Edge Sampling and Graph Parameter Estimation via Vertex Neighborhood Accesses

TL;DR

This paper introduces new models and algorithms for sublinear-time edge and triangle counting in massive graphs, providing lower bounds, upper bounds, and efficiency improvements across different neighborhood access settings.

Contribution

It proposes a novel hash-ordered neighbor access model, establishes matching bounds for edge sampling, and demonstrates improved algorithms for triangle counting in various access models.

Findings

Lower bounds in full neighborhood access model for edge sampling

Matching upper bounds in hash-ordered neighbor access model

Improved algorithms for triangle counting with full neighborhood access

Abstract

In this paper, we consider the problems from the area of sublinear-time algorithms of edge sampling, edge counting, and triangle counting. Part of our contribution is that we consider three different settings, differing in the way in which one may access the neighborhood of a given vertex. In previous work, people have considered indexed neighbor access, with a query returning the $i$-th neighbor of a given vertex. Full neighborhood access model, which has a query that returns the entire neighborhood at a unit cost, has recently been considered in the applied community. Between these, we propose hash-ordered neighbor access, inspired by coordinated sampling, where we have a global fully random hash function, and can access neighbors in order of their hash values, paying a constant for each accessed neighbor. For edge sampling and counting, our new lower bounds are in the most powerful full neighborhood access model. We provide matching upper bounds in the weaker hash-ordered neighbor access model. Our new faster algorithms can be provably implemented efficiently on massive graphs in external memory and with the current APIs for, e.g., Twitter or Wikipedia. For triangle counting, we provide a separation: a better upper bound with full neighborhood access than the known lower bounds with indexed neighbor access. The technical core of our paper is our edge-sampling algorithm on which the other results depend.