  Bypass rewiring improves connectivity and robustness of networks against
removal of nodes including failures and attacks. A concept of bypass rewiring
on directed networks is proposed, and random bypass rewiring on infinite
directed random networks is analytically and numerically investigated with
simulations. As a result, random bypass rewiring makes infinite directed
(undirected) random networks extremely robust for arbitrary occupation
probabilities if and only if in-degree of every node except a fixed number of
nodes is equal to the out-degree (every node except a finite number of nodes
has even degree); bypass rewiring makes the percolation threshold 0.
Consequently, a finite network has a strongly connected spanning subnetwork
which has an Eulerian path or cycle if and only if there exists an way of
bypass rewiring to make the finite network extremely robust for every
combination of removed nodes; Eulerian networks are extremely robust for every
combination of removed nodes.
