Question

Given an undirected graph with n nodes, where each of the nodes has a value. The node value may also be called as a weight in this case(note that I am assigning a weight to the node and not the edge), then given a node ‘a’ in the graph, is it possible to compute any metric/measure which tells me the ‘importance’ of the node in the graph?

I am not sure whether Centrality is what i am looking at. I think Centrality is more related to the degree of the node; whereas I am looking at how a ‘important’ node is from the other important nodes(which are calculated using weights) in the graph.

Just to expatiate on my Q(and taking a slightly different view) : Assume a graph which initially has no weights/value assigned for any of the nodes. Now, i identify a few nodes and assign a weight/value 1 to them. Now, given a node(which has not been assigned a weight/value), i want to calculate its weight/value, based on its ‘weighted’ distance from each of the other nodes(which have been assigned weights). You can probably take the distance as the number of edges and multiple it by a weighting factor to calculate the weight/value of a node.

Answer 1

Did you take a look at Eigenvector centrality? From Wikipedia (emphasis mine):

Eigenvector centrality is a measure of
the importance of a node in a network.
It assigns relative scores to all
nodes in the network based on the
principle that connections to
high-scoring nodes contribute more to
the score of the node in question than
equal connections to low-scoring
nodes.