Today I was trying to make a theoretical model for my peer to peer network simulator. Basically its a network of peers which is modified each loop according to certain rules (either random peer sampling or according to similarity-based clustering). The network is modeled as an adjacency matrix and to modify this matrix I may multiply it by a certain matrix; since the network is closed then it is possible to utilize a permutation matrix for that (peers exchange neighbors with each others). I also use a randomization matrix between each two operations to account for randomness in choice. So if we denote the network (adjacency matrix of the network graph) at cycle $n$ as $G(n)$, then the operation that happens in each cycle of the simulation is $$G(n+1) \leftarrow E \times R \times G(n) .$$ Where $R$ is the random permutation matrix and $E$ is the exchange matrix.
Why would this be interesting ? well, one reason that might occur to me is that if we were able to explicitly construct an algebra for $P$ and another for $NP$ then we might have another way to attack the famous $P=NP$ problem, which would then be equivalent to whether the two algebras are equivalent. This might be an interesting research direction for me, and in fact, I wrote this article as a sort of a TODO to remind me of this idea later when I finish my PhD and be able to pursue a research direction of my own. Thanks for reading and any comments or references are welcome!