MO412/2021s1 - Tiago da Silva Almeida

Considering the degree correlation coefficient, which is defined as $r = \sum_{jk} \frac{jk(e_{jk}-q_jq_k)}{\sigma^2}$ with  $\sigma^2=\sum_k k^2q_k-\left[ \sum_k kq_k \right]^2$. And considering Net A and Net B in the figures below. Which topology do the Net A and B present? Net A is disassortative and Net B is assortative;  Net A is neutral and Net B is disassortative; Net A is assortative and Net B is neutral; Net A is assortative and Net B is disassortative;  None of the above;

In a free-scale network, the degrees of the biggest hub increases linearly with the system size in an anomalous regime (AR). Another regime predicts that hub size increases as $\ln{\ln{N}}$. In this case, it is an ultra-small world regime (USWR). However, may happens a critical point regime (CPR), where the double logarithm correction shrinks the distance compared to a random network. When this CPR is transpassed the size of hubs is reduced and they are less numerous, i.e., the network is now random. Based on this, which of the following networks are not distinguished between free-scale or a random network: Actor network ($\gamma=2.12$, $k_{min}=54$, P-VALUE=$0.00$); Science collaboration ($\gamma=3.35$, $k_{min}=25$, P-VALUE=$0.001$); Yeast protein interactions ($\gamma=2.89$, $k_{min}=7$, P-VALUE=$0.67$); WWW (in) ($\gamma=3.42$, $k_{min}=72$, P-VALUE=$0.13$); None of the above;

The clustering coefficient is an important metric for a network. For example, the metric allows to measuring how many people are related and the potential transmission of some disease. Regarding this, consider an undirected network with five nodes (A, B, C, D, and E), and the following links: (A, B), (A, C), (A, D), (A, E), (B, C), (C, D), (C, E), and (D, E). Also, consider the following sentences: I) The clustering coefficient of a node takes count the degree of that node. The degree of a link v in a network G is the number of links of G incident with v , each loop counting as two edges. For example, node C has a clustering coefficient of 2/3. II) The average clustering coefficient is just an arithmetic average of the clustering coefficient of all nodes. For this network is 21/30. III) The global clustering coefficient is based on triplets of nodes. A triplet is three nodes that are connected by either two (open triplet) or three (closed triplet) undirected ties. For this network i...

Consider the following theorem: "Let f be a definite and continuous function in a closed interval (a, b) and derivable at the internal points. Then there is at least one point c , comprised between a and b , such that, f (b) - f (a ) = f '(c) (ba) ". How is this theorem known? Chain rule; Mean value Theorem; Rolle's Theorem; Extreme Value Theorem; None of above;