2021-051

Random networks evolve in a dynamic process when \( \langle k \rangle \) grows from 0 to infinity. They start with isolated nodes, which are progressively linked in this random process. As a consequence, a giant component can emerge and change the network topology. The appearance of the giant component is only one of the transitions that occur as we change \( \langle k \rangle \). Four topologically distinct regimes are usually studied, namely: subcritical regime, critical point, supercritical regime and connected regime. Consider the following statements:

(I) In the subcritical regime, the network consists of numerous small, isolated components.

(II) At the critical point, most nodes are located in numerous small components. These numerous small components are mainly trees, while the giant component may contain loops.

(III) In the supercritical regime, numerous isolated components coexist with the giant component. These small components are trees, while the giant component contains only cycles. The supercritical regime lasts until the giant component absorbs all nodes.

(IV) In the connected regime, there are isolated nodes or unconnected components.

Which alternative lists the correct statements about the regimes?

1. I and II.
2. II and III.
3. I, II, III.
4. III and IV.
5. None of the above.

Original idea by: Bruno Moritani