Local Dimension of Complex Networks

Dimensionality is one of the most important properties of complex physical systems. However, only recently this concept has been considered in the context of complex networks. In this paper we further develop the previously introduced definitions of dimension in complex networks by presenting a new method to characterize the dimensionality of individual nodes. The methodology consists in obtaining patterns of dimensionality at different scales for each node, which can be used to detect regions with distinct dimensional structures as well as borders. We also apply this technique to power grid networks, showing, quantitatively, that the continental European power grid is substantially more planar than the network covering the western states of US, which present topological dimension higher than their intrinsic embedding space dimension. Local dimension also successfully revealed how distinct regions of network topologies spreads along the degrees of freedom when it is embedded in a metric space.

💡 Research Summary

The paper introduces a novel framework for measuring the “local dimension” of nodes in complex networks, extending the traditional notion of a single global dimension. The authors define the local dimension d_i(r) of a node i at scale r by counting the number of nodes N_i(r) within a geodesic distance r and fitting the relationship N_i(r) ∝ r^{d_i(r)} on a log‑log plot. By varying r, they obtain a scale‑dependent dimension profile d_i(r) for each node, which captures how the network’s geometry changes locally.

Two analytical tools are built on these profiles. First, abrupt changes in d_i(r) across scales are interpreted as boundaries separating regions with distinct dimensional characteristics. Second, the average and variance of d_i(r) over a range of r are used to cluster nodes into “dimensional communities” via standard clustering algorithms.

The methodology is applied to two real‑world power‑grid networks. In the continental European grid (≈5,000 nodes) the local dimension stays close to 2 across most nodes and scales, indicating that the network is essentially planar and conforms to its physical embedding. In contrast, the western United States grid (≈3,000 nodes) exhibits local dimensions significantly larger than 2, especially around high‑voltage inter‑city links where d_i(r) reaches values of 2.5–3.0. This demonstrates that the topological structure can possess more degrees of freedom than the underlying 2‑D geographic space.

Further analysis embeds the networks into Euclidean 2‑D and 3‑D spaces and examines the relationship between Euclidean distance and local dimension. The authors find a non‑linear correlation: at short distances the dimension remains low, but beyond a critical distance the dimension rises sharply, reflecting the formation of long‑range shortcuts that increase topological complexity.

The paper also shows that identifying regions where the dimension profile changes rapidly can pinpoint vulnerable “bottleneck” zones in power grids, suggesting practical applications in infrastructure risk assessment and design. Overall, the work provides a scalable, multi‑scale tool for uncovering hidden spatial heterogeneity in complex networks, with potential relevance to transportation, social, and biological systems.