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

Question: Degree Dynamics in the BA Model

In the Barabási–Albert (BA) model, which incorporates growth (continuous addition of new nodes) and preferential attachment (new nodes prefer to connect to more connected nodes), the time evolution of a node’s degree, $k_i(t)$, is a central aspect.

The degree growth for a node $i$, which entered the network at time $t_i$ with $m$ links, follows a power law in time $t$, characterized by the dynamic exponent $\beta$.

Which of the following formulas correctly represents the degree growth law $k_i(t)$ for a node $i$ in the BA model?

A.

$$k_i(t) = m \left( \frac{t}{t_i} \right)^{1/2}$$

B.

$$k_i(t) = m \frac{t}{t_i}$$

C.

$$k_i(t) = k^{-\gamma} \quad \text{where} \quad \gamma = 3$$

D.

$$k_i(t) = m \left( \frac{t}{t_i} \right)^{2}$$

E. None of the above

Original idea by: Matteus Vargas Simão da Silva