We study differentially private (DP) algorithms for recovering clusters in well-clustered graphs, which are graphs whose vertex set can be partitioned into a small number of sets, each inducing a subgraph of high inner conductance and small outer conductance. Such graphs have widespread application as a benchmark in the theoretical analysis of spectral clustering. We provide an efficient ($\epsilon$,$\delta$)-DP algorithm tailored specifically for such graphs. Our algorithm draws inspiration from the recent work of Chen et al., who developed DP algorithms for recovery of stochastic block models in cases where the graph comprises exactly two nearly-balanced clusters. Our algorithm works for well-clustered graphs with $k$ nearly-balanced clusters, and the misclassification ratio almost matches the one of the best-known non-private algorithms. We conduct experimental evaluations on datasets with known ground truth clusters to substantiate the prowess of our algorithm. We also show that any (pure) $\epsilon$-DP algorithm would result in substantial error.

Fifty years ago, mathematician Paul Erds posed a problem to friends at one of his regular tea parties. The trio thought they would be able to come up with a solution the same afternoon. It took 49 years for other mathematicians to provide an answer. The Erds-Faber-Lovász conjecture focused on a familiar question in mathematics, one of graph coloring. However, this was not on a conventional graph, but on another more-complex structure: a hypergraph.

Walking rows of soybeans in the mid-summer heat is an exhausting but essential chore in breeding new cultivars. Researchers brave the heat daily during crucial parts of the growing season to look for plants showing desirable traits, such as early pod maturity. But without a way to automate detection of these traits, breeders can't test as many plots as they'd like in a given year, elongating the time it takes to bring new cultivars to market. In a new study from the University of Illinois, researchers predict soybean maturity date within two days using drone images and artificial intelligence, greatly reducing the need for boots on the ground. "Assessing pod maturity is very time consuming and prone to errors. It's a scoring system based on the color of the pod, so it is also subject to human bias," says Nicolas Martin, assistant professor in the Department of Crop Sciences at Illinois and co-author on the study.