Connectivity in Random Forests and Credit Networks

Recent work has highlighted credit networks as an effective mechanism for modeling trust in a network: agents issue their own currency and trust each other for a certain amount of each other’s currency, allowing two nodes to transact if there is a chain of sufficient residual trust between them. Under a natural model of repeated transactions, the probability that two agents can successfully transact in a credit network (i.e. the liquidity between these two agents) is the same as the probability that they are connected to each other in a uniformly random forest of the network. Motivated by this connection, we define the RFconnectivity between a pair of nodes in a graph G as the probability that the two nodes belong to the same connected component in a uniformly random forest of G. Our first result is that for an arbitrary subset S of nodes in G, the average RF-connectivity between pairs of nodes in S is at least 1− 2/h(GS), where h(GS) is the edge expansion of the subgraph GS induced by S. Informally, this implies that a well-connected “community” of nodes S in a credit network will have high liquidity among themselves, regardless of the structure of the remaining network. We extend this result to show that in fact every node in S has good average RFconnectivity to other nodes in S whenever S has good edge expansion. We also show that our results are nearly tight by proving an upper bound on the liquidity of regular graphs. For our motivating application, it is important that we relate the average RF-connectivity in S to the expansion inside S and not merely to expansion of G since we would like to assert that a well-connected community has high liquidity even if the graph as a whole is not well-connected. This naturally leads to a monotonicity conjecture: the RFconnectivity of two nodes can not decrease when a new edge is added to G. We show that the monotonicity conjecture is equivalent to showing negative correlation between inclusion of any two edges in a random forest, a long-standing open problem. Our result about the average RF-connectivity of nodes in S may be viewed as establishing a weak version of the monotonicity conjecture.