imichaelnorris
commented
9 years ago - Yes, a graph valued random variable is one posible graph out of a set of graphs. Just like "normal" valued random variables are one number out of a range of numbers, there is a set of possible graphs and the random variable will be one of these graphs or it will represent a "distribution" on the graphs.
- I assume that depends on how you define the graph valued random variable. I recall our professor saying that we will always be dealing with the number of nodes being fixed in this class, so we could either define the graph-valued random variable as taking on any graph with N nodes and E edges, or we could define it as taking any graph with N nodes, which could have from 0 to n(n-1)/2 edges.
Further, we could define a distribution on the number of edges. So maybe a graph with N nodes and 1 total edge is just as likely as a graph with N edges. This would be a uniform distribution. But we could also have Gaussian distribution centered on some N that is close to a model that we would like to represent.
Or we could define a distribution on the degree of each node, which would mean that every node's degree will be a random variable according to a statistical distribution. (I think in this case, given a large enough amount of nodes whose degree follows any distribution, we can apply the Central Limit Theorem to show that the graph will have its total number of edges following a normal distribution that is related to the distribution of the degree of each node.)
kristinmg
jovo
Does graph valued random variable mean that it is just one possible graph out of a set of graphs? Are the number of nodes and edges fixed?