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

For static networks, box-covering provides the most popular approach to test discrete scale invariance. Given a network $\mathcal{G}=\{\mathcal{V},\mathcal{E}\}$, a box-cover of size $l_{b}$ can be defined as $C_{l_{b}}(\mathcal{G})=\{c_{i}|\bigcup_{i}c_{i}=V,c_{i}\cap c_{j}=\phi\}$, such that, the minimum distance between any two nodes in $c_{i}$ is at most $l_{b}$. The fractal dimension of the network is defined as,

$$d_{b}=\lim_{\epsilon\to 0}\frac{\log(N_{b}/N)}{\log(\epsilon)}$$

(1)

where $N$ is the number of nodes in the network, $|\mathcal{V}|$; $N_{b}$ is the number of boxes of size $l_{b}$ required to cover the network, $|C_{l_{b}}(\mathcal{G})|$, and $\epsilon=\frac{1}{l_{b}}$. A network is fractal if $d$ is finite. Furthermore, if $d_{b}$ remains constant as $l_{b}$ varies, then the network is length-scale invariant or self-similar. The term $N_{b}/N$ typically defines the average number of nodes in each box, or its mass; for a scale-invariant network, the mass of a box follows a power-law variation with its size [6, 7].

Temporal interaction networks naturally extend the spatial scale transformation to the temporal domain, yet it remains unexplored in the literature. A temporal interaction network can be formalized as $\mathcal{G}^{\prime}=\{\mathcal{V}^{\prime},\mathcal{E}^{\prime}\}$, where each edge $e_{k}\in\mathcal{E}^{\prime}$ is a triplet $(v_{i},v_{j},t_{k})$ for $v_{i},v_{j}\in\mathcal{V}^{\prime}$, and $t_{k}\in[t_{\text{start}},t_{\text{end}}]$ is the timestamp of the corresponding interaction. A popular approach of temporal graph learning is to represent $\mathcal{G}^{\prime}$ as a sequence of $M$ static snapshots $[G_{0},G_{1},\cdots,G_{M}]$ by sampling interactions with a given time window $\Delta t$; each $G_{m}=\{V_{m}\subseteq\mathcal{V}^{\prime},E_{m}\subseteq\mathcal{E}^{\prime}\}$ is constructed such that for any $e_{k}=(v_{i},v_{j},t_{k})\in\mathcal{E}^{\prime}$, if $t_{k}\in[t_{\text{start}}+(m-1)\Delta t,t_{\text{start}}+m\Delta t]$, $v_{i},v_{j}$ are added to $V_{m}$ and the edge $(v_{i},v_{j})$ is added to $E_{m}$. As shown in Figure 1, the size of the sampling window dictates the granularity of approximation: the smaller the window, the closer the snapshots are to the actual interaction network. Therefore, the sampling time window size $\Delta t$ provides a temporal analogue of the box size $l_{b}$. Putting them together, we define the modified box counting for a temporal interaction network $\mathcal{G}^{\prime}$. For a given box size $l_{b}$ and a sampling time size $\Delta t$, the box-cover is defined as:

$$C_{l_{b},\Delta t}(\mathcal{G}^{\prime})=\bigcup_{m\in[1,M]}C_{l_{b}}(G_{m})$$

(2)

For brevity, we will denote $|C_{l_{b},\Delta t}(\mathcal{G}^{\prime})|$ as $N_{l_{b},\Delta t}$ henceforth. Furthermore, we define the total number of vertices in the span $[t_{\text{start}},t_{\text{end}}]$ for time-window size $\Delta t$ as $N_{0,\Delta t}=\sum_{m}\lvert V_{m}\rvert$. In Figure 1, we show a working example. For the given network, $N_{3,\Delta t_{1}}=10$, whereas $N_{2,\Delta t_{2}}=9$.

Therefore, if we define the volume of the spatio-temporal boxes as $v=l_{b}\Delta t$, then one can redefine Equation 1 in case of temporal networks as,

$$d_{v}=\lim_{v\to\infty}\frac{\log(N_{v}/N_{0,\Delta t})}{-\log(v)}$$

(3)

Similar to its static counterpart, a temporal network is scale-invariant if $d_{v}$ is constant for all scale sizes $v$, and a fractal for a finite limiting value of $d_{v}$. For a temporal scale size of $\Delta t$, the term $N_{v}/N_{0,\Delta t}$ again defines the number of nodes within scale volume $v$. A detailed description of flow-scale transformation is presented in Methods.

Equipped with spatio-temporal scale transformation defined in Equation 2, we analyze six different temporal interaction networks arising from various real-world interactions – Enron email interactions [28] (ia-email), Reddit hyperlink interactions [29] (reddit-hyperlink), Arxiv high-energy physics papers citations [31] (ca-cit), protein-protein interactions [30] (DPPIN-Babu), Wikipedia talkpage interactions [32] (wiki-talk) and user interactions via posts and comments on the Superuser forum [32] (superuser) (see Table 1 for various statistics of these datasets). The choice of networks encompasses a wide variety of interactions: from user interactions over social platforms to protein-protein interactions. Furthermore, the list includes very slow-growing (e.g., citation networks) to very fast-growing (e.g., Reddit or Superuser) networks. We need to set different sizes for the time window $\Delta t$ in different networks to cope with this variation of interaction density.

In Figure 2, we summarize the scale transformation properties of these networks. Equation 3 indicates that there can be three possible patterns for a network to exhibit: i) a linear variation of $\log(N_{v}/N_{0})$ vs $\log(v)$ denoting scale-invariance, ii) a linear variation of $\log(N_{v}/N_{0})$ with $\log(v)$ for larger values of $v$, denoting a finite fractal dimension but not scale-invariance, and iii) no linear relationship between $\log(N_{v}/N_{0})$ and $\log(v)$, denoting a non-fractal network. The box-counting estimation of fractality and scale invariance is inherently statistical. Since the networks in consideration are finite, a small degree of noise can skew the presence (or absence) of linear relationships among observed samples. To deal with this, we fit two regression curves, one assuming $\log(N_{v}/N_{0})\propto\log(v)$ (i.e., a finite fractal dimension) and another assuming $\log(N_{v}/N_{0})\propto v$ (i.e., $d_{v}\to\infty$ as $v\to\infty$). As Figure 2 suggests, all the networks except DPPIN-Babu and superuser have a finite fractal dimension in the limiting case. We can calculate the value of the corresponding fractal dimensions from the slope of the regression lines – $1.96$ for ia-email, $3.76$ for reddit-hyperlink, $2.45$ for ca-cit, and $1.74$ for wiki-talk. Among these fractal networks, only ia-email shows scale-invariant characteristics under flow-scale transformations.

We further experiment on the scaling properties of some of these networks under purely spatial or temporal scale transformations (see Supplementary, Figures S1 to S4). As we can observe, a finite fractal dimension under flow-scale transformation necessitates scale-invariance under both length- and time-scale transformations. Failure in either of these conditions would lead to non-fractal behaviour (like DPPIN-Babu).

One can intuitively identify the proposed flow-scale boxes as a potential volume of influence of a cluster of nodes over both space as well as time. As we go from smaller to larger time-scales as well length-scales, the chances of a node interacting with other nodes increase. This eludes one to draw an analogy between a temporal interaction network and a system of point particles moving randomly in space; the chance of two particles interacting with each other increases as they come closer in space. Such an analogy naturally presumes a latent geometry of the point-particle system. It has been shown previously that an assumption of underlying non-Euclidean space explains the emergence of self-similarity in complex networks [19, 33]. Following the precedence of linking hyperbolic geometry to the evolution of complex networks [34, 35], we investigate the spatial and temporal scale transformation properties of discrete objects on hyperbolic space to seek parallels with the temporal information networks. Please see Methods for constructing necessary structures of hyperbolic geometry in terms of the Lorentzian model.

By definition of hyperbolic geometry, points closer to the origin have a shorter geodesic distance compared to points further away (Figure 3 b.). This provides us with a natural analogy to complex networks. For example, a hub, which is reachable from other nodes with a shorter path, can be represented as a point near the origin (and peripheral nodes can be mapped further away). Furthermore, the Lorentzian model equips us with the property that one can define operations on the Euclidean tangent space with minimal complexity and map them back to the hyperbolic space using the exponential map. With this, we proceed to define the point-particle motion over $\mathbb{H}^{n,K}$.

We define the process over the timespan $T:=[0,1]$. A point-particle is characterized by its position and velocity vectors on the tangent space at the origin, $\mathcal{T}_{\mathbf{O}}\mathbb{H}^{n,K}$. At any timestamp $t\in T$, a point-particle is initialized at position $\mathbf{u}(t)$ randomly over $\mathcal{T}_{\mathbf{O}}\mathbb{H}^{n,K}$. The ‘true’ or latent trajectory of the particle is governed by a parametric estimation of the velocity given by

$$\mathbf{\tilde{v}}(t)=\sum_{i=0}^{p}\mathbf{v}_{i}t^{i}$$

(4)

where $\mathbf{v}_{i}$ are random parameters that characterize $\mathbf{\tilde{v}}(t)$. This latent trajectory is intrinsic to the particle-system and, therefore, can not be observed directly. Instead, the system is realized in discrete timestamps $t_{i}\in T$, as shown in Figure 3 c. The position of a particle over $\mathcal{T}_{\mathbf{O}}\mathbb{H}^{n,K}$ at time $t_{i}$ can be calculated as

$$\mathbf{u}(t_{i})=\mathbf{u}(t_{i-1})+\mathbf{\tilde{v}}(t_{i-1})(t_{i}-t_{i-1})$$

(5)

and then mapped to $\mathbb{H}^{n,K}$ using the exponential map. With equally spaced timestamps $t_{i}$’s and $\Delta t=t_{i}-t_{i-1}$, the sequence of particle positions approach the true trajectory as $\Delta t\to 0$. This $\Delta t$ serves the purpose of the time-scale transformation similar to Equation 2. Since in the hyperboloid model, $\mathbb{H}^{n,K}$ is immersed in $\mathbb{R}^{n+1}$, the natural choice of defining the length-scale transformation is via covering the particle system with $n+1$-dimensional grids of size $l_{b}$, as shown in Figure 3 b. With these two definitions, we can directly define the fractal dimension of this particle system as the same as in Equation 3.

The point-particle motion described in the previous section can be majorly characterized by two parameters: the curvature of the space and the choice of the distribution from which the initial particle positions are sampled.

We experiment with three different scenarios in terms of curvature. Our procedures defined in the previous section already demonstrate the constant negative curvature scenario: the particle position at any time $t$ is found via mapping $\mathbf{u}(t)$ to $\mathbb{H}^{n,K}$ using the exponential map (Equation 11). An analogous phenomenon over a zero curvature (i.e., Euclidean) space can be achieved if $\mathbf{u}(t)$ and $\mathbf{\tilde{v}}(t)$ are defined over $\mathbb{R}^{n+1}$ instead of $\mathcal{T}_{\mathbf{O}}\mathbb{H}^{n,K}$ and scale transformations are applied over them directly instead of applying an exponential map. Finally, we experiment with a space with negative curvature varying with time. This can be achieved using the following definition of the exponential map:

	$\displaystyle\mathbf{h}(t_{i})=\operatorname{exp}^{K(t_{i})}_{\mathbf{O}(t_{i})}(\mathbf{u}(t_{i}))=\operatorname{cosh}\left(\frac{|\mathbf{u}(t_{i})|_{\mathcal{L}}}{\sqrt{K(t_{i})}}\right)\mathbf{O}(t_{i})$
	$\displaystyle+\sqrt{K(t_{i})}\operatorname{sinh}\left(\frac{|\mathbf{u}(t_{i})|_{\mathcal{L}}}{\sqrt{K(t_{i})}}\right)\frac{\mathbf{u}(t_{i})}{|\mathbf{u}(t_{i})|_{\mathcal{L}}}$		(6)

Here, the origin $\mathbf{O}$ shifts with time since it is defined explicitly using the curvature of the space.

Additionally, we experiment with the particle positions initially sampled from a Gaussian vs a uniform distribution, both centred at the origin of $\mathbb{R}^{n+1}$. We find that the three distinctive patterns of scale transformation that we observed in the case of the real networks emerge with specific choices of curvature and distributions in the point-particle motion as well, as shown in Figure 4. When the particle positions are sampled uniformly over a constant curvature space, we observe that the fractal dimension, albeit non-constant, converges to a finite value as $v\to\infty$. We conclude that a constant negative curvature space almost always guarantees a finite fractal dimension but not a scale-invariant one (Figure 4 a. shows the variation with $K=1$ and $n=3$). Scale-invariance exclusively arises when we move to space with time-varying negative curvature, specifically, $-\frac{1}{K_{0}}e^{at}$ for some positive constants $K_{0}$ and $a$; in the example shown in Figure 4 b., $K_{0}=1$, $a=1$, $n=3$. Finally, a particle set initialized from a Gaussian over the flat Euclidean space fails to exhibit a finite fractal dimension (Figure 4 c. with $n=3$ and a zero mean unit standard deviation Gaussian).

The definition of flow-scale boxes requires simultaneous scale transformation over space as well as time. To define the time-scale transformation, we use successive coarsening of the temporal measure that instantiates the interaction timestamps in the following manner.

Let a temporal interaction network be defined as a chronologically ordered sequence of time-stamped interactions $I(t_{0},t_{N})=\{(v_{i},v_{j},t_{k})\}$, where $t_{0}$ and $t_{N}$ are the start and end times of the interactions, respectively, and $v_{i}$ and $v_{j}$ are any two nodes interacting at time $t_{k}$. The discrete nature of these timestamps presumes that the observation was done at a certain degree of coarsening, i.e., they are measured to the nearest integer value on a specific unit of time like seconds, hours, or days. To illustrate further with an example, if the timestamps are reported in integer seconds, and if an interaction actually occurred at time $3.05$ seconds, then the timestamp that will be associated with the said interaction will be $3$ seconds. Therefore, it will be treated as contemporary with another interaction happening at a time of $3.14$ seconds. In this example, the chosen scale of time is $1$ second. We extend this to any arbitrary time-scale to renormalize the network. Given a time-scale $\Delta t$, the renormalized interaction network can be defined as:

$$I_{\Delta t}(t_{0},t_{N})=\{(v_{i},v_{j},t_{0}+m\Delta t)|m\in\{0,\cdots,\lfloor\frac{N}{\Delta t}\rfloor\}$$

(7)

for any $(v_{i},v_{j},t_{k})\in I(t_{0},t_{N})$ such that $t_{k}\in[t_{0}+m\Delta t,t_{0}+(m+1)\Delta t)$. Such a time-scale renormalization treats any interaction happening within the $\Delta t$ window as contemporary, thereby segmenting $I(t_{0},t_{N})$ into $\lfloor\frac{N}{\Delta t}\rfloor$ static snapshots. Each of these snapshots can be considered a static network constructed by nodes interacting with each other simultaneously (under the time-scale renormalization). The $m$-th snapshot is then defined as $G_{m}=\{V_{m},E_{m}\}$ with $V_{m}$ and $E_{m}$ being the set of nodes and edges, respectively such that, for any $e=(v_{i},v_{j})\in E_{m}$, there exists at least one interaction $(v_{i},v_{j},t_{0}+m\Delta t)\in I_{\Delta t}(t_{0},t_{N})$.

These snapshots now bear a geometric structure (i.e., the distance between two nodes can be defined based on the path length) conditioned upon the concept of simultaneity introduced by the coarsening of the time-scale. This allows us to apply box-counting to initiate length-scale renormalization.

Next, we apply box-covering to perform length-scale renormalization on the snapshots. We use the Maximum Excluded Mass Burning (MEMB) algorithm [42] to count the number of boxes of a given size needed to cover a given snapshot $G_{m}$. MEMB defines boxes with their radius instead of diameter. A box of radius $r_{b}$ is a connected subgraph with a root node of arbitrary choice such that any node in the subgraph is at most $r_{B}$ distance away from the root node. The number of boxes to cover the interaction network is then,

$$N_{v}=\sum_{m}\operatorname{MEMB}(G_{m})$$

(8)

where $N_{v}$ is the number of flow-scale boxes of time-scale $\Delta t$ and length-scale $l_{b}=2r_{b}$ needed to cover the temporal interaction network $I(t_{0},t_{N})$. Details of the choices of parameter sizes for different datasets are described in Supplementary, Table S1.

We start with the $n$-dimensional, Lorentzian Hyperboloid model of the hyperbolic space, denoted by $\mathbb{H}^{n,K}$ of constant negative curvature $-\frac{1}{K}$, embedded in an $(n+1)$-dimensional space (i.e., $\mathbb{H}^{n,K}\subset\mathbb{R}^{n+1}$). For any two points $\mathbf{h},\mathbf{g}\in\mathbb{R}^{n+1}$, we have the Minkowski inner product:

$$\langle\mathbf{h},\mathbf{g}\rangle_{\mathcal{L}}=-h_{0}g_{0}+\sum_{i=1}^{n}h_{i}g_{i}$$

(9)

For any $\mathbf{h}\in\mathbb{H}^{n,K}$, we have $\langle\mathbf{h},\mathbf{h}\rangle_{\mathcal{L}}=-K$. The tangent space to $\mathbb{H}^{n,K}$ at point $\mathbf{x}\in\mathbb{H}^{n,K}$ is defined as,

$$\mathcal{T}_{\mathbf{x}}\mathbb{H}^{2,K}=\{\mathbf{v}|\langle\mathbf{v},\mathbf{x}\rangle_{\mathcal{L}}=0\}$$

(10)

For any point $\mathbf{u}\in\mathcal{T}_{\mathbf{x}}\mathbb{H}^{n,K}$, the Minkowski norm is defined by $|\mathbf{u}|_{\mathcal{L}}=\langle\mathbf{u},\mathbf{u}\rangle_{\mathcal{L}}$. The origin of $\mathbb{H}^{n,K}$ is defined as $\mathbf{O}$ where $O_{0}=\sqrt{K}$ and $O_{i}=0$ for all $i\neq 0$. One can define a bijective map between $\mathbb{H}^{n,K}$ and $\mathcal{T}_{\mathbf{x}}\mathbb{H}^{n,K}$, also called the exponential map, as,

$$\mathbf{h}=\operatorname{exp}^{K}_{\mathbf{x}}(\mathbf{u})=\operatorname{cosh}\left(\frac{|\mathbf{u}|_{\mathcal{L}}}{\sqrt{K}}\right)\mathbf{x}+\sqrt{K}\operatorname{sinh}\left(\frac{|\mathbf{u}|_{\mathcal{L}}}{\sqrt{K}}\right)\frac{\mathbf{u}}{|\mathbf{u}|_{\mathcal{L}}}$$

(11)

for $\mathbf{h},\mathbf{x}\in\mathbb{H}^{n,K}$ and $\mathbf{u}\in\mathcal{T}_{\mathbf{x}}\mathbb{H}^{n,K}$. The inverse of the exponential map is called the logarithmic map and is defined as,

$$\mathbf{u}=\operatorname{log}^{K}_{\mathbf{x}}(\mathbf{h})=d^{K}_{\mathcal{L}}(\mathbf{x},\mathbf{h})\frac{\mathbf{h}+\frac{1}{K}\langle\mathbf{h},\mathbf{x}\rangle_{\mathcal{L}}\mathbf{x}}{|\mathbf{h}+\frac{1}{K}\langle\mathbf{h},\mathbf{x}\rangle_{\mathcal{L}}\mathbf{x}|_{\mathcal{L}}}$$

(12)

where $d^{K}_{\mathcal{L}}(\mathbf{x},\mathbf{h})=\sqrt{K}\operatorname{arccosh}(-\langle\mathbf{x},\mathbf{h}\rangle_{\mathcal{L}}/K)$ is the distance (length of the geodesic curve) between $\mathbf{x}$ and $\mathbf{h}$.