# How to test whether the majority of vertices belong to one giant component?

I have a sample of 200 independent networks and I want to test the hypothesis that majority of vertices belong to one giant component. I wonder what is the appropriate approach to do that.

More formally, suppose we have two variables, $A$ and $B$, where $A$ denotes number of vertices belonging to a giant component in a network $i$ and $B$ denotes number of vertices not belonging to giant component in network $i$.

The task is to test the hypothesis $\mu_A > \mu_B$. Because $A$ and $B$ are clearly dependent here, I doubt that classical statistical tests (e.g., Z-test) are appropriate here.

Any suggestion would be greatly appreciated.

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Based on the wikipedia definition of a giant component, it merely need be connected a majority of the nodes in the network. Is this the precise interpretation that you are considering, i.e. that the giant component is connected to >.5 of N where N is the number of nodes in the network? – mpacer Jun 16 '11 at 18:23
@ThisIstheId Yes, using threshold value I can derive if particular network contains giant component or not. This way I obtain 200 binary values (i.e., 1 = network has giant component and 0 = network hasn't giant component). But what's next step? – Andrej Jun 16 '11 at 18:59

If you do not have underlying data, but just have the network then you might try a different approach. If you believe structural equivalence is more important than geodesic distance in terms of clustering the nodes, then it makes sense to bootstrap the vertex labels (i.e., sample the vertex labels with replacement and if in the original network there is an edge from $i \to j$, then if you draw vertices $i$ and $j$, then connect them in the bootstrap sample as $i \to j$. Do this 10,000 times and see how many bootstrapped networks have a larger giant component.