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Networks topology can be represented over Riemannian manifolds (i.e., curved surfaces), given the symmetric positive definite (SPD) property of their spectral graphs. Moreover, maximizing flow rate of a baseline network topology through relay placement can be equivalent to finding the relay location that maximizes the geodesic distance (i.e., Riemannian metric) between the representations of a relay-assisted network topology and the baseline one over Riemannian manifolds. Therefore in this paper, we propose two complementary approaches to find relay locations that maximize Riemannian metrics, such as Log-Euclidean metric (LEM), and hence maximize the network flow rate. First, we propose a Riemannian multi-armed bandit (RMAB) reinforcement learning model to track the relay positions, which increase the LEM towards the baseline network. Particularly, selecting a possible relay location is considered as an action, whereas the LEM represents the reward of the RMAB model. Second, we propose a Riemannian Particle Swarm Optimization (RPSO) algorithm that iteratively attempts to find the representation of relay-assisted network topology with maximum LEM towards that of the baseline network over the Riemannian manifold. Simulation results show that both the RMAB and RPSO approaches converge to near-optimum solutions, which in the case of single relay placement achieve 94.3% and 90.6%, respectively, of the maximum possible network flow rate.

Relay Placement for Maximum Flow Rate via Learning and Optimization over Riemannian Manifolds