Computing the connected components of a graph is a fundamental problem in algorithmic graph theory. A major question in this area is whether we can compute connected components in o(log n) parallel time. Recent works showed an affirmative answer in the ...

In data-parallel optimization of machine learning models, workers collaborate to improve their estimates of the model: more accurate gradients allow them to use larger learning rates and optimize faster. In the decentralized setting, in which workers ...

In this paper, we study a continuously flocking Cucker--Smale model with noise, which has isotropic and polarized stationary solutions depending on the intensity of the noise. The first result establishes the threshold value of the noise parameter which ...

We study the long-time behavior of the growth-fragmentation equation, a nonlocal linear evolution equation describing a wide range of phenomena in structured population dynamics. We show the existence of a spectral gap under conditions that generalize those ...

The motivation of this work is to extend the techniques of higher order random walks on simplicial complexes to analyze mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second eigenvalue of the down-...

Time-inhomogeneous queueing models play an important role in service systems modeling. Although the transient solutions of corresponding continuous-time Markov chains (CTMCs) are more precise than methods using stationary approximations, most authors ...

This paper considers a Bayesian approach to graph-based semi-supervised learning. We show that if the graph parameters are suitably scaled, the graph-posteriors converge to a continuum limit as the size of the unlabeled data set grows. This consistency ...

Massively parallel computation (MPC) algorithms for graph problems have witnessed a resurgence of interest in recent years. Despite major progress for numerous graph problems however, the complexity of the sparse graph connectivity problem in this model ...

The question of whether quantum spatial search in two dimensions can be made optimal has long been an open problem. We report progress towards its resolution by showing that the oracle complexity for target location can be made optimal, by increasing ...

We analyse the eigenvalue and eigenvector structure of the flip-flop quantum walk on regular graphs, explicitly demonstrating how it is quadratically faster than the classical random walk. Then we use it in a controlled spatial search algorithm with ...

We study how the spectral gap of the normalized Laplacian of a random graph changes when an edge is added to or removed from the graph. There are known examples of graphs where, perhaps counter-intuitively, adding an edge can decrease the spectral gap, ...

We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of small relative ...

A new coercivity estimate on the spectral gap of the linearized Boltzmann collision operator for multiple species is proved. The assumptions on the collision kernels include hard and Maxwellian potentials under Grad's angular cutoff condition. Two proofs are ...

We consider a broad class of walk-based, parameterized node centrality measures for network analysis. These measures are expressed in terms of functions of the adjacency matrix and generalize various well-known centrality indices, including Katz and subgraph ...

Consider $N$ points in ${R}^d$ and $M$ local coordinate systems that are related through unknown rigid transforms. For each point, we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each ...

In machine learning, a popular tool to analyze the structure of graphs is the hitting time and the commute distance (resistance distance). For two vertices u and v, the hitting time H_uv is the expected time it takes a random walk to travel from u to v. ...

We prove that the spectral gap of the Swendsen--Wang dynamics for the random-cluster model is larger than the spectral gap of a single-bond dynamics, which updates only a single edge per step. For this we give a representation of the algorithms on the joint (...

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our ...

This paper investigates the synchronization behavior of a class of flocks modeled by the nearest neighbor rules. While connectivity of the associated dynamical neighbor graphs is crucial for synchronization, it is well known that the verification of such ...

We discuss the spectrum of the one-dimensional Schrödinger operator which possesses three point interactions in the basic period cell and investigate its band structure. We suppose that each point interaction on a lattice is given by a rotation or is ...