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We present a general framework, applicable to a broad class of random walkson complex networks, which provides a rigorous lower bound for the meanfirst-passage time of a random walker to a target site averaged over itsstarting position, the so-called global mean first-passage time (GMFPT). Thisbound is simply expressed in terms of the equilibrium distribution at thetarget, and implies a minimal scaling of the GMFPT with the network size. Weshow that this minimal scaling, which can be arbitrarily slow for a properchoice of highly connected target, is realized under the simple condition thatthe random walk is transient at the target site, and independently of thesmall-world, scale free or fractal properties of the network. Last, we putforward that the GMFPT to a specific target is not a representative property ofthe network, since the target averaged GMFPT satisfies much more restrictivebounds, which forbid any sublinear scaling with the network size.

Global mean first-passage times of random walks on complex networks