Open Access

Many real-world processes can naturally be modeled as systems of interacting agents. However, the long-term simulation of such agent-based models is often intractable when the system becomes too large. In this paper, starting from a stochastic spatio-temporal agent-based model (ABM), we present a reduced model in terms of stochastic PDEs that describes the evolution of agent number densities for large populations. We discuss the algorithmic details of both approaches; regarding the SPDE model, we apply Finite Element discretization in space which not only ensures efficient simulation but also serves as a regularization of the SPDE. Illustrative examples for the spreading of an innovation among agents are given and used for comparing ABM and SPDE models.

In particle systems, flocking refers to the phenomenon where particles’ individual velocities eventually align. The Cucker-Smale model is a well-known mathematical framework that describes this behaviour. Many continuous descriptions of the Cucker-Smale model use PDEs with both particle position and velocity as independent variables, thus providing a full description of the particles mean-field limit (MFL) dynamics. In this paper, we introduce a novel reduced inertial PDE model consisting of two equations that depend solely on particle position. In contrast to other reduced models, ours is not derived from the MFL, but directly includes the model reduction at the level of the empirical densities, thus allowing for a straightforward connection to the underlying particle dynamics. We present a thorough analytical investigation of our reduced model, showing that: firstly, our reduced PDE satisfies a natural and interpretable continuous definition of flocking; secondly, in specific cases, we can fully quantify the discrepancy between PDE solution and particle system. Our theoretical results are supported by numerical simulations.

The problem of decomposing networks into modules (or clusters) has gained much attention in recent years, as it can account for a coarsegrained description of complex systems, often revealing functional subunits of these systems. A variety of module detection algorithms have been proposed, mostly oriented towards finding hard partitionings of undirected networks. Despite the increasing number of fuzzy clustering methods for directed networks, many of these approaches tend to neglect important directional information. In this paper, we present a novel random walk based approach for finding fuzzy partitions of directed, weighted networks, where edge directions play a crucial role in defining how well nodes in a module are interconnected. We will show that cycle decomposition of a random walk process connects the notion of network modules and information transport in a network, leading to a new, symmetric measure of node communication. Finally, we will use this measure to introduce a communication graph, for which we will show that although being undirected it inherits all necessary information about modular structures from the original network.

We investigate the problem of finding modules (or clusters, communities) in directed networks. Until now, most articles on this topic have been oriented towards finding complete network partitions despite the fact that this often is unwanted. We present a novel random walk based approach for non-complete partitions of the directed network into modules in which some nodes do not belong to only one of the modules but to several or to none at all. The new random walk process is reversible even for directed networks but inherits all necessary information about directions and structure of the original network. We demonstrate the performance of the new method in application to a real-world earthquake network.