From Centralized to Distributed Entropy: Long-Term Resilience and Structural Evolution of Regional Innovation...

Understanding how regional innovation networks maintain functionality under disruption is critical for regional economic sustainability. This study investigates the structural evolution and topological resilience of inter-city patent cooperation networks in China's Yangtze River Delta (YRD) from 2005 to 2024. We construct weighted networks of 41 cities and simulate targeted attack scenarios to quantify network resilience as the area under the robustness curve. Using degree distribution entropy to quantify the spatial distribution of innovation activity, we demonstrate that the system's innovation activity redistributes from a Shanghai-dominated, concentrated, single-core configuration to a more distributed, polycentric architecture. The resilience index increases from 0.0836 in 2005 to 0.5000 in 2015, and full connectivity is achieved by 2024. Notably, removing the top-ranked node in 2010 reduces the largest connected component by 5.3%, whereas removing four core nodes simultaneously in 2015 produces a 9.8% reduction, indicating that the redistribution of innovation activity is associated with systemic robustness. Correlation analysis further reveals strong associations between resilience and network density, weighted-degree entropy, and short path lengths, reflecting a hub-dominated topology that evolved from a single hub (Shanghai) to multiple co-existing hubs (Shanghai, Nanjing, Hangzhou, Hefei). These findings provide empirical evidence consistent with the view that polycentric configurations are associated with innovation ecosystem resilience and offer actionable insights for regional sustainability policies.