The Bulk and the Extremes of Minimal Spanning Acycles and Persistence Diagrams of Random Complexes

Frieze showed that the expected weight of the minimum spanning tree (MST) of the uniformly weighted graph converges to $\zeta(3)$. Recently, this result was extended to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analogue -- the Minimum Spanning Acycle (MSA). In this work, we go beyond and look at the histogram of the weights in this random MSA -- both in the bulk and in the extremes. In particular, we focus on the `incomplete' setting, where one has access only to a fraction of the potential face weights. Our first result is that the empirical distribution of the MSA weights asymptotically converges to a measure based on the shadow -- the complement of graph components in higher dimensions. As far as we know, this result is the first to explore the connection between the MSA weights and the shadow. Our second result is that the extremal weights converge to an inhomogeneous Poisson point process. A interesting consequence of our two results is that we can also state the distribution of the death times in the persistence diagram corresponding to the above weighted complex, a result of interest in applied topology.Comment: 15 pages, 5 figures, Corrected Typo

Online Learning with Adversaries: A Differential-Inclusion Analysis

We introduce an observation-matrix-based framework for fully asynchronous online Federated Learning (FL) with adversaries. In this work, we demonstrate its effectiveness in estimating the mean of a random vector. Our main result is that the proposed algorithm almost surely converges to the desired mean $\mu.$ This makes ours the first asynchronous FL method to have an a.s. convergence guarantee in the presence of adversaries. We derive this convergence using a novel differential-inclusion-based two-timescale analysis. Two other highlights of our proof include (a) the use of a novel Lyapunov function to show that $\mu$ is the unique global attractor for our algorithm's limiting dynamics, and (b) the use of martingale and stopping-time theory to show that our algorithm's iterates are almost surely bounded.Comment: 6 pages, 2 figure