the connectivity threshold
The connectivity threshold is the edge density at which a random graph stops being broken into pieces and becomes one connected whole. It sits at a much higher density than the giant-component threshold: a graph can have a giant containing 99 percent of vertices yet still fail to be connected, because a few isolated vertices linger. The question it answers is: how many edges does it take not merely to have a big cluster, but to tie every single vertex into one component?
The theorem (Erdos-Renyi) is sharp and elegant. Set p = (log n + c)/n for a constant c. Then P(G(n,p) is connected) tends to e^(-e^(-c)) as n grows. In particular, if p = (log n - omega(n))/n with omega tending to infinity, the graph is whp disconnected; if p = (log n + omega(n))/n the graph is whp connected. So the threshold is p* = (log n)/n, and it is sharp with a window of width 1/n. The mechanism is that connectivity is essentially controlled by the last obstruction to disappear, which is the isolated vertex. The expected number of isolated vertices is n (1-p)^(n-1), which is about n e^(-np) = e^(log n - np) = e^(-c) at p = (log n + c)/n; the number of isolated vertices is asymptotically Poisson with mean e^(-c) (by the Stein-Chen method), so P(no isolated vertex) tends to e^(-e^(-c)). The deep part is that the LAST thing standing between disconnection and connection really is isolated vertices: above (log n)/n, whp every component other than one giant has already been absorbed, so 'connected' and 'no isolated vertex' coincide in the limit.
This threshold matters wherever one needs a network that is fully reachable, not just mostly: random key-distribution schemes, fault-tolerant networks, and the coupon-collector-like phenomenon that the cost of total coverage is a log n factor above the cost of macroscopic coverage. The honest subtlety is that the equivalence of 'connected' and 'no isolated vertex' is itself a theorem and is special to this dense-enough regime; in sparser regimes the obstruction to connectivity is genuinely the small components, not just singletons. Note also that a closely parallel statement holds for the disappearance of vertices of degree below any fixed k: G(n,p) has minimum degree at least k exactly at the threshold p = (log n + (k-1) log log n + c)/n, again with a Poisson limit for the number of low-degree vertices.
With n = 10000, the connectivity threshold is p* = (log n)/n which is about 9.2/10000 = 0.00092, i.e. average degree about log n = 9.2. Below average degree 9.2 there are whp still some isolated vertices; the chance of being exactly connected at p = (log n + c)/n follows e^(-e^(-c)): at c = 0 it is about 0.368, at c = 3 about 0.951.
Connectivity arrives at average degree log n, a log-factor above the giant; the last obstacle is the isolated vertex.
The giant (degree 1) and connectivity (degree log n) thresholds are far apart: a giant is not the whole graph. The equivalence 'connected iff no isolated vertex' is itself a theorem of this regime, not a tautology.