Question 5 (Degree Correlations)

Consider a Barbell graph consisting of two cliques \(K_n\) connected by a path of length \(L\). Assume both \(n\) and \(L\) may grow asymptotically.

Evaluate the following statements regarding the assortativity coefficient \(r\):

I. If \(L = o(n^2)\), the assortative contribution of the cliques asymptotically dominates the bridge structure.

II. If \(L = \Theta(n^3)\), the graph necessarily becomes strongly disassortative.

III. When \(L \to \infty\) with fixed \(n\), the relative contribution of hub-hub edges becomes asymptotically negligible.

IV. A large number of degree-\(2\) to degree-\(2\) edges does not necessarily imply \(r \to 1\).

  1. T, F, T, T
  2. T, T, F, T
  3. F, F, T, T
  4. T, F, F, T
  5. None of the above

Original idea by: Antonio De Cesare Del Nero

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