164 research outputs found
The Total s-Energy of a Multiagent System
We introduce the "total s-energy" of a multiagent system with time-dependent
links. This provides a new analytical lens on bidirectional agreement dynamics,
which we use to bound the convergence rates of dynamical systems for
synchronization, flocking, opinion dynamics, and social epistemology
Inertial Hegselmann-Krause Systems
We derive an energy bound for inertial Hegselmann-Krause (HK) systems, which
we define as a variant of the classic HK model in which the agents can change
their weights arbitrarily at each step. We use the bound to prove the
convergence of HK systems with closed-minded agents, which settles a conjecture
of long standing. This paper also introduces anchored HK systems and show their
equivalence to the symmetric heterogeneous model
A Sharp Bound on the -Energy and Its Applications to Averaging Systems
The {\em -energy} is a generating function of wide applicability in
network-based dynamics. We derive an (essentially) optimal bound of on the -energy of an -agent symmetric averaging system, for any
positive real , where~ is a lower bound on the nonzero weights.
This is done by introducing the new dynamics of {\em twist systems}. We show
how to use the new bound on the -energy to tighten the convergence rate of
systems in opinion dynamics, flocking, and synchronization
Toward a Theory of Markov Influence Systems and their Renormalization
Nonlinear Markov chains are probabilistic models commonly used in physics, biology, and the social sciences. In "Markov influence systems" (MIS), the transition probabilities of the chains change as a function of the current state distribution. This work introduces a renormalization framework for analyzing the dynamics of MIS. It comes in two independent parts: first, we generalize the standard classification of Markov chain states to the dynamic case by showing how to "parse" graph sequences. We then use this framework to
carry out the bifurcation analysis of a few important MIS families.
In particular, we show that irreducible MIS are almost always
asymptotically periodic. We also give an example of "hyper-torpid" mixing, where a stationary distribution is reached in super-exponential time, a timescale that cannot be achieved by any Markov chain
Self-Sustaining Iterated Learning
An important result from psycholinguistics (Griffiths & Kalish, 2005) states that no language can be learned iteratively by rational agents in a self-sustaining manner. We show how to modify the learning process slightly in order to achieve self-sustainability. Our work is in two parts. First, we characterize iterated learnability in geometric terms and show how a slight, steady increase in the lengths of the training sessions ensures self-sustainability for any discrete language class. In the second part, we tackle the nondiscrete case and investigate self-sustainability for iterated linear regression. We discuss the implications of our findings to issues of non-equilibrium dynamics in natural algorithms
Simplex range reporting on a pointer machine
AbstractWe give a lower bound on the following problem, known as simplex range reporting: Given a collection P of n points in d-space and an arbitrary simplex q, find all the points in P ∩ q. It is understood that P is fixed and can be preprocessed ahead of time, while q is a query that must be answered on-line. We consider data structures for this problem that can be modeled on a pointer machine and whose query time is bounded by O(nδ + r), where r is the number of points to be reported and δ is an arbitrary fixed real. We prove that any such data structure of that form must occupy storage Ω(nd(1 − δ)− ε), for any fixed ε > 0. This lower bound is tight within a factor of nε
A Connectivity-Sensitive Approach to Consensus Dynamics
The paper resolves a long-standing open question in network dynamics. Averaging-based consensus has long been known to exhibit an exponential gap in relaxation time between the connected and disconnected cases, but a satisfactory explanation has remained elusive. We provide one by deriving nearly tight bounds on the s-energy of disconnected systems. This in turn allows us to relate the convergence rate of consensus dynamics to the number of connected components. We apply our results to opinion formation in social networks and provide a theoretical validation of the concept of an Overton window as an attracting manifold of "viable" opinions
- …