Shortest-path percolation on random networks

TL;DR

The paper introduces the shortest-path-percolation (SPP) model on Erdős–Rényi networks, where edges on minimum-length origin–destination paths are removed once paths of length $Q_t\le C$ exist for demands $o_t\to d_t$. For finite budget $C$, SPP belongs to ordinary percolation universality, while $C\to\infty$ yields an abrupt, nonstandard transition with distinct critical exponents, due to path-based correlations rather than competitive selection. Finite-size scaling in both conventional and event-based ensembles confirms standard exponents for finite $C$, but reveals a weakly discontinuous, explosive-like character for infinite $C$, including a redefined mapping between control parameters and distinct scaling regimes. The work quantifies how network dismantling depends on budget and demonstrates robust methodological approaches for simulating path-based percolation, including efficient shortest-path sampling, DAG-based path enumeration, and adaptive strategies to handle large graphs. These findings highlight how percolation phenomena can change qualitatively under path-centric rules, with implications for modeling resource exhaustion in transport and infrastructure networks and for exploring range-dependent universality in networked systems.

Abstract

We propose a bond-percolation model intended to describe the consumption, and eventual exhaustion, of resources in transport networks. Edges forming minimum-length paths connecting demanded origin-destination nodes are removed if below a certain budget. As pairs of nodes are demanded and edges are removed, the macroscopic connected component of the graph disappears, i.e., the graph undergoes a percolation transition. Here, we study such a shortest-path-percolation transition in homogeneous random graphs where pairs of demanded origin-destination nodes are randomly generated, and fully characterize it by means of finite-size scaling analysis. If budget is finite, the transition is identical to the one of ordinary percolation, where a single giant cluster shrinks as edges are removed from the graph; for infinite budget, the transition becomes more abrupt than the one of ordinary percolation, being characterized by the sudden fragmentation of the giant connected component into a multitude of clusters of similar size.

Figures (16)

• Figure 1: The shortest-path-percolation model. (a) There are two possible shortest paths connecting the origin-destination pair $o_t \to d_t$ demanded by the agent $t$, denoted by orange dotted edges and purple dashed edges, respectively. The length of such shortest paths is $Q_t = 4$. (b) If the maximum length allowed in the SPP model is $C \geq 4$, one of the two shortest paths is selected at random, then all of its edges are removed from the graph. Here, all purple dashed edges are deleted, and the graph fragments into four clusters.
• Figure 2: Shortest-path percolation transition in the conventional ensemble. (a) Percolation strength $P$ as a function of fraction of removed edges $p$ with different values of $C$. The SPP model is applied to Erdős-Rényi (ER) graphs with size $N=2^{20}$. (b) Pseudocritical point $p_c(N)$ as a function of $N$ for different values of $C$. The horizontal line denotes the estimated critical point $p_c$ for each $C$. (c) Average cluster size $S$ as a function of $p$ for ER graphs for different network sizes $N$. Here $C=1$. (d) Similar to (c) but for $C=N$.
• Figure 3: Shortest-path-percolation transition in the event-based ensemble. Here we use $C=N$. (a) The distribution of ${P}_{c}(N) \, N^{\beta/\bar{\nu}}$ with $\beta/\bar{\nu} = 0.21$. Different curves correspond to different network sizes $N$. (b) Same as in (a) but for $S_{c}(N) \,N^{-\gamma/\bar{\nu}}$ with $\gamma/\bar{\nu} = 0.55$.
• Figure 4: Control parameters in the shortest-path percolation model. (a) The fraction of removed edges $p$ is plotted as a function of number of demanding agents $t$ for different values of $C$. Results are valid for Erdős-Rényi graphs with size $N=2^{20}$. (b) Curve collapse for $C=1$ and different $N$ values. The abscissas values are rescaled as $t N^{-\theta}$, with $\theta = 2.01$, to obtain a collapse between the various curves. (c) Same as in (b), but for $C=N$. The collapse is obtained by rescaling the abscissas as $t N^{-\theta^{\ominus}}$, with $\theta^{\ominus} = 0.86$. (d) Same as in (c), but rescaling the abscissas as $t N^{-\theta^{\oplus}}$, with $\theta^{\oplus} = 2.07$. (e) Pseudocritical threshold $t_c(N)$ as a function of $N$ for different values of $C$. The full black line indicates the scaling $t_{c}(N) \sim N^{2}$, whereas the dashed yellow line stands for $t_{c}(N) \sim N^{0.86}$. (f) Number of demanded pairs required to fully dismantle the network $t_{max}(N)$vs.$N$ for different values of $C$. The black line indicates the scaling $t_{\max}(N) \sim N^{2}$.
• Figure 5: Abruptness of the shortest-path-percolation transition. (a) We plot $\Delta p$vs.$N$ for different $C$ values. Displayed lines represent the best fits of the scaling $\Delta p \sim N^{-\alpha}$. We get $\alpha = 0.02 \pm 0.01$ for $C=1$, $\alpha = 0.03 \pm 0.01$ for $C=2$, $\alpha = 0.03 \pm 0.01$ for $C=3$, and $\alpha = 0.13 \pm 0.01$ for $C=N$. (b) $\Delta \tilde{t}$vs.$N$. Lines stand for best fits of the scaling $\Delta \tilde{t} \sim N^{- \alpha'}$. We find $\alpha' = 0.03 \pm 0.01$ for $C=1$, $\alpha' = 0.03 \pm 0.01$ for $C=2$, $\alpha' = 0.04 \pm 0.01$ for $C=3$, and $\alpha' = 0.88 \pm 0.01$ for $C=N$.