advent VS CoinBLAS

A more interesting example: https://www.youtube.com/watch?v=i_zDbInYOpQ

AoC solutions here: https://github.com/lukechampine/advent/tree/master/2022

(The language has builtin commands for fetching inputs and submitting solutions)

FWIW I finished AoC 2020 in Go -- but I had 5 years' worth of helper functions to prop me up: https://github.com/lukechampine/advent

When you consider that a graph and a matrix are isomorphic, doing vector matrix multiplication takes a vector with a set value, say row 4, and multiplies it by a matrix where row 4 has values present that represent edges to the nodes that are adjacent to it (ie "adjacency" matrix). The result is a vector with the next "step" in a BFS across the graph, do that in a loop and you step across the whole graph.

A cool result of this is, for example, taking an adjacency matrix and squaring it is the "Friend of a Friend" graph. It takes every node/row and multiplies it by itself, returning a matrix that are adjacent to the adjacencies of each node, ie, the friends (adjacencies of the adjacencies) of friends (adjacencies) of the nodes.

Deeper traversal are just higher nodes, a matrix cubed are the friends of the friends of the friends.

A picture is worth a thousand words, see figure 7 of this paper:

https://arxiv.org/pdf/1606.05790.pdf

Also check out figure 8, this shows how incidence matrices can work to represent hyper and multi graphs. An pair of incidence matrices reprsent two graphs, one from nodes to edges and the other from edges to nodes, these are n by m and m by n. When you multiply them, you get a square adjacency matrix that "projects" the incidence into an adjacency. This can be used to collapse hypergraphs into simple graphs that use different semirings to combine the multiple edges.

For some pretty pictures of this kind of stuff, check out CoinBLAS (note I am not a crypto-bro, it was just a very handy extremely large multi-graph that I could easily download in chunks to play with):

https://github.com/Graphegon/CoinBLAS/

Python wrapper around The GraphBLAS API:

https://github.com/michelp/pygraphblas

For an upcoming paper we've open sourced using pygraphblas to analyse the bitcoin graph using the GAP benchmarks on a server with 1TB of RAM:

https://github.com/Graphegon/CoinBLAS