On Statistical Estimation of Edge-Reinforced Random Walks

Reinforced random walks (RRWs), including vertex-reinforced random walks (VRRWs) and edge-reinforced reinforced random walks (ERRWs), model phenomena where transition probabilities evolve based on prior visitation history [5], [8], [15], [16]. These models have found applications in various areas, such as network embedding [18], reinforced PageRank [6], and modeling animal behaviors [14], among others. However, statistical estimation of the parameters governing RRWs remains underexplored. This work focuses on estimating the initial edge weights of ERRWs using observed trajectory data. Leveraging the connections between ERRW and random walks in a random environment (RWRE) [9], [10], we propose an estimator based on the generalized method of moments and the “magic formula”. To analyze the sample complexity, we exploit the hyperbolic Gaussian structure embedded in the random environment to bound the order of the random conductance, and hence derive sample complexity bounds. These findings contribute to the theoretical foundation of promising statistical and algorithmic applications of ERRWs.