Complexity: What Is It?

Abstract

On the basis of the correlations of distant—in time and/or in space—elements of a many-element system, we propose the hard core of a mathematical definition of what may be referred to as complexity. This definition consistently leads to the concept of degree of complexity.

1. Introduction

Years ago, chatting with Murray Gell-Mann at the Santa Fe Institute, New Mexico, I suggested that we use the word “complexity” in the title of some manuscript that we were writing. He promptly answered “Oh, no, defining complexity is very hard!” I then suggested to use “complex systems” instead. He then said “That’s OK, that’s easier.” Indeed, like beauty, complexity is easy to identify, although it is definitively hard to define [1].

In the present paper, we outline a possible mathematical definition of complexity, at least in the sense that it is usually referred to when focusing on diversified natural, technological and social complex systems. Characteristics that are frequently mentioned include intermediate situations between extreme disorder and extreme order (e.g., glasses or spin-glasses, which emerge between crystal-like states and fluid-like ones [2], as well as critical points emerging in various many-body thermostatistical or geometrical systems), which would correspond to so-called “simple” systems, as opposed to “complex” ones. Or, in nonlinear dynamical systems, weak chaos, lying between strong chaos and regular orbits (e.g., at the Feigenbaum–Coullet–Tresser point of the z-logistic map [3]). Let us also mention that “complexity” definitively is a concept frequently addressed in human sciences as well (see, for instance, [4,5]).

The scientific and technological importance of these issues explains the emergence and dynamism of many centers around the world dedicated to the study of complex systems, such as the Santa Fe Institute, New Mexico, Dresden (Max Planck Institute for the Physics of Complex Systems), Gorlitz (Institute CASUS—Center for Advanced Systems UnderStanding), Rio de Janeiro (National Institute for Science and Technology of Complex Systems), Vienna (Complexity Science Hub), Catania (Dottorato in Sistemi Complessi per le Scienze Fisiche, Socio-economiche e della Vita, University of Catania), Sydney (Complex Systems Research, University of Sydney) L’Aquila (International Research Center for Mathematics and Mechanics of Complex Systems, University of L’Aquila), Hong Kong (Research Centre for Language and Human Complexity, Hong Kong, Peking University), Bristol (Centre for Complexity Sciences, University of Bristol), Warwick (Centre for Complexity Science, University of Warwick), Southampton (Institute for Complex Systems Simulation, University of Southampton), Dublin (Complex and Adaptive Systems Laboratory, University College Dublin), Tempe (Centre for Social Dynamics and Complexity, Arizona State University), Evanston (Northwestern Institute on Complex Systems, Northwestern University), among many others.

2. Possible Mathematical Definition of Complexity

We believe that a satisfactory mathematical definition of the complexity of a system should be grounded on the correlation functions of its observables (we use here the word “observable” to refer to all quantities that are measurable in terms of Karl Popper’s falsifiability). We refer more precisely to the influence of (time and/or space) distance on the correlation functions between elements of the system. Our purpose here is not to propose a widely general definition of complexity but to illustrate the issue with the simplest, though typical, particular instances.

Let us consider an infinitely long classical system (assumed, for simplicity, to be one-dimensional) constituted by many elements ($i=1,2,\dots ,N$). A (real) variable $m_i(x_i,t)$ is associated with each of them, where $(x_i,t)$ respectively are its position and time. We consider two of them, namely $m_i(x_i,t)$ and $m_j(x_j,t^{\prime })$, and define some relevant correlation function $c_{i,j}$, which will then depend on $(x_i,x_j,t,t^{\prime })$. Let us focus on the simple case where, through say averaging, $c_{i,j}$ only depends on $(\xi ,\tau )$, where $\xi =x_j-x_i$ and $\tau =t^{\prime }-t$ (i.e., translational symmetry). We then now have the function $c(\xi ,\tau )$, which we assume has been defined such that $c(0,0)$ is finite and positive, and that, for simplicity, it monotonically vanishes when $\xi$ and/or $\tau$ diverge. Our main focus now is on how quickly or slowly it vanishes with increasing $\xi$ or $\tau$. We consider two paradigmatic particular cases, namely $c(\xi ,0)$ and $c(0,\tau )$. Let us address now $c(\xi ,0)$ (the discussion of $c(0,\tau )$ precisely follows the very same path). We define the nth moment of $c(\xi ,0)$ as follows:

$${\mu }_n\equiv {\int }_0^{\infty }d\xi {\xi }^{n}c(\xi ,0)(n=0,1,2,\dots ),$$

(1)

or, somewhat more generally,

$$\mu \left(\nu \right)\equiv {\int }_0^{\infty }d\xi {\xi }^{\nu }c(\xi ,0)(\nu \in \mathbb{R},\nu \ge 0).$$

(2)

The system is said to be simple (in the sense of being non-complex) in what concerns this particular correlation function if $\mu \left(\nu \right)$ is finite for all values of $\nu$. The system is instead said to be complex if a value ${\nu }_{max}$ exists such that $\mu \left(\nu \right)$ diverges for all momenta whose value of $\nu$ is above ${\nu }_{max}$, i.e., for $\nu \ge {\nu }_{max}$. In other words, the system is simple when ${\nu }_{max}$ diverges; it is complex otherwise.

We can consistently define the degree of complexity $\kappa$ simply as the following (we have added a $(+1)$ in the denominator to avoid a divergence in the extreme case of ${\nu }_{max}=0$. Another agreeable possibility would be to alternatively define, as the degree of complexity, ${\kappa }^{\prime }\equiv e^{-{\nu }_{max}}\in [0,1]$).

$$\kappa \equiv {\displaystyle \frac{1}{{\nu }_{max}+1}}\in [0,1].$$

(3)

It immediately follows that ${\nu }_{max}\to \infty$; hence, $\kappa \to 0$ if $c(\xi ,0)$ decays exponentially (or stretched-exponentially) with $\xi$, or if it has a cut-off, i.e., if it is nonzero only within a finite vicinity of $\xi =0$.

If $c(\xi ,0)$ decays from say a finite value $c(0,0)$ to zero, like say a power-law $1/{\xi }^{\alpha }$$(\alpha \ge 0)$, we then have ${\nu }_{max}=0$ for $0\le \alpha \le 1$ and ${\nu }_{max}=\alpha -1$ for $\alpha >1$. Consistently, $\kappa =1$ for $0\le \alpha \le 1$ and $\kappa ={\displaystyle \frac{1}{\alpha }}$ for $\alpha >1$. The simplicity limit $\kappa =0$ is attained only for $\alpha \to \infty$.

The above discussion implies that if $c(\xi ,0)$ decays like say the q-exponential function $e_q^{-\beta \xi }\equiv {\displaystyle \frac{1}{{[1+(q-1)\beta \xi ]}^{1/(q-1)}}}(\beta >0;q\ge 1)$, then $\alpha ={\displaystyle \frac{1}{q-1}}$, which decreases from ∞ to 1 when q increases from $q=1$ to $q=2$, the maximal value below which a q-exponential distribution with $\beta >0$ is normalizable. Consistently, the degree of complexity $\kappa$ increases when $(q-1)$ increases.

3. Remarks

Summarizing, we have outlined here a mathematical definition for complexity grounded on the nature of the correlations of distant (in space and/or in time) elements of a many-element system. This standpoint leads naturally to a definition for the degree of complexity (or—paraphrasing Murray Gell-Mann’s “plectics”—plectic degree [6]) $\kappa$, which varies from $\kappa =0$ (a lack of complexity, which we may as well refer to as simplicity) to $\kappa =1$ (extreme complexity). This definition is based on the range (not to be confused with the intensity! (A typical expression would be $A/{\xi }^{\alpha }(A\ge 0;\alpha \ge 0)$. A characterizes the intensity, and $\alpha$ characterizes the range: increasing A corresponds to stronger intensity, and increasing $\alpha$ corresponds to a shorter range)) of the correlations. More precisely, if all the correlations (which decay to zero for diverging space $\xi$ and/or time $\tau$) of all the system’s observables are local (sufficiently short-ranged), we have $\kappa =0$; if such correlations are nonlocal (sufficiently long-ranged), i.e., if they are global, we have $0<\kappa \le 1$.

The most typical correlation decays in simple systems are asymptotically exponential (or stretched exponential, or including a cut-off), whereas the most typical decays in complex systems are asymptotically power-laws or logarithmic. Inanimate matter frequently is of the first kind, whereas living matter (this comparison was, for the first time in my knowledge, advanced (to me, many years ago) by Ricardo Ferreira) (or living-like systems such as biological, linguistic, growing urban and computer networks, aging phenomena, complex networks, economic, seismic, ecological ones) frequently is of the second kind.

Let us focus at this point on the appropriate entropic functionals for generic simple and complex systems.

Consistently, the BG additive entropic functional (as well as its quantum and classical counterparts)

$$S_{BG}\left(\left\{p_i\right\}\right)\equiv -k\sum _{i=1}^W{p}_i\ln p_i\left(\sum _{i=1}^W{p}_i=1\right),$$

(4)

and its associated statistical mechanics typically are legitimate and fully satisfactorily applicable to simple systems; k is a positive constant chosen once forever (the typical choice in physical sciences is $k=k_B$, the Boltzmann constant; the typical choice in computational sciences is $k=1$). As an aside comment, let us mention that the most general additive entropic functional is the Renyi one, namely

$$S_q^R\left(\left\{p_i\right\}\right)\equiv k{\displaystyle \frac{\ln {\sum }_{i=1}^W{p}_i^q}{1-q}}\left(q\in \mathbb{R};\sum _{i=1}^W{p}_i=1;S_1^R=S_{BG}\right),$$

(5)

whose equal-probability value is given by one and the same expression for all values of q, namely $S_q^R=k\ln W$. This strong property forces $S_q^R\left(\left\{p_i\right\}\right)$ to lose concavity for $q>1$, while it preserves it for $q\in (0,1]$. This is in contrast with $S_q\left(\left\{p_i\right\}\right)$ introduced here just below, which is concave for all $q>0$.

For complex systems, we typically need instead nonadditive entropic functionals such as [7]

$$\begin{array}{ccc}\hfill S_q\left(\left\{p_i\right\}\right)& \equiv & k{\displaystyle \frac{1-{\sum }_{i=1}^W{p}_i^q}{q-1}}=-k\sum _{i=1}^W{p}_i^q{\ln }_q{p}_i=-k\sum _{i=1}^W{p}_i{\ln }_{2-q}{p}_i\hfill \\ & =& k\sum _{i=1}^W{p}_i{\ln }_q{\displaystyle \frac{1}{p_i}}\left(q\in \mathbb{R};\sum _{i=1}^W{p}_i=1;S_1=S_{BG}\right),\hfill \end{array}$$

(6)

where ${\ln }_{q}z\equiv {\displaystyle \frac{z^{1-q}-1}{1-q}}(q\in \mathbb{R};z>0;{\ln }_{1}z=\ln z)$, or [8]

$$S_{\delta }\left(\left\{p_i\right\}\right)\equiv k\sum _{i=1}^W{p}_i{\left[\ln {\displaystyle \frac{1}{p_i}}\right]}^{\delta }\left(\delta >0;\sum _{i=1}^W{p}_i=1;S_1=S_{BG}\right),$$

(7)

or even others (see, for instance, [9,10,11,12,13]). The equal-probability expression of $S_q$ (Equation (6)) is given by $S_q=k{\ln }_{q}W$, and that of $S_{\delta }$ (Equation (7)) is given by $S_{\delta }=k{\left(\ln W\right)}^{\delta }$. The generalized statistical mechanics associated with nonadditive entropic functionals is currently referred to as nonextensive statistical mechanics [14] the characterization through the word “nonextensive” comes from the fact that this generalized statistical mechanical theory is frequently related to long-range interactions, which in turn lead to a superextensive total energy, and not so the total entropy, which remains extensive in all cases). (See Bibliography in [15].) Illustrative analytical, experimental and computational validations of $S_q$ and $S_{\delta }$ can be seen in [3,13,16,17,18,19,20,21,22,23,24,25,26,27,28,29] and elsewhere.

In what concerns many-body (classical, quantum or relativistic) Hamiltonian systems (generically, conservative ones), long-range correlations are not to be confused with long-range interactions. Indeed, long-range interactions necessarily yield long-range correlations. Short-ranged interactions can instead yield either short-ranged (e.g., out of criticality in continuous phase transitions) or long-ranged (e.g., precisely at criticality, to be distinguished from close to criticality) correlations. In other words, short-ranged interactions are necessary but not sufficient to guarantee short-ranged correlations. Indeed, the range of the correlations depends not only on the range of the interactions but also on the initial and/or boundary conditions of the system (the Ortega y Gasset famous sentence “Yo soy yo y mi circunstancia” [I am me and my circumstance] definitively is applicable here). It is possible, for a given system, to be simple (i.e., to be typically grounded on the BG additive entropic functional and concomitantly follow BG statistical mechanics) in what concerns equilibrium properties, and simultaneously be complex (i.e., to to be typically grounded on nonadditive entropic functionals and concomitantly follow generalized statistical mechanics) in what concerns out-of-equilibrium ones (such as in heat, charge and mass transport phenomena). In other words, the nature of a system can simultaneously be simple or complex depending on the properties (as well as on boundary or initial conditions) focused on by the observer. A paradigmatic illustration can be the same large quantity of water in a calm little pond, or in a turbulently running river.

Finally, let us also emphasize that, in possibly all cases involving well-defined energies and/or analogous quantities, we expect the Legendre-transformations structure of classical thermodynamics to be preserved (see [14,30] and references therein). This preservation mandates the entropy to be extensive in the thermodynamical sense, i.e., $S\left(N\right)\propto N(N\to \infty )$. Assuming we have the case of equal probabilities, if $W\left(N\right)$ diverges like ${\mu }^N(\mu >1)$, then $S_{BG}$ certainly is adequate; if $W\left(N\right)$ diverges like $N^{\rho }(\rho >0)$, then $S_{q=1-1/\rho }$ is extensive; if $W\left(N\right)$ diverges like ${\nu }^{N^{\gamma }}(\nu >1;\gamma >0)$, then $S_{\delta =1/\gamma }$ is extensive.

A detailed comparison of the present mathematical definition of complexity with other (qualitative or quantitative) approaches existing in the literature would certainly be enriching (see, for instance, [31,32]); it remains, however, outside of the present effort.