Finite-Budget SPP Universality Class

• Finite-budget SPP is a network model where a finite removal budget triggers the deletion of entire shortest paths, leading to continuous, mean-field-like transitions.
• The model exhibits universal scaling independent of network degree heterogeneity, as the homogenization effect erases local structural differences before percolation.
• Mathematical analysis and Monte Carlo simulations reveal that critical exponents remain constant across different network architectures when using finite budgets.

The finite-budget shortest-path percolation (SPP) universality class encompasses a set of critical phenomena arising in stochastic network processes where the destruction of network resources is governed by a parameterized, finite upper bound on the length of removed shortest paths. Finite-budget SPP, fundamentally distinct from both standard percolation and unbounded-budget SPP, exhibits critical exponents and scaling properties that are robust under wide variations in the network’s microscopic structure—even erasing the effects of strong degree heterogeneity present in scale-free networks. This universality parallels deep results established in disordered systems, random matrices, and interface growth, where universal scaling behavior emerges despite model-dependent constraints, provided the appropriate scaling limits are observed.

1. Shortest-Path Percolation Model Definition

The SPP model simulates resource consumption in networks by a stochastic process parameterized by a budget $C$. At each iteration, two random nodes are selected and the shortest path between them is computed (if such a path exists). If the geodesic path length $Q_t$ satisfies $Q_t \leq C$, all edges along the path are simultaneously removed; if $Q_t > C$, no modification occurs. This dynamic is repeated until the network fragments into microscopic disconnected components, marking a percolation-like structural transition (Kim et al., 11 Sep 2025).

• For $C=1$, SPP reduces to ordinary bond percolation: a single edge is removed per event, and correlations between removed edges are absent.
• For $C>1$ and finite, entire correlated edge paths are excised, imposing topological correlations absent in bond percolation.
• For unbounded $C$, e.g. $C=N$ where $N$ is the number of nodes, the model exhibits a pronouncedly more abrupt transition and defines a separate universality class.

This model is studied on both Erdős–Rényi (ER) random graphs and scale-free networks (SFNs) with degree distributions $P(k)\sim k^{-\lambda}$.

2. Universality Class Structure and the Role of Budget

Numerical and analytical studies establish that the SPP model realizes two distinct universality classes determined by the budget parameter $C$:

Universality Class	Budget $C$	Critical Exponents	Dependence on Degree Exponent $\lambda$	Transition Profile
Finite-budget SPP	$1 < C < \infty$	Mean-field: $\beta/\bar\nu \approx 1/3$	None; independent of $\lambda$	Continuous (mean-field-like)
Unbounded-budget SPP	$C=N$ or $\infty$	$\beta/\bar\nu\approx 0.22$, $\gamma/\bar\nu\approx 0.54$	None; independent of $\lambda$	More abrupt, non-mean-field

The defining feature of the finite-budget SPP universality class is that critical exponents, such as the ratio $\beta/\bar\nu$ describing the order parameter scaling, are independent of the underlying network’s degree distribution exponent $\lambda$. Thus, even networks with strong degree heterogeneity—such as SFNs with $2<\lambda<3$—exhibit the same scaling behavior as homogeneous random graphs (Kim et al., 11 Sep 2025).

3. Homogenization Mechanism and Topological Correlations

A characteristic mechanism responsible for this universality is the “homogenization” induced by SPP dynamics. In SFNs, hubs (high-degree nodes) are traversed by a disproportionate number of geodesics due to their central role in network connectivity ($g\sim k^{(\lambda-1)/(\eta-1)}$, where $g$ is betweenness centrality). As shortest paths are sequentially removed, hubs are hit early and disproportionately, leading to the rapid depletion of their connecting edges. This process destroys the SFN’s inherent structural heterogeneity (“topological smoothing”) well before the percolation threshold is reached.

Consequently, at the percolation transition, the network’s residual structure closely resembles that of a homogeneous random graph, regardless of the original $P(k)$, leading to mean-field criticality (Kim et al., 11 Sep 2025).

4. Mathematical Formulation and Critical Exponents

The critical behavior of finite-budget SPP is characterized using standard percolation observables, with definitions and scalings adopted from percolation theory:

• Order Parameter: $P = s_Z / \sum_{z=1}^Z s_z$, where $s_Z$ is the size of the largest component.
• Average Cluster Size: $S = \sum_{z=1}^{Z-1} s_z^2 / \sum_{z=1}^{Z-1} s_z$.
• Finite-Size Scaling: The pseudo-critical point $r_c(N)$ (fraction of removed edges at the largest drop in $P$) obeys $r_c(N) = r_c + b N^{-1/\bar\nu_1}$, with $\bar\nu_1$ as the scaling exponent.

Monte Carlo simulations confirm that for finite $C$ (e.g., $C=2,3$), the exponent ratios $\beta/\bar\nu$ and $\gamma/\bar\nu$ approach $\approx 1/3$, independent of $\lambda$. For $C=1$ (bond percolation), these exponents revert to established percolation universality, which is $\lambda$-dependent—for SFNs with $\lambda < 3$ or $\lambda < 4$, non-mean-field values emerge. For infinite $C$ the transition is discontinuous, as indicated by smaller critical exponent ratios (Kim et al., 11 Sep 2025).

5. Broader Context: Connection to Universality Theory

The emergence of universality in finite-budget SPP mirrors canonical results in disordered systems and stochastic growth phenomena. In the context of the KPZ universality class (Corwin, 2011), models with explicit resource constraints (finite energy, finite cost) converge to the same scaling exponents and limiting distributions as their unconstrained counterparts when correct scaling limits are imposed. Similarly, random block matrix ensembles with finite-rank blocks (interpreted as a “finite-budget” of interactions per degree of freedom) exhibit deterministic spectral densities and universality via measure concentration (Cicuta et al., 2022).

Thus, in both stochastic network processes and stochastic physical processes (“SPP” in the sense of “stochastic physical process”), finite-budget constraints do not generally destroy universality, provided correlations and large-scale limits are correctly accounted for.

6. Implications and Applications

The findings have broad implications for the analysis of networked systems with consumption-like dynamics: transportation, communication, and infrastructure networks may all experience critical failures characterized by finite-budget SPP universality. The independence from degree heterogeneity simplifies modeling, as critical thresholds and scaling predictions are robust with respect to microscopic structure.

In addition, the SPP mechanism highlights a general paradigm in complex systems: global topological properties may be fundamentally altered not simply by the details of local connectivity, but by correlated, resource-limited stochastic processes that reshape system properties before transitions are even approached. This shift suggests avenues for designing network resilience and targeted interventions by controlling resource allocation parameters.

7. Summary and Outlook

The finite-budget SPP universality class is defined by mean-field critical exponents that are invariant under diverse network architectures, a phenomenon traceable to the homogenization effect of shortest-path resource depletion. This universality is robust, arising whenever budgeted stochastic removal processes induce sufficient topological smoothing pre-transition. The conceptual and mathematical parallels between this class and universality phenomena in KPZ-type models and random matrix ensembles underline the central role of large-scale correlation buildup and limit theorems in dictating critical behavior, even under strong model constraints (Corwin, 2011, Cicuta et al., 2022, Kim et al., 11 Sep 2025).