Device-to-device (D2D) communication in cellular networks allows direct transmission between two cellular devices with local communication needs. Due to the increasing number of autonomous heterogeneous devices in future mobile networks, an efficient resource allocation scheme is required to maximize network throughput and achieve higher spectral efficiency. In this paper, performance of network-integrated D2D communication under channel uncertainties is investigated where D2D traffic is carried through relay nodes. Considering a multi-user and multi-relay network, we propose a robust distributed solution for resource allocation with a view to maximizing network sum-rate when the interference from other relay nodes and the link gains are uncertain. An optimization problem is formulated for allocating radio resources at the relays to maximize end-to-end rate as well as satisfy the quality-of-service (QoS) requirements for cellular and D2D user equipments under total power constraint. Each of the uncertain parameters is modeled by a bounded distance between its estimated and bounded values. We show that the robust problem is convex and a gradient-aided dual decomposition algorithm is applied to allocate radio resources in a distributed manner. Finally, to reduce the cost of robustness defined as the reduction of achievable sum-rate, we utilize the \textit{chance constraint approach} to achieve a trade-off between robustness and optimality. The numerical results show that there is a distance threshold beyond which relay-aided D2D communication significantly improves network performance when compared to direct communication between D2D peers.