Dynamics of hot random hyperbolic graphs

Abstract

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 $T^{\text{th}}$ moment. Otherwise, the contact distribution depends on the expected degree distribution and we show that if the latter is a power law with exponent $\gamma \in (2, T+1]$, then the former decays as a power law with exponent $\gamma+1 > 3$. On the other hand, the intercontact distribution exhibits power-law decays with exponent $2-T \in (0, 1)$ for $T \in (1,2)$, 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 $T^{\text{th}}$ moment if $T \in (1,2)$, 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 $\gamma \in (2, 3)$, then the former decays as a power law with exponent $3-\gamma \in (0,1)$. 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 $T \to \infty$, these results also suggest that this is not an adequate null temporal network model.