  Credit networks represent a way of modeling trust between entities in a
network. Nodes in the network print their own currency and trust each other for
a certain amount of each other's currency. This allows the network to serve as
a decentralized payment infrastructure---arbitrary payments can be routed
through the network by passing IOUs between trusting nodes in their respective
currencies---and obviates the need for a common currency. Nodes can repeatedly
transact with each other and pay for the transaction using trusted currency. A
natural question to ask in this setting is: how long can the network sustain
liquidity, i.e., how long can the network support the routing of payments
before credit dries up? We answer this question in terms of the long term
failure probability of transactions for various network topologies and credit
values.
  We prove that the transaction failure probability is independent of the path
along which transactions are routed. We show that under symmetric transaction
rates, the transaction failure probability in a number of well-known graph
families goes to zero as the size, density or credit capacity of the network
increases. We also show via simulations that even networks of small size and
credit capacity can route transactions with high probability if they are
well-connected. Further, we characterize a centralized currency system as a
special type of a star network (one where edges to the root have infinite
credit capacity, and transactions occur only between leaf nodes) and compute
the steady-state transaction failure probability in a centralized system. We
show that liquidity in star networks, complete graphs and Erd\"{o}s-R\'{e}nyi
networks is comparable to that in equivalent centralized currency systems; thus
we do not lose much liquidity in return for their robustness and decentralized
properties.
