Consensus Strikes Back in the Hegselmann-Krause Model of Continuous Opinion Dynamics Under Bounded Confidence

The agent-based bounded confidence model of opinion dynamics of Hegselmann and Krause (2002) is reformulated as an interactive Markov chain. This abstracts from individual agents to a population model which gives a good view on the underlying attractive states of continuous opinion dynamics. We mutually analyse the agent-based model and the interactive Markov chain with a focus on the number of agents and onesided dynamics. Finally, we compute animated bifurcation diagrams that give an overview about the dynamical behavior. They show an interesting phenomenon when we lower the bound of confidence: After the first bifurcation from consensus to polarisation consensus strikes back for a while.Continuous Opinion Dynamics, Bounded Confidence, Interactive Markov Chain, Bifurcation, Number of Agents, Onesided Dynamics

A Definition Scheme for Quantitative Bisimulation

FuTS, state-to-function transition systems are generalizations of labeled transition systems and of familiar notions of quantitative semantical models as continuous-time Markov chains, interactive Markov chains, and Markov automata. A general scheme for the definition of a notion of strong bisimulation associated with a FuTS is proposed. It is shown that this notion of bisimulation for a FuTS coincides with the coalgebraic notion of behavioral equivalence associated to the functor on Set given by the type of the FuTS. For a series of concrete quantitative semantical models the notion of bisimulation as reported in the literature is proven to coincide with the notion of quantitative bisimulation obtained from the scheme. The comparison includes models with orthogonal behaviour, like interactive Markov chains, and with multiple levels of behavior, like Markov automata. As a consequence of the general result relating FuTS bisimulation and behavioral equivalence we obtain, in a systematic way, a coalgebraic underpinning of all quantitative bisimulations discussed.Comment: In Proceedings QAPL 2015, arXiv:1509.0816

Compositional Verification and Optimization of Interactive Markov Chains

Interactive Markov chains (IMC) are compositional behavioural models extending labelled transition systems and continuous-time Markov chains. We provide a framework and algorithms for compositional verification and optimization of IMC with respect to time-bounded properties. Firstly, we give a specification formalism for IMC. Secondly, given a time-bounded property, an IMC component and the assumption that its unknown environment satisfies a given specification, we synthesize a scheduler for the component optimizing the probability that the property is satisfied in any such environment

Agent Behavior Prediction and Its Generalization Analysis

Machine learning algorithms have been applied to predict agent behaviors in real-world dynamic systems, such as advertiser behaviors in sponsored search and worker behaviors in crowdsourcing. The behavior data in these systems are generated by live agents: once the systems change due to the adoption of the prediction models learnt from the behavior data, agents will observe and respond to these changes by changing their own behaviors accordingly. As a result, the behavior data will evolve and will not be identically and independently distributed, posing great challenges to the theoretical analysis on the machine learning algorithms for behavior prediction. To tackle this challenge, in this paper, we propose to use Markov Chain in Random Environments (MCRE) to describe the behavior data, and perform generalization analysis of the machine learning algorithms on its basis. Since the one-step transition probability matrix of MCRE depends on both previous states and the random environment, conventional techniques for generalization analysis cannot be directly applied. To address this issue, we propose a novel technique that transforms the original MCRE into a higher-dimensional time-homogeneous Markov chain. The new Markov chain involves more variables but is more regular, and thus easier to deal with. We prove the convergence of the new Markov chain when time approaches infinity. Then we prove a generalization bound for the machine learning algorithms on the behavior data generated by the new Markov chain, which depends on both the Markovian parameters and the covering number of the function class compounded by the loss function for behavior prediction and the behavior prediction model. To the best of our knowledge, this is the first work that performs the generalization analysis on data generated by complex processes in real-world dynamic systems

Distributed Synthesis in Continuous Time

We introduce a formalism modelling communication of distributed agents strictly in continuous-time. Within this framework, we study the problem of synthesising local strategies for individual agents such that a specified set of goal states is reached, or reached with at least a given probability. The flow of time is modelled explicitly based on continuous-time randomness, with two natural implications: First, the non-determinism stemming from interleaving disappears. Second, when we restrict to a subclass of non-urgent models, the quantitative value problem for two players can be solved in EXPTIME. Indeed, the explicit continuous time enables players to communicate their states by delaying synchronisation (which is unrestricted for non-urgent models). In general, the problems are undecidable already for two players in the quantitative case and three players in the qualitative case. The qualitative undecidability is shown by a reduction to decentralized POMDPs for which we provide the strongest (and rather surprising) undecidability result so far

What\u27s in a Name? The Matrix as an Introduction to Mathematics

In my classes on the nature of scientific thought, I have often used the movie The Matrix (1999) to illustrate how evidence shapes the reality we perceive (or think we perceive). As a mathematician and self-confessed science fiction fan, I usually field questions related to the movie whenever the subject of linear algebra arises, since this field is the study of matrices and their properties. So it is natural to ask, why does the movie title reference a mathematical object? Of course, there are many possible explanations for this, each of which probably contributed a little to the naming decision. First off, it sounds cool and mysterious. That much is clear, and it may be that this reason is the most heavily weighted of them all. However, a quick look at the definitions of the word reveals deeper possibilities for the meaning of the movie’s title. Consider the following definitions related to different fields of study taken from Wikipedia on January 4, 2010: • Matrix (mathematics), a mathematical object generally represented as an array of numbers. • Matrix (biology), with numerous meanings, often referring to a biological material where specialized structures are formed or embedded. • Matrix (archeology), the soil or sediment surrounding a dig site. • Matrix (geology), the fine grains between larger grains in igneous or sedimentary rocks. • Matrix (chemistry), a continuous solid phase in which particles (atoms, molecules, ions, etc.) are embedded. All of these point to an essential commonality: a matrix is an underlying structure in which other objects are embedded. This is to be expected, I suppose, given that the word is derived from the Latin word referring to the womb — something in which all of us are embedded at the beginning of our existence. And so mathematicians, being the Latin scholars we are, have adapted the term: a mathematical matrix has quantities (usually numbers, but they could be almost anything) embedded in it. A biological matrix has cell components embedded in it. A geological matrix has grains of rock embedded in it. And so on. So a second reason for the cool name is that we are talking, in the movie, about a computer system generating a virtual reality in which human beings are embedded (literally, since they are lying down in pods). Thus, the computer program forms a literal matrix, one that bears an intentional likeness to a womb. However, there are other ways to connect the idea of a matrix to the film’s premise. These explanations operate on a higher level and are explicitly relevant to the mathematical definition of a matrix as well as to the events in the trilogy of Matrix movies. They are related to computer graphics, Markov chains, and network theory. This essay will explore each of these in turn, and discuss their application to either the events in the film’s story-line or to the making of the movie itself