Complex networks embedded in space: Dimension and scaling relations between mass, topological distance and Euclidean distance

Many real networks are embedded in space, where in some of them the links length decay as a power law distribution with distance. Indications that such systems can be characterized by the concept of dimension were found recently. Here, we present further support for this claim, based on extensive numerical simulations for model networks embedded on lattices of dimensions $d_e=1$ and $d_e=2$. We evaluate the dimension $d$ from the power law scaling of (a) the mass of the network with the Euclidean radius $r$ and (b) the probability of return to the origin with the distance $r$ travelled by the random walker. Both approaches yield the same dimension. For networks with $\delta < d_e$, $d$ is infinity, while for $\delta > 2d_e$, $d$ obtains the value of the embedding dimension $d_e$. In the intermediate regime of interest $d_e \leq \delta < 2 d_e$, our numerical results suggest that $d$ decreases continously from $d = \infty$ to $d_e$, with $d - d_e \sim (\delta - d_e)^{-1}$ for $\delta$ close to $d_e$. Finally, we discuss the scaling of the mass $M$ and the Euclidean distance $r$ with the topological distance $\ell$. Our results suggest that in the intermediate regime $d_e \leq \delta < 2 d_e$, $M(\ell)$ and $r(\ell)$ do not increase with $\ell$ as a power law but with a stretched exponential, $M(\ell) \sim \exp [A \ell^{\delta’ (2 - \delta’)}]$ and $r(\ell) \sim \exp [B \ell^{\delta’ (2 - \delta’)}]$, where $\delta’ = \delta/d_e$. The parameters $A$ and $B$ are related to $d$ by $d = A/B$, such that $M(\ell) \sim r(\ell)^d$. For $\delta < d_e$, $M$ increases exponentially with $\ell$, as known for $\delta=0$, while $r$ is constant and independent of $\ell$. For $\delta \geq 2d_e$, we find power law scaling, $M(\ell) \sim \ell^{d_\ell}$ and $r(\ell) \sim \ell^{1/d_{min}}$, with $d_\ell \cdot d_{min} = d$.

💡 Research Summary

The paper investigates how spatial constraints affect the structural and dynamical properties of complex networks that are embedded in Euclidean space. The authors consider two embedding dimensions, (d_e=1) (a linear chain) and (d_e=2) (a square lattice), and generate networks by assigning each node a fixed average degree (k) (typically 4). Links are added with a probability that decays as a power law of the Euclidean distance, (p(r)\propto r^{-\delta}), where (\delta) is the distance‑decay exponent. This construction generalises the Watts–Strogatz model (recovered for (\delta=0)) and mimics real‑world systems such as the Internet, airline routes, or protein interaction maps, where long‑range connections become progressively rarer.

Two independent methods are used to define an effective network dimension (d). First, the mass‑radius relation (M(r)\sim r^{d}) is measured by counting the number of nodes inside a Euclidean hypersphere of radius (r) centred on a randomly chosen origin. Second, a random walker is released from the origin; the probability (P_{0}(r)) that the walker returns to its starting site after having travelled an average Euclidean distance (r) scales as (P_{0}(r)\sim r^{-d}). Both approaches yield consistent values of (d).

The results fall into three regimes determined by the distance exponent (\delta):

1. Weak constraint ((\delta<d_e)) – The characteristic length scales (\bar r_n) (moments of the distance distribution) and the maximal link length (r_{\max}) both grow linearly with the system size (L). Consequently the network behaves like a mean‑field random graph; the effective dimension diverges ((d\to\infty)). Random‑walk return probabilities decay extremely slowly, reflecting the high connectivity.

2. Strong constraint ((\delta>2d_e)) – Both (\bar r_n/L) and (r_{\max}/L) vanish as (L\to\infty). The network becomes locally similar to the underlying lattice, and the measured dimension equals the embedding dimension, (d=d_e). In this regime the usual power‑law scaling of topological distance (\ell) with Euclidean distance (r) is recovered.

3. Intermediate regime ((d_e\le\delta<2d_e)) – Here the dimension decreases continuously from infinity to (d_e) as (\delta) increases. Numerical data show a divergence of the form (d-d_e\sim(\delta-d_e)^{-1}) near the lower bound, and an empirical fit \