Hierarchical synchronization and distortion scaling in social media networks: A fractal-like topology theory

The rapid proliferation of social media as a dominant channel for information dissemination has intensified concerns over systemic information distortion—whereby content is progressively altered through successive layers of transmission. While prior studies have explored such distortion qualitatively, the quantitative interplay between propagation topology and stochastic cognitive perturbations remains insufficiently understood. In this work, we propose a novel fractal-inspired directed hierarchical network (DHN) model to capture the structural patterns of propagation, and introduce a Noise-Frustrated Hegselmann–Krause (NFHK) framework to model opinion dynamics under noise. Analytical results, supported by group and graph theory, reveal that noise accumulation leads to increasing opinion distortion and the emergence of intra-layer synchronization. Multi-agent simulations confirm these effects, showing that noise intensity shapes both convergence rates and weak intra-layer clustering. Empirical validation using a representative retweet cascade demonstrates that the proposed model reproduces real-world distortion patterns and synchronization behaviors, even without direct links. This work uncovers a unified mechanism for information distortion in digital platforms and offers topology-aware insights for public opinion governance and platform regulation.