  The vast majority of real-world networks are scale-free, loopy, and sparse,
with a power-law degree distribution and a constant average degree. In this
paper, we study first-order consensus dynamics in binary scale-free networks,
where vertices are subject to white noise. We focus on the coherence of
networks characterized in terms of the $H_2$-norm, which quantifies how closely
agents track the consensus value. We first provide a lower bound of coherence
of a network in terms of its average degree, which is independent of the
network order. We then study the coherence of some sparse, scale-free
real-world networks, which approaches a constant. We also study numerically the
coherence of Barab\'asi-Albert networks and high-dimensional random Apollonian
networks, which also converges to a constant when the networks grow. Finally,
based on the connection of coherence and the Kirchhoff index, we study
analytically the coherence of two deterministically-growing sparse networks and
obtain the exact expressions, which tend to small constants. Our results
indicate that the effect of noise on the consensus dynamics in power-law
networks is negligible. We argue that scale-free topology, together with loopy
structure, is responsible for the strong robustness with respect to noisy
consensus dynamics in power-law networks.
