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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 preservation of structural controllability under simultaneous failure in both the communication links and the agents is investigated. The effects of the loss of agents or communication links on the controllability of an information flow graph have been the subject of two previous studies. This work expands the corresponding results 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. A method is subsequently presented for investigating the problem of joint t-controllability in a directed information flow graph, using polynomial-time algorithms. The proposed methods are applied to a number of well-known directed graphs to clarify the results.