  We consider first and second order consensus algorithms in networks with
stochastic disturbances. We quantify the deviation from consensus using the
notion of network coherence, which can be expressed as an $H_2$ norm of the
stochastic system. We use the setting of fractal networks to investigate the
question of whether a purely topological measure, such as the fractal
dimension, can capture the asymptotics of coherence in the large system size
limit. Our analysis for first-order systems is facilitated by connections
between first-order stochastic consensus and the global mean first passage time
of random walks. We then show how to apply similar techniques to analyze
second-order stochastic consensus systems. Our analysis reveals that two
networks with the same fractal dimension can exhibit different asymptotic
scalings for network coherence. Thus, this topological characterization of the
network does not uniquely determine coherence behavior. The question of whether
the performance of stochastic consensus algorithms in large networks can be
captured by purely topological measures, such as the spatial dimension, remains
open.
