Hendrik Schawe

We study analytically the relaxation eigenmodes of a simple Monte Carlo algorithm, corresponding to a particle in a box which moves by uniform random jumps. Moves outside of the box are rejected. At long times, the system approaches the equilibrium probability density, which is uniform inside the box. We show that the relaxation towards this equilibrium is unusual: for a jump length comparable to the size of the box, the number of relaxation eigenmodes can be surprisingly small, one or two. We provide a complete analytic description of the transition between these two regimes. When only a single relaxation eigenmode is present, a suitable choice of the symmetry of the initial conditions gives a localizing decay to equilibrium. In this case, the deviation from equilibrium concentrates at the edges of the box where the rejection probability is maximal. Finally, in addition to the relaxation analysis of the Master equation, we also describe the full eigen-spectrum of the Master equation including its sub-leading eigen-modes.

We present an extensive study of the joint effects of heterogeneous social agents and their heterogeneous social links in a bounded confidence opinion dynamics model. The full phase diagram of the model is explored for two different network's topologies and compared to two opposed extreme cases: on one hand heterogeneous agents which constitute a mixed population and on the other, their interactions are modeled by a lattice. The results show that when agents prone to compromising coexist with close minded ones, the steady state of the dynamics shows coexistent phases. In particular, unlike the homogeneous case in networks, or the heterogeneous case in fully mixed population, it is possible that the society ends up in consensus around one extreme opinion. Moreover, during the dynamics, the consensus may be overturned from one extreme to the other of the opinion space. We also show that the standard order parameter, the normalized average size of the largest opinion cluster, may be misleading in this case, as it hides the existence of these phases. The phase where the opinion of the society is overturned does not require the presence of agents with special characteristics, (stubborn, extremists, etc.); it results from the interplay of agents which have agreed on an extreme opinion with the remaining group that holds the opposite one. Among the former, some may be prone to compromising with other agents which are out of the majority group, these agents, according to their location in the network, may act like bridges between the two groups and slowly attract the whole society to the other extreme.

We investigate the statistics of the convex hull for a single run-and-tumble particle (RTP) in two dimensions. RTP, also known as the persistent random walker, has gained significant interest in the recent years due to its biological application in modelling the motion of bacteria. We consider two different statistical ensembles depending on whether (i) the total number of tumbles n or (ii) the total observation time t is kept fixed. Benchmarking the results on the perimeter, we study the statistical properties of the area of the convex hull for a RTP. Exploiting the connections to extreme value statistics, we obtain exact analytical expressions for the mean area for both ensembles. For fixed- t ensemble, we show that the mean area possesses a scaling form in γt (with γ being the tumbling rate) and the corresponding scaling function is exactly computed. Interestingly, we find that it exhibits a crossover from ∼ t ³ scaling at small times t ≪ γ − 1 to ∼ t scaling at large times t ≫ γ − 1 . On the other hand, for fixed- n ensemble, the mean expectedly grows linearly with n for n ≫ 1. All our analytical findings are supported with the numerical simulations.

Most opinion dynamics models are based on pairwise interactions. However in many real situations, discussions take place within groups of people. Here, we define a higher order Deffuant model by generalizing the original pairwise interaction model for bounded-confidence opinion-dynamics to interactions involving a group of agents of size k. The generalized model is naturally encoded in a hypergraph. We study this dynamics in different hypergraph topologies, from random hypergraph ensembles, to spatially embedded hyper-lattices. We show that including higher order interactions induces a drastic change in the onset of consensus for random hypergraphs; instead of the sharp phase transition, characteristic of the dyadic Deffuant model, the system undergoes a smooth size independent crossover to consensus, as the confidence value increases. This phenomenon is absent from regular hypergraphs, which conserve a phase transition.

We investigate, both analytically and with numerical simulations, a Monte Carlo dynamics at zero temperature, where a random walker evolving in continuous space and discrete time seeks to minimize its potential energy, by decreasing this quantity at each jump. The resulting dynamics is universal in the sense that it does not depend on the underlying potential energy landscape, as long as it admits a unique minimum; furthermore, the long time regime does not depend on the details of the jump distribution, but only on its behavior for small jumps. We work out the scaling properties of this dynamics, as embodied by the walker probability density. Our analytical predictions are in excellent agreement with direct Monte Carlo simulations.

In this work we present results to the problem of the Hegselmann-Krause dynamics in networks obtained by an extensive study of the behavior of the standard order parameter sensitive to the onset of consensus: the normalized size of the giant cluster. This order parameter reveals the nontrivial effect of the network topology on the steady states of the dynamics, overlooked by previous works, which concentrated on the onset of unanimity, and allows to detect regions of polarization between the fragmented and the consensus phases. While the previous results on unanimity are confirmed, the consensus threshold shifts in the opposite direction compared to the threshold for unanimity. A detailed finite-size scaling analysis shows that, in general, consensus is easier to obtain in networks than in mixed populations. At a difference with previous studies, we show that the network topology is relevant beyond the finiteness of the average degree with increasing system size. In particular, in pure random networks (either uniform random graphs or scale-free networks), the consensus threshold seems to vanish in the thermodynamic limit. A detailed analysis of the time evolution of the dynamics reveals the role of bridges in the network, which allow for the interaction between agents belonging to clusters of very different opinions, after several repeated interaction steps. These bridges are at the origin of the shift of the confidence threshold to lower values in networks as compared to lattices or the mixed population.

In this work we present novel results to the problem of the Hegselmann-Krause dynamics in networks obtained by an extensive study of the behavior of the standard order parameter sensitive to the onset of consensus: the normalized size of the giant cluster. This order parameter reveals the non trivial effect of the network topology on the steady states of the dynamics, overlooked by previous works, which concentrated on the onset of unanimity, and allows to detect regions of polarization between the fragmented and the consensus phases. While the previous results on unanimity are confirmed, the consensus threshold shifts in the opposite direction compared to the threshold for unanimity. A detailed finite size scaling analysis shows that, in general, consensus is easier to obtain in networks than in mixed populations. At a difference with previous studies, we show that the network topology is relevant beyond the finitness of the average degree with increasing system size. In particular, in pure random networks (either uniform random graphs or scale free networks), the consensus threshold seems to vanish in the thermodynamic limit. A detailed analysis of the time evolution of the dynamics reveals the role of bridges in the network, which allow for the interaction between agents belonging to clusters of very different opinions, after several repeated interaction steps. These bridges are at the origin of the shift of the confidence threshold to lower values in networks as compared to lattices or the mixed population.