2026-354

Consider the following adjacency matrix of an undirected network

$$A=
  \begin{bmatrix}
  0&1&1&0&0&0&0\\
  1&0&1&1&0&0&0\\
  1&1&0&1&1&0&0\\
  0&1&1&0&0&1&0\\
  0&0&1&0&0&1&1\\
  0&0&0&1&1&0&1\\
  0&0&0&0&1&1&0
  \end{bmatrix}$$

Consider the candidate communities 

$$C_1=\{1,2,3,4\}
  \qquad
  C_2=\{5,6,7\}$$

Which statement is correct?

1. \(C_1\) is a clique and \(C_2\) is a strong community
2. \(C_1\) is a strong community but not a clique, while \(C_2\) is a clique
3. Both \(C_1\) and \(C_2\) are strong communities but not cliques
4. \(C_1\) is a weak but not strong community and \(C_2\) is a clique
5. None of the above

Original idea by: Antonio De Cesare Del Nero