Papadopoulos Fragkiskos, Zambirinis Sofoclis
Department of Electrical Engineering, Computer Engineering and Informatics, Cyprus University of Technology, 3036 Limassol, Cyprus.
Phys Rev E. 2022 Feb;105(2-1):024302. doi: 10.1103/PhysRevE.105.024302.
We derive the most basic dynamical properties of random hyperbolic graphs (the distributions of contact and intercontact durations) in the hot regime (network temperature T>1). We show that for sufficiently large networks the contact distribution decays as a power law with exponent 2+T>3 for durations t>T, while for t<T it exhibits exponential-like decays. This result holds irrespective of the expected degree distribution, as long as it has a finite Tth moment. Otherwise, the contact distribution depends on the expected degree distribution and we show that if the latter is a power law with exponent γ∈(2,T+1], then the former decays as a power law with exponent γ+1>3. However, the intercontact distribution exhibits power-law decays with exponent 2-T∈(0,1) for T∈(1,2), while for T>2 it displays linear decays with a slope that depends on the observation interval. This result holds irrespective of the expected degree distribution as long as it has a finite Tth moment if T∈(1,2), or a finite second moment if T>2. Otherwise, the intercontact distribution depends on the expected degree distribution and if the latter is a power law with exponent γ∈(2,3), then the former decays as a power law with exponent 3-γ∈(0,1). Thus, hot random hyperbolic graphs can give rise to contact and intercontact distributions that both decay as power laws. These power laws, however, are unrealistic for the case of the intercontact distribution, as their exponent is always less than one. These results mean that hot random hyperbolic graphs are not adequate for modeling real temporal networks, in stark contrast to cold random hyperbolic graphs (T<1). Since the configuration model emerges at T→∞, these results also suggest that this is not an adequate null temporal network model.
我们推导了热态（网络温度T>1）下随机双曲图的最基本动力学性质（接触持续时间和两次接触之间持续时间的分布）。我们表明，对于足够大的网络，接触分布在持续时间t>T时按指数为2+T>3的幂律衰减，而在t<T时呈现指数型衰减。只要期望度分布具有有限的T阶矩，该结果就与期望度分布无关。否则，接触分布取决于期望度分布，并且我们表明，如果后者是指数为γ∈(2,T+1]的幂律，则前者按指数为γ+1>3的幂律衰减。然而，对于T∈(1,2)，两次接触之间的分布按指数为2-T∈(0,1)的幂律衰减，而对于T>2，它呈现线性衰减，其斜率取决于观察区间。只要期望度分布在T∈(1,2)时有有限的T阶矩，或者在T>2时有有限的二阶矩，该结果就与期望度分布无关。否则，两次接触之间的分布取决于期望度分布，如果后者是指数为γ∈(2,3)的幂律，则前者按指数为3-γ∈(0,1)的幂律衰减。因此，热随机双曲图可以产生都按幂律衰减的接触分布和两次接触之间的分布。然而，这些幂律对于两次接触之间的分布情况并不现实，因为它们的指数总是小于1。这些结果意味着热随机双曲图不足以对真实的时间网络进行建模，这与冷随机双曲图（T<1）形成鲜明对比。由于配置模型在T→∞时出现，这些结果还表明这不是一个合适的零时间网络模型。
