Diffusion bank networks and capital flows

We study how bank networks can be driven, via diffusion, to a state where they exhibit greater resistance to a systemic shock. Firstly without making any assumption about the dynamics which drives the interbank lending in the network we prove that the state in which the banks exhibit greater resistance to a systemic shock is the state S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}$$\end{document} in which they have equal leverages. Then we introduce a diffusion–like law which drives the interbank lending dynamics in the network. We prove that the steady state of the system is precisely the state S.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}.$$\end{document} In the general case, where the bank network is modelled by a directed, weighted graph, we define two projection operators, one acts on the space of flows of loans and the other one acts on the space of the leverages. The projection operator projects the vector of the initial flows (the vector of the initial leverages) to the vector of the steady state where, in both cases, all the leverages are equal. Examples are given where the networks are driven to their optimal and suboptimal states; when the steady state can only be achieved by reversion of the flows of loans a suboptimal state can be determined.