Let's say we have an undirected graph of 50 millions of nodes (let's say a first approximation of a country with closed borders).
Let's say there is a link/edge between node $a$ and node $b$ if person $a$ and $b$ have met with distance < 1 meter today.
On each node, you have a boolean value True/False (or "contaminated+recovered+dead" / "not contaminated").
Which part of graph theory and which theorems give us asymptotic information for a question like:
We put value True for 1000 random nodes on the graph, and False everywhere else. Let's say that after 1 iteration, there is a probability $p=0.5$ that if $a$ has a link with $b$ who has True, then $a$ becomes True too (contamination).
Question: after $n$ days, how many people have True?
Of course this will depend on the topology of the graph. Are there ways to answer to this question, given a few constants that describe the graph's topology, like:
$$\alpha = \text{average number of edges for nodes}$$
?
What are the relevant keywords in graph theory to search about this? (sadly, graph theory wasn't taught where I studied maths at university).
Numerical simulation: I more or less know how to simulate such a process with code (e.g. Python), and how to observe such things, but I was interested in the part of graph theory that can give answers.