Sampling methods and estimation of triangle count distributions in large networks

Abstract

This paper investigates the distributions of triangle counts per vertex and edge, as a means for network description, analysis, model building, and other tasks. The main interest is in estimating these distributions through sampling, especially for large networks. A novel sampling method tailored for the estimation analysis is proposed, with three sampling designs motivated by several network access scenarios. An estimation method based on inversion and an asymptotic method are developed to recover the entire distribution. A single method to estimate the distribution using multiple samples is also considered. Algorithms are presented to sample the network under the various access scenarios. Finally, the estimation methods on synthetic and real-world networks are evaluated in a data study.

Using a Bayesian approach to reconstruct graph statistics after edge sampling

“…These works have aimed also at designing sampling schemes specifically to preserve a given quantity. Other works have used sampling to estimate quantities on graphs that are prohibitively large to work with in their entirety, with a focus on triangle counting [25,2,23] or other motifs [16,6].…”

Reconstructing degree distribution and triangle counts from edge-sampled graphs

“…These works have aimed also at designing sampling schemes specifically to preserve a given quantity. Other works have used sampling to estimate quantities on graphs that are prohibitively large to work with in their entirety, with a focus on triangle counting [20,2,18] or other motifs [13,5].…”

“…The second by Ganguly et al [11] uses a range of estimators for individual vertex degrees in node-sampled networks; simple scale-up estimators, risk minimisation estimators and Bayes posterior estimates. Antunes et al [2] whose work was on sampling methods for estimating the triangle distribution, studied the n = 1 sample size problem as restricted access scenario as a case study. Other than this, little attention has been given specifically to these restricted access problems, noted in [24].…”

“…First, it is possible to estimate the prior using a Monte Carlo method to minimise the 2 norm of the error. In this approach, we find a degree sequence {κ i } which minimises min ||pκ i − k i || 2 2 , with the restrictions that the degree is an integer number and the sum of the degrees is equal to twice the number of links. To do this, we start with κ i = k i /p .…”

Introduction to the special issue on COMPLEX NETWORKS 2019

“…In Antunes et al (2021), the authors consider the problem of estimating the probability distribution of triangles. To this end, they propose a flexible sampling-based scheme with three sampling designs (without replacement, with replacement, and Bernoulli sampling) motivated by various network access scenarios (full access, restricted access, and streaming edges).…”