Learning Erdős-Rényi Random Graphs via Edge Detecting Queries
This work provides efficient algorithms for learning random graphs in query-based settings, which is incremental as it builds on group testing methods to address a specific graph learning bottleneck.
The paper tackles the problem of learning an unknown Erdős-Rényi random graph using edge-detecting queries, showing that it can be done with asymptotically vanishing error probability using only O(bar{k} log n) tests, compared to the harder case of arbitrary graphs which requires Ω(min{k^2 log n, n^2}) tests.
In this paper, we consider the problem of learning an unknown graph via queries on groups of nodes, with the result indicating whether or not at least one edge is present among those nodes. While learning arbitrary graphs with $n$ nodes and $k$ edges is known to be hard in the sense of requiring $Ω( \min\{ k^2 \log n, n^2\})$ tests (even when a small probability of error is allowed), we show that learning an Erdős-Rényi random graph with an average of $\bar{k}$ edges is much easier; namely, one can attain asymptotically vanishing error probability with only $O(\bar{k}\log n)$ tests. We establish such bounds for a variety of algorithms inspired by the group testing problem, with explicit constant factors indicating a near-optimal number of tests, and in some cases asymptotic optimality including constant factors. In addition, we present an alternative design that permits a near-optimal sublinear decoding time of $O(\bar{k} \log^2 \bar{k} + \bar{k} \log n)$.