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In this paper, we consider the problems from the area of sublinear-timealgorithms of edge sampling, edge counting, and triangle counting. Part of ourcontribution is that we consider three different settings, differing in the wayin 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 aquery that returns the entire neighborhood at a unit cost, has recently beenconsidered in the applied community. Between these, we propose hash-orderedneighbor access, inspired by coordinated sampling, where we have a global fullyrandom 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 powerfulfull neighborhood access model. We provide matching upper bounds in the weakerhash-ordered neighbor access model. Our new faster algorithms can be provablyimplemented efficiently on massive graphs in external memory and with thecurrent APIs for, e.g., Twitter or Wikipedia. For triangle counting, we providea separation: a better upper bound with full neighborhood access than the knownlower bounds with indexed neighbor access. The technical core of our paper isour edge-sampling algorithm on which the other results depend.

Edge Sampling and Graph Parameter Estimation via Vertex Neighborhood Accesses