The telling phrase is "removing this edge disconnects the network." That is the definition of a bridge.
An edge (u,v) is a bridge if there is no other path from u to v. Tarjan's bridge algorithm finds all such edges in one DFS pass.
This is an undirected graph problem. Do not confuse bridges with articulation points (critical nodes, not edges). Bridges disconnect the graph when removed. Articulation points disconnect it when the node and all its edges are removed.