  We connect the Grover walk with sinks to the Grover walk with tails. The
survival probability of the Grover walk with sinks in the long time limit is
characterized by the centered generalized eigenspace of the Grover walk with
tails. The centered eigenspace of the Grover walk is the attractor eigenspace
of the Grover walk with sinks. It is described by the persistent eigenspace of
the underlying random walk whose support has no overlap to the boundaries of
the graph and combinatorial flow in the graph theory.
