How large is your graph?

TL;DR

This paper investigates the query complexity of estimating graph size from local access, establishing tight bounds for undirected graphs and demonstrating the difficulty of the problem in directed graphs without strong structural assumptions.

Contribution

It provides tight bounds for undirected graph size estimation and introduces a new parameter for directed graphs that characterizes when sublinear query algorithms are possible.

Findings

Query complexity for undirected graphs is tight and depends on stationary distribution and average degree.

Estimating size in directed graphs generally requires linear queries unless strong structural properties are known.

A new graph parameter generalizing conductance determines when sublinear algorithms are feasible.

Abstract

We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a \emph{seed} node and is allowed to make queries about neighbours of nodes that have already been seen. In the case of undirected graphs, an estimator of Katzir et al. (2014) based on a sample from the stationary distribution $\pi$ uses $O\left(\frac{1}{\|\pi\|_2} + \text{davg}\right)$ queries, we prove that this is tight. In addition, we establish this as a lower bound even when the algorithm is allowed to crawl the graph arbitrarily, the results of Katzir et al. give an upper bound that is worse by a multiplicative factor $t_\text{mix} \cdot \log(n)$. The picture becomes significantly different in the case of directed graphs. We show that without strong assumptions on the graph structure, the number of nodes cannot be predicted to within a constant multiplicative factor without using a number of queries that are at least linear in the number of nodes, in particular, rapid mixing and small diameter, properties that most real-world networks exhibit, do not suffice. The question of interest is whether any algorithm can beat breadth-first search. We introduce a new parameter, generalising the well-studied conductance, such that if a suitable bound on it exists and is known to the algorithm, the number of queries required is sublinear in the number of edges, we show that this is tight.