Fig. 2: Impact of empty hyperedges on the RHM ensemble.
From: Entropy-based models to randomise real-world hypergraphs
More in detail, the trends of \(\langle {N}_{{{\emptyset}}}\rangle /L={p}_{{{\emptyset}}}={(1-p)}^{N}{\to }^{N\to +\infty }{e}^{-h}\), i.e., the probability for the generic hyperedge to be emtpy is represented in (a) and the one of \(P({N}_{{{\emptyset}}} > 0)=1-{[1-{(1-p)}^{N}]}^{L}{\to }^{N\to +\infty }1-{(1-{e}^{-h})}^{L}\), i.e., the probability of observing at least one empty hyperedge is represented in (b). Evaluating them in correspondence of \({p}_{f}^{\,{\mbox{RHM}}}={h}_{f}^{{\mbox{RHM}}\,}/N=\ln L/N\simeq 0.023\) (vertical line) returns, respectively, the values 1/L = 10−3 and 1 − (1−1/L)L ≃ 0.6323. The dense (sparse) regime is recovered for large (small) values of p. Each dot represents an average taken over an ensemble of 103 configurations (explicitly sampled from the RHM) and is accompanied by the corresponding 95% confidence interval.
