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Graphs in many applications, such as social networks and IoT, are inherently streaming, involving continuous additions and deletions of vertices and edges at high rates. Constructing random walks in a graph, i.e., sequences of vertices selected with a specific probability distribution, is a prominent task in many of these graph applications as well as machine learning (ML) on graph-structured data. In a streaming scenario, random walks need to constantly keep up with the graph updates to avoid stale walks and thus, performance degradation in the downstream tasks. We present Wharf, a system that efficiently stores and updates random walks on streaming graphs. It avoids a potential size explosion by maintaining a compressed, high-throughput, and low-latency data structure. It achieves (i) the succinct representation by coupling compressed purely functional binary trees and pairing functions for storing the walks, and (ii) efficient walk updates by effectively pruning the walk search space. We evaluate Wharf, with real and synthetic graphs, in terms of throughput and latency when updating random walks. The results show the high superiority of Wharf over inverted index- and tree-based baselines.