From 3905b719c0d0b5f5a41b3bc5176ae1b5429dfb4c Mon Sep 17 00:00:00 2001 From: Pagwin Date: Sun, 25 May 2025 22:11:26 -0400 Subject: [PATCH] tolerant graph synchronization --- posts/tolerant-graph-synchronization.md | 59 +++++++++++++++++++++++++ 1 file changed, 59 insertions(+) create mode 100644 posts/tolerant-graph-synchronization.md diff --git a/posts/tolerant-graph-synchronization.md b/posts/tolerant-graph-synchronization.md new file mode 100644 index 0000000..95b4e46 --- /dev/null +++ b/posts/tolerant-graph-synchronization.md @@ -0,0 +1,59 @@ +--- +title: "The Tolerant Graph Synchronization Problem" + +description: "Trying to formalize an interesting comp sci problem with real world applications." + +date: "2025-05-25" + +draft: false + +tags: [] +--- + +I'm going to assume that you know what a [graph](https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)) is coming into this. + +## The Semi-Formal Generic Graph Synchronization Problem Description + +The problem goes as follows + +You have a connected undirected graph with all nodes but 1 colored black. +The one non-black node can be any non-black color. + +The objective is to come up with a set of rules, within some constraint(s), where a node will update to a new color depending on the colors of it's neighbors (nodes with a direct edge connection) such that in a finite number of steps the entire graph will be red optimizing for some criteria. + +At minimum we have the constraint that a node with all black neighbors will remain black. + +Note: the baseline graph synchronization problem is contained within the generic version with. The baseline problem has the additional constraint that a solution is only valid if at all steps either all nodes of the graph are red or no nodes of the graph are red with no optimization criteria. + +## The Semi-Formal Tolerant Graph Synchronization Problem + +The tolerant version does not have the additional constraint the baseline problem has. +Instead the tolerant version has optimization criteria to minimize. + +Those criteria are + +1) For each red node minimize the time spent neighboring of non-red nodes, this being the multiple of the number of ticks and the number of neighbors under this criteria +2) Minimize the number of ticks taken before all nodes are red + +There are probably other optimization criteria, constraints and confounding factors for the real world problem(s) that lead to me wanting to investigate this. + +## Real World Cases/Who cares + +You know how a lot of people feel stuck on social media platforms due to networking effects. +Yeah, having a strategy to coordinate moving away where people aren't stuck at the new place for long would be nice. + +Likewise for irl cities when those networking effects kick in but cost of living is a bitch. + +That said you can probably see how in both those examples there are other factors at work beyond communication and moving over. +The reason I don't include those yet is because I'm not sure what needs to be included or what the best way to include it would be. + +## Plausible optimal solution for the problem as written + +I suspect (with my gut) the optimal solution is just to have the initial non-black node be red and then have the rule "if any neighbors are red turn red". + +I don't know if exponential graphs like social graphs mess that solution up though or if all the alternatives do no better. + +## Conclusion + +If you have thoughts/a proof on the actual solution for the problem as written I'd like to hear about them. +Likewise if you have ideas on formalizing the problem for those real world cases with irl constraints, optimization criteria and confounding factors let me know.