Hierarchy and Chirality: Explicit Results for Random Digraphs and Active Elastic Filaments

Analytic descriptions of complex systems often rest on symmetries. In this talk, I will discuss two problems lacking such symmetries, one discrete and one continuous, in which an explicit description is nevertheless possible. The first concerns nonreciprocal random networks, directed networks that model hierarchical systems such as barter or ecological networks: the nodes are ranked, and links pointing up and down the ranking occur with differing probabilities. Using self-consistent integral equations, I will obtain the sizes of the giant strongly connected, in- and out-components of these networks, together with the probability that a given node belongs to each, and show that a giant strongly connected component exists precisely when the logarithmic mean of the two link rates exceeds one, in agreement with the rigorous theory of inhomogeneous random digraphs. Since the logarithmic mean never exceeds the arithmetic mean, which sets the threshold in the absence of hierarchy, hierarchy suppresses percolation. The second concerns active elastic filaments, driven by chiral forces which derive from no energy. Formulated intrinsically, in terms of the metric and curvature of the filament, their overdamped dynamics yields coupled partial differential equations whose stationary solutions include straight, translating conformations and curved, rotating ones. I will discuss the stability of these solutions and compare them with simulations, which confirm the predicted shape multi-stability while revealing further conformations, and with experimental microtubule gliding assays, with which they agree in order of magnitude.