How large is your graph?
Varun Kanade, Frederik Mallmann-Trenn, Victor Verdugo

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 uses 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 . 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…
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Taxonomy
TopicsMarkov Chains and Monte Carlo Methods · Bayesian Modeling and Causal Inference · Stochastic processes and statistical mechanics
