6 that, for a given, very large n , our probability distri- butions p k n must be able, as k increases, to unbound- edly increase the entropy of the whole system to reach the global value H ( X ( n )), which lies in the interval (( µ − ϵ ) log n, ( µ + ϵ ) log n ). Furthermore, scale invari- ance condition depicted in eq. (34) forces that, as k increases, contributions to the entropy never go neither to 0 nor to log( k + 1 / log k ), but lie within this interval. In other words, the sum defined by the entropies must diverge as k increases over a system where n is arbitrary large, whereas the sequence of its normalized versions must converge to µ . The above derivations concerning the convergence properties of the entropy -see also Ap- pendix A- clearly state that those properties hold if p$_{n}$ satisfies, on one hand, for large i ’s, to avoid that h ( n ) → 1. On the other hand, if we want to avoid that h ( n ) → 0, the following inequality must hold: Therefore, the solution of our problem lies in the range defined by: From the study of the entropies of a power law performed in the previous section, we know that δ can be arbitrarily small if the size of the system is large enough. Thus: which leads us to Zipf’s law as the unique asymptotic solution: Complex, far from equilibrium systems involve a ten- sion between amplifying mechanisms and negative feed- backs able to buffer the impact of fluctuations. In this paper we have considered the consequences of such ten- sion in terms of one of its most well known outcomes: the presence of an inverse scaling law connecting the size of observed events and its rank. The commonality of Zipf’s law in both natural and man-made systems has been a puzzle that attracted for years the attention of scien- tists, sociologists and economists alike. The fact that such a plethora of apparently unrelated systems display the same statistical pattern points towards some funda- mental, unifying principle. In this paper we treat complex systems as stochas- tic systems describable in terms of algorithmic complex- ity and thus statistical entropy. A general result from the algorithmic complexity theory is that eq. (3) holds for stochastic systems. Taking this general result as the starting point, we define a characterization of a wide class of complex systems which grasps the open nature of many complex systems, summarized in eq. (9). The main achievement of this equation is that it encodes the concepts of growing and, even most important, the stabi- lization of complexity properties in an intermediate point between order and disorder, a feature observed in many systems displaying Zipf’s-like statistics. From this equa- tion we derived Zipf’s law as the natural outcome of sys- tems belonging to this class of stochastic systems. Our development avoids the classical procedures based on maximization (minimization) of some functional in order to find the most probable configuration of states, since in far from equilibrium the ensemble formalism, to- gether with Jaynes’ maximum entropy principle [60] can fail due to the open, non-reversible behavior of the sys- tems considered here. Thus we do not introduce mo- ment constraints, as it is usual in equilibrium statisti- cal mechanics [63], but instead a constraint on the value achieved by the normalized entropy, no matter the scale we observe the system. Both a scaling ansatz and a more general scale invariance assumption lead to Zipf’s law as the unique solution for this problem. We observe that the finite size effects define an interval of exponents around 1, namely (1 − δ, 1 + δ ), which could partly explain the variation observed in finite, natural systems. However, it is true that a system satisfying eq. (9) does not nec- essarily exhibit Zipf’s law. Further work should explore in depth the physically relevant conditions leading the evolution of Zipf’s like systems to remove the mathemat- ical assumptions made in this paper, thereby obtaining a complete description of them from a completely general, theoretical viewpoint. Consider a system whose behavior is described by the random variable X ( n ) taking values on the set Σ = { s$_{1}$, ..., s$_{n}$ } , | Σ | = n , according to the probabilty dis- tribution p$_{n}$ ( s$_{i}$ ). The labeling ’ i ’ of the state is chosen in such a way that p$_{n}$ ( s$_{1}$ ) ≥ p$_{n}$ ( s$_{2}$ ) ≥ ... ≥ p$_{n}$ ( s$_{i}$ ) ≥ ... ≥ p$_{n}$ ( s$_{n}$ ) . The Shannon entropy of our system of n states, to be noted H ( X ( n )), is defined as [58]: The normalized entropy of the system, to be written, h ( n ), is defined as: IV. DISCUSSION Appendix A: Entropic Properties of Power-Law distributed systems f$_{n}$ ( i, i + 1) < ( i + 1 i ) (1 − δ ) , (35) f$_{n}$ ( i, i + 1) > ( i + 1 i ) (1+ δ ) . (36) ( i + 1 i ) (1 − δ ) > f$_{n}$ ( i, i + 1) > ( i + 1 i ) (1+ δ ) . (37) f$_{n}$ ( i, i + 1) = p$_{n}$ ( s$_{i}$ ) p$_{n}$ ( s$_{i}$$_{+1}$ ) ≈ i + 1 i (38) p$_{n}$ ( s$_{i}$ ) ∝ i − $^{1}$. (39) H ( X ( n )) = − ∑ k ≤ n p$_{n}$ ( s$_{k}$ ) log p$_{n}$ ( s$_{k}$ ) . (A1) h ( n ) ≡ H ( X ( n )) log n . (A2)