Self-similarity of temporal interaction networks arises from hyperbolic geometry with time-varying curvature

TL;DR

This work proposes a novel technique of scale transformation that dissects temporal interaction networks under spatio-temporal scales, namely, flow scales, and relates the emergence of flow-scale self-similarity to the latent geometry of such networks.

Abstract

The self-similarity of complex systems has been studied intensely across different domains due to its potential applications in system modeling, complexity analysis, etc., as well as for deep theoretical interest. Existing studies rely on scale transformations conceptualized over either a definite geometric structure of the system (very often realized as length-scale transformations) or purely temporal scale transformations. However, many physical and social systems are observed as temporal interactions among agents without any definitive geometry. Yet, one can imagine the existence of an underlying notion of distance as the interactions are mostly localized. Analysing only the time-scale transformations over such systems would uncover only a limited aspect of the complexity. In this work, we propose a novel technique of scale transformation that dissects temporal interaction networks under spatio-temporal scales, namely, flow scales. Upon experimenting with multiple social and biological interaction networks, we find that many of them possess a finite fractal dimension under flow-scale transformation. Finally, we relate the emergence of flow-scale self-similarity to the latent geometry of such networks. We observe strong evidence that justifies the assumption of an underlying, variable-curvature hyperbolic geometry that induces self-similarity of temporal interaction networks. Our work bears implications for modeling temporal interaction networks at different scales and uncovering their latent geometric structures.

Figures (10)

• Figure 1: Schematic description of flow-scale transformation on example temporal interaction network.Top: A temporal interaction network with timestamped interaction edges; here $t_0<t_1<\cdots <t_6$. A snapshot of the network within the time interval $(t_i, t_j)$ would contain edges corresponding to interactions within that interval. Middle: Box-covering of the network with box size $=3$ and time size $=\Delta t_1$; here $t_0<t_1\leq t_0+\Delta t_1$, $t_0+\Delta t_1<t_2<t_3\leq t_0+2\Delta t_1$, $t_0+2\Delta t_1<t_4\leq t_0+3\Delta t_1$, and $t_0+3\Delta t_1<t_5<t_6\leq t_0+4\Delta t_1$. In this case, the total number of boxes covering the network is $10$. Similarly in the bottom, a box covering with box size $2$ and time size $\Delta t_2$ is shown, where $t_0<t_1<t_2\leq t_0+\Delta t_2$, $t_0+\Delta t_2<t_3<t_4\leq t_0+2\Delta t_2$, and, $t_0+2\Delta t_2<t_5<t_6\leq t_0+3\Delta t_2$. The required number of boxes is $9$.
• Figure 2: Scale transformation characteristics of the real temporal interaction networks. We show the values of $\log(N_v/N_0)$ vs. $\log(v)$ for different real networks. To check for the finiteness of the fractal dimension $d_v$, we fit two regression lines: one assuming a linear relationship between the last four data points of $\log(N_v/N_0)$ vs $\log(v)$ (blue dashed line), another assuming the same between $\log(N_v/N_0)$ vs $v$ (red dashed line). Evidently, the ia-email network exhibits scale-invariance regarding Equation \ref{['eq:temporal-fractal-dim']}, with a fractal dimension of $1.96$. For DPPIN-Babu and superuser networks, $d_v$ does not show any bounded variation. The rest of the networks, though have finite fractal dimensions, do not show scale-invariance.
• Figure 3: Point-particle trajectories on the hyperbolic manifold.a. The Lorentzian model for representing the 2-D Hyperbolic space $\mathbb{H}^{2,K}$ with constant negative curvature $-\frac{1}{K}$ as a hyperboloid; any point $\mathbf{u}$ on the tangent vector space at the origin, $\mathcal{T}_{\mathbf{O}}\mathbb{H}^{2,K}$ can be mapped to $\mathbb{H}^{2,K}$ via the exponential map (and back to the tangent space using the logarithmic map). b. Box-counting of point-set over the hyperbolic plane. In this example, a 2-d hyperbolic set is covered using 17 3-d grids of size $l_b$. c. Temporal scale transformation over the hyperboloid model; the blue curve on the leftmost figure represents the actual trajectory of a point-particle over the hyperboloid in time $[0, 1]$; middle and rightmost figures show the discrete time-scale transformation of the trajectory with $\Delta t=0.25$ and $0.125$, respectively.
• Figure 4: Scale transformation characteristics of point-particle motion.a. This plot corresponds to a 3-dimensional space of constant curvature $-1$, where the particle positions are sampled uniformly over the tangent space; though not scale-invariant, it has a finite fractal dimension at the limiting case. b. This plot corresponds to a 3-dimensional space with the negative curvature increasing exponentially with time, and particles are sampled uniformly; it shows a scale-invariant characteristic against the transformation of $v$. c. This plot corresponds to a flat 3-dimensional space with the particle positions sampled from a Gaussian; evidently, the fractal dimension is not bounded, and therefore, the said structure does not possess fractal properties.
• Figure S1: Scale transformation characteristics of the ia-email network. The top panel shows the variation in the number of boxes with fixed-sized boxes as we change the time-scale. The bottom panel shows the variation of the number of boxes vs the box size for different time-scales.