best algorithm for finding distance for all pairs where edges' weight is 1
I would refer you to the following paper: "Sub-cubic Cost Algorithms for the All Pairs Shortest Path Problem" by Tadao Takaoka. There a sequential algorithm with sub-cubic complexity for graphs with unit weight (actually max edge weight = O(n ^ 0.624)) is available.
As the title said, I'm trying to implement an algorithm that finds out the distances between all pairs of nodes in given graph. But there is more: (Things that might help you)

The graph is unweighted. Meaning that all the edges can be considered as having weight of 1.
|E| <= 4*|V|
The graph is pretty big (at most ~144 depth)
The graph is directed
There might be cycles 
I'm writing my code in python (please if you reference algorithms, code would be nice too :))

I know about Johnson's algorithm, Floyd-Warshal, and Dijkstra for all pairs. But these algorithms are good when the graph has weights.
I was wondering if there is a better algorithm for my case, because those algorithms are intended for weighted graphs.
Thanks!