8 otherwise, the sum diverges. Furthermore, it is also true that the above condition also holds for the following se- ries: Indeed, note that, given an arbitrary δ > 0 there exists a finite number i ∗ such that: and, if we define the following exponent, β ( i $^{∗}$): there exists a finite constant, Ψ( δ ), defined as: such that: With the above properties, it is clear that, if there exists a constant φ (1 + δ ) < ∞ such that: then, the entropy of a power law with exponent higher than 1 is bounded. As we shall see, it is straightforward by checking directly the behavior of H ( n, 1 + δ ): Since H ( n, γ ) is an increasing function on n , and we can define a constant φ (1 + δ ), (where ϵ is any positive, finite constant). Clearly, Thus, Consequence If an unknown probability distribution is dominated from some k by some power-law with expo- nent higher than 1 + δ , our entropy will be bounded. Consequence If an unknown probability distribution is dominated from some k by some power-law with expo- nent higher than 1 + δ , our normalized entropy will tend to 0 . 5 . The normalized entropy of a power-law distribu- tion in a system with n different states, p$_{n}$ with exponent lower than 1 converges to 1. Let us suppose that we have the following probability distribution, with 0 < δ < 1: Note that [66]: Applying directly the definition of entropy, If we compute the limit of h ( n, 1 − δ ): Consequence If our (unknown) probability distribu- tion is not dominated from some k by a power law with exponent higher than 1 − δ , our normalized entropy will converge to 1 . We thank D. Jou, S. Manrubia, J. Fortuny and our colleagues at the Complex Systems Lab for their useful comments. We also acknowledge the helpful comments provided by two anonymous referees. This work has been founded by the McDonnell Foundation (BCM) and by the Santa Fe Institute (RS). Acknowledgments ∞ ∑ k log k k 1+ δ . (A17) i ∗ ≡ min { i : ( δ − log(log i ) log i ) > 0 } (A18) β ( i $^{∗}$) ≡ 1 + δ − log(log i $^{∗}$) log i ∗ , (A19) Ψ( δ ) ≡ ∑ i<i ∗ ( log i i 1+ δ − 1 i β ( i ∗ ) ) + ζ ( β ( i $^{∗}$)) , (A20) ∞ ∑ k log k k 1+ δ < Ψ( δ ) . (A21) lim n $_{→∞}$H ( n, 1 + δ ) < φ (1 + δ ) , (A22) 1 + δ ζ (1 + δ ) ∞ ∑ i =1 log i i 1+ δ + log( ζ (1 + δ )) < ∞ , (A23) φ (1 + δ ) ≡ lim n $_{→∞}$H ( n, 1 + δ ) + ϵ (A24) p$_{n}$ ( s$_{i}$ ) = 1 Z i − (1 − δ $^{)}$. (A26) ∑ k ≤ n 1 k 1 − δ = ∫ n 1 1 x 1 − δ + O (1) ≈ n δ δ . (A27) H ( n, 1 − δ ) = δ (1 − δ ) n δ ∑ k ≤ n log k k 1 − δ + δ log n − log δ. (A28) lim n $_{→∞}$h ( n, 1 − δ ) = lim n →∞   δ (1 − δ ) log n · n δ ∑ k ≤ n log k k 1 − δ + δ   = lim n →∞ 1 − δ log n ( log n − 1 δ ) + δ = 1 − δ + δ = 1 . H ( n, 1 + δ ) < φ (1 + δ ) . (A25) lim n $_{→∞}$h ( n, 1 + δ ) = lim n →∞ H ( n, 1 + δ ) log n ≤ lim n →∞ φ (1 + δ ) log n = 0 . lim n $_{→∞}$H ( n, 1 + δ ) = 1 + δ ζ (1 + δ ) ∞ ∑ i =1 log i i 1+ δ + log( ζ (1 + δ )) .