Variables of interest to describe the topology of a random graph? I'm starting to study random graphs (I read a few things about Erdös-Rényi model here). 
What are the most useful variables to describe the topology of a random graph?
I guess:
$$\alpha = \text{average (over all nodes) of the number of edges from a given node}$$
$$\beta = \text{standard deviation (over all nodes) of the number of edges from a given node}$$
$$\gamma = \text{average (over all pairs of nodes) of shortest distance between two nodes}$$
are useful, what are the typical quantitative metrics that we use to distinguish, say:
x_1 -- x_2 -- ... -- x_N (just a long chain) from a dense graph with many links between nodes,
or which quantitative indicators an measure the presence or absence of "clusters"?
 A: Two fundamental notions along these lines are the vertex connectivity and the edge connectivity of $G$, which tell us how many vertices or edges respectively need to be deleted to disconnect $G$.
A fancier answer is the Cheeger constant/isoperimetric number of the graph. To find it, we minimize
$$
    \frac{e(S, V(G) \setminus S)}{|S|}
$$
over all subsets $S \subseteq V(G)$ with $|S| \le \frac12|V(G)|$, where $e(A,B)$ denotes the number of edges between vertices in $A$ and vertices in $B$. The idea is that the set $S$ minimizing this ratio is a cluster in the graph: it's a relatively large set of vertices with few edges leaving it. 
Of course, computing this quantity requires checking lots of subsets $S$ (and computing the connectivity of a graph is hard, too), and there's a more algebraic approach: the algebraic connectivity of the graph $G$. This is the second-smallest eigenvalue of the Laplacian matrix of $G$. (The smallest eigenvalue is always $0$, corresponding to the all-$1$'s eigenvector.)
This is related to the isoperimetric number by an inequality: the isoperimetric number is at least half the algebraic connectivity. A rough idea of the connection between the two is that the indicator vector of the set $S$ is a good approximation of an eigenvector of the Laplacian matrix with a small eigenvalue. The algebraic connectivity is also at most the vertex connectivity.
