The intersection of connection graphs and discrete optimal transport presents a novel paradigm for understanding complex graphs and node interactions. In this paper, we delve into this unexplored territory by focusing on the Beckmann problem within the context of connection graphs. Our study establishes feasibility conditions for the resulting convex optimization problem on connection graphs. Furthermore, we establish strong duality for the conventional Beckmann problem, and extend our analysis to encompass strong duality and duality correspondence for a quadratically regularized variant. To put our findings into practice, we implement the regularized problem using gradient descent, enabling a practical approach to solving this complex problem. We showcase optimal flows and solutions, providing valuable insights into the real-world implications of our theoretical framework.

翻译：连接图与离散最优传输的交汇为理解复杂图结构及节点交互提供了新颖的研究范式。本文聚焦连接图上的Beckmann问题，深入探索这一未充分研究的领域。我们建立了连接图上相关凸优化问题的可行性条件，继而证明了经典Beckmann问题的强对偶性，并将分析扩展到二次正则化变体，给出了该变体的强对偶性及对偶对应关系。为将理论成果付诸实践，我们采用梯度下降法实现正则化问题，为求解这一复杂问题提供了可行方案。通过展示最优流与最优解，我们揭示了理论框架在实际应用中的重要价值。