Physics and Society 5 JAN, 2015

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

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 embedding and distance‑dependent link probabilities shape the effective dimensionality of complex networks. The authors consider model networks built on regular lattices of embedding dimension (d_e = 1) and (d_e = 2). A pair of nodes separated by Euclidean distance (r) is connected with probability (p(r) \propto r^{-\delta}), where (\delta) controls the strength of the distance decay. Two independent definitions of a network dimension (d) are employed: (a) the scaling of the mass (the number of nodes) within a Euclidean radius (r), (M(r) \sim r^{d}); and (b) the scaling of the return probability of a random walker to its origin after traveling a Euclidean distance (r), (P_{0}(r) \sim r^{-d}). Extensive Monte‑Carlo simulations show that both methods give identical values of (d) for any (\delta), confirming the robustness of the dimension concept.

Three regimes emerge as a function of (\delta):

  1. (\delta < d_e) – Long‑range links are abundant, the network behaves like a small‑world or even a complete graph. The mass grows faster than any power of (r) and the measured dimension diverges, i.e., (d = \infty).

  2. (\delta > 2 d_e) – Links are effectively short‑ranged; the network reproduces the geometry of the underlying lattice. Consequently (M(r) \sim r^{d_e}) and (d = d_e).

  3. Intermediate regime (d_e \le \delta < 2 d_e) – The most interesting behavior occurs here. The dimension decreases continuously from infinity to the embedding dimension as (\delta) increases. Near the lower bound the authors find a scaling law (d - d_e \sim (\delta - d_e)^{-1}), indicating a rapid crossover. In this regime the usual power‑law relations between topological distance (\ell) (the minimal number of hops) and both mass and Euclidean distance break down.

The paper then examines the relationship between the topological distance (\ell) and the two geometric quantities (M(\ell)) and (r(\ell)). For (\delta < d_e) the mass grows exponentially, (M(\ell) \sim \exp(c \ell)), while the Euclidean distance remains essentially constant, reflecting the “ultra‑small‑world” nature of the network. For (\delta \ge 2 d_e) conventional scaling is recovered: (M(\ell) \sim \ell^{d_{\ell}}) and (r(\ell) \sim \ell^{1/d_{\min}}) with the product (d_{\ell} d_{\min} = d).

In the intermediate regime the authors discover stretched‑exponential growth: \