arXiv subjects

William Pickering

Publications and source records attributed to William Pickering.

7 recordsLinked to original sources

arXiv2025

DeepFleet: Multi-Agent Foundation Models for Mobile Robots

We introduce DeepFleet, a suite of foundation models designed to support coordination and planning for large-scale mobile robot fleets. These models are trained on fleet movement data, including robot positions, goals, and interactions, from hundreds of thousands of robots in Amazon warehouses worldwide. DeepFleet consists of four architectures that each embody a distinct inductive bias and collectively explore key points in the design space for multi-agent foundation models: the robot-centric (RC) model is an autoregressive decision transformer operating on neighborhoods of individual robots; the robot-floor (RF) model uses a transformer with cross-attention between robots and the warehouse floor; the image-floor (IF) model applies convolutional encoding to a multi-channel image representation of the full fleet; and the graph-floor (GF) model combines temporal attention with graph neural networks for spatial relationships. In this paper, we describe these models and present our evaluation of the impact of these design choices on prediction task performance. We find that the robot-centric and graph-floor models, which both use asynchronous robot state updates and incorporate the localized structure of robot interactions, show the most promise. We also present experiments that show that these two models can make effective use of larger warehouses operation datasets as the models are scaled up.

cs.RO↗

arXiv2018

A Nonlinear Transform for the Diagonalization of the Bernoulli-Laplace Diffusion Model and Orthogonal Polynomials

The Bernoulli-Laplace model describes a diffusion process of two types of particles between two urns. To analyze the finite-size dynamics of this process, and for other constructive results we diagonalize the corresponding transition matrix and calculate explicitly closed-form expressions for all eigenvalues and eigenvectors of the Markov transition matrix $T_{BL}$. This is done by a new method based on mapping the eigenproblem for $T_{BL}$ to the associated problem for a linear partial differential operator $L_{BL}$ acting on the vector space of homogeneous polynomials in three indeterminates. The method is applicable to other Two Urns models and is relatively easy to use compared to previous methods based on orthogonal polynomials or group representations.

math-ph↗

arXiv2016

Analysis of the high dimensional naming game with committed minorities

The naming game has become an archetype for linguistic evolution and mathematical social behavioral analysis. In the model presented here, there are $N$ individuals and $K$ words. Our contribution is developing a robust method that handles the case when $K = O(N)$. The initial condition plays a crucial role in the ordering of the system. We find that the system with high Shannon entropy has a higher consensus time and a lower critical fraction of zealots compared to low-entropy states. We also show that the critical number of committed agents decreases with the number of opinions and grows with the community size for each word. These results complement earlier conclusions that diversity of opinion is essential for evolution; without it, the system stagnates in the status quo [S. A. Marvel et al., Phys. Rev. Lett. 109, 118702 (2012)]. In contrast, our results suggest that committed minorities can more easily conquer highly diverse systems, showing them to be inherently unstable.

physics.soc-ph↗

arXiv2015

Solution of the multi-state voter model and application to strong neutrals in the naming game

We consider the voter model with $M$ states initially in the system. Using generating functions, we pose the spectral problem for the Markov transition matrix and solve for all eigenvalues and eigenvectors exactly. With this solution, we can find all future probability probability distributions, the expected time for the system to condense from $M$ states to $M-1$ states, the moments of consensus time, the expected local times, and the expected number of states over time. Furthermore, when the initial distribution is uniform, such as when $M=N$, we can find simplified expressions for these quantities. In particular, we show that the mean and variance of consensus time for $M=N$ is $\frac{1}{N}(N-1)^2$ and $\frac{1}{3}(π^2-9)(N-1)^2$ respectively.

physics.soc-ph↗

arXiv2015

Spectral solution of urn models for interacting particle systems

Using generating function methods for diagonalizing the transition matrix in 2-Urn models, we provide a complete classification into solvable and unsolvable subclasses, with further division of the solvable models into the Martingale and non-Martingale subcategories, and prove that the stationary distribution is a Gaussian function in the latter. We also give a natural condition related to the symmetry of the random walk in which the non-Martingale Urn models lead to an increase in entropy from Gaussian states. The condition also shows that universal symmetry in the macro-state is equivalent to increasing entropy. Certain models of social opinion dynamics, treated as Urn models, do not increase in entropy, unlike isolated mechanical systems.

physics.soc-ph↗

arXiv2014

Information Sharing for Strong Neutrals on Social Networks - Exact Solutions for Consensus Times

To analyze the nuances of the root concept of neutral in social networks, we focus on several related interpretations and suggest corresponding mathematical models for each of them from the family of information-sharing multi-agents network games known as Voter models and the Naming Games (NG). We solve the case of the strong neutrals known as the middle-roaders for global quantities such as expected times to consensus and local times. By using generating functions and treating the two extreme and middle opinions in this modification as a two balls, three urns version of the Voter model, we give closed-form expressions for the eigenvalues and eigenvectors of its Markov propagator. This modification of the two-opinions Naming Games is applicable to the roles and behaviour of neutrals in social forums or blogs, and represent a significant departure from the linguistic roots of the original NG.

physics.soc-ph↗

arXiv2014

Solution of the Voter Model by Spectral Analysis

An exact spectral analysis of the Markov Propagator for the Voter model is presented for the complete graph, and extended to the complete bipartite graph and uncorrelated random networks. Using a well-defined Martingale approximation in diffusion-dominated regions of phase space, which is almost everywhere for the Voter model, this method is applied to compute analytically several key quantities such as exact expressions for the $m$ time step propagator of the Voter model, all moments of consensus times, and the local times for each macrostate. This spectral method is motivated by a related method for solving the Ehrenfest Urn problem and by formulating the Voter model on the complete graph as an Urn model. Comparisons of the analytical results from the spectral method and numerical results from Monte-Carlo simulations are presented to validate the spectral method.

math.PR↗