Fig. 5: Impact of empty hyperedges on the HCM ensemble.
From: Entropy-based models to randomise real-world hypergraphs
More in detail, trends of \(\langle {N}_{{{\emptyset}}}\rangle /L=\overline{{p}_{{{\emptyset}}}}\), i.e., the probability for the generic hyperedge to be empty and \(P({N}_{{{\emptyset}}} > 0)\simeq 1-{\prod }_{\alpha =1}^{L}(1-{e}^{-{h}_{\alpha }})\), i.e., the probability of observing at least one empty hyperedge, are represented as functions of the connectance ρ, respectively in (a and b). Evaluating the latter in correspondence of the filling threshold, reading \({p}_{f}^{\,{\mbox{HCM}}\,}\simeq 0.032\) (vertical line), returns the value ≃ 0.642. The dense (sparse) regime is recovered for large (small) values of z. Each dot represents an average taken over an ensemble of 103 configurations (explicitly sampled from the HCM) and is accompanied by the corresponding 95% confidence interval.
