Question: Analysis of Degree Correlations in Complex Networks

A network engineer is analyzing the topology of a large technological network. After data collection and a complete analysis, he calculated the Degree Correlation Coefficient $r$ for the network, obtaining the value: $r=-\mathrm{0.19.}$

The Degree Correlation Coefficient ($r$) is defined as:

$$r \;=\; \frac{\sum_{j,k} jk\,\bigl(e_{jk} - q_j q_k\bigr)}{\sigma_r^2}$$

where $e_{jk}$ represents the probability of finding nodes with degrees $j$ and $k$ at the ends of a randomly selected edge; $q_k$ is the probability that the end of an edge has degree $k$$(q_k=k{p}_k\mathrm{/}⟨k⟩);\; and$$\sigma_r^2$ is the normalization term.

Based on this result ($r = -0.19$) and on the theory of degree correlations, mark the correct statement about the topology of this network:

A) The value $r = -0.19$ classifies the network as neutral. Neutral networks are those in which nodes connect to each other with the probabilities expected in random networks, implying that the difference $\sum_{j,k} jk\,\bigl(e_{jk} - q_j q_k\bigr)$is zero. The negative value obtained should be dismissed as a statistical artifact or a “structural cutoff”.

B) The value $r = -0.19$ indicates that the network is disassortative. In disassortative networks, hub nodes tend to avoid linking to each other. This means that, for pairs of high-degree nodes (both $j$ and $\mathrm{k\; large),\; the\; observed\; probability }$$e_{jk}$ is lower than the expected probability $q_j q_k$ of an uncorrelated network.

C) The negative value of $r$ points to an assortative network, since this type of network typically exhibits hubs connecting to low-degree nodes. The condition for $r$ to be positive ($r \ge 0$) is that the magnitude of the correlation captured by $\langle jk\rangle - \langle j\rangle \langle k\rangle$ (the numerator of the formula) is positive.

D) Technological networks are classically assortative. The $r = -0.19$ is atypical, but if it were truly disassortative, the impact would be increased robustness against targeted attacks, since removing a hub would not result in large cascades of disconnections among other hubs.

E) None of the above.

Original idea by: Matteus Vargas Simão da Silva