Good ways to compare graphs of different sizes?

Sorry if this is a basic question but I was wondering if anyone could help me find out the class of problems this specific question of mine falls into. I was looking for any standard metrics that can be used to compare graphs of different size and connectivity. Specifically, consider the following example:

      G1                             G2
      2                              D
      |                             /  \
4 --- 1 --- 3                 C -- A1 - A2 -- E
      |
      5

What I am interested in is to capture the notion of stability inside one graph (intra-stability) and relative to another graph (inter-stability). For instance,

Intra-Stability:

In G1, in my hypothetical metric, 2,3,4,5 all have the same effect were they to be removed from the graph. In G2, C,E would have the same effect but D would have more effect. However, A1,A2 would have more effect were they to be removed. What I am looking for here is a notion of stability of a graph. I am guessing I could just use the degree of each node to capture the effect of a specific node but am not sure how to compute it for the whole graph per-se.

Inter-Stability:

Can we say something about G1 and G2 in a relative sense i.e. something like because G1 h开发者_运维百科as a stability metric X and G2 has Y and because X < Y, we conclude G1 is less stable than G2? The definition of stable itself is left open but I am trying to capture how unreliable a graph is or how dependent is it on one node.

Can someone point me in the right direction in order to be able to quantify this or at least what this problem is referred to as?

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in graph theory, a cut or cut-set seems to describe your maximal instability description.

as a metric, you may be talking about 'connectivity'