2025-274

Two undirected networks, Network A and Network B, have identical degree distributions \(p_k\)​ but distinct degree correlation patterns.

Regarding the average nearest-neighbor degree function $k_{nn}(k)$ and the degree correlation matrix $e_{jk}$ of each network, we have:

• In Network A, $k_{nn}(k)$ is approximately constant across $k$, and $e_{jk}$ roughly factorizes into $q_j{q}_k$.

• In Network B, $k_{nn}(k)$ decreases systematically with $k$, and the $e_{jk}$ matrix shows high values in the upper-left and lower-right corners rather than along the diagonal.

Consider the following statements:

I. Network A is neutral, while Network B is disassortative.
II. Network B is expected to have a negative Newman correlation coefficient $r$.
III. If a new Network C were highly assortative, its $e_{jk}$ would concentrate along the diagonal, and $k_{nn}(k)$ would increase with $k$.
IV. Because Networks A and B share the same $p_k$, their correlation coefficient $r$ must also be the same.

Which of the statements above are correct?

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

Original idea by: Mateus de Padua Vicente