Abstract
We consider networks formed by the union of $M$ random $1$ -regular directed graphs. These graphs are also called permutation models in the literature. We first present a proof showing that the expansion factor of such graphs is greater than or equal to $\log\; N$ a.a.s when $M> 4\;\log\; N,$ where $N$ is the number of nodes in the network. The reason for considering such random graph models is their applicability in the design of peer-to-peer networks and data center networks of switches. Assuming that each node in the network has upload and download capacities greater than $8\; \log\; N,$ we also show that the above result implies that all-to-all communication is possible in such a network, if the total incoming data rate and the total outgoing data rate at each node are both less than or equal to $1.$
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