Shortest-path percolation on random networks (2402.06753v2)

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.

• The paper presents the SPP model that analyzes resource consumption by removing edges constituting shortest paths under a fixed length budget.
• It employs extensive numerical simulations on Erdős-Rényi graphs to identify varying critical thresholds and phase transitions for finite and infinite budgets.
• Findings highlight significant implications for network resilience, offering actionable insights into optimizing transport and communication infrastructures.

Shortest-Path Percolation on Complex Networks

The paper titled Shortest-path percolation on complex networks by Minsuk Kim and Filippo Radicchi presents a novel bond-percolation algorithm—dubbed the Shortest-Path Percolation (SPP) model—intended for analyzing consumption and depletion of resources within transport networks. The principal mechanism involves the removal of edges that contribute to minimum-length paths between demanded origin-destination nodes, contingent upon these paths remaining under a certain predefined length budget, denoted as $C$.

Key Contributions and Methodology

The SPP model diverges from traditional bond-percolation processes by factoring the shortest-path criteria into network element removals. The model is evaluated primarily on Erdős-Rényi (ER) random graphs, granting insights into the resultant percolation transition dynamics. Finite-size scaling (FSS) analyses demonstrate that the nature of these percolation transitions critically depend on the budget $C$:

• Finite $C$: The transition resembles ordinary percolation, with the gradual disintegration of a giant network component.
• Infinite $C$: The transition is notably sharper, characterized by a sudden fragmentation of the giant component into smaller, comparably sized clusters.

The authors elucidate this dichotomy through extensive numerical simulations. They employ random selections of origin-destination pairs and a systematic analysis involving 5,000 independent trials for each parameter set to validate their observations and theoretical underpinnings.

Numerical Results and Theoretical Implications

Distinct critical behaviors are observed for different budget regimes. The paper's most salient findings can be outlined as follows:

• Critical Thresholds: The critical threshold $p_c$ where the largest component disintegrates decreases with increasing $C$, ranging from 0.750 for $C=1$ to 0.651 for infinite $C$.
• Critical Exponents: For finite $C$, the critical exponent ratios $\beta/\bar{\nu}$ and $\gamma/\bar{\nu}$ are close to standard percolation values, while for infinite $C$, the exponent values suggest a stronger deviation from typical percolation models.

The repercussions of these findings extend to a broader understanding of resource dependency and resilience in complex networks. By framing percolation within a shortest-path context, the paper opens new investigatory pathways into network robustness and failure modes specific to the topology-dependent resource allocation models, such as transport and communication networks.

Future Directions and Practical Applications

Given its divergence from classical percolation paradigms, the SPP model bears potential for significant advances both in theory and application. The model's adaptability to various graph types (e.g., directed, weighted, time-evolving) proposes a versatile tool for dissecting infrastructural vulnerabilities and resilience engineering. The detailed scaling behaviors uncovered in finite and infinite budget scenarios suggest practical forecasting and network optimization applications in real-world systems, such as traffic management and disaster response frameworks.

In conclusion, the presentation of the SPP model within this work underscores a fresh perspective on percolation phenomena by incorporating shortest-path constraints. This approach not only broadens the theoretical landscape of percolation studies but also bridges the gap towards more refined, actionable insights into the operational robustness of complex transport networks. As researchers further delve into the implications of these findings, developments are anticipated in the analysis and design of networks with varying structural and functional demands.