EF45 – Multi-Agent Social Systems



Coherent Movements in Co-evolving Agent–Message Systems

Project Heads

Felix Höfling, Robert I.A. Patterson (until 02/2024)

Project Members

Zahra Mokhtari (01/21-05/22; since 01/23 part-time)

Project Duration

First funding period: 01.01.2021 − 31.12.2021; Second funding period: 01.01.2022 − 31.10.2024

Located at

FU Berlin


The dynamics of social systems emerges from the behaviour and interaction of individual agents. A key question in the field is which basic rules and mechanisms can generate a certain observed phenomenon on the population scale? For example, in the context of opinion dynamics classical voter models show that neighbour contacts can lead to clustering, termed as opinion alignment. However often, the direct exchange of opinions between individuals is an idealisation, opinions are typically not black or white, and they change rarely by a single event (e.g., an election campaign). Instead, opinions form gradually by consuming a large number of small messages such as direct chats with peers, but nowadays with increasing importance also due to short texts (tweets, whatsapp, news) focussed on few specific topics.
In particular, each message has the potential to influence many individuals over an extended period of time.
Such a message passing requires coupling the agent dynamics to a co-evolving secondary field encoding the messages. Specifically, agents leave many small and volatile messages behind that can influence other agents until they disappear. Here, we aim to investigate the formation and fragmentation of opinions under a message-based communication. Our ultimate goal is to understand what makes a group of people sharing similar opinions change their opinion coherently under otherwise unchanged external conditions?
Animal systems and cell cultures exhibit a closely related message-based interaction known as auto-chemotaxis. There, the agents move in real space and preferentially towards high concentrations of a pheromone (a signalling chemical) that they produce themselves. This leads to a delayed and long-range coupling between agents and intriguing collective behaviour, such as trail formation. Continuum mathematical models of auto-chemotaxis are traditionally based on the Patlak–Keller–Segel equations (PKS), a system of two partial differential equations for the density of agents and the concentration of a pheromone. This is a very reasonable phenomenological model, but it offers little insight into how agents (e.g. ants) might take decisions.
We introduce an agent-based model for self-propelled particles that make oriented deposits of pheromones and also sense such deposits to which they then respond with gradual changes of their direction of motion. Based on extensive off-lattice computer simulations aiming at the scale of insects, e.g. ants, we identify a number of emerging stationary patterns and obtain qualitatively the non-equilibrium state diagram of the model, spanned by the strength of the agent–pheromone interaction and the number density of the population.

Furthermore, we have derived a set of hydrodynamic limit equation within a mean-field approximation. The equations describe the evolution of probability densities for agent position and orientation and are coupled to the co-evolving pheromone field. Currently, we’re conducting a stability analysis to gain insights into the formation of trails. In the hydrodynamic limit we show that the system of agents and pheromones evolves according to an explicit non-linear evolution equation with up to four dimensionless parameters. This equation exhibits a rich variety of behaviour depending on the values of these parameters, in particular the formation of trails is expected to be connected to the instability of the uniform solution, which we can detect with linear stability analysis as the density of agents rises above a critical value.

External Website

Related Publications

Mokhtari Z, Patterson RI, Höfling F. Spontaneous trail formation in populations of auto-chemotactic walkers. New Journal of Physics. 2022 Jan 5;24(1):013012.
Irani E, Mokhtari Z, Zippelius A. Dynamics of bacteria scanning a porous environment. Physical Review Letters. 2022 Apr 5;128(14):144501.
Fazelzadeh M, Irani E, Mokhtari Z, Jabbari-Farouji S. Effects of inertia on conformation and dynamics of tangentially driven active filaments. Physical Review E. 2023 Aug 11;108(2):024606.

Related Pictures

The aligning interaction between agents (close-up: disks) and their deposited pheromones (dots) gives rise to macroscopic structures such as system-spanning trails and rotating clusters.
Representative configurations of agent positions for different values of crowdedness and the relative alignment strengths.
Trail solution of the PDE system obtained as the hydrodynamic limit of the particle system.