Degree heterogeneity in spatial networks with total cost constraint

Recently, In [Phys. Rev. Lett. 104, 018701 (2010)] the authors studied a spatial network which is constructed from a regular lattice by adding long-range edges (shortcuts) with probability $P_{ij}\sim r_{ij}^{-\alpha}$, where $r_{ij}$ is the Manhattan length of the long-range edges. The total length of the additional edges is subject to a cost constraint ($\sum r=C$). These networks have fixed optimal exponent $\alpha$ for transportation (measured by the average shortest-path length). However, we observe that the degree in such spatial networks is homogenously distributed, which is far different from real networks such as airline systems. In this paper, we propose a method to introduce degree heterogeneity in spatial networks with total cost constraint. Results show that with degree heterogeneity the optimal exponent shifts to a smaller value and the average shortest-path length can further decrease. Moreover, we consider the synchronization on the spatial networks and related results are discussed. Our new model may better reproduce the features of many real transportation systems.

💡 Research Summary

The paper revisits the spatial network model introduced by Li et al. (Phys. Rev. Lett. 104, 018701, 2010), in which a regular $d$‑dimensional lattice is augmented with long‑range shortcuts. In the original formulation each potential shortcut between nodes $i$ and $j$ is added with probability $P_{ij}\propto r_{ij}^{-\alpha}$, where $r_{ij}$ is the Manhattan distance, and the total length of all added shortcuts is constrained by a global cost $C=\sum r_{ij}$. Under this constraint the average shortest‑path length $\langle\ell\rangle$ is minimized when the exponent equals $\alpha^{*}=d+1$, a result that is independent of the cost $C$. While this model captures the trade‑off between distance‑dependent wiring cost and navigation efficiency, it produces networks with almost homogeneous degree distributions, contrary to many real transportation systems (airline, railway, logistics) that exhibit pronounced hub‑centric heterogeneity.

To address this discrepancy the authors propose a simple yet powerful modification: they bias the attachment probability by the current degrees of the two endpoints. The new probability reads
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