Colloquia

We utilize transport systems daily to commute, e.g. via road networks, or bring energy to our houses through the power grid. Our body needs transport networks, such as the lymphatic, arterial or venous system, to distribute nutrients and remove waste. If the transported quantity is information, for example carried by an electrical signal, then even the internet and the brain can be thought of as members of this broad class of webs. Despite our daily exposure to transport networks, their function and physics can still surprise us. This is exemplified by the Braess paradox, where the addition of an extra road in a network worsens rather than improves traffic contrary to a naïve prediction.

In this talk we will explore two cases that highlight the importance of time in load transmission in transport networks. In the first problem, we will discuss how short timescale dynamics in the flow alters the topology of the network in longer timescales, and shapes its morphology. We will first present the system of phenomenological adaptation equations that govern the structural evolution of vascular networks. We will then demonstrate how implicit of explicit dynamics in the boundary conditions (the power supply, or heart) can drastically alter the network topology, and discuss the implications for the development and function of human circulation. Moving to a larger system, will provide evidence that the similar dynamical developmental rules to the ones that are thought to control vascular remodeling in humans also shape tidal delta geomorphology. 

In the second problem, we consider stochastic transport in geometrically embedded graphs. Intuition suggests that providing a shortcut between a pair of nodes improves the time it takes to randomly explore the entire graph. Counterintuitively, we find a Braess' paradox analog. For regular diffusion, shortcuts can worsen the overall search efficiency of the network, although they bridge topologically distant nodes. We propose an optimization scheme under which each edge adapts its conductivity to minimize the graph's search time. The optimization reveals a relationship between the structure and diffusion exponent and a crossover from dense to sparse graphs as the exponent increases.