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Most real-world networks are too large to be measured or studied directly andthere is substantial interest in estimating global network properties fromsmaller sub-samples. One of the most important global properties is the numberof vertices/nodes in the network. Estimating the number of vertices in a largenetwork is a major challenge in computer science, epidemiology, demography, andintelligence analysis. In this paper we consider a population random graph G =(V;E) from the stochastic block model (SBM) with K communities/blocks. A sampleis obtained by randomly choosing a subset W and letting G(W) be the inducedsubgraph in G of the vertices in W. In addition to G(W), we observe the totaldegree of each sampled vertex and its block membership. Given this partialinformation, we propose an efficient PopULation Size Estimation algorithm,called PULSE, that correctly estimates the size of the whole population as wellas the size of each community. To support our theoretical analysis, we performan exhaustive set of experiments to study the effects of sample size, K, andSBM model parameters on the accuracy of the estimates. The experimental resultsalso demonstrate that PULSE significantly outperforms a widely-used methodcalled the network scale-up estimator in a wide variety of scenarios. Weconclude with extensions and directions for future work.

Estimating the Size of a Large Network and its Communities from a Random Sample