Shortest-Path Percolation Model

• Shortest-Path Percolation (SPP) is a stochastic process on graphs that removes edges along shortest paths selected within a defined path-length budget, modeling resource depletion in networks.
• The model couples edge removal across optimal routes, introducing strong topological correlations that lead to abrupt fragmentation transitions.
• Large-scale simulations demonstrate that SPP on scale-free networks rapidly homogenizes hub structures, yielding percolation behavior identical to that of Erdős–Rényi graphs.

The shortest-path percolation (SPP) model is a stochastic process defined on a graph in which edges are progressively removed along shortest paths connecting randomly chosen origin–destination node pairs, subject to a tunable path-length budget parameter. Initially conceived to model resource depletion (e.g., in transport, communication, or supply networks), SPP extends traditional bond percolation by coupling edge removal across optimal routes rather than independently, thereby introducing strong topological correlations. Recent research has focused on its critical behavior, universality class, and transition dynamics, particularly on scale-free networks (SFNs) characterized by power-law degree distributions (Kim et al., 11 Sep 2025).

1. Formal Definition and Process Dynamics

Consider an undirected, unweighted graph $\mathcal{M} = ({\mathcal{V}}, {\mathcal{E}}_t)$, with ${\mathcal{V}}$ the set of nodes, and ${\mathcal{E}}_t$ the edge set at time $t$. At each discrete time step $t$, a pair $(o_t, d_t)$ is uniformly selected from ${\mathcal{V}}$. If a shortest path (geodesic) of length $Q_t$ exists between $o_t$ and $d_t$, and $Q_t \leq C$ (budget parameter), a chosen shortest path is supplied and every edge along it is removed: $${\mathcal{E}}_{t+1} = {\mathcal{E}}_t \setminus \{ \text{edges along one chosen shortest path between } o_t, d_t \}$$ If $Q_t > C$ or such a path does not exist, no edges are removed at that step.

When $C = 1$, SPP reduces to ordinary bond percolation, wherein random individual edges are removed. For $C > 1$, removal is correlated: entire optimal paths are deleted in a single update, introducing structural dependencies specific to the graph's topology and usage patterns.

2. Percolation Transition and Universal Behavior

The central phenomenon in SPP is the fragmentation of the initially macroscopic connected component into many microscopic clusters—akin to a percolation transition. The transition is tracked via the order parameter $P$: $P = \frac{s_Z}{\sum_{z=1}^{Z} s_z}$ where $s_Z$ is the size of the largest cluster and $\{s_z\}$ labels component sizes after fragmentation. The mean cluster size, excluding the largest, is given by: $$S = \frac{\sum_{z=1}^{Z-1} s_z^2}{\sum_{z=1}^{Z-1} s_z}$$

A pseudo-critical point $r_c^*(N)$ is operationally defined as the fraction of edges removed at the largest drop in $P$. Ensemble average $r_c(N)$ follows finite-size scaling: $r_c(N) = r_c + b N^{-1/\bar{\nu}_1}$ with fluctuations

$\sigma[r_c^*] \sim N^{-1/\bar{\nu}_2}$

At criticality, $P$ and $S$ obey scaling laws: $$P_c(N) \sim N^{-\beta / \bar{\nu}}, \quad S_c(N) \sim N^{\gamma / \bar{\nu}}$$

3. Universality Class and Independence from Degree Exponent

Classical percolation on SFNs exhibits marked dependence on the degree exponent $\lambda$, with anomalous behavior for $\lambda < 3$ (nonpercolating phase only after removal of nearly all edges). For SPP with finite budget ($C > 1$), large-scale simulations reveal that the transition on SFNs is identical to that on Erdős–Rényi (ER) graphs; the critical exponents ($\beta/\bar{\nu}, \gamma/\bar{\nu}$) match mean-field values typical of ordinary percolation.

This universality—remarkably independent of $\lambda$—emerges because SPP, through frequent shortest-path removal of edges traversing the hubs, rapidly destroys the heterogeneous degree structure prior to transition. This homogenization mechanism contrasts strongly with independent edge deletion of ordinary percolation, where hub preservation underpins $\lambda$-dependent criticality.

For $C = 1$, SPP recovers the anomalous, $\lambda$-dependent percolation transition.

4. Large-Scale Simulation Methodology

Monte Carlo simulations utilize the uncorrelated configuration model to generate SFNs with $P(k) \sim k^{-\lambda}$ for controllable $\lambda$. For each realization, the event-based ensemble records observables at the point of maximal $P$-drop, yielding high-accuracy estimates for scaling exponents and finite-size corrections. Data collapses for $P_c(N)$ and $S_c(N)$ confirm the universality with ER graphs for all tested $\lambda > 2$ with $C > 1$.

Tabular summary of principal observables:

Observable	Mathematical Definition	Physical Meaning
Order parameter $P$	$s_Z / \sum_{z=1}^{Z} s_z$	Fraction in largest cluster
Cluster size $S$	$(\sum_{z=1}^{Z-1} s_z^2) / (\sum_{z=1}^{Z-1} s_z)$	Mean finite cluster size
Critical fraction $r_c$	$r_c(N) = r_c + b N^{-1/\bar{\nu}_1}$	Threshold for percolation transition

5. Impact of Degree Heterogeneity and Role of Hubs

In SFNs, hubs possess high betweenness centrality ($g \sim k^{(\lambda-1)/(\eta-1)}$), serving as frequent conduits for shortest paths. The SPP process with $C > 1$ preferentially strips edges attached to hubs, rapidly reducing degree variance and effectively erasing the original network heterogeneity before the critical point. This leads to a percolation regime indistinguishable from ER graphs.

This mechanism is absent in ordinary percolation, where hubs retain connections longer, preserving the $\lambda$-sensitive critical properties until depletion.

6. Mathematical Formulations and Scaling Behavior

Key formulae used for finite-size scaling, event-based ensemble analysis, and centrality highlight the model's quantitative underpinnings:

• Percolation strength: $P = s_Z / \sum_{z=1}^Z s_z$
• Mean cluster size: $$S = (\sum_{z=1}^{Z-1} s_z^2) / (\sum_{z=1}^{Z-1} s_z)$$
• Pseudo-critical threshold:

$r_c^*(N) = \arg\max_r [P(r, N) - P(r + 1/E, N)]$

• Finite-size scaling of transition:

$r_c(N) = r_c + b N^{-1/\bar{\nu}_1}$

• Scaling of pseudo-critical fluctuations:

$\sigma[r_c^*](N) \sim N^{-1/\bar{\nu}_2}$

• Betweenness centrality scaling:

$g \sim k^{(\lambda - 1) / (\eta - 1)}$

7. Implications for Real-world Network Robustness

The homogenization effect of SPP has significant consequences for resource-limited supply, transport, and communication networks characterized by strong heterogeneity. Even in scale-free systems, optimal-route resource depletion can induce abrupt fragmentation transitions typical of regular graphs, implying that attempts to enhance robustness via hub proliferation may be ineffective when critical resources are consumed along shortest paths.

A plausible implication is that network intervention strategies must anticipate the correlated nature of resource depletion in SPP-like dynamics, possibly favoring distributed redundancy over hub-centric design. Conversely, intentional targeted fragmentation via SPP could be leveraged for controlled network partitioning.

Conclusion

The shortest-path percolation model on scale-free networks reveals that path-based, budget-constrained resource consumption radically transforms the transition dynamics, erasing topology-induced heterogeneity and producing a universality class aligned with ER graphs for $C>1$. This finding, supported by large-scale simulations and scaling theory (Kim et al., 11 Sep 2025), underlines the necessity of re-evaluating resilience in systems subject to optimal-path-driven depletion, with immediate relevance for transport, communication, and supply networks exhibiting scale-free architecture.