2017/02/13 by Varun Kanade, Kanade, Varun, Frederik Mallmann-Trenn +4
Computer Science · Mathematics · #Bayesian Modeling and Causal Inference #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #cs.DS
paper · pdf · doi:10.48550/arxiv.1702.03959
arxiv created 2017/02/13 · openalex publication_date 2017/02/13 · arxiv updated 2017/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
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 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 π uses O((1)/(‖π‖2) + davg) 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 tmix ⋅ 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.