  In this paper, structural controllability of a leader-follower multi-agent
system with multiple leaders is studied from a graph-theoretic point of view.
The problem of preservation of structural controllability under simultaneous
failures in both the communication links and the agents is investigated. The
effects of the loss of agents and communication links on the controllability of
an information flow graph are previously studied. In this work, the
corresponding results are exploited to introduce some useful indices and
importance measures that help characterize and quantify the role of individual
links and agents in the controllability of the overall network. Existing
results are then extended by considering the effects of losses in both links
and agents at the same time. To this end, the concepts of joint
(r,s)-controllability and joint t-controllability are introduced as
quantitative measures of reliability for a multi-agent system, and their
important properties are investigated. Lastly, the class of jointly critical
digraphs is introduced and it is stated that if a digraph is jointly critical,
then joint t-controllability is a necessary and sufficient condition for
remaining controllable following the failure of any set of links and agents,
with cardinality less than t. Various examples are exploited throughout the
paper to elaborate on the analytical findings.
