Explaining Snapshots of Network Diffusions: Structural and Hardness Results

Much research has been done on studying the diffusion of ideas or technologies on social networks including the \textit{Influence Maximization} problem and many of its variations. Here, we investigate a type of inverse problem. Given a snapshot of the diffusion process, we seek to understand if the snapshot is feasible for a given dynamic, i.e., whether there is a limited number of nodes whose initial adoption can result in the snapshot in finite time. While similar questions have been considered for epidemic dynamics, here, we consider this problem for variations of the deterministic Linear Threshold Model, which is more appropriate for modeling strategic agents. Specifically, we consider both sequential and simultaneous dynamics when deactivations are allowed and when they are not. Even though we show hardness results for all variations we consider, we show that the case of sequential dynamics with deactivations allowed is significantly harder than all others. In contrast, sequential dynamics make the problem trivial on cliques even though it's complexity for simultaneous dynamics is unknown. We complement our hardness results with structural insights that can help better understand diffusions of social networks under various dynamics.Comment: 14 pages, 3 figure

Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness

Understanding the singular value spectrum of an n x n matrix A is a fundamental task in countless numerical computation and data analysis applications. In matrix multiplication time, it is possible to perform a full SVD of A and directly compute the singular values sigma_1,...,sigma_n. However, little is known about algorithms that break this runtime barrier. Using tools from stochastic trace estimation, polynomial approximation, and fast linear system solvers, we show how to efficiently isolate different ranges of A\u27s spectrum and approximate the number of singular values in these ranges. We thus effectively compute an approximate histogram of the spectrum, which can stand in for the true singular values in many applications. We use our histogram primitive to give the first algorithms for approximating a wide class of symmetric matrix norms and spectral sums faster than the best known runtime for matrix multiplication. For example, we show how to obtain a (1 + epsilon) approximation to the Schatten 1-norm (i.e. the nuclear or trace norm) in just ~ O((nnz(A)n^{1/3} + n^2)epsilon^{-3}) time for A with uniform row sparsity or tilde O(n^{2.18} epsilon^{-3}) time for dense matrices. The runtime scales smoothly for general Schatten-p norms, notably becoming tilde O (p nnz(A) epsilon^{-3}) for any real p >= 2. At the same time, we show that the complexity of spectrum approximation is inherently tied to fast matrix multiplication in the small epsilon regime. We use fine-grained complexity to give conditional lower bounds for spectrum approximation, showing that achieving milder epsilon dependencies in our algorithms would imply triangle detection algorithms for general graphs running in faster than state of the art matrix multiplication time. This further implies, through a reduction of (Williams & William, 2010), that highly accurate spectrum approximation algorithms running in subcubic time can be used to give subcubic time matrix multiplication. As an application of our bounds, we show that precisely computing all effective resistances in a graph in less than matrix multiplication time is likely difficult, barring a major algorithmic breakthrough

Robust ADMM-based wavefield reconstruction inversion with phase retrieval

International audienceWavefield Reconstruction Inversion (WRI) extends the linear regime of Full Waveform Inversion (FWI) by relaxing the wave equation with a feedback term to the data during wavefield simulation before updating the subsurface model by minimizing the source residuals this relaxation generates. Wavefield reconstruction and parameter estimation are efficiently nested with the alternating-direction method of multipliers (ADMM), leading to the so-called iteratively-refined WRI (IR-WRI). Capitalizing on the bilinearity of the wave equation, the alternating-direction strategy recasts the parameter estimation as a linear subproblem, which can be easily regularized with bound constraints and nonsmooth regularizations in the ADMM framework. Although the promise of regularized IR-WRI to image large-contrast media has been shown, the robustness of the parameter estimation subproblem can be further improved by phase retrieval during the early stages of IR-WRI, that is when the phase of the reconstructed wavefields is inaccurate in the deep part of the velocity model. Then, the velocity model inferred from phase retrieval at low frequencies is used as initial model for the subsequent steps of classical IR-WRI. This new workflow successfully reconstructs the BP salt model from a sparse fixed-spread acquisition using a 3 Hz starting frequency and a homogeneous initial velocity model

Faster eigenvector computation via shift-and-invert preconditioning

We give faster algorithms and improved sample complexities for the fundamental problem of estimating the top eigenvector. Given an explicit matrix A € Rn×d, we show how to compute an e approximate top eigenvector of ATA in time O (jnnz(A) + • log l/ϵ). Here nnz(A) is the number of nonzeros in A, sr(A) is the stable rank, and gap is the relative eigengap. We also consider an online setting in which, given a stream of i.i.d. samples from a distribution V with covariance matrix E and a vector xq which is an O(gap) approximate top eigenvector for E, we show how to refine xo to an € approximation using O j samples from V. Here v(P) is a natural notion of variance. Combining our algorithm with previous work to initialize xo, we obtain improved sample complexities and runtimes under a variety of assumptions on V. We achieve our results via a robust analysis of the classic shift-and-invert preconditioning method. This technique lets us reduce eigenvector computation to approximately solving a scries of linear systems with fast stochastic gradient methods

Phase reconstruction for time-frequency inpainting

International audienceWe address the problem of phase inpainting, i.e. the reconstruction of partially-missing phases in linear measurements. We thus aim at reconstructing missing phases of some complex coefficients assuming that the phases of the other coefficients as well as the modulus of all coefficients are known. The mathematical formulation of the inverse problem is first described and then, three methods are proposed: a first one based on the well known Griffin and Lim algorithm and two other ones based on positive semidefinite programming (SDP) optimization methods namely PhaseLift and PhaseCut, that are extended to the case of partial phase knowledge. The three derived algorithms are tested with measurements from a short-time Fourier transform (STFT) in two situations: the case where the missing data are distributed uniformly and indepedently at random and the case where they constitute holes with a given width. Results show that the knowledge of a subset of phases contributes to improve the signal reconstruction and to shorten the convergence of the optimization process

Modelling Interacting Epidemics in Overlapping Populations

Abstract. Epidemic modelling is fundamental to our understanding of biological, social and technological spreading phenomena. As conceptual frameworks for epidemiology advance, it is important they are able to elucidate empirically-observed dynamic feedback phenomena involving interactions amongst pathogenic agents in the form of syndemic and counter-syndemic effects. In this paper we model the dynamics of two types of epidemics with syndemic and counter-syndemic interaction ef-fects in multiple possibly-overlapping populations. We derive a Markov model whose fluid limit reduces to a set of coupled SIR-type ODEs. Its numerical solution reveals some interesting multimodal behaviours, as shown in our case studies