Questão 4 - When Identical Degree Distributions Hide Distinct Correlation Patterns

When Identical Degree Distributions Hide Distinct Correlation Patterns

“Two networks may share the same degree distribution and yet encode profoundly different organizational principles.”

 

Context
Consider two undirected networks, A and B, each with the same degree distribution $P(k)$ and identical average degree $\langle k \rangle = 6.2$.

Despite this structural similarity, empirical analysis shows that the average neighbor degree function $k_{nn}(k)$ behaves differently for each network:

• For Network A, $k_{nn}(k)$ increases approximately as a power law $k_{nn}(k) \sim a,k^{\mu}$ with $\mu > 0$.

• For Network B, $k_{nn}(k)$ decreases with $k$, following $k_{nn}(k) \sim a,k^{\mu}$ with $\mu < 0$.

Definition
The Pearson correlation coefficient of degree correlation $r$ is defined as:

$$r = \frac{\sum_{j,k} j k \,\big(e_{jk} - q_j q_k\big)}{\sigma^2}$$

where:

• $e_{jk}$ is the joint probability that a randomly chosen edge connects nodes of degrees $j$ and $k$;

• $q_k = \dfrac{k\,P(k)}{\langle k \rangle}$ is the excess degree distribution;

• $\sigma^2 = \sum_k k^2 q_k - \left( \sum_k k q_k \right)^2$.

where: …

$e_{jk}$ is the joint probability that a randomly chosen edge connects nodes of degrees $j$ and $k$; 

$q_k = \frac{k\,P(k)}{\langle k \rangle}$ is the excess degree distribution; 

$\sigma^2 = \sum_k k^2 q_k - \left( \sum_k k q_k \right)^2$.

Question

Which statement best describes the structural difference between networks A and B?

A. Both networks are neutral, since identical degree distributions imply a random mixing pattern.
B. Network A is assortative, where high-degree nodes tend to connect to other high-degree nodes, while Network B is disassortative, with hubs linking preferentially to low-degree nodes.
C. Both networks are perfectly assortative, as indicated by $r = 1$.
D. Network B has no degree correlation since $k_{nn}(k)$ decreases with $k$.

E. None of the above.

         Original idea by: Luiza Barguil

        Tags:Degree correlations • Assortativity • Disassortativity • Network topology • Graph analysis