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Geometric organization and inference of shortest path nodes in soft random geometric graphs

Zhihao Qiu1, Sámuel G. Balogh1,2, Xinhan Liu1, Piet Van Mieghem1, and Maksim Kitsak1,3,*

  • *Contact author: maksim.kitsak@gmail.com
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Phys. Rev. Research 8, 033149 – Published 7 August, 2026

DOI: https://doi.org/10.1103/dq2j-z8k6

Abstract

The shortest path problem is related to many dynamic processes on networks, ranging from routing in communication networks to signaling in molecular interaction networks. When the network is fully known, the shortest path problem can be solved precisely and in polynomial time. If, however, the network of interest is only partially observable, the shortest path problem is no longer straightforward. Inspired by the shortest path problem in partially observable networks, we investigate the geometric properties of shortest paths in Euclidean soft random geometric graphs (SRGGs). We find that shortest paths are aligned along geodesic curves connecting shortest path end points. The strength of the shortest path alignment, as quantified by the average distance to geodesic from shortest path nodes and the average path stretch, is higher for larger SRGGs with short-range connections. In addition, we find that the strength of the shortest path alignment is nonmonotonic with respect to the average degree of the SRGG. Based on these observations, we establish the conditions under which the alignment of shortest paths may be sufficiently strong to allow the identification of shortest path nodes based on their proximity to geodesic curves. We show that in partially observable networks with uncertain node positions, our geometric approach can outperform network-based shortest path algorithms. In practical settings, our findings may have applications to navigation, wireless routing, and flow characterization in infrastructure networks.

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Phys. Rev. Research 8, 033149– Published 7 August, 2026
  • Received 9 February 2026
  • Accepted 17 June 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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