Infinite-Budget SPP Universality Class

• Infinite-budget SPP universality class is defined by the abrupt fragmentation of networks via unlimited shortest-path removals.
• It exhibits unique scaling properties with critical exponents (β/ν ≈ 0.22, γ/ν ≈ 0.54) that differ from finite-budget cases.
• The model underscores structural homogenization, offering practical insights for network resilience and cascading failure analysis.

The infinite-budget SPP universality class describes the critical behavior and scaling properties of shortest-path percolation (SPP) processes on random networks when the budget parameter C is unbounded. In SPP, sequential random requests for connectivity between node pairs are fulfilled by removing all edges along a shortest path if its length is no greater than C. As the process continues, repeated path removals lead to a percolation-like transition where the network fragments into microscopic components. The universality class concerns how the critical exponents and transition characteristics relate across different underlying network topologies and budget regimes, with particular emphasis on scale-free networks (SFNs) and their structural transformation under SPP.

1. Shortest-Path Percolation (SPP) Model: Principles and Dynamics

The SPP model is constructed to analyze resource consumption in networked systems driven by routing requests. At each discrete time step, a network $\mathcal{G}_t=(\mathcal{V},\mathcal{E}_t)$ undergoes the following operations:

• Random selection of an origin-destination pair $(o_t, d_t)$.
• Computation of the shortest path $Q_t$ between $o_t$ and $d_t$ in $\mathcal{G}_t$.
• If $Q_t \leq C$ (budget constraint), all edges along one such shortest path are removed from $\mathcal{G}_t$.

For $C=1$, SPP reduces precisely to ordinary bond percolation, with single-edge removals. For $C>1$, the removal of entire correlated sets of edges (shortest paths) is a distinguishing factor, introducing new dependencies and transition features.

2. Universality Classes: Finite Versus Infinite Budget

The paper identifies two distinct SPP universality classes based on the value of C:

• Finite-budget SPP universality class ($C$ fixed): The critical exponents (ratios such as $\beta/\bar{\nu}$ and $\gamma/\bar{\nu}$) coincide quantitatively with those of mean-field bond percolation on Erdős–Rényi (ER) networks, with $\beta/\bar{\nu} \approx 1/3$. Notably, even on SFNs with degree exponent $\lambda < 4$, the exponents lose sensitivity to $\lambda$, indicating strong universality.
• Infinite-budget SPP universality class ($C$ unbounded): For $C=N$, where $N$ is the number of nodes, the transition becomes significantly sharper: exponent estimates shift to $\beta/\bar{\nu} \approx 0.22$ and $\gamma/\bar{\nu} \approx 0.54$. The break of the giant component occurs more abruptly, separating this class not only from finite-budget SPP and ordinary percolation but also defining new scaling behaviors.

The SPP process homogenizes the network topology by systematically removing high-betweenness and high-degree hub nodes. This mechanism causes the transition to lose its dependence on $\lambda$ and renders the scaling identical across ER and SFN topologies for finite budget, but distinctive for the infinite budget.

3. Structural Homogenization Under SPP

A key theoretical finding is the dynamic "flattening" of network heterogeneity under SPP. The removal of shortest paths disproportionately affects nodes with high degree and centrality, quickly reducing the degree spectrum. This effect is quantified by analyzing the ratio

$$\phi(N) = \langle k_{\max}^{\dagger}(N) / k_{\max}(N) \rangle = \phi_0 + aN^{-\delta},$$

where $k_{\max}$ and $k_{\max}^\dagger$ are the original and pre-transition maximum degrees, respectively. As $N$ increases, $\phi_0 \rightarrow 0$ signals the loss of scale-freeness and hence the suppression of degree-based heterogeneity.

This homogenization is fundamental to the observed universality, aligning SFN transitions with ER transitions for finite budget, and enabling the emergence of a distinct regime under infinite budget conditions.

4. Critical Exponents and Finite-Size Scaling Analysis

Monte Carlo simulations on SFNs generated by the uncorrelated configuration model underpin finite-size scaling (FSS) investigations. The two main observables are:

• Order parameter $P$: Relative size of the largest component.
• Average cluster size $S$: Excludes the largest component, $$\displaystyle S = \frac{\sum_{z=1}^{Z-1} s_z^2}{\sum_{z=1}^{Z-1} s_z}$$.

Observables are sampled at the pseudo-critical point $r_c^*$ (largest drop in $P$ after a single edge removal), and FSS ansatzes yield critical exponent ratios. For finite $C$, all tested $\lambda$ values ($2.1$, $2.7$, $3.5$, $4.5$) show mean-field behavior. For infinite $C$, the scaling exponents are reduced, manifesting the sharper nature of the transition.

Correlation exponents $(1/\bar{\nu}_1, 1/\bar{\nu}_2)$ are extracted via scaling relations:

$$r_c(N) = r_c + bN^{-1/\bar{\nu}_1}, \quad \sigma_{r_c^*}(N) \sim N^{-1/\bar{\nu}_2}.$$

Differences in these exponents across budget and degree exponent regimes reveal nuanced finite-size effects and explosive transition mechanisms unique to infinite-budget processes.

5. Analytical Framework and Scaling Relations

SPP transitions are characterized by a set of formulas central to their universality analysis:

Observable	Formula	Description
Order parameter	$P = s_Z / \sum_{z=1}^Z s_z$	Size of giant component
Average cluster size	$$S = \big(\sum_{z=1}^{Z-1} s_z^2\big)/\big(\sum_{z=1}^{Z-1} s_z\big)$$	Mean size excluding largest component
Degree scaling ratio	$$\phi(N) = \langle k^\dagger_{\max}(N)/ k_{\max}(N) \rangle$$	Homogenization indicator
FSS ansatz (critical)	$r_c(N) = r_c + bN^{-1/\bar{\nu}_1}$	Pseudo-critical point scaling
FSS fluctuation	$\sigma_{r_c^*}(N) \sim N^{-1/\bar{\nu}_2}$	Fluctuation scaling
Betweenness scaling	$g \sim k^{(\lambda-1)/(\eta-1)}$	Centrality exponent relation

These formulas provide quantitative means to compare transition scaling and network evolution for both finite and infinite-budget regimes.

6. Implications, Applications, and Broader Significance

The identification of the infinite-budget SPP universality class has significant consequences for network science and statistical mechanics. It demonstrates that dynamical resource consumption (via correlated shortest-path removal) fundamentally alters the critical phenomena on complex networks. The loss of degree dependence and emergence of sharper transitions suggest robust mechanisms that override the typical heterogeneity-induced behavior of SFNs. Applications include resource allocation in communication and transportation networks, resilience analysis, and modeling of cascading failures.

Beyond practical domains, these findings inform theoretical perspectives on universality, showing that correlated dynamics and macroscopic parameters (such as the budget constraint) can drive topological phase transitions into new regimes. SPP provides a testbed for examining how constrained or unconstrained resource depletion models can transform universality classes, especially in complex and heterogeneous systems.

7. Connections to Related Universality Classes

While the infinite-budget SPP universality class is an outgrowth of and, on certain regimes, analogous to the KPZ universality class (in its own infinite-budget form (Corwin, 2011)), it is distinguished by its network context and correlated path-removal dynamics. The SPP-induced homogenization and critical scaling contrast with phenomena observed in long-range correlated percolation models (Chalhoub et al., 27 Mar 2024), configuration model phase transitions with infinite second moment (Dhara et al., 2019), and infinite-degree sparse block matrix random ensembles (Cicuta et al., 2022), yet all reflect the overarching principle that universality can arise from global constraints or mechanisms that erase microscopic detail.

In summary, the infinite-budget SPP universality class is defined by the abruptness and scaling features of the transition induced by unlimited shortest-path removal on random networks, with critical properties that are largely independent of degree distribution and network heterogeneity due to the homogenization inherent in the dynamical process. These insights contribute fundamentally to understanding universality in networked systems under correlated resource exhaustion dynamics (Kim et al., 11 Sep 2025).