Who Replaces Whom? Local versus Non-local Replacement in Social and Evolutionary Dynamics

In this paper, we inspect well-known population genetics and social dynamics models. In these models, interacting individuals, while participating in a self-organizing process, give rise to the emergence of complex behaviors and patterns. While one main focus in population genetics is on the adaptive behavior of a population, social dynamics is more often concerned with the splitting of a connected array of individuals into a state of global polarization, that is, the emergence of speciation. Applying computational and mathematical tools we show that the way the mechanisms of selection, interaction and replacement are constrained and combined in the modeling have an important bearing on both adaptation and the emergence of speciation. Differently (un)constraining the mechanism of individual replacement provides the conditions required for either speciation or adaptation, since these features appear as two opposing phenomena, not achieved by one and the same model. Even though natural selection, operating as an external, environmental mechanism, is neither necessary nor sufficient for the creation of speciation, our modeling exercises highlight the important role played by natural selection in the interplay of the evolutionary and the self-organization modeling methodologies.Comment: 14 pages, 11 figure

Aggregation and Emergence in Agent-Based Models: A Markov Chain Approach

We analyze the dynamics of agent--based models (ABMs) from a Markovian perspective and derive explicit statements about the possibility of linking a microscopic agent model to the dynamical processes of macroscopic observables that are useful for a precise understanding of the model dynamics. In this way the dynamics of collective variables may be studied, and a description of macro dynamics as emergent properties of micro dynamics, in particular during transient times, is possible.Comment: 5 pages, 1 figur

Opinion Dynamics and Communication Networks

This paper examines the interplay of opinion exchange dynamics and communication network formation. An opinion formation procedure is introduced which is based on an abstract representation of opinions as k-dimensional bitstrings. Individuals interact if the difference in the opinion strings is below a defined similarity threshold dI. Depending on dI, different behaviour of the population is observed: low values result in a state of highly fragmented opinions and higher values yield consensus. The first contribution of this research is to identify the values of parameters dI and k, such that the transition between fragmented opinions and homogeneity takes place. Then, we look at this transition from two perspectives: first by studying the group size distribution and second by analysing the communication network that is formed by the interactions that take place during the simulation. The emerging networks are classified by statistical means and we find that non-trivial social structures emerge from simple rules for individual communication.

Opinion Polarization by Learning from Social Feedback

We explore a new mechanism to explain polarization phenomena in opinion dynamics in which agents evaluate alternative views on the basis of the social feedback obtained on expressing them. High support of the favored opinion in the social environment, is treated as a positive feedback which reinforces the value associated to this opinion. In connected networks of sufficiently high modularity, different groups of agents can form strong convictions of competing opinions. Linking the social feedback process to standard equilibrium concepts we analytically characterize sufficient conditions for the stability of bi-polarization. While previous models have emphasized the polarization effects of deliberative argument-based communication, our model highlights an affective experience-based route to polarization, without assumptions about negative influence or bounded confidence.Comment: Presented at the Social Simulation Conference (Dublin 2017

Diffusion maps for Lagrangian trajectory data unravel coherent sets

Dynamical systems often exhibit the emergence of long-lived coherent sets, which are regions in state space that keep their geometric integrity to a high extent and thus play an important role in transport. In this article, we provide a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. Our method can be seen as an extension of diffusion maps to trajectory space, and it allows us to construct “dynamical coordinates,” which reveal the intrinsic low-dimensional organization of the data with respect to transport. The only a priori knowledge about the dynamics that we require is a locally valid notion of distance, which renders our method highly suitable for automated data analysis. We show convergence of our method to the analytic transfer operator framework of coherence in the infinite data limit and illustrate its potential on several two- and three-dimensional examples as well as real world data. One aspect of the coexistence of regular structures and chaos in many dynamical systems is the emergence of coherent sets: If we place a large number of passive tracers in a coherent set at some initial time, then macroscopically they perform a collective motion and stay close together for a long period of time, while their surrounding can mix chaotically. Natural examples are moving vortices in atmospheric or oceanographic flows. In this article, we propose a method for extracting coherent sets from possibly sparse Lagrangian trajectory data. This is done by constructing a random walk on the data points that captures both the inherent time-ordering of the data and the idea of closeness in space, which is at the heart of coherence. In the rich data limit, we can show equivalence to the well-established functional-analytic framework of coherent sets. One output of our method are “dynamical coordinates,” which reveal the intrinsic low- dimensional transport-based organization of the data

Transition manifolds of complex metastable systems: Theory and data-driven computation of effective dynamics

We consider complex dynamical systems showing metastable behavior but no local separation of fast and slow time scales. The article raises the question of whether such systems exhibit a low-dimensional manifold supporting its effective dynamics. For answering this question, we aim at finding nonlinear coordinates, called reaction coordinates, such that the projection of the dynamics onto these coordinates preserves the dominant time scales of the dynamics. We show that, based on a specific reducibility property, the existence of good low-dimensional reaction coordinates preserving the dominant time scales is guaranteed. Based on this theoretical framework, we develop and test a novel numerical approach for computing good reaction coordinates. The proposed algorithmic approach is fully local and thus not prone to the curse of dimension with respect to the state space of the dynamics. Hence, it is a promising method for data-based model reduction of complex dynamical systems such as molecular dynamics