The complexity of a physical system or a dynamical process expresses the degree to which components engage in organized structured interactions. High complexity is achieved in systems that exhibit a mixture of order and disorder (randomness and regularity) and that have a high capacity to generate emergent phenomena.
Complexity across Scientific Disciplines
Despite the importance and ubiquity of the concept of complexity in modern science and society, no general and widely accepted means of measuring the complexity of a physical object, system, or process currently exists. The lack of any general measure may reflect the nascent stage of our understanding of complex systems, which still lacks a general unified framework that cuts across all natural and social sciences. While a general measure has remained elusive until now, there is a broad spectrum of measures of complexity that apply to specific types of systems or problem domains. The justification for each of these measures often rests on their ability to produce intuitively correct values at the extremes of the complexity spectrum. Despite the heterogeneous approaches taken to defining and measuring complexity, the belief persists that there are properties common to all complex systems, and new proposals are continually being produced and tested by an interdisciplinary community of physicists, biologists, mathematicians, computer scientists, economists and social theorists.
General Features of Complexity
Quantitative or analytic definitions of complexity often disagree about the precise formulation, or indeed even the formal framework within which complexity is to be defined. Yet, there is an informal notion of complexity based upon some of the ingredients that are shared by most if not all complex systems and processes. Herbert Simon was among the first to discuss the nature and architecture of complex systems (Simon, 1981) and he suggested to define complex systems as those that are “made up of a large number of parts that have many interactions.” Simon goes on to note that “in such systems the whole is more than the sum of the parts in the […] sense that, given the properties of the parts and the laws of their interaction, it is not a trivial matter to infer the properties of the whole.”
Components. Many complex systems can be decomposed into components (elements, units) that are structurally distinct, and may function as individuals, generating some form of local behavior, activity or dynamics. Components of complex systems may themselves be decomposable into subcomponents, resulting in systems that are complex at multiple levels of organization, forming hierarchies of nearly, but not completely decomposable components.
Interactions. Components of complex systems engage in dynamic interactions, resulting in integration or binding of these components across space and time into an organized whole. Interactions often modulate the individual actions of the components, thus altering their local functionality by relaying global context. In many complex systems, interactions between subsets of components are mediated by some form of communication paths or connections. Such structured interactions are often sparse, i.e. only a small subset of all possible interactions is actually realized within the system. The specific patterns of these interactions are crucial in determining how the system functions as a whole. Some interactions between components may be strong, in the sense that they possess great efficacy with respect to individual components. Other interactions may be considered weak, as their effects are limited in size or sporadic across time. Both, strong and weak interactions can have significant effects on the system as a whole.
Emergence. Interactions between components in integrated systems often generate phenomena, functions, or effects that cannot be trivially reduced to properties of the components alone. Instead these functions emerge as a result of structured interactions and are properties of the system as a whole. In many cases, even a detailed and complete examination of the individual components will fail to predict the range of emergent processes that these components are capable of if allowed to interact as part of a system. In turn, dissection of an integrated system into components and interactions generally results in a loss of the emergent process.
In summary, systems with numerous components capable of structured interactions that generate emergent phenomena may be called complex. The observation of complex systems poses many challenges, as an observer needs simultaneously to record states and state transitions of many components and interactions. Such observations require that choices be made concerning the definition of states, state space and time resolution. How states are defined and measured can impact other derived measures such as those of system complexity.
Measures of Complexity
Measures of complexity allow different systems to be compared to each other by applying a common metric. This is especially meaningful for systems that are structurally or functionally related. Differences in complexity among such related systems may reveal features of their organization that promote complexity.
Some measures of complexity are algorithmic in nature and attempt to identify a minimal description length. For these measures complexity applies to a description, not the system or process per se. Other measures of complexity take into account the time evolution of a system and often build on the theoretical foundations of statistical information theory.
Most extant complexity measures can be grouped into two main categories. Members of the first category (algorithmic information content and logical depth) all capture the randomness, information content or description length of a system or process, with random processes possessing the highest complexity since they most resist compression. The second category (including statistical complexity, physical complexity and neural complexity) conceptualizes complexity as distinct from randomness. Here, complex systems are those that possess a high amount of structure or information, often across multiple temporal and spatial scales. Within this category of measures, highly complex systems are positioned somewhere between systems that are highly ordered (regular) or highly disordered (random). Figure 1 shows a schematic diagram of the shape of such measures, however, it should be emphasized again that a generally accepted quantitative expression linking complexity and disorder does not currently exist.
Complexity as Randomness
Algorithmic information content was defined (Kolmogorov, 1965; Chaitin, 1977) as the amount of information contained in a string of symbols given by the length of the shortest computer program that generates the string. Highly regular, periodic or monotonic strings may be computed by programs that are short and thus contain little information, while random strings require a program that is as long as the string itself, thus resulting in high (maximal) information content. Algorithmic information content (AIC) captures the amount of randomness of symbol strings, but seems ill suited for applications to biological or neural systems and, in addition, has the inconvenient property of being uncomputable. For further discussion see algorithmic information theory.
Logical Depth (Bennett, 1988) is related to AIC and draws additionally on computational complexity defined as the minimal amount of computational resources (time, memory) needed to solve a given class of problem. Complexity as logical depth refers mainly to the running time of the shortest program capable of generating a given string or pattern. Similarly to AIC, complexity as logical depth is a measure of a generative process and does not apply directly to an actually existing physical system or dynamical process. Computing logical depth requires knowing the shortest computer program, thus the measure is subject to the same fundamental limitation as AIC.
Effective measure complexity (Grassberger, 1986) quantifies the complexity of a sequence by the amount of information contained in a given part of the sequence that is needed to predict the next symbol. Effective measure complexity can capture structure in sequences that range over multiple scales and it is related to the extensivity of entropy (see below).
Thermodynamic depth (Lloyd and Pagels, 1988) relates the entropy of a system to the number of possible historical paths that led to its observed state, with “deep” systems being all those that are “hard to build”, whose final state carries much information about the history leading up to it. The emphasis on how a system comes to be, its generative history, identifies thermodynamic depth as a complementary measure to logical depth. While thermodynamic depth has the advantage of being empirically calculable, problems with the definition of system states have been noted (Crutchfield and Shalizi, 1999). Although aimed at distinguishing complex systems as different from random ones, its formalism essentially captures the amount of randomness created by a generative process, and does not differentiate regular from random systems.
Complexity as Structure and Information
A simple way of quantifying complexity on a structural basis would be to count the number of components and/or interactions within a system. An examination of the number of structural parts (McShea, 1996) and functional behaviors of organisms across evolution demonstrates that these measures increase over time, a finding that contributes to the ongoing debate over whether complexity grows as a result of natural selection. However, numerosity alone may only be an indicator of complicatedness, but not necessarily of complexity. Large and highly coupled systems may not be more complex than those that are smaller and less coupled. For example, a very large system that is fully connected can be described in a compact manner and may tend to generate uniform behavior, while the description of a smaller but more heterogeneous system may be less compressible and its behavior may be more differentiated.
Effective complexity (Gell-Mann, 1995) measures the minimal description length of a system’s regularities. As such, this measure is related to AIC, but it attempts to distinguish regular features from random or incidental ones and therefore belongs within the family of complexity measures that aim at capturing how much structure a system contains. The separation of regular features from random ones may be difficult for any given empirical system, and it may crucially depend on criteria supplied by an external observer.
Physical complexity (Adami and Cerf, 2000) is related to effective complexity and is designed to estimate the complexity of any sequence of symbols that is about a physical world or environment. As such the measure is particularly useful when applied to biological systems. It is defined as the Kolmogorov complexity (AIC) that is shared between a sequence of symbols (such as a genome) and some description of the environment in which that sequence has meaning (such as an ecological niche). Since the Kolmogorov complexity is not computable, neither is the physical complexity. However, the average physical complexity of an ensemble of sequences (e.g. the set of genomes of an entire population of organisms) can be approximated by the mutual information between the ensembles of sequences (genomes) and their environment (ecology). Experiments conducted in a digital ecology (Adami, 2002) demonstrated that the mutual information between self-replicating genomes and their environment increased along evolutionary time. Physical complexity has also been used to estimate the complexity of biomolecules. The structural and functional complexity of a set of RNA molecules were shown to positively correlate with physical complexity (Carothers et al., 2004), indicating a possible link between functional capacities of evolved molecular structures and the amount of information they encode.
Statistical complexity (Crutchfield and Young, 1989) is a component of a broader theoretic framework known as computational mechanics, and can be calculated directly from empirical data. To calculate the statistical complexity, each point in the time series is mapped to a corresponding symbol according to some partitioning scheme, so that the raw data is now a stream of consecutive symbols. The symbol sequences are then clustered into causal states according to the following rule: two symbol sequences (i.e. histories of the dynamics) are contained in the same causal state if the conditional probability of any future symbol is identical for these two histories. In other words, two symbol sequences are considered to be the same if, on average, they predict the same distribution of future dynamics. Once these causal states have been identified, the transition probabilities between causal states can be extracted from the data, and the long-run probability distribution over all causal states can be calculated. The statistical complexity is then defined as the Shannon entropy of this distribution over causal states. Statistical complexity can be calculated analytically for abstract systems such as the logistic map, cellular automata and many basic Markov processes, and computational methods for constructing appropriate causal states in real systems, while taxing, exist and have been applied in a variety of contexts.
Predictive information (Bialek et al., 2001), while not in itself a complexity measure, can be used to separate systems into different complexity categories based on the principle of the extensivity of entropy. Extensivity manifests itself, for example, in systems composed of increasing numbers of homogeneous independent random variables. The Shannon entropy of such systems will grow linearly with their size. This linear growth of entropy with system size is known as extensivity. However, the constituent elements of a complex system are typically inhomogeneous and interdependent, so that as the number of random variables grows the entropy does not always grow linearly. The manner in which a given system departs from extensivity can be used to characterize its complexity.
Neural complexity (Tononi et al., 1994), which may be applied to any empirically observed system including brains, is related to the extensivity of a system. One of its building blocks is integration (also called multi-information), a multivariate extension of mutual information that estimates the total amount of statistical structure within an arbitrarily large system. Integration is computed as the difference between the sum of the component’s individual entropies and the joint entropy of the system as a whole. The distribution of integration across multiple spatial scales captures the complexity of the system. Consider the following three cases (Figure 2). (1) A system with components that are statistically independent will exhibit globally disordered or random dynamics. Its joint entropy will be exactly equal to the sum of the component entropies, and system integration will be zero, regardless of which spatial scale of the system is examined. (2) Any statistical dependence between components will lead to a contraction of the system’s joint entropy relative to the individual entropies, resulting in positive integration. If the components of a system are highly coupled and exhibit statistical dependencies as well as homogeneous dynamics (i.e. all components behave identically) then estimates of integration across multiple spatial scales of the system will, on average, follow a linear distribution. (3) If statistical dependencies are inhomogeneous (for example involving groupings of components, modules, or hierarchical patterns) the distribution of integration will deviate from linearity. The total amount of deviation is the system’s complexity. The complexity of a random system is zero, while the complexity of a homogeneous coupled (regular) system is very low. Systems with rich structure and dynamic behavior have high complexity.
Applied to neural dynamics, complexity as defined by Tononi et al. (1994) was found to be associated with specific patterns of neural connectivity, for example those exhibiting attributes of small-world networks (Sporns et al., 2000). Descendants of this complexity measure have addressed the matching of systems to an environment as well as their degeneracy, i.e. their capacity to produce functionally equivalent behaviors in different ways.
Traditionally, complex systems have been analyzed using tools from nonlinear dynamics and statistical information theory. Recently, the analytical framework of complex networks has led to a significant reappraisal of commonalities and differences between complex systems found in different scientific domains (Amaral and Ottino, 2004). A key insight is that network topology, the graph structure of the interactions, places important constraints on the system's dynamics, by directing information flow, creating patterns of coherence between components, and by shaping the emergence of macroscopic system states. Complexity is highly sensitive to changes in network topology (Sporns et al., 2000). Changes in connection patterns or strengths may thus serve as modulators of complexity. The link between network structure and dynamics represents one of the most promising areas of complexity research in the near future.
Why does complexity exist in the first place, especially among biological systems? A definitive answer to this question remains elusive. One perspective is based on the evolutionary demands biological systems face. The evolutionary success of biological structures and organisms depends on their ability to capture information about the environment, be it molecular or ecological. Biological complexity may then emerge as a result of evolutionary pressure on the effective encoding of structured relationships which support differential survival.
Another clue may be found in the emerging link between complexity and network structure. Complexity appears very prominently in systems that combine segregated and heterogeneous components with large-scale integration. Such systems become more complex as they more efficiently integrate more information, that is, as they become more capable to accommodate both the existence of specialized components that generate information and the existence of structured interactions that bind these components into a coherent whole. Thus reconciling parts and wholes, complexity may be a necessary manifestation of a fundamental dialectic in nature.
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- Author's homepage
- Jim Crutchfield's homepage
- Seth Lloyd's homepage
- Chris Adami's homepage
- Dan McShea's homepage
- Giulio Tononi's homepage
- A movie showing connectivity and dynamics for the three cases in Fig.2