Peer grading is the process of students reviewing each others' work, such as homework submissions, and has lately become a popular mechanism used in massive open online courses (MOOCs). Intrigued by this idea, we used it in a course on algorithms and data structures at the University of Hamburg. Throughout the whole semester, students repeatedly handed in submissions to exercises, which were then evaluated both by teaching assistants and by a peer grading mechanism, yielding a large dataset of teacher and peer grades. We applied different statistical and machine learning methods to aggregate the peer grades in order to come up with accurate final grades for the submissions (supervised and unsupervised, methods based on numeric scores and ordinal rankings). Surprisingly, none of them improves over the baseline of using the mean peer grade as the final grade. We discuss a number of possible explanations for these results and present a thorough analysis of the generated dataset.

We consider the problem of vector quantization of i.i.d. samples drawn from a density p on \mathbbR^d. It is desirable that the representatives selected by the quantization algorithm have the same distribution p as the original sample points. However, quantization algorithms based on Euclidean distance, such as k-means, do not have this property. We provide a solution to this problem that takes the unweighted k-nearest neighbor graph on the sample as input. In particular, it does not need to have access to the data points themselves. Our solution generates quantization centers that are “evenly spaced". We exploit this property to downsample geometric graphs and show that our method produces sparse downsampled graphs. Our algorithm is easy to implement, and we provide theoretical guarantees on the performance of the proposed algorithm.

Consider an unweighted k-nearest neighbor graph on n points that have been sampled i.i.d. from some unknown density p on R^d. We prove how one can estimate the density p just from the unweighted adjacency matrix of the graph, without knowing the points themselves or their distance or similarity scores. The key insights are that local differences in link numbers can be used to estimate some local function of p, and that integrating this function along shortest paths leads to an estimate of the underlying density.

We study the family of p-resistances on graphs for p ≥ 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p=1, the p-resistance coincides with the shortest path distance, for p=2 it coincides with the standard resistance distance, and for p → ∞ it converges to the inverse of the minimal s-t-cut in the graph. Secondly, we consider the special case of random geometric graphs (such as k-nearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phase-transition takes place. There exist two critical thresholds p^* and p^** such that if p < p^, then the p-resistance depends on meaningful global properties of the graph, whereas if p > p^, it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p^ = 1 + 1/(d-1) and p^** = 1 + 1/(d-2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p^* and p^** is an artifact of our proofs. We also relate our findings to Laplacian regularization and suggest to use q-Laplacians as regularizers, where q satisfies 1/p^* + 1/q = 1.

We consider the problem of local graph clustering where the aim is to discover the local cluster corresponding to a point of interest. The most popular algorithms to solve this problem start a random walk at the point of interest and let it run until some stopping criterion is met. The vertices visited are then considered the local cluster. We suggest a more powerful alternative, the multi-agent random walk. It consists of several ``agents'' connected by a fixed rope of length l. All agents move independently like a standard random walk on the graph, but they are constrained to have distance at most l from each other. The main insight is that for several agents it is harder to simultaneously travel over the bottleneck of a graph than for just one agent. Hence, the multi-agent random walk has less tendency to mistakenly merge two different clusters than the original random walk. In our paper we analyze the multi-agent random walk theoretically and compare it experimentally to the major local graph clustering algorithms from the literature. We find that our multi-agent random walk consistently outperforms these algorithms.