Geometric Organization and Inference of Shortest Path Nodes in Soft Random Geometric Graphs

TL;DR

The paper investigates how shortest paths in Euclidean Soft Random Geometric Graphs align along geodesic curves between endpoints and how this geometry can be exploited to identify shortest-path nodes under partial observability. By introducing and quantifying metrics such as the distance to geodesics $d(q,\gamma(i,j))$ and path stretch $S_{ij}$, the authors show that shortest-path nodes cluster near geodesics, with alignment strengthening as $N$ and $\beta$ increase and displaying a nonmonotonic dependence on $\langle D\rangle$. They propose a geometry-based inference method (distance to geodesic) and compare it to network-reconstruction approaches, finding robust performance especially when links are unknown or coordinates are noisy. The findings have potential practical impact for navigation, wireless routing, and infrastructure-flow characterization in partially observable geometric networks, and the results extend to Waxman-type SRGGs, highlighting the utility of latent-space geometry for path discovery.

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 {\it Euclidean} Soft Random Geometric Graphs (SRGGs). We find that shortest paths are aligned along geodesic curves connecting shortest path endpoints. 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 non-monotonic 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.

Figures (12)

• Figure 1: (a) shows an example of how the clustering coefficient changes with the input inverse temperature parameter in SRGGs. (b) shows an example of how the real average degree changes with the input expected degree in SRGGs. The x-axis is the input expected degree, while the y-axis is the real average degree.
• Figure 2: (a) A toy example of a geodesic, distance from a node to the geodesic and a shortest path (highlighted by green) with the corresponding shortest path stretch indicated by the black dashed line along the green path. (b) The distribution of the average (avg), maximum (max) and minimum (min) distances between shortest path nodes and geodesics connecting shortest path endpoints. We compare the resulting distributions to (ran) those between randomly chosen nodes and the geodesic. Within an SRGG $G$, we randomly select $10^{6}$ shortest path endpoint pairs and for each $(i,j)$ pair, we find shortest path set $\mathcal{P}_{ij}^*$, geodesic $\gamma(i,j)$. For shortest path nodes in $\mathcal{P}_{ij}^*$, we compute the minimum (min), maximum (max) and average (avg) distance to the geodesic. In addition, for each shortest path set $\mathcal{P}_{ij}^*$, we created a random set $\mathcal{P}_{ij}^{\rm rand}$ of the same cardinality containing randomly selected nodes. Using $\mathcal{P}_{ij}^{\rm rand}$, we computed average distances to the geodesic (ran). All parameters used in the experiments are indicated in the respective panels.
• Figure 3: (a) The average distance to the geodesic $\langle d \rangle$ as a function of inverse temperature $\beta$ for SRGGs with $\mathbb{E}[D] = 10$ and variable size $N$. The inset displays the average distance to the geodesic $\langle d \rangle$ as a function of the number of nodes $N$ for SRGGs with expected degree $\mathbb{E}[D] = 10$ and the inverse temperature $\beta=8$. (b) The average distance to the geodesic $\langle d \rangle$ as a function of the average degree $\langle D \rangle$ for SRGGs of variable size $N$ and the inverse temperature $\beta = 4$. The inset depicts the three phases discussed in the text. (c) The average shortest path stretch $\langle S \rangle$ as a function of the inverse temperature $\beta$ for SRGGs of $\mathbb{E}[D] = 10$ and variable size $N$. The inset displays the average shortest path stretch $\langle S \rangle$ as a function of the number of nodes $N$ for SRGGs with expected degree $\mathbb{E}[D] = 10$ and the inverse temperature $\beta=8$. (d) The average shortest path stretch $\langle S \rangle$ as a function of the average degree $\langle D \rangle$ for SRGGs of variable size $N$ and the inverse temperature parameter $\beta = 128$. All plots correspond to the average of $1,000$ randomly selected shortest paths, while the error bars quantify the standard deviations.
• Figure 4: (a) The average shortest path stretch $\langle S \rangle$ as a function of the average degree $\langle D \rangle$ for inverse temperatures $\beta=1$, $\beta=8$, and $\beta=128$. The plots for beta $\beta = 8$ and $\beta=128$ are scaled by multiplicative factors $10^{-1}$ and $10^{-2}$, respectively, to avoid their overlap. The error bars depict standard deviation values. (b) Critical degree values $\langle D \rangle_c$ corresponding to the onset of the GCC as a function of expected degrees $\langle D\rangle_{\rm max}$ corresponding to the largest distance to geodesic. The scatter plot includes $20$ points corresponding to different inverse temperature values $\beta$ selected uniformly at random from the $(2,128]$ interval. (c) The average link length $\langle d \rangle_{\ell}$ as a function of the average degree $\langle D \rangle$ in SRGGs with different inverse temperatures $\beta$. The solid lines are power-law curves with exponents $\tau = {\rm min} \left(1/2,\beta/2 - 1\right)$. (d) The average shortest path hopcount $\langle h \rangle$ as a function of the average degree $\langle D \rangle$ in SRGGs with different inverse temperatures $\beta$. The solid lines are empirical fits for $\beta =2.5$ and $\beta=128$ values. (e,f) The average shortest path stretch $\langle S \rangle$ as a function of the average degree $\langle D \rangle$ for inverse temperatures (e) $\beta=2.5$ and (f) $\beta=128$. (e) The left inset depicts $\langle S \rangle \langle D \rangle^{-\tau}$ as a function of ${\rm ln} \langle D\rangle^{-1}$. Note that $\langle S \rangle \langle D \rangle^{-\tau} \propto {\rm ln} \langle D \rangle ^{-1}$, confirming the low-degree behavior of the shortest path stretch $\langle S \rangle$. The right inset depicts $\langle S \rangle \langle D \rangle^{-\tau}$ as a function of $\langle D \rangle$. Note that $\langle S \rangle \langle D \rangle^{-\tau}$ reaches a horizontal asymptote, confirming the power-law scaling of the shortest path stretch for large $\langle D \rangle$ values. (f) The inset depicts $\langle S \rangle^{-2} \langle D \rangle$ as a function of $\langle D \rangle -\langle D \rangle_c$, testing the low-degree behavior of the shortest path stretch. Note that the shortest path stretch $\langle S \rangle$ is nearly independent of $\langle D \rangle$ for a large range of $\langle D \rangle$ values.
• Figure 5: Precision of shortest path finding using (i) distance to geodesic (Geo-based) and (ii) SRGG (SRGG+network) and (iii) RGG (RGG+network) reconstructions. Panels display average precision as a function of the coordinate noise amplitude $\alpha$ and panels correspond to different combinations of $\mathbb{E}[D]$ and $\beta$ parameters. All experiments are performed using SRGG networks of $N=10^{4}$ nodes. Each bar is the average precision computed for $100$ randomly chosen node pairs and the error bars depict standard deviations.