Whether real-world complex networks are scale free or not has long been controversial. Recently, in Broido and Clauset [A. D. Broido, A. Clauset, Nat. Commun. 10, 1017 (2019)], it was claimed that the degree distributions of real-world networks are rarely power law under statistical tests. Here, we attempt to address this issue by defining a fundamental property possessed by each link, the degree–degree distance, the distribution of which also shows signs of being power law by our empirical study. Surprisingly, although full-range statistical tests show that degree distributions are not often power law in real-world networks, we find that in more than half of the cases the degree–degree distance distributions can still be described by power laws. To explain these findings, we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted, two-way selection process. The model does not always produce solid power-law distributions but predicts that the degree–degree distance distribution exhibits stronger power-law behavior than the degree distribution of a finite-size network, especially when the network is dense. We test the strength of our model and its predictive power by examining how real-world networks evolve into an overly dense stage and how the corresponding distributions change. We propose that being scale free is a property of a complex network that should be determined by its underlying mechanism (e.g., preferential attachment) rather than by apparent distribution statistics of finite size. We thus conclude that the degree–degree distance distribution better represents the scale-free property of a complex network.

长期以来，现实世界中的复杂网络是否是无规模的一直存在争议。最近，在Broido和Clauset [AD Broido，A。Clauset，Nat。社区10，1017（2019）]，有人声称，根据统计测试，现实网络的度分布很少是幂律。在这里，我们试图通过定义每个链接所拥有的基本属性（度-度距离）来解决这个问题，根据我们的经验研究，其分布也显示出幂律的迹象。令人惊讶的是，尽管全方位统计测试表明，度分布在现实世界网络中通常不是幂律，但我们发现，在超过一半的情况下，度-度距离分布仍可以由幂律来描述。为了解释这些发现，我们引入了双向优先选择模型，其中链接配置是随机加权的双向选择过程。该模型并不总是产生可靠的幂律分布，而是预测度-度距离分布比有限尺寸网络的度分布表现出更强的幂律行为，尤其是在网络密集的情况下。通过检查现实世界网络如何演变成过密的阶段以及相应的分布如何变化，我们测试了模型的强度及其预测能力。我们认为，无标度是复杂网络的一个属性，应由其基本机制（例如，优先连接）决定，而不是由有限大小的表观分布统计决定。因此，我们得出结论，度-度距离分布更好地表示了复杂网络的无标度特性。