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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 thispaper, we study first-order consensus dynamics in binary scale-free networks,where vertices are subject to white noise. We focus on the coherence ofnetworks characterized in terms of the $H_2$-norm, which quantifies how closelyagents track the consensus value. We first provide a lower bound of coherenceof a network in terms of its average degree, which is independent of thenetwork order. We then study the coherence of some sparse, scale-freereal-world networks, which approaches a constant. We also study numerically thecoherence of Barab\'asi-Albert networks and high-dimensional random Apolloniannetworks, which also converges to a constant when the networks grow. Finally,based on the connection of coherence and the Kirchhoff index, we studyanalytically the coherence of two deterministically-growing sparse networks andobtain the exact expressions, which tend to small constants. Our resultsindicate that the effect of noise on the consensus dynamics in power-lawnetworks is negligible. We argue that scale-free topology, together with loopystructure, is responsible for the strong robustness with respect to noisyconsensus dynamics in power-law networks.

Scale-free Loopy Structure is Resistant to Noise in Consensus Dynamics in Complex Networks