Latent position models for random graphs attribute a vector to each vertex in the graph, and suppose that edge formation is a function of these unobserved latent positions. This leads to the natural incorporation of vertex-attributed data, since we may append this to the vertex latent position vectors, once these have been appropriately estimated. To extend this idea to the case of edge-attributed network information, we define edge latent positions and show how to estimate them in the stochastic blockmodel setting using spectral decompositions of the line graph adjacency matrix. We find that line graphs of stochastic blockmodels are stochastic blockmodel-like, and our concentration bounds on the spectra overcome the hurdles of random matrix size and lack of spectral gap which make naive study of random line graphs challenging. See more here: arXiv: 2103.14726 

If we only observe the interactions between entities in an organization, and nothing about the entities themselves, can we still learn about their underlying dynamics? We consider a model for time series of networks where the latent position for each entity corresponds to a trajectory of a stochastic process. In this setting, learning about the dynamics in the organization corresponds to understanding the evolution of this stochastic process. We prove that a measure of distance on the estimated latent positions between times concentrates around a corresponding distance on the latent position random variables at these times. When we approximate these distances with a curve in Euclidean space, or mirror, we obtain a visualization for these dynamics, and learn more about the underlying stochastic process. In this picture, we see the results of our approach for the case of email communication networks in a large organization. We see a significant departure in the spring and summer of 2020 from previous dynamics, likely corresponding to pandemic work-from-home changes. See more here:  arXiv:2205.06877 

To estimate the eigenvectors and eigenvalues of an unobserved parameter matrix P, one naturally considers the eigenvectors and eigenvalues of its observed random perturbation, A. Typically, the eigensystem of A is itself approximated using some iterative algorithm. But given that A is some distance from P, some amount of statistical error will always appear in the final estimate: Even if one chooses a very small numerical tolerance for the approximation of the eigensystem of A, the total error of the estimate cannot be decreased below this amount. So rather than wasting this additional computation, we might instead terminate the approximation algorithm once the size of the numerical error is smaller than the typical size of the statistical error. In the particular setting of a low-rank positive semidefinite mean matrix P, we show that "terminating early" in this way does not negatively impact the final estimation error, saving computational resources and time. See more here: arXiv: 1608.00451