Motivated by cooperative distributed networks in which users dynamically alternate between transmit and receive modes under half-duplex constraints, this paper studies the Age of Information (AoI) in a distributed multi-user network using an ALOHA-based protocol. We derive closed-form expressions for the average AoI and formulate an optimization problem over transmission probabilities. After proving the convexity of the problem, we leverage the derived optimality conditions to characterize optimal policies for general network graphs, obtain closed-form solutions for $d$-regular topologies, and derive tractable optimality conditions for star topologies. Numerical results confirm that the proposed mechanism can effectively and adaptively determine user-specific optimal transmission probabilities across varying network topologies. These findings contribute to the design of adaptive and efficient distributed networks with enhanced information freshness.