Dynamics of hot random hyperbolic graphs
Journal
Physical Review E
Date Issued
February 3, 2022
Author(s)
DOI
10.1103/PhysRevE.105.024302
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.
(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.