Temporal self-similarity reveals percolation universality classes in complex networks

Abstract

Catastrophic fragmentation and structural transitions are ubiquitous in real-world complex systems, yet their underlying universality classes remain largely elusive due to strong structural heterogeneity and the absence of well-defined critical thresholds. Here, we discover a robust phenomenon of temporal self-similarity governing the dynamic percolation process across diverse complex networks. By tracking the full statistics of incremental growth events, we reveal that fragmentation dynamics are governed by two independent Fisher-type critical exponents, τc and τs. These exponents uniquely characterize the system’s universality class, from which all other standard critical exponents can be derived through newly established scaling relations. After rigorously validating this framework on canonical network models, we apply it to extensive empirical datasets. Strikingly, our analyses across biological, social, and infrastructural systems demonstrate that real-world networks systematically exhibit universality classes distinct from those predicted by idealized network models, reflecting the influence of higher-order structural features. Our findings establish a dynamic paradigm that bridges statistical physics and real-world resilience, offering a parameter-free, highly scalable approach to classify and characterize structural vulnerabilities in inherently heterogeneous systems. Without prior knowledge of the critical point, classifying network collapse is a challenge. Here, authors uncover temporal self-similarity by tracking the ordered sequence of bond addition events, and demonstrate that this framework identifies percolation universality for both model and empirical networks, using only a single system size and requiring no exact threshold.