Optimal transport and synchronization of complex networks

PLANARY SESSION

Friday 26th of September

Abstract:

What is the best way to add shortcuts to a transport system? Typically transport in disordered systems is dominated by a few channels. In fact, the concept of optimal path through a complex geometry plays an important role in physics, ranging from Laplacian flow to navigation. In the first part of the talk, we propose strategies for the optimal design of complex transport networks. Our network is built from a regular two-dimensional (d = 2) square lattice to be improved by adding long-range connections (shortcuts) with probability Pij ∼ r−α ij , where rij is the Euclidean distance between sites i and j, and α is a variable exponent. We then introduce a cost constraint on the total length of the additional links and find optimal transport in the system for α = d + 1. Remarkably, this condition remains optimal, regardless of the strategy used for navigation, being based on local or global knowledge of the network structure, in sharp contrast with results previously obtained for unconstrained navigation using global or local information, where the optimal conditions are α = 0 and α = d, respectively. In the second part of the talk, we start by introducing the concept of bear- ings. These are mechanical dissipative systems that, when perturbed, relax toward a synchronized (bearing) state. We then show that such structures can be perceived as physical realizations of com- plex networks of oscillators with asymmetrically weighted couplings. Accordingly, these networks can exhibit optimal synchronization properties through fine tuning of the local interaction strength as a function of node degree. In analogy, the synchronizability of bearings can be maximized by counterbalancing the number of contacts and the inertia of their constituting rotor disks through the mass-radius relation, m ∼ rβ, with an optimal exponent β = β× which converges to unity for a large number of rotors. Under this condition, and regardless of the presence of a long-tailed distribution of disk radii, our results show that the energy dissipation rate is homogeneously distributed among elementary rotors.