TITLE: What nodes of a graph should be vaccinated first?
QUESTION [23 upvotes]: Consider a graph, choose some  "p: 0<p<1" (probability to infect the neighbor node).
Choose some random number "K" of nodes which are "infected" initially.
So we can generate epidemic model on a graph - each time tick - neighbors of already infected nodes become also "infected" with probability "p".
Now assume we have an option to "vaccinate" some number of nodes "V", that means they will never be infected. It is clear intuitively that it is more reasonable to "vacciante" those nodes with many neighbors, i.e. high degree nodes, however, graph theory suggests that there are several more sophisticated  measures of "importance" of nodes - i.e. "centrality measures". Intuitively it corresponds to the situation that node may have many neigbours, but its neigbours have low number of neigbours, that means second step of epidemy will not be so harmful. That is why it is not obvious that direct choice of nodes with highest degrees is optimal.
Question: Is there any research on choosing relevant nodes for vaccination ? What are the outcomes ?
If yes, how less effective would be the naive strategy to vaccinate nodes with the highest degrees comparing to more advanced ones ?
PS
There can be several ways to measure "effectiveness" of vaccination ,  as well as epidemic model may contain more parameters like - how long is node "infected", as well as answer might depend on type of a random graph considered - any information is welcome. Well, keeping real world situation in mind would be also nice.

REPLY [2 votes]: This is a much studied question, at the crossroad of several fields: epidemiology, mathematics, statistical mechanics, and computer science, at least.
The model you consider is known as SI with parameter p, of which many variants exist.
The vaccination problem is very similar to the robustness problem: how many nodes may be removed from a network without breaking connectivity for most nodes. Here, the main criterion is the size of the largest connected component, seen as a bound for the size of any epidemics.
The problem is also related to the influence maximization problem, where one seeks for (sets of) nodes in a network that may effectively spread an information.
Mathematical approaches to all these problems are strongly related to the study of connected components in various kinds of random graphs. A much regarded question is the influence of the graph degree distribution, with various interesting threshold effects.
An approach that I like claims that, if you have no global knowledge of your network, then you should choose random nodes and ask them to cite a random person. This person is likely to have many contacts (the probability that it is cited is proportional to the number of persons knowing her), more than random individuals, and so she is a good vaccination target.
It is hard to choose among the many references I may cite on this topic, but I suggest the 2008 book "Dynamical Processes on Complex Networks" by Barrat, Barthélémy and Vespignani, Cambridge University Press, as well as our survey paper "Impact of random failures and attacks on Poisson and power-law random networks" published in ACM Computing Surveys. You may also search the web for the topics above.
Many recent approaches leverage mobility traces and try to incorporate temporal information in addition to graph information, see for instance
the COVID-19 Mobility Network Modeling project at Stanford.