Broadcast with Limited Spreading Let consider well-known problem in distributed computing Broadcast Problem in network with stopping crashes.

The most appropriate solution to the broadcast problem is flooding algorithm, when the source of the message just send it to all available nodes of the network on the first round, if there are crashes on the edges some number of message will fail. Then on the second round, the nodes that received messages from the previous round send it to all available nodes and so on.

Let consider slightly different model, when on  every round the source node with message is capable to send a message to only one single node of course stopping failure might occur. It's very useful model, which can represent a sort of controllable or limited spreading. 

Unfortunately I don't know the formal name of the model. Do you familiar with result in the model of "limited spreading", I am sure there are must some work in this direction.
Canonical solutions to the broadcast problem include, as you mention, flooding (controlled &amp; uncontrolled, although uncontrolled flooding is untenable for a variety of reasons), as well as spanning-tree based approaches and N-way unicast (which is basically just to point out that broadcast protocols don't represent fundamentally more powerful communication mechanisms).

What you're asking about sounds somewhat similar to a sort of controlled flooding with fault tolerance. Alternatively, you might conceive of your scheme as a spanning-tree approach where multiple redundant spanning trees are defined for each node. In fact, a hybrid of these approaches would help avoid loops, explosion, etc.

Kurose and Ross provide a good explanation in their book "Computer Networks - A Top Down Approach", which you might use to develop your own broadcast protocol.