  We consider leader-follower multi-agent systems that have many leaders,
defined on any connected weighted undirected graphs, and address the leader
selection and demotion problems. The leader selection problem is formulated as
a minimization problem for the $H^2$ norm of the difference between the
transfer functions of the original and new agent systems, under the assumption
that the leader agents to be demoted are fixed. The leader demotion problem is
that of finding optimal leader agents to be demoted, and is formulated using
the global optimal solution to the leader selection problem. We prove that a
global optimal solution to the leader selection problem is the set of the
original leader agents except for those that are demoted to followers. To this
end, we relax the original problem into a differentiable problem. Then, by
calculating the gradient and Hessian of the objective function of the relaxed
problem, we prove that the function is convex. It is shown that zero points of
the gradient are global optimal solutions to the leader selection problem,
which is a finite combinatorial optimization problem. Furthermore, we prove
that any set of leader agents to be demoted subject to a fixed number of
elements is a solution to the leader demotion problem. By combining the
solutions to the leader selection and demotion problems, we prove that if we
choose new leader agents from the original ones except for those specified by
the set of leader agents to be demoted, then the relative $H^2$ error between
the transfer functions of the original and new agent systems is completely
determined by the numbers of original leader agents and leader agents that are
demoted to follower agents. That is, we reveal that the relative $H^2$ error
does not depend on the number of agents on the graph. Finally, we verify the
solutions using a simple example.
