Factors Influencing Fractional-Order Dynamics in Large, Scale-Free Robotics Swarms

When studying or designing a large-scale network system, a simpler representation of the dynamics of the system is often needed. This is because, for a very large scale system where the dynamics of all the nodes are coupled, the system of differential equations describing the system are very high order. In such cases, engineers are faced with determining a "lower-order" model that provides a good enough representation of the large-scale dynamics. Prior work by the author has identified fractional-order dynamics in large, scale-free networks and noted that fractional-order models often better match the dynamic response of the network when the stiffness relationship (a spring constant) in the relationship between nodes in the network is larger. This work extends those results by systematically determining what parameters in the network are statistically significantly correlated with fractional-order models better matching the large-scale dynamics than integer-order models. Specifically we find a weak correlation between the degree of connectivity of the network, the spring constant, the distance in the network between the two nodes, and the size of the network to be statistically significantly correlated to whether fractional-models are better. Interestingly, in contrast to the stiffness, the damping is statistically uncorrelated with fractional-dynamics.