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 \({\mathcal {S}}\) 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 \({\mathcal {S}}.\) 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.

我们研究了如何通过扩散将银行网络驱动到对系统性冲击表现出更大抵抗力的状态。首先未做关于其驱动同业拆借网络中的动态任何假设我们证明了在其中银行表现出系统性休克更大的阻力的状态是状态\（{\ mathcal {S}} \） ，其中它们具有平等的杠杆作用。然后，我们引入了类似扩散的定律，该定律驱动了网络中银行间同业拆借的动态。我们证明系统的稳定状态恰好是状态\（{\ mathcal {S}}。\}。在一般情况下，通过有向加权图对银行网络进行建模，我们定义了两个投影算子，一个作用于贷款的流动空间，另一个作用于杠杆的空间。投影算子将初始流量的向量（初始杠杆的向量）投影到稳态的向量，在这两种情况下，所有杠杆都是相等的。给出了将网络驱动到最佳状态和次最佳状态的示例；当只能通过恢复贷款流量来达到稳定状态时，可以确定为次优状态。