Diffusion in dynamic networks with continuous inputs to allocate responsibility

Responsibility in complex networks extends beyond direct actions: players should also bear responsibility for the indirect effects within their supply chains or network. We introduce a novel framework to allocate responsibility for indirect environmental, social, and economic impacts across a dynamic network. Unlike static approaches, our framework accounts for the evolving structure of supply chains, financial systems, and other interconnected systems, where relationships change over time. We use the time-dependent Laplacian matrix to capture how responsibility propagates through the network, revealing a diffusion process that aligns with key axioms of fairness: linearity, efficiency, symmetry, and the independent player property. We show that approximating the responsibility measure preserves these properties, supporting the use of our framework as a rigorous and practical method to allocate responsibility in real-world networks