Ls(d) (30) which is a modification of our supervised loss, whereDl and Du denote the set of labeled and unlabeleddata,respectively.

Projecting this differential equation on the last coordinate givesdHe+1t = dt, that is, He+1t = t. Finally,let (a(n))n N beaCauchysequencein T . Straightforward calculations yield the equality,valid for any x R, tanh(x)=2σ(2x) 1. But,foranyn 1, Next, it is clear that the signature of a constant path is equal to1 = (1,0,...,0,...) which is the nullelementinT . More precisely, fork = 1, C(1;0) = 1 00 = 1 and C(1;1) = 0 01 = 0. Assume that the formula is true at orderk. Then, at order k + 1, there are two cases.

In this paper, we propose to estimate the invariant subspace across heterogeneous multiple networks using a novel bias-corrected joint spectral embedding algorithm. The proposed algorithm recursively calibrates the diagonal bias of the sum of squared network adjacency matrices by leveraging the closed-form bias formula and iteratively updates the subspace estimator using the most recent estimated bias. Correspondingly, we establish a complete recipe for the entrywise subspace estimation theory for the proposed algorithm, including a sharp entrywise subspace perturbation bound and the entrywise eigenvector central limit theorem. Leveraging these results, we settle two multiple network inference problems: the exact community detection in multilayer stochastic block models and the hypothesis testing of the equality of membership profiles in multilayer mixed membership models. Our proof relies on delicate leave-one-out and leave-two-out analyses that are specifically tailored to block-wise symmetric random matrices and a martingale argument that is of fundamental interest for the entrywise eigenvector central limit theorem.

Densest subgraph detection is a fundamental graph mining problem, with a large number of applications. There has been a lot of work on efficient algorithms for finding the densest subgraph in massive networks. However, in many domains, the network is private, and returning a densest subgraph can reveal information about the network. Differential privacy is a powerful framework to handle such settings. We study the densest subgraph problem in the edge privacy model, in which the edges of the graph are private. We present the first sequential and parallel differentially private algorithms for this problem. We show that our algorithms have an additive approximation guarantee. We evaluate our algorithms on a large number of real-world networks, and observe a good privacy-accuracy tradeoff when the network has high density.