In this lecture, we introduce a short unit on complex systems approaches that intersect with computation and algorithms. We start with Interacting Particle Systems (IPS), which are a class of mathematical models describing systems that interact based on location or contact with each other. There are "Self-Organizing Particle Systems (SOPS)" that describe hypothetical agents that might be designed to build or maintain structures in engineered systems. Or there are more generic IPS models meant to understand phenomena in population dynamics or community ecology. We focus on the "Voter Model", which can be viewed as a model of consensus in agents that randomly trade opinions or (equivalently) fixation in evolutionary systems that are not under selection. We analyze the Voter Model as a non-ergodic Markov chain with absorbing states and then show how a dual process that represents a sort of "contact tracing" of consensus backward in time is ergodic and can thus be analyzed with a suite of mathematical tools. One of those tools helps us prove the possibly counterintuitive result that when the voter model operates in fewer than two dimensions, it will reach consensus/fixation with probability one but will have non-zero probability of never reaching consensus/fixation in three or more dimensions. Thus, the dimensionality of the (translation invariant) neighborhoods matters. We then prepare for our next lecture, where we'll introduce Cellular Automata (CA), a deterministic interacting particle system that nevertheless can produce fascinating patterns that help demonstrate how computation can be embodied in space.
Whiteboard notes for this lecture can be found at: https://www.dropbox.com/s/5bt6uinpkidw2p4/IEE598-Lecture8A-2022-04-21-Complex_Systems_Approaches_to_Computation-Interacting_Particle_Systems_and_the_Voter_Model.pdf?dl=0
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