This paper presents a distributed optimization algorithm based on dual averaging of subgradients, analyzing its convergence rates and network scaling properties. The algorithm is designed for optimizing a global objective function that is the sum of local convex functions over a network, with only local computation and communication allowed. The method maintains and forms weighted averages of subgradients across the network, enabling a clear separation between the algorithm's convergence and the effects of network constraints. The algorithm's convergence is analyzed in terms of the network's spectral properties, showing that the number of iterations required scales inversely with the network's spectral gap. This result is confirmed by both theoretical lower bounds and simulations for various network structures. The algorithm handles both deterministic and stochastic optimization scenarios, as well as communication constraints. The paper provides convergence rates for different network topologies, including cycles, grids, random geometric graphs, and expanders. It shows that the convergence rate depends on the network's structure, with well-connected networks (like expanders) achieving faster convergence. The analysis also extends to stochastic communication and gradient noise, demonstrating that the algorithm remains effective under these conditions. Key contributions include a new subgradient algorithm for distributed constrained optimization, a detailed analysis of network scaling effects, and the derivation of convergence rates for various network structures. The results show that the algorithm's performance is significantly influenced by the network's topology, with faster convergence on well-connected networks. The analysis also highlights the importance of the spectral gap in determining convergence rates, and the algorithm's robustness to communication and gradient noise.This paper presents a distributed optimization algorithm based on dual averaging of subgradients, analyzing its convergence rates and network scaling properties. The algorithm is designed for optimizing a global objective function that is the sum of local convex functions over a network, with only local computation and communication allowed. The method maintains and forms weighted averages of subgradients across the network, enabling a clear separation between the algorithm's convergence and the effects of network constraints. The algorithm's convergence is analyzed in terms of the network's spectral properties, showing that the number of iterations required scales inversely with the network's spectral gap. This result is confirmed by both theoretical lower bounds and simulations for various network structures. The algorithm handles both deterministic and stochastic optimization scenarios, as well as communication constraints. The paper provides convergence rates for different network topologies, including cycles, grids, random geometric graphs, and expanders. It shows that the convergence rate depends on the network's structure, with well-connected networks (like expanders) achieving faster convergence. The analysis also extends to stochastic communication and gradient noise, demonstrating that the algorithm remains effective under these conditions. Key contributions include a new subgradient algorithm for distributed constrained optimization, a detailed analysis of network scaling effects, and the derivation of convergence rates for various network structures. The results show that the algorithm's performance is significantly influenced by the network's topology, with faster convergence on well-connected networks. The analysis also highlights the importance of the spectral gap in determining convergence rates, and the algorithm's robustness to communication and gradient noise.