Achilles' heel of football passing networks.

We can construct passing networks when we regard a player as a node and a pass as a link in football games. Thus, we can analyze the networks by using tools developed in network science. Among various metrics characterizing a network, centrality metrics have often been used to identify key players in a passing network. However, a tolerance to continuous incapacitation of players in a passing network, namely the robustness of the network, has been poorly understood so far. Because the robustness of a passing network can be connected to the increase of ball possession, it would be deeply related to the outcome of a game. Here, we developed position-dependent passing networks of 45 matches by 18 teams belonging to the Japan Professional Football League. Then, nodes were continuously removed from the passing networks by two removal methods so that we could evaluate the robustness of these networks against the cascading failures. The results show that these passing networks commonly contain hubs (key players making passes) and the robustness of the networks has similar properties observed in scale-free networks. Then, we analyzed the most robust networks in detail and found that their full backs increase the robustness by often invoking a heavier emphasis on attack. Moreover, we showed that the robustness of the passing networks and the team performance have a positive correlation.