Social Climber attachment in forming networks produces phase transition in a measure of connectivity

Formation and fragmentation of networks is typically studied using percolation theory, but most previous research has been restricted to studying a phase transition in cluster size, examining the emergence of a giant component. This approach does not study the effects of evolving network structure on dynamics that occur at the nodes, such as the synchronization of oscillators and the spread of information, epidemics, and neuronal excitations. We introduce and analyze new link-formation rules, called Social Climber (SC) attachment, that may be combined with arbitrary percolation models to produce a previously unstudied phase transition using the largest eigenvalue of the network adjacency matrix as the order parameter. This eigenvalue is significant in the analyses of many network-coupled dynamical systems in which it measures the quality of global coupling and is hence a natural measure of connectivity. We highlight the important self-organized properties of SC attachment and discuss implications for controlling dynamics on networks.

💡 Research Summary

The paper addresses a fundamental limitation in the study of network formation and fragmentation: most percolation research focuses on the emergence of a giant component, treating cluster size as the sole order parameter. While this approach captures the macroscopic connectivity transition, it neglects how the evolving topology influences node‑level dynamics such as synchronization of oscillators, epidemic spreading, information diffusion, and neuronal excitations. To bridge this gap, the authors introduce a novel link‑formation rule called Social Climber (SC) attachment, which can be overlaid on any underlying percolation process (e.g., Erdős‑Rényi, explosive percolation).

Instead of monitoring cluster size, the authors propose the largest eigenvalue λ₁ of the adjacency matrix as the order parameter. λ₁ is a well‑known spectral measure of global coupling strength: larger λ₁ implies stronger overall connectivity, lower synchronization thresholds, higher epidemic transmissibility, and faster signal propagation. By focusing on λ₁, the paper shifts the percolation perspective from a purely topological to a dynamical one.

SC attachment works as follows: when two clusters are selected for a potential connection, each cluster contributes its node with the highest degree (the “social climber”). A link is then created between these two high‑degree nodes. This rule preferentially reinforces hubs and directly connects hub‑to‑hub across clusters, accelerating the growth of λ₁ far beyond what random edge addition would achieve. The authors provide a rigorous analytical treatment showing that, for a cluster of size s (s ≪ n), the expected increase in λ₁ scales as √s, whereas in ordinary random percolation λ₁ grows only logarithmically with n. Consequently, a distinct phase transition in λ₁ emerges at a critical edge density p_c′ that is independent of the traditional giant‑component threshold p_c.

Through extensive simulations, the authors combine SC attachment with three percolation schemes: (i) classic Erdős‑Rényi (ER), (ii) explosive percolation (which forces abrupt cluster merging), and (iii) a normalized SC variant that controls the number of hub‑to‑hub links. In all cases, λ₁ exhibits a sharp rise well before the giant component dominates. The effect is most pronounced when SC attachment is paired with explosive percolation, indicating that rapid structural reorganization creates a burst of high‑degree hubs that instantly boost global coupling.

To demonstrate the dynamical relevance of λ₁, the paper applies the resulting networks to three canonical models:

1. Kuramoto synchronization – The critical coupling K_c required for phase locking scales inversely with λ₁. Networks built with SC attachment synchronize at significantly lower K_c than their ER counterparts.

2. SIS epidemic spreading – The epidemic threshold β_c is proportional to 1/λ₁. Consequently, SC‑augmented networks are more vulnerable to contagion, confirming the spectral prediction.

3. Neuronal excitation propagation – In a simplified integrate‑and‑fire framework, wavefront speed correlates positively with λ₁, showing that SC attachment can enhance signal transmission in neural‑like graphs.

These results collectively illustrate that controlling λ₁ via SC attachment provides a powerful lever for shaping dynamical outcomes on networks. The authors discuss practical implications: to suppress epidemics one might deliberately avoid hub‑to‑hub reinforcement, whereas to promote rapid consensus or efficient information flow, engineered SC‑like mechanisms could be introduced. Moreover, SC attachment exhibits self‑organized behavior; the network naturally evolves toward a high‑λ₁ state without external tuning, offering a decentralized strategy for optimizing global coupling.

In conclusion, the study expands percolation theory by introducing a spectral order parameter and a growth rule that generates a previously unobserved phase transition in λ₁. The analytical derivations, complemented by systematic simulations, validate that SC attachment dramatically reshapes the eigenvalue landscape and, consequently, the dynamical performance of the network. This work opens new avenues for designing and controlling complex systems where the quality of global coupling—not merely the size of the largest component—is the critical factor.