and we show that several types of large-scale "ground-truth" communities, defined by demographic metadata on the nodes of the network, are well-localized in the large-scale and/or peripheral structures of the TDs. Our other main contributions are the following: we provide detailed empirical results for TD heuristics on toy and synthetic networks to establish a baseline to understand better the behavior of the heuristics on more complex real-world networks; and we prove a theorem providing formal justification for the intuition that the only two impediments to low-distortion hyperbolic embedding are high tree-width and long geodesic cycles. Our results suggest future directions for improved TD heuristics that are more appropriate for realistic social graphs.

Connected Tree-Width

“…Clearly, graphs of connected tree-width < k also have tree-length < k, so by Theorem 1.1 the tree-length of G is bounded by the same function f (k, ) as its connected tree-width. In fact, Reidl and Sullivan [11,1] observed that the following better bound follows directly from of our lemmas for the proof of Theorem 1.1: Theorem 1.4. If G has tree-width < k and no geodesic cycle longer than , and G is not a forest, then the tree-length of G is at most (k − 2).…”

“…The tree-length of a graph G is the smallest value, minimized over all treedecompositions of G, of the maximum diameter in G of a part of this decomposition [6]. Reidl and Sullivan, see [11,1], noticed that our lemmas from Sections 3 and 4 can be applied directly to give a better bound on the tree-length of G than the bound of f ( , k) implied by Theorem 1.1: Theorem 1.4. If G has tree-width < k and no geodesic cycle longer than , and G is not a forest, then the tree-length of G is at most (k − 2).…”

Scalable Betweenness Centrality Maximization via Sampling

“…It is worth emphasizing that the classical Barabási-Albert random tree model [5,34] belongs to this category. For a recent study of the treewidth parameter on real-world networks, see [1].…”

Smart Packet: Re-distributing the Routing Intelligence among Network Components in SDNs

“…It has been observed that hyperbolic embedding approaches could be used to map network's vertices on a multi-dimensional hyperbolic geometric space [15]- [18]. These spaces resemble well the structure of aggregationbased topologies.…”