To address cyber security issues in multi-agent type systems, we formulate a game theoretic problem on resilient graphs, where the utilities of the players are mainly related to the connectivity of the network. An attacker is capable to disconnect part of the edges of the graph by emitting jamming signals while, in response, the defender recovers some of them by increasing the transmitted power for the communication over the corresponding edges. The players' actions are constrained by their energy for transmissions and they play this two-stage game repeatedly over time. Their strategies are characterized by the edge connectivity and the timings to start/stop their actions. A numerical example is provided to demonstrate the efficacy of the results.