The introduction of a {0,1}-valued game associated to a connected graph allows us to assign to each node a value of weighted connectivity to the different solutions that for the cooperative games are obtained by means of the semivalues. The marginal contributions of each node to the coalitions differentiate an active connectivity from another reactive connectivity, according to whether the node is essential to obtain the connection or it is the obstacle for the connection between the nodes in the coalition. Diverse properties of this concept of connectivity can be derived.