Complexity - 2020 - Siegenfeld - An Introduction to Complex Systems Science and Its Applications.pdf

Hindawi Complexity Volume 2020, Article ID 6105872, 16 pages https://doi.org/10.1155/2020/6105872 Review Article An Introduction to Complex Systems Science and Its Applications 1,2 Alexander F. Siegenfeld and Yaneer Bar-...

Hindawi Complexity Volume 2020, Article ID 6105872, 16 pages https://doi.org/10.1155/2020/6105872 Review Article An Introduction to Complex Systems Science and Its Applications 1,2 Alexander F. Siegenfeld and Yaneer Bar-Yam2 1 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA 2 New England Complex Systems Institute, Cambridge, MA, USA Correspondence should be addressed to Alexander F. Siegenfeld; [email protected] Received 11 February 2020; Accepted 22 May 2020; Published 27 July 2020 Academic Editor: Carlos Gershenson Copyright © 2020 Alexander F. Siegenfeld and Yaneer Bar-Yam. )is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. )e standard assumptions that underlie many conceptual and quantitative frameworks do not hold for many complex physical, biological, and social systems. Complex systems science clarifies when and why such assumptions fail and provides alternative frameworks for understanding the properties of complex systems. )is review introduces some of the basic principles of complex systems science, including complexity profiles, the tradeoff between efficiency and adaptability, the necessity of matching the complexity of systems to that of their environments, multiscale analysis, and evolutionary processes. Our focus is on the general properties of systems as opposed to the modeling of specific dynamics; rather than provide a comprehensive review, we pedagogically describe a conceptual and analytic approach for understanding and interacting with the complex systems of our world. )is paper assumes only a high school mathematical and scientific background so that it may be accessible to academics in all fields, decision-makers in industry, government, and philanthropy, and anyone who is interested in systems and society. 1. Introduction complexity of systems to that of their environments. Section 3 considers the analysis of complex systems, attending to the How can we scientifically approach the study of complex oft-neglected question of when standard assumptions do systems—physical, biological, and social? Empirical studies, and—more importantly—do not apply. Section 4 discusses while useful, are by themselves insufficient, since all ex- principles for effectively intervening in complex systems periments require a theoretical framework in which they can given that their full descriptions are often beyond the limits be interpreted. While many such frameworks exist for of human comprehension. Section 5 provides further understanding particular components or aspects of systems, reading. Section 6 concludes the work. the standard assumptions that underlie most quantitative studies often do not hold for systems as a whole, resulting in a mischaracterization of the causes and consequences of 2. Basic Principles of Complex Systems Science large-scale behavior. )is paper provides an introduction to complex systems 2.1. Why Complex Systems Science? Complex systems science, demonstrating a few of its applications and its science considers systems with many components. )ese capacity to help us make more effective decisions in the systems could be physical, biological, or social. Given this complex systems of our world. It focuses on some general diversity of systems, it may seem strange to study them all properties of complex systems, rather than on the modeling under one framework. But while most scientific disciplines of specific dynamics as in the subfields of dynamical systems, tend to focus on the components themselves, complex sys- agent-based modeling and cellular automata, network sci- tems science focuses on how the components within a system ence, and chaos theory. Section 2 introduces key concepts, are related to one another. For instance, while most ac- including complexity profiles, the tradeoff between efficiency ademic disciplines would group the systems in Figure 1 by and adaptability, and the necessity of matching the column, complex systems science groups them by row.  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 2 Complexity Examples of Behaviors Gas Pond Crowds life Random Cannonball Infections Armies Coherent Snowflake Humans Corporations Correlated Physical Biological Social Figure 1: Each column contains three examples of systems consisting of the same components (from left to right: molecules, cells, and people) but with different relations between them. Each row contains systems representing a certain kind of relationship between components. For random systems, the behavior of each component is independent from the behavior of all other components. For coherent systems, all components exhibit the same behavior; for example, the behavior (location, orientation, and velocity) of one part of the cannonball completely determines the behavior of the other parts. Correlated systems lie between these two extremes, such that the behaviors of the system’s components do depend on one another, but not so strongly that every component acts in the same way; for example, the shape of one part of a snowflake is correlated with but does not completely determine the shape of the other parts. Implicit in these descriptions is the necessity of specifying the set of behaviors under consideration, as discussed in Section 2.2. (Image source: ). Systems may differ from each other not because of system components [3–7]. Other examples of self-organized differences in their parts but because of differences in how behaviors include the spontaneous formation of conversa- these parts depend on and affect one another. For example, tion groups at a party, the allocation of goods in a decen- steam and ice are composed of identical water molecules but, tralized economy, the evolution of ecosystems, and the due to differences in the interactions between the molecules, flocking of birds. Such large-scale behaviors and patterns have very different properties. Conversely, all gases share cannot be determined by examining each system part in many behaviors in common despite differences in their isolation. By instead considering general properties of sys- constituent molecules. )e same holds for solids and liquids. tems as wholes, complex systems science provides an in- )e behaviors that distinguish solids from liquids from gases terdisciplinary scientific framework that allows for the are examples of emergence: they cannot be determined from discovery of new ideas, applications, and connections. a system’s parts individually. Fluid turbulence, as one might A full description of all the small-scale details of even observe in a flowing river, is an example of how the rela- relatively simple systems is impossible; therefore, sound tionships between parts can give rise to emergent large-scale analyses must describe only those properties of systems that behaviors and patterns that are self-organized, meaning that do not depend on all these details. )at such properties exist they arise not from some external or centralized control but is due to universality, a phenomenon that will be discussed in rather autonomously from the interactions between the Section 3. Statistical physics provides an underlying insight  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Complexity 3 that allows for the discovery of such properties: namely, that system, we think of the behaviors arising from the ordered while attempting to characterize the behavior of a particular arrangement of molecules in a human, not the behaviors state of a system (e.g., a gas) may be entirely intractable, arising from the maximally disordered arrangement of characterizing the set of all possible states of the system may molecules in a gas. It therefore may be tempting to conclude not only be tractable but may also provide us with a model of that complex systems are those with reduced disorder. But the relevant information (e.g., the pressure, temperature, the systems with the least disorder are those in which all density, and compressibility). In other words, taking a step components exhibit the same behavior (coherent systems in back and considering the space of possible behaviors provides Figure 1), and such behavior is easy to describe and thus not a powerful analytical lens that can be applied not only to intuitively complex. physical systems but also to biological and social ones. To resolve this apparent paradox, we must consider that the length of a system’s description depends on the level of detail used to describe it. )us, complexity depends on scale. 2.2. What Is Complexity? We define the complexity of a On a microscopic scale, it really is more difficult to describe behavior as equal to the length of its description. )e length the positions and velocities of all the molecules of the gas of a description of a particular system’s behavior depends than it is to do the same for all the molecules of the human. on the number of possible behaviors that system could But at the scale of human perception, the behaviors of a gas exhibit. For example, a light bulb that has two possible are determined by its temperature and pressure, while the states—either on or off—can be described by a single bit: 0 behaviors of a human remain quite complex. Entropy or 1. Two bits can describe four different behaviors (00, 01, corresponds to the amount of complexity at the smallest 10, or 11), three bits can describe eight behaviors, and so scale, but characterizing a system requires understanding its on. Mathematically, we can write C log2 N, where C is complexity across multiple scales. A system’s complexity the complexity of a system and N is its number of possible profile is a plot of the system’s complexity as a function of behaviors (technically, log2 N is actually an upper bound scale. In the examples below, scale will be taken to be for the system’s complexity since if some behaviors are length, but fundamentally, the scale of a behavior is equal to more likely than others, the average length of the system’s the number of coordinated components involved in the description can be reduced by using shorter descriptions behavior, for which physical length is a proxy. A gas is very for the more common behaviors and longer descriptions simple at the scale of human perception because at this scale, for the less common ones—lossless compression algo- only behaviors involving trillions of molecules are relevant, rithms rely on this logic), but for our purposes here, it is and there are relatively few distinguishable behaviors of a gas sufficient to state that the greater the number of possible involving so many molecules. behaviors, the greater the complexity. As shown in Figure 2, random, coherent, and correlated It is important to note that one must carefully define the systems (see Figure 1) have qualitatively different complexity space of possible behaviors. For instance, if we are interested profiles. Random systems have the most complexity at the in a light bulb already in a socket, the light bulb has two smallest scale (finest granularity/most detail), but the possible behaviors, as above, but if we are instead interested amount of complexity rapidly drops off as the scale is in- in the complexity of building a light bulb, the space of creased and the random behaviors of the individual com- possible behaviors might include all of the ways in which its ponents are averaged out. A coherent system has the same parts could be arranged. As another example, consider amount of complexity at small scales as it does at larger programming a computer to correctly answer a multiple- scales because describing the overall behavior of the system choice question with four choices. At first glance, this task is (e.g., the position and velocity of a cannonball) also describes very simple: since there are four possible behaviors, only two the behavior of all the components (e.g., the positions and bits are required. Nonetheless, we have the sense that velocities of all the atoms). Note that complexity tends to programming a computer to score perfectly on a multiple- increase (or remain the same) as the scale decreases, since choice test would be quite difficult. )is apparent paradox is looking at a system in more detail (while still including the resolved, however, when we recognize that such a task is whole system in the description) tends to yield more in- difficult only because we do not a priori know what questions formation. For a correlated system, various behaviors occur will be on the test, and thus, the true task is to be able to at various scales, and so the complexity gradually increases correctly answer any multiple-choice question. )is task is as one examines the system in greater and greater detail. For quite complex, given the large number of possible ways the instance, from very far away, a human, being barely visible, program could respond to a string of arbitrary multiple- has very little complexity. As the level of detail is gradually choice questions. increased, the description will first include the overall po- sition and velocity of the human and then the positions and 2.3. Complexity and Scale. Consider a human, and then velocities of each limb, followed by the movement of hands, consider a gas containing the very same molecules that are in fingers, facial expressions, as well as words that the human the human but in no particular arrangement. Which system may be saying. Continuing to greater levels of detail, the is more complex? )e gas possesses a greater number of organs and then tissues and patterns within the human brain possible arrangements of the molecules (i.e., has more en- become relevant, and eventually so do the individual cells. At tropy, or disorder) and thus would take longer to describe at scales smaller than that of a cell, complexity further increases a microscopic level. However, when we think of a complex as one sees organelles (cellular substructures), followed by  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 4 Complexity Random Many types of goods but only a few of each type Complexity Complexity Correlated Coherent Many copies of a few types of goods Scale Scale Figure 2: Representative complexity profiles for random, coherent, Figure 3: )e complexity profile of a factory that can produce a large and correlated systems (see Figure 1). Any given system may have number of copies of a few types of goods, and the complexity profile aspects of each at various scales. of a factory that can produce many types of goods but not in large numbers. )e number of copies of a good produced is a proxy for large molecules such as proteins and DNA, and then scale since, given a fixed technology, mass production requires eventually smaller molecules and individual atoms. At each larger-scale coordinated action in the factory (e.g., an assembly line). )e number of different types of goods that can be produced at a level, the length of the description grows longer. )is in- given scale is a proxy for the number of different possible behaviors credible multiscale structure with gradually increasing of the factory—and thus its complexity—at that scale. complexity is a defining characteristic of complex systems. 2.4. Tradeoffs between Complexity and Scale. )e intuition many different types of products, but none at scale. Of that complex systems require order is not unfounded: for course, a factory may be able to increase both the complexity there to be complexity at larger scales, there must be be- and scale of its production by adding new machinery or haviors involving the coordination of many smaller-scale more workers; the precise tradeoff between complexity and components. )is coordination suppresses complexity at scale applies only when considering a fixed set of compo- smaller scales because the behaviors of the smaller-scale nents with a fixed set of individual behaviors. A subtle point components are now limited by the interdependencies be- to be made here is that introducing interactions between two tween them. )e tension between small-scale and large-scale parts of a system may in some cases increase the set of complexity can be made precise: given a fixed set of com- relevant individual behaviors of each part, thereby in- ponents with a fixed set of potential individual behaviors, the creasing the total area under the complexity profile. For area under the complexity profile will be constant, regardless example, if two people enter into communication with each of the interdependencies (or lack thereof ) between the other, the communication itself (e.g., speech) may now be a components. More precisely, the sum of a system’s com- relevant behavior of each individual person that was not plexity at each scale (i.e., the area under its complexity there before. profile) will equal the sum of each individual component’s A corollary of the tradeoff between complexity and scale complexity. )us, for any system, there is a fundamental is the tradeoff between adaptability and efficiency [10–15]. tradeoff between the number of behaviors a system can have Adaptability arises when there are many possible actions and the scale of those behaviors. happening in parallel that are mostly independent from one For instance, consider a factory consisting of many another, i.e., when the system has high complexity. Effi- workers. )e output of the factory can be characterized ciency, on the other hand, arises when many parts of a using a complexity profile (Figure 3). )e number of dif- system are all working in concert, so that the system can ferent types of goods that the factory can produce at a given perform the task for which it was designed at the largest scale is a proxy for the factory’s complexity at that scale, with possible scale. Due to the tradeoff between complexity and the number of copies of the same type of good that the scale, a system with more adaptability will have a complexity factory can produce in a given amount of time being a proxy profile with greater complexity but predominantly at smaller for scale. )e fundamental tradeoff is evident in the fact that scales, while a system with more efficiency will have a if the factory wants to be able to churn out many copies of a complexity profile with lower complexity but extending to single type of good in a short amount of time, it will have to larger scales. )us, a very efficient system will, due to its coordinate all of its workers (perhaps having them work on necessarily lower complexity, not be as adaptable to un- an assembly line), thereby reducing their individual freedom foreseen variations within itself or its environment, while a to make many different kinds of goods. )e factory’s pro- very adaptable system, designed to handle all sorts of shocks, duction would then have low complexity but at a large scale will necessarily have to sacrifice some larger-scale behaviors. (e.g., churning out many identical Model-T Fords—“Any )e Soviets thought they could have their cake and eat it, too: customer can have a car painted any color that he wants so they originally believed that their economy would outper- long as it is black”). On the other hand, if the factory’s form capitalist ones because capitalist economies have so employees work independently, they will be able to create much waste related to multiple businesses competing to do  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Complexity 5 the same thing (Chapter 16 in ). It would be far more As another example, healthcare involves both small-scale efficient to coordinate all economic production. But in tasks with high overall complexity such as case management creating such large-scale economic structures, lower-scale and large-scale, lower-complexity tasks such as manufacturing complexity was sacrificed, resulting in a nonadaptive system. and delivering vaccines. Vaccinations are lower com- Improperly regulated capitalist systems may also sacrifice plexity but larger scale because essentially the same actions are redundancy and adaptability for efficiency, resulting in, for performed for nearly every patient. Large-scale top-down instance, excessive concentrations of market power, harmful organizations and initiatives are suited for large-scale, lower- feedback loops, and herd-like behaviors [17–21]. complexity tasks, but tasks like case management require Due to the tradeoff between complexity and scale, any health systems with a high degree of small-scale (i.e., local) mechanism that creates larger-scale complexity—whether complexity. market or government or otherwise—will necessarily reduce )e eurozone provides a potential illustration of a individual complexity. )is is not to say that larger-scale multiscale complexity mismatch. Fiscal policy is made complexity is always harmful; it is often worth trading some predominantly at the scale of individual countries and thus individual-level freedoms for larger-scale cooperation. has a higher complexity at the country scale but relatively When, then, is complexity at a particular scale desirable? little complexity at the scale of the entire eurozone, while monetary policy is made at the scale of the entire eurozone and thus has some complexity at the scale of the eurozone 2.5. Why Be Complex? A determination of when complexity but lacks the ability to vary (i.e., lacks complexity) at the scale is desirable is provided by the Law of Requisite Variety : of individual countries. Many have argued that economic to be effective, a system must be at least as complex as the difficulties within the eurozone have arisen because this environmental behaviors to which it must differentially mismatch has precluded effective interactions between fiscal react. If a system must be able to provide a different response and monetary policy [25–29]. to each of 100 environmental possibilities and the system has Problems arise not from too much or too little com- only 10 possible actions, the system will not be effective. At plexity (at any scale) per se but rather from mismatches the very least, the system would need 100 possible actions, between the complexities of a task to be performed and the one for each scenario it could encounter. ()e above con- complexities of the system performing that task. (Inciden- dition is necessary but of course not sufficient; a system with tally, human emotions appear to reflect this principle: we are sufficiently many actions may still not take the right actions bored when our environment is too simple and over- in the right circumstances.) Note that the environment to whelmed when it is too complex.) Note that the system which a system must react is itself also a system and will in one scenario may be the task/environment in another; for sometimes be referred to as such. instance, the same complexity that helps a system interact Since complexity is defined only with respect to a par- with its environment may prevent its effective management ticular scale, we can refine the Law of Requisite Variety: to be by other systems. In none of the above examples have the effective, a system must match (or exceed) the complexity of complexity profiles been precisely calculated, nor have scales the environmental behaviors to which it must differentially been precisely defined. Instead, proxies for scale are used and react at all scales for which these behaviors occur. To estimated comparisons of complexity made. Such an ap- illustrate this multiscale version of the Law of Requisite proach cannot yield precise results (indeed, no approach Variety, we consider military conflict (see Figure 4). can, given the complexity a full description of such systems Here, one military can be considered as the system, while the would require), but additional precision is not needed when other military is part of the environment with which the even the approximate analysis reveals large mismatches in system must interact. For two militaries of equal complexity, complexity. (To remedy the diagnosed mismatches, more i.e., with the same number of behaviors, but with one detailed analyses may be required.) While it may be military operating at a larger scale (e.g., two very tightly tempting to attribute the problems arising from a complexity controlled armies, but with one army larger than the other), mismatch to particular proximate causes and chains of the larger-scale military will likely win. For two militaries of events, problems of one form or another will be inevitable equal scale but unequal complexity (e.g., two equally sized unless the underlying mismatch is addressed. and equally powered fleets, but with one being more ma- neuverable than the other), the higher-complexity military will likely win, since the high-complexity military has an 2.6. Subdivided Systems. Even if the complexity of the action for every action of the lower-complexity military but system matches that of its environment at the appropriate not vice versa. When a military with high complexity at a scales, there is still the possibility of a complexity mismatch. smaller scale (e.g., a guerrilla force) conflicts with a military Consider two pairs of friends—four people total, each of with larger-scale behavior but lower complexity (e.g., the US whom can lift 100 pounds—and consider two 200-pound army in Vietnam or the Soviet army in Afghanistan), the couches that need to be moved. Furthermore, assume that terrain, which constrains the scale of the conflict, plays an each person is able to coordinate with her friend but not with important role. In an open field, or in open waters, the either of the other two people. Overall then, the system of military that has more complexity at the larger scales is people has sufficient complexity at the appropriate scales to favored, while in the jungle or in the mountains, higher move both couches since each pair of friends can lift one of complexity at smaller scales is favored. the 200-pound couches. However, were one person from  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 6 Complexity Higher complexity Complexity Complexity Larger scale Scale Scale (a) (b) Higher complexity at smaller scales Complexity Higher complexity at larger scales Scale (c) Figure 4: Schematic complexity profiles of militaries in conflict. (a) If two armies are operating with the same number of possible behaviors but at different scales, the larger-scale one is favored. (b) If two armies are operating at the same scale but with different numbers of possible behaviors, the higher-complexity one is favored. (c) If two armies are operating at different scales and with different numbers of possible behaviors, which one is favored depends on the terrain (see text). Note that these profiles are simplified to highlight the key concepts; actual militaries operate at multiple scales. More generally, (a) and (b) depict conflicts in which one army has at least as much complexity as the other at every scale. each pair of friends to be assigned to each couch, they would with each subdivision of problems requiring particular types not be able to lift the couches because the two people lifting of coordinated knowledge and effort in order to be solved. each couch would not belong to the same pair of friends and Academia’s complexity across multiple scales allows it to thus would not be able to coordinate their actions. )e effectively work on many of these problems. However, there problem here is that while the pairs of friends possess may exist problems that academia, despite having sufficient enough overall complexity at the right scales to lift the overall multiscale complexity, is nonetheless unable to solve couches, the subdivision within the system of friends is not because the subdivisions within the problem do not match matched to the natural subdivision within the system of the subdivisions within academia. )e increase in inter- couches. )e mismatch in complexity can be seen if we focus disciplinary centers and initiatives over the past few decades our attention on just a single couch: while the couch requires suggests the perception of such a mismatch; however, the coordinated action at the scale of 200 pounds, the two people structure of the academic system as a whole may still hinder lifting it are capable only of two independent actions, each at progress on problems that do not fall neatly within a dis- the scale of 100 pounds. cipline or subdiscipline [31–36]. )e way in which academic departments are organized )e above examples provide an illustration of the provides a more realistic example of the potential of sub- principle that in order for a system to differentially react to a division mismatch. Academia has multiple levels of subdi- certain set of behaviors in its environment, not only must the vision (departments, subfields, etc.) in order to organize system as a whole have at least as much complexity at all knowledge and coordinate people, resulting in a high overall scales as this set of environmental behaviors (as described in degree of complexity across multiple scales, where scale Section 2.5.), but also each subset of the system must have at could refer to either the number of coordinated people or the least as much complexity at all scales as the environmental amount of coordinated knowledge, depending on which behaviors corresponding to that subset. A good rule of aspect of the academic system is under consideration. thumb for applying this principle is that decisions con- Similarly, there are multiple levels of natural subdivision in cerning independent parts or aspects of a system should be the set of problems that academia can potentially address, able to be made independently, while decisions concerning  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Complexity 7 dependent parts of the system should be made dependently. It follows that the organizations that make such decisions should be subdivided accordingly, so that their subdivisions Lower levels given some autonomy match the natural divisions in the systems with which they interact. )e subdivisions present in the human brain and Complexity the analysis of subdivisions in neural networks more gen- erally (Chapters 2.4-2.5 in ) demonstrate how systems that are subdivided so as to match the natural subdivisions in Tightly controlled their environments outperform those with more internal connectivity. Scale 2.7. Hierarchies. A common way in which systems are or- Figure 5: Complexity profiles of two hierarchies, each with the ganized is through hierarchies. In an idealized hierarchy, same number of people. Here, the scale is the number of coor- there are no lateral connections: any decision that involves dinated man-hours. In one hierarchy, all decisions, regardless of the multiple components of the hierarchy must pass through a scale, are made by a single person, while in the other, different common node under whose control these components all decisions are made at various levels of the hierarchy. (directly or indirectly) lie. )e complexity profile of such a hierarchy depends on the rigidity of the control structure (Figure 5). At one extreme, every decision, no matter how likewise, perhaps some of the powers that some argue should large or small, is made by those at the top of the hierarchy. be devolved from the federal to the state level should in fact )is hierarchy has the same amount of complexity across all be devolved to the local level). its scales: namely, the complexity of whatever decisions are It is important to distinguish between the complexity of a being made at the top. At the other extreme, there is no hierarchy and the complexity of the decisions that the people communication within the hierarchy, and every individual within the hierarchy are capable of making. For instance, acts independently. )is hierarchy has very little complexity one could design a tightly controlled hierarchy that could beyond the individual level. Between these two extremes is a take a large number of large-scale actions (i.e., high com- typical hierarchy, in which different decisions are made at plexity at its largest scale), but since the decision-making different levels. abilities of even the most capable humans are of finite No type of hierarchy is inherently better than any other. complexity, the individuals at the top may be fundamentally For a particular environment, the best hierarchy is one for unable to correctly choose from among these actions. )is which the complexity profile matches that of the tasks brings us to an important limitation of hierarchies: the needed to be performed. A tightly controlled (top-heavy) complexity of the decisions concerning the largest-scale hierarchy is not well suited to environments in which there is behaviors of a hierarchy—the behaviors involving the entire a lot of variation in the systems with which the lower levels of organization—is limited by the complexity of the group of the hierarchy must interact; neither is a very loosely con- people at the top. )us, a hierarchy will necessarily fail trolled hierarchy well suited to environments that require when the complexity of matching its largest-scale behaviors large-scale coordinated action. For example, centralizing too to those of its environment is higher than the complexity of much power within the US governance system at the federal decision-making that is achievable by any individual or (as opposed to the local or state) level would not allow for committee. (Note that the complexity of deciding in which sufficient smaller-scale complexity to match the variation behaviors of a system should correspond to which behaviors among locales; too decentralized a governance system would of its environment is generally much greater than the not allow for sufficient larger-scale complexity to engage complexity of either the system or the environment alone: with problems that require nationally coordinated re- for example, if both the system and environment have 10 sponses. Assigning decisions to higher levels in hierarchies possible behaviors, the system has enough complexity to allows for more efficiency and scale but less adaptability and match the environment, but properly deciding which be- variation. haviors of the system should correspond to which envi- We should also consider not just the overall complexity ronmental conditions requires correctly choosing one profile of governance systems but how well the subdivisions option out of a space of 10 factorial or 3,628,800 possibil- in governance systems match those within their territories ities.) )e failure of command economies provides a stark (Section 2.6.). Metropolitan areas are in some ways more example: the allocation of resources and labor is too complex similar to one another than they are to the rural areas of their a problem for any one person or group of people to un- respective states. So while dividing the US into 50 states derstand. Markets allocate resources via a more networked provides substantial lower-scale governmental complexity, system: decisions regarding how to allocate resources are this complexity is not necessarily well matched to natural made without any individual making them, just as decisions urban-rural divides. To the extent that such a mismatch are made in the human brain without any neuron making exists, there may be issues currently handled at the state level them. (Whether or not these market allocations are desirable that would be better handled at the local level, thereby depends in part on the way in which the market is structured allowing for different policies in urban and rural areas (and and regulated.)  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 8 Complexity We began by considering idealized hierarchies with only 3.2. When Mean-Field 7eory Breaks Down. )e systems for vertical connections, but lateral connections provide another which mean-field theory applies exhibit large-scale be- mechanism for enabling larger-scale behaviors. For instance, haviors that are the average of the behaviors of their cities can interact with one another (rather than interacting components. )ey must possess a separation of scales, only with their state and national governments) in order to which arises when the statistical fluctuations of their copy good policies and learn from each other’s mistakes. components are sufficiently independent from one an- )rough these sorts of evolutionary processes (described other above a certain scale. Mean-field theory may hold further in Section 4), large-scale decisions (large-scale because even in the presence of strong interactions, so long as the policies may be copied by multiple cities) that are more effect of those strong interactions can be captured by the complex than any individual component can be made. Such average behavior of the system—that is, so long as each lateral connections can exist within a hierarchical framework in component of the system can be modeled as if it were which the top of the hierarchy (in this example, the national interacting with the average (i.e., mean field) of the sys- government) maintains significant control, or they can exist tem. For example, the large-scale motion of solids is well outside of a strictly hierarchical structure, as in the human described by mean-field theory, even though the mole- brain. Furthermore, these lateral connections can vary in cules in a solid interact with one another quite strongly, strength. Overly strong connections lead to herd-like behaviors because the main effect of these interactions is to keep with insufficient smaller-scale variation, such as groupthink each molecule at a certain distance and orientation from [37–39] (no system is exempt from the tradeoff described in the average location (center of mass) of the solid. Like- Section 2.4.), while overly weak connections result in mostly wise, under some (but certainly not all) conditions, independent behavior with little coordination. economic markets can be effectively described by mod- eling each market actor as interacting with the aggregate 3. Analyzing Complex Systems forces of supply and demand rather than with other in- dividual market actors. )e previous section has examined some of the general However, when there are sufficiently strong correla- properties of systems with many components. But how do tions between the components of the system, i.e., when we study particular systems? How do we analyze data from the interactions between a component of the system and a complex systems, and how do we choose which data to specific set of other components (as opposed to its general analyze? interaction with the rest of the system) cannot be neglected, mean-field theory will break down. )ese systems will instead exhibit large-scale behaviors that 3.1. How Do We Understand Any System? In a sense, it is arise not solely from the properties of individual com- surprising that we can understand any macroscopic system ponents but also from the relationships between com- at all, as even a very simple mechanical system has trillions ponents. For example, while the behavior of a muscle can upon trillions of molecules. We are able to understand such be roughly understood from the behavior of an individual systems because they possess a separation of scales , muscle cell, the behavior of the human brain is funda- meaning that the macroscopic behavior we are interested in mentally different from that of individual neurons, be- occurs at a far larger scale than the behavior of the individual cause cognitive behaviors are determined largely by molecules, with not much behavior occurring in between variations in the synapses between neurons. Similarly, the these two scales (see Figure 6). )is separation allows us to complex ecological behaviors of a forest cannot be de- treat the macroscopic and microscopic behaviors separately: termined by the behaviors of its constituent organisms in for mechanical systems, we treat the macroscopic behavior isolation. explicitly with Newtonian mechanics, while the microscopic Because their small-scale random occurrences are not behavior is considered in aggregate using thermodynamics. statistically independent, complex systems often exhibit More generally, the approach described above is an large-scale fluctuations not predicted by mean-field theory, example of a mean-field theory , in which the average such as forest fires, viral content on social media, and crashes behaviors of a system’s components are explicitly modeled in economic markets. Sometimes, these large-scale fluctu- and the deviations of the individual components from this ations are adaptive: they enable a system to collectively average are treated as statistically independent random respond to small inputs. For instance, humans respond fluctuations. )is approach works very well for systems such strongly to minor disturbances in the density of air, such as as computers, cars, airplanes, and buildings, in which the the sound of their own names. However, some large-scale motions of individual molecules are—apart from some fluctuations pose systemic risks. mostly uncorrelated fluctuations—well described by the motion of the piece of material to which they belong. Mean- field assumptions are also often employed in analyses of 3.3. Fat-Tailed Distributions and Systemic Risk. When the biological, social, and economic systems; these assumptions components of a system are independent from one another work well in many cases, but, as we will see, they are not above a certain scale, then at much larger scales, the mag- always appropriate for complex systems. It is important, nitudes of the fluctuations of the system follow a normal therefore, to determine under what conditions mean-field distribution (bell curve), for which the mean and standard theory holds. deviation are well defined and for which events many  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Complexity 9 1.2 1.0 Probability density in-tailed Complexity 0.8 0.6 0.4 0.2 Fat-tailed s0 Scale 0 2 4 6 8 Figure 6: A complexity profile of a system with a separation of Scale scales. A separation of scales implies that the behaviors occurring Figure 7: A normal distribution (thin-tailed) and a distribution below a certain scale (s0 in the above figure) are at larger scales with a power-law decay (fat-tailed). )e fat-tailed distribution may mostly independent from one another, and that therefore, at these appear more stable, due to the lower probability of small-scale larger scales, only the average effects of the small-scale behaviors fluctuations and the fact that samples from the distribution may not are relevant. contain any extreme events. However, sooner or later, a fat-tailed distribution will produce an extreme event, while one could wait standard deviations above the mean are astronomically im- thousands of lifetimes of the universe before a normal distribution probable. Interdependencies, however, can lead to a distri- produces a similarly extreme event. Note that the axes of this graph are truncated; the illustrated fat-tailed distribution can, with small bution of fluctuations in which the probability of an extreme but nonnegligible probability (0.04%), produce events with a scale event, while still small, is not astronomically so. Such dis- of one million or more. tributions are characterized as fat-tailed—see Figure 7. For example, while human height follows a thin-tailed distribu- tion, with no record of anyone over twice as tall as the average 3.4. Understanding Complex Systems. Because it is usually human, human wealth—due to the complex economic in- easier to collect data regarding components of a system teractions between individuals—follows a fat-tailed distri- than it is to collect data regarding interactions between bution, with multiple individuals deviating from the average components, studies often fail to capture the information by factors of more than one million. relevant to complex systems, since complex large-scale One danger of interdependencies is that they may make behaviors critically depend on such interactions. Fur- systems appear more stable in the short term by reducing the thermore, as discussed in Section 3.3., data analysis can extent of small-scale fluctuations, while actually increasing the severely underestimate the probability of extreme events probability of catastrophic failure [44–47]. )is danger is (tail risk). Finally, analyses often (implicitly) assume lin- compounded by the fact that when underlying probability earity, i.e., they assume that the total impact of a set of distributions have fat tails (a situation made more likely by factors is equal to the sum of the impacts of each individual interdependencies), standard statistical methods often break factor, an assumption that often breaks down for complex down, leading to potentially severe underestimates of the systems, which may possess feedback loops, abrupt tran- probabilities of extreme events. As a thought experiment, sitions (tipping points), and other highly nonlinear be- imagine 100 ladders, each with a 1/10 probability of falling. If haviors [57–64]. the ladders are independent from one another, the probability How can we understand the systems for which these that all of them fall is astronomically low (literally so: there is standard approaches do not apply? Our understanding of about a 1020 times higher chance of randomly selecting a all systems with many components depends on universality particular atom out of all of the atoms in the known universe). , i.e., the existence of large-scale behaviors that do not If we tie all the ladders together, we will have made them safer, depend on the microscopic details. )e standard ap- in the sense that the probability of any individual ladder proaches are predicated on the assumption of sufficient falling will be much smaller, but we will have also created a independence between components, which allows large- nonnegligible chance that all of the ladders might fall down scale behaviors to be determined without a full accounting together. Other examples include the interconnectedness of of the system’s details via mean-field theory and/or normal our financial systems resulting in the possibility of global distributions. But mean-field theory is just one example of market crashes [49–54] and the interconnectedness of travel universality. routes increasing the probability of pandemics such as the Sound is another example: all materials, regardless of their Spanish flu and COVID-19 [55, 56]. When such crises do composition, allow for the propagation of sound waves. Sound occur, they are often attributed to proximate causes or chains behaves so similarly in all materials because at the length scales of events, and measures are then implemented to ensure that relevant to sound waves, which are far larger than the sizes of those particular chains of events will not occur again. But individual atoms and molecules, the effect of the microscopic unless the underlying systemic instabilities are addressed, parameters is merely to set the speed of the sound. Note that another crisis is bound to happen sooner or later, even if its sound waves cannot be understood as a property of the average precise form cannot be predicted. behavior—in this case, average density—of a material, since it is  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 10 Complexity precisely the systematic correlations in the deviations from that elements that were not considered ahead of time. It should average that give rise to sound. Nor is sound best understood also be noted that in a functional system with a high degree by focusing on the small-scale details of atomic motion: sci- of complexity, the potential positive impact of a change is entists understood sound even before they learned what atoms generally much smaller than its potential negative impact. are. )e key to understanding sound waves is to recognize that For example, a small change to the wiring in a computer is they have a multiscale structure—with larger-scale fluctuations unlikely to dramatically improve the computer’s perfor- corresponding to lower frequencies and smaller-scale fluctu- mance, but it could cause the computer to crash. Airplanes ations corresponding to higher frequencies—and to model are another example. )is phenomenon is a consequence of them accordingly. the fact that, by definition, a high degree of complexity Lim et al. apply this approach to studying ethnic violence implies that there are many system configurations that will. )ey built a predictive model to analyze where ethnic not work for every one configuration that will. violence has the potential to occur and applied their model Given the absence of perfect knowledge, how can the to India and to what was Yugoslavia. Ethnic violence has success of systems we design or are part of be assured? many causes, but rather than focusing on specific, culturally While the success of many systems rests on the assumption dependent mechanisms or on the average properties of that good decisions will be made, some systems do not regions, such as demographic or economic statistics, the depend on individual understanding and can perform well authors instead considered the multiscale patterns in how in spite of the fallibility of decision-makers (whether due to ethnic groups were geographically distributed (Figure 8). corruption, subconscious bias, or the fundamental limi- )ey found that ethnic violence did not occur when the tations of human minds). )e study of complex systems ethnic groups were either well mixed or well separated but approaches this observation scientifically by (implicitly or rather occurred only when ethnic groups separated into explicitly) considering the decision-makers themselves as geographic patches (this separation falls into the same part of the system and of limited complexity/decision- universality class as the separation of oil and water), with the making ability. )e question thus becomes: how do we violence most likely to occur for geographic patches of a design systems that exceed the complexity of the decision- particular size. )is analysis implies that ethnic violence can makers within them? be prevented by the use of well-placed political boundaries, as in Switzerland. Although not explicitly included in the analysis, specific details of a region are relevant insofar as 4.1. Evolutionary Processes. While uncertainty makes most they are either a cause or an effect (or both) of the patch systems weaker, some systems benefit from uncertainty and size—for instance, animosity between two ethnic groups, variability [69–72]. )e common characteristic of these though not explicitly considered, may be a cause as well as a systems is their embodiment of some sort of evolutionary consequence of geographic segregation. process, i.e., a process in which successful changes are copied Understanding all the details of any complex system is (and further modified) while unsuccessful changes are not. impossible, just as it is for most systems with a separation of )e classic evolutionary processes are biological: due to scales; there is just too much complexity at the smallest scale. variability introduced by random mutations, organisms with However, unlike the behaviors of systems with a separation the complexity and scale of humans evolved from single- of scales, the important large-scale behaviors of complex celled organisms. Furthermore, humans themselves have the systems are not simply the average of their small-scale be- property of benefiting from exposure to random shocks haviors. )e interdependencies at multiple scales can make it (provided the shocks are not too strong). Immune system difficult or impossible to precisely understand how small- performance is improved by early exposure to nonlethal scale behaviors give rise to larger-scale ones, but even for pathogens [73, 74]; muscles and bones are strengthened by complex systems, there is much less complexity at the larger microtears and microfractures, respectively; we learn by scales than there is at the smaller scales. )us, there will exposure to new information and problem-solving; and our always be large-scale behaviors that do not depend on most psychologies are strengthened by exposure to adversity, of the system’s details (see Figure 9). )e key to analyzing provided the adversity is not too severe [75, 76]. these behaviors is to find the appropriate mathematical (or Competitive market economies provide another exam- conceptual) description, i.e., to identify variables that de- ple of how systems can thrive on uncertainty. Due to our scribe the relevant space of possible (large-scale) behaviors, ignorance of which will succeed, many potential innovations which for complex systems is neither a simple average nor a and businesses must be created and improved upon in full account of all the details. For additional examples of this parallel, the successful ones expanding and the unsuccessful multiscale approach, see. ones failing. )e successful among these can then be im- proved upon in the same manner—with many approaches 4. Complex Systems and Uncertainty being applied at once—and so on. (However, without ef- fectively regulated multiscale cooperative frameworks—see Although the principles discussed throughout Sections 2 and Section 4.2.—large-scale parts of the economic system may 3 help us recognize the fundamental properties and limi- optimize for the wrong goals, settling into harmful societal tations of systems, our understanding of most complex equilibria [77, 78].) systems will inevitably be imperfect. And regardless of how Likewise, the internal processes of large organizations well considered a plan is, a truly complex system will present may follow an evolutionary pattern in which small parts of  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Complexity 11 Hungarian, 1.9 Other, 3.9 Montenegrin, 2.5 Serb, 36.3 Yugoslav, 5.4∗ Macedonia, 5.9 Albanian, 7.7 Slovene, 7.8 Muslim, 8.9 Croat, 19.7 ∗ Self-identified as Yugoslavs. Dispersed around country. Albanian Muslim Bulgarian Serb Croat Slovak Serbs Muslims Hungarian Slovene Croats Albanians Macedonian No majority Montenegrin present Based on opstina data from 1991 census 0 100 kilometers 0 100 miles (a) (b) 0 1 Conflict Muslims Serbs Albanians Croats (c) (d) Figure 8: A figure from Lim et al.’s paper on ethnic violence. )e sites where their model predicts a potential for ethnic violence are shown in red in (c) and (d) with confirmed reports of ethnic violence depicted by the yellow dots in (d). the organization can fail and thus be improved upon; without In order to thrive in uncertainty and exceed the com- such flexibility, the entire organization may fail at once in the plexity of individual decision-making, systems can incor- face of a changing internal or external environment. In some porate evolutionary processes so that they, even if very cases, the failure of the entire organization makes room for limited at first, will naturally improve over time. )e first more effective organizations to take its place (assuming the step is to allow for enough variation in the system, so that the economy is sufficiently decentralized and competitive so that system can explore the space of possibilities. Since a large the organization in question is not “too big to fail”). )e amount of variation means a lot of complexity and com- collapse of government is generally not one of those cases, plexity trades off with scale (Section 2.4.), such variation however , so it is especially important that governance must occur at smaller scales (in both space and time). For systems possess the flexibility to internally benefit from example, in the case of governance, enabling each city to randomness and uncertainty. Perhaps counterintuitively, not experiment independently allows for many plans to be tried allowing small failures to occur may weaken systems in the out in parallel and to be iterated upon. )e opposite strategy long run by halting evolutionary processes and by creating would be to enact one national plan, the effects of which will interdependencies that lead to systemic risk (Section 3.3.). not be able to be comparatively evaluated.  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License 12 Complexity Full description may cooperate with each other to form groups, which in turn of the system may cooperate to form even larger groups, and so on. )us, a complex network of cooperation and competition among groups of various sizes (scales) can naturally evolve. In order to promote effective group cooperation, com- Complexity petition must be properly structured. A soccer team in which the players compete with their own team members to score Description of the system’s goals will not be effective, but one in which the players most important compete for the title of the most fit may be. )e framework (largest scale) in which competition occurs must be structured so that the behaviors competitors are incentivized to take actions that are net good Scale for the group; otherwise, a kind of tragedy-of-the-commons situation occurs. )e potential for competition to go awry Figure 9: A representative complexity profile of a complex system. highlights the importance of having a multiscale structure Understanding all the details (i.e., all of the small-scale behaviors) is with competition occurring on multiple levels, rather than impossible and unnecessary; the most important information is contained in the large-scale behaviors. However, for systems for having everyone in the system compete with everyone else. which mean-field theory does not apply, characterizing these be- With the multiscale structure, groups with unhealthy evo- haviors will involve more than a simple average. lutionary dynamics are selected against, while groups with a healthy mix of competition and cooperation that benefits the entire group are selected for. )ere is evidence that the )e second step is to allow for a means of communi- geographic nature of evolution—in which organisms evolve cation between various parts of the system so that successful in somewhat separated environments and mean-field theory choices are adopted elsewhere and built upon (e.g., cities does not apply—has resulted in precisely this multiscale copying the successful practices of other cities). Plans will structure and has therefore allowed for the evolution of always have unintended consequences; the key is to allow genuine (e.g., not reciprocal) altruistic behavior [81, 82]. unintended consequences to work for rather than against the Likewise, market economic systems are successful not be- system as a whole. Systems can explicitly design only systems cause free markets produce optimal outcomes (real-world of lesser complexity since an explicit design is itself a be- markets often sharply deviate from the assumptions of free- havior of the first system. However, systems that evolve over market models, and externalities abound) but rather be- time can become more complex than their designers. )e cause, at their best, appropriately regulated market systems desire for direct control must therefore be relinquished in allow for multiscale evolutionary processes to naturally arise, order to allow complexity to autonomously increase over resulting in innovations and complexity far beyond what time. anyone could have imagined, let alone designed. 5. Further Reading 4.2. Multiscale Evolutionary Processes. Successful evolu- tionary processes generally do not consist of unbridled Complex systems science, also known as complexity science, competition but rather contain both competition and co- contains many subfields. One starting point for exploring operation, each occurring at multiple scales. For ex- complex systems more broadly is this clickable map of ample, cells cooperate within multicellular organisms in complex systems science and related fields. Encyclopedias order to more effectively compete with other organisms, and [84, 85] and textbooks [1, 86–90] provide a range of per- organisms cooperate both within and between species in spectives. In addition to the topics and references discussed order to more effectively compete with other species. throughout this introduction, we provide a selection among Competition at larger scales naturally breeds cooperation at the many works applying complex systems science to social smaller scales because in order for a group to effectively systems and policy [91–105] and management [106–109]. compete with another group (large-scale competition), there Complex systems science includes, among others, the fields must be cooperation within the group. Cooperation can also of system dynamics , evolutionary dynamics breed competition since sometimes the best way for the [4, 111, 112], network science , fractals and scaling group to achieve its shared goals is to facilitate some healthy [114–117], urban science , pattern formation [119, 120], competition among its subgroups. )ose subgroups must econophysics , and nonlinear dynamics and chaos foster cooperation within themselves in order to effectively [122, 123]. Book series on complex systems topics include compete with each other, and they too may be able to in- the Santa Fe Institute Series and Unifying 7emes in Complex crease the effectiveness of their internal cooperation by Systems. introducing some healthy competition among their mem- bers (Figure 10 provides an example). If these members are 6. Summary themselves groups, the process of competition begetting cooperation that begets more competition can continue to Systems with many components often exhibit emergent large- even smaller scales. )is process can work in reverse as well: scale behaviors that cannot be directly inferred from the in order for individuals to compete more effectively, they behaviors of their components. However, an early insight of  8503, 2020, 1, Downloaded from https://onlinelibrary.wiley.com/doi/10.1155/2020/6105872 by Test, Wiley Online Library on [16/07/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License Complexity 13 Competition between sports Team collaboration Competition between sports enables the sport for fan attention and money to exist and compete increases team collaboration Collaboration Competition between teams between teams Collaboration of Competition between teams players enables causes selection of teams teams to compete with collaborating players Collaboration Competition between players between players Figure 10: An illustration from Chapter 7 in , showing the interplay between cooperation and competition in the context of sports teams and leagues. statistical physics is that in spite of the impossibility of de- importance of interdependencies and the complexity that scribing the details of trillions of molecules, the macroscopic arises from these interdependencies. To some extent, these properties of the molecules can be well understood by ana- problems can be mitigated by matching the data analysis or lyzing their space of possible behaviors, rather than their organizational structure to natural divisions within the specific configurations and motions. While many macro- system of interest. Since complex systems are those for scopic properties can be described in terms of the average which behaviors occur over multiple scales, successful or- behaviors of the molecules, the macroscopic properties of ganizations and analyses for complex systems must also be certain physical phenomena, such as phase transitions, cannot multiscale in nature. However, even when armed with all the be understood by averaging over system components; ac- proper information and tools, human understanding of cordingly, physicists were forced to develop new, multiscale most complex systems will inevitably fall short, with un- methods. Likewise, while standard statistical methods—which predictability being the best prediction. To confront this infer the average properties of a system’s many compo- reality, we must design systems that are robust to the ig- nents—can successfully model some biological and social norance of their designers and that, like evolution, are systems, they fail for others, sometimes spectacularly so. strengthened rather than weakened by unpredictability. Taking a systemic view by considering the space of Such systems are flexible with multiple processes occurring possible behaviors can yield insights that cannot be gleaned by in parallel; these processes may compete with one another considering only the proximate causes and effects of par- within a multiscale cooperative framework such that ef- ticular problems or crises. A system’s complexity—which fective practices are replicated. Only these systems—that depends on its number of distinct potential behaviors (i.e., on grow in complexity over time from trial and error and the the space of possibilities)—is a starting point from which to input of many—exhibit the necessary complexity to solve get a handle on its large-scale properties, in the same way that problems that exceed the limits of human comprehension. entropy is the starting point for statistical physics. Because the number of distinct behaviors of a system depends on the level Conflicts of Interest of detail (behaviors that appear the same at lower resolution may be distinct at higher resolution), complexity depends on )e authors declare that they have no conflicts of interest. scale. Interdependencies between components reduce com- plexity at smaller scales by restricting the freedom of indi- Acknowledgments vidual components while creating complexity at larger scales by enabling behaviors that involve multiple components )is material is based upon work supported by the National working together. )us, for systems that consist of the same Science Foundation Graduate Research Fellowship Program components, there is a fundamental tradeoff between the under Grant no. 1122374 and by the Hertz Foundation. )e number of behaviors at smaller and larger scales. )is tradeoff authors thank Uyi Stewart for discussions that led to the among scales is related to the tradeoff between a system’s writing of this paper, Gwendolyn Towers for editing early adaptability, which depends on the variety of different re- drafts of the manuscript, and Robi Bhattacharjee for helpful sponses it has to internal and external disturbances, and its discussions regarding complexity and scale. efficiency, which depends on its operating scale. )ere is no ideal scale at which a system should possess complexity; References rather, the most effective systems are those that at each scale Y. 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