Multi-layered consensus protocols for edge dynamics in modular networked agents

This paper addresses the edge consensus problem in modular multi-agent systems by introducing a hierarchical protocol that combines line-graph transformations with community detection via the Label Propagation Algorithm (LPA). In contrast to classical ‘flat’ consensus schemes, which often exhibit slow convergence and limited scalability in large or sparse networks, the proposed m-layer framework exploits intra- and inter-cluster interactions through generalized lifting operators. Rigorous spectral analysis establishes explicit step-size conditions under which the hierarchical update is a strict contraction on the disagreement subspace, guaranteeing asymptotic consensus. To quantify efficiency, we provide closed-form complexity bounds, showing polynomial overall growth and near-linear scaling under sparsity, with strong potential for parallel implementation. Numerical experiments on both illustrative examples and large-scale Barabási–Albert and stochastic block networks demonstrate accelerated convergence and reduced communication compared to flat protocols. While our design ensures asymptotic convergence, we also discuss links with finite-time and quantized consensus, noting that extensions to single-bit or logarithmic quantizers can yield practical consensus with bounded error. The integration of LPA-driven modular decomposition into the edge-dynamics framework provides a scalable, structurally adaptive consensus mechanism with provable guarantees, opening avenues toward directed, time-varying, stochastic, and event-triggered extensions in large-scale cyber-physical systems.