Python: Create a graph with defined number of edges per node
It seems to me you should

decide how many nodes you will have
generate the number of links per node in your desired distribution - make sure the sum is even
start randomly connecting pairs of nodes until all link requirements are satisfied

There are a few more constraints - no pair of nodes should be connected more than once, no node should have more than (number of nodes - 1) links, maybe you want to ensure the graph is fully connected - but basically that's it.
How I can create a graph with 
-predefined number of connections for each node, say 3
-given distribution of connections (say Poisson distribution with given mean) 
Thanks