全局层次与局部结构：无标度网络中的虚假自反馈
Global hierarchy vs. local structure: spurious self-feedback in scale-free networks
摘要：
具有厚尾度分布的网络在许多科学学科中无处不在。这种系统的特征在于所谓的集线器，即与其他节点有大量连接的特定节点。通过这个属性，它们被认为是集体网络行为的关键，例如，在伊辛模型上的复杂拓扑。这特别适用于到全局有序网络状态的转变，从而以分层方式进行，并且具有非平凡的局部结构。标准的无标度网络伊辛模型的平均场论低估了枢纽节点的存在，但却为全球秩序的出现提供了非常可靠的估计。在这里，我们揭示了一个虚假的自反馈效应，固有的平均场论，这一明显的悖论的基础。更具体地说，我们证明了高阶相互作用的影响，精确地取消了自我反馈的枢纽，我们揭示了枢纽的重要性，在网络中的本地与全球秩序的不同发病。由于我们的论点的一般性质，我们希望我们为巴拉巴西-阿尔伯特类型的伊辛网络的原型案例所揭示的机制也与其他具有强层次底层网络结构的系统相关。
英文摘要：
Networks with fat-tailed degree distributions are omnipresent across many scientific disciplines. Such systems are characterized by so-called hubs, specific nodes with high numbers of connections to other nodes. By this property, they are expected to be key to the collective network behavior, e.g., in Ising models on such complex topologies. This applies in particular to the transition into a globally ordered network state, which thereby proceeds in a hierarchical fashion, and with a non-trivial local structure. Standard mean-field theory of Ising models on scale-free networks underrates the presence of the hubs, while nevertheless providing remarkably reliable estimates for the onset of global order. Here, we expose that a spurious self-feedback effect, inherent to mean-field theory, underlies this apparent paradox. More specifically, we demonstrate that higher order interaction effects precisely cancel the self-feedback on the hubs, and we expose the importance of hubs for the distinct onset of local versus global order in the network. Due to the generic nature of our arguments, we expect the mechanism that we uncover for the archetypal case of Ising networks of the Barabási-Albert type to be also relevant for other systems with a strongly hierarchical underlying network structure.
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