We consider a Stackelberg game in a network, where a leader minimizes the cost of interdicting arcs and a follower seeks the shortest distance between given origin and destination nodes under uncertain arc traveling cost. In particular, we consider a risk-averse leader, who aims to keep high probability that the follower's traveling distance is longer than a given threshold, interpreted by a chance constraint. For the case with a wait-and-see follower, i.e., the follower selects a shortest path after seeing realizations of the random arc cost, we propose a branch-and-cut algorithm and apply lifting techniques to exploit the combinatorial structure of the risk-averse leader's interdiction problem. For the case with a here-and-now follower, we formulate a monolithic mixed-integer linear programming formulation for benchmark. We demonstrate the computational efficacy of our approaches, risk-averse interdiction solution patterns, and result sensitivity, via testing instances of randomly generated grid networks and real-world transportation networks.