Small-world graphs have characteristically low average distance and thus cause force-directed methods to generate drawings that look like hairballs. This is by design as the inherent objective of these methods is a globally uniform edge length or, more generally, accurate distance representation. The problem arises, for instance, with graphs of high density or high conductance, or in the presence of high-degree vertices, all of which tend to pull vertices together and thus result in clutter overspreading variation in local density.We here propose a method specifically for a class of small-world graphs that are typical for online social networks. The method is based on a spanning subgraph that is sparse but connected and consists of strong ties holding together communities. To identify these ties we propose a novel criterion for structural embeddedness. It is based on a weighted accumulation of triangles in quadrangles and can be determined efficiently. An evaluation on empirical and generated networks indicates that our approach improves upon previous methods using other edge indices. Although primarily designed to achieve more informative drawings, our spanning subgraph may also serve as a sparsifier that trims a small-world graph prior to the application of a clustering algorithm.

We show that most algorithms from the literature on listing the triangles of a graph have a common abstraction. Our unifying framework highlights that these seemingly different algorithms are in fact instantiations of a single generic procedure, and even suggests some additional variants. More importantly, it yields parsimonious implementations that are in general more efficient than those described in the original works. In addition, we show that the running time of nearly every triangle listing variant is in O(a(G)m), where a(G) is the arboricity of the graph and m the number of edges. So far this bound has been proven only for Chiba and Nishizeki's (SIAM J. Computing, 1985) triangle listing algorithm. Finally, algorithmic experimentation reveals that an improved implementation of this algorithm outperforms all subsequently proposed algorithms.

Force-directed layout methods are among the most common approaches for drawing general graphs. Among them, stress minimization produces layouts of comparatively high quality while also imposing comparatively high computational demands. We propose a speed-up method based on the aggregation of terms in the objective function. It is akin to aggregate repulsion from far-away nodes during spring embedding but transfers the idea from the layout space into a preprocessing phase. An initial experimental study informs a method to select representatives, and subsequent more extensive experiments indicate that our method yields better approximations of minimum-stress layouts in less time than related methods.

The prevalence of select substructures is an indicator of network effects in applications such as social network analysis and systems biology. Moreover, subgraph statistics are pervasive in stochastic network models, and they need to be assessed repeatedly in MCMC sampling and estimation algorithms. We present a new approach to count all induced and non-induced four-node subgraphs (the quad census) on a per-node and per-edge basis, complete with a separation into their non-automorphic roles in these subgraphs. It is the first approach to do so in a unified manner, and is based on only a clique-listing subroutine. Computational experiments indicate that, despite its simplicity, the approach outperforms previous, less general approaches.By way of the more presentable triad census, we additionally show how to extend the quad census to directed graphs. As a byproduct we obtain the asymptotically fastest triad census algorithm to date.

Abstract. Force-directed layout methods constitute the most common approach to draw general graphs. Among them, stress minimization produces layouts of comparatively high quality but also imposes comparatively high computational demands. We propose a speed-up method based on the aggregation of terms in the objective function. It is akin to aggregate repulsion from far-away nodes during spring embedding but transfers the idea from the layout space into a preprocessing phase. An initial experimental study informs a method to select representatives, and subsequent more extensive experiments indicate that our method yields better approximations of minimum-stress layouts in less time than related methods.