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 r r is defined as: r = ∑ j , k j k ( e j k − q j q k ) σ 2 r = \frac{\sum_{j,k} j k \,\big(e_{jk} - q_j q_k\big)}{\sigma^2} r = σ 2 ∑ j , k jk ( e jk − q j q k ) where: ejk e_{jk} e ...