无标度环状结构在复杂网络共识动力学中对噪声具有抗性
Scale-free Loopy Structure is Resistant to Noise in Consensus Dynamics in Complex Networks
AI总结：
研究无标度二元网络中白噪声对共识动力学的影响，发现网络的H2范数相干性趋于常数，证明无标度拓扑和环状结构增强了噪声下的鲁棒性。
摘要：
大多数现实网络都是无标度、环状且稀疏的，具有幂律度分布和恒定的平均度。本文研究了二元无标度网络中的一阶共识动力学，其中顶点受白噪声影响。我们关注以H2范数表征的网络相干性，该范数量化了代理跟踪共识值的紧密程度。我们首先给出了网络相干性的下界，该下界与平均度有关，与网络阶数无关。然后研究了一些稀疏的无标度现实网络的相干性，其趋于常数。我们还研究了Barabási-Albert网络和高维随机Apollonian网络的相干性，当网络增长时也趋于常数。最后，基于相干性与基尔霍夫指数的联系，我们研究了两种确定性增长稀疏网络的相干性，并获得精确表达式，这些表达式趋于小常数。我们的结果表明，幂律网络中噪声对共识动力学的影响可以忽略不计。我们认为无标度拓扑与环状结构共同负责了幂律网络在噪声共识动力学中的强鲁棒性。
英文摘要：
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 Barabási-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.
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