An Introduction to Complexity Science for the Social Sciences

[For a version of this post with footnotes and references, see here.]


In both social sciences and policy-making, researchers and practitioners tackle multifaceted phenomena. Examples are (armed) conflicts, migration, the emergence of populism, the automation of professions, financial crises, international trade, or social integration. Efforts in research and practice have led to various approaches and processes to analyse these phenomena and make decisions under uncertainty.

I argue that the toolbox used to tackle these real-world problems could benefit a lot from the growing field of complexity science, i.e. the study of complex systems in the physical, biological, and social worlds. I believe complexity science is a vital tool that yields a more honest and granular understanding of social phenomena. Complexity science is not revolutionary, it is the middle-ground between the assemblage of (a) insights and methods from many scientific disciplines and (b) the dilution of disciplinary boundaries.

The hierarchy of evidence rarely applies to social phenomena, given that some cannot be evaluated by randomized controlled trials because of their complex nature, unique sequence, or inherent lack of data. Therefore, the degree of uncertainty present in social science is enormous and increases with rising interconnectedness. Internet, transport technologies, and softer social rules allow denser social interaction which complexifies causality chains. It is crucial to articulate a vocabulary that enables researchers and practitioners to conceptualize such realities.

However, long and detailed descriptions of complex phenomena are not a panacea. They are static, subject to interpretation, and do not allow for data integration. Therefore, one needs dynamic formalism that enables to capture complexity with clarity and parsimony, while relying on the limited available data. Dynamic formalism refers to the use of computational models to understand social phenomena.

Complexity science offers the needed theoretical vocabulary and puts forward computational models to study complex systems. This short introduction presents the field of complexity science and provides a first overview for researchers and practitioners that are interested in expanding their analytical toolbox.


Dynamics and resemblance

In the world, one may observe different systems - physical, biological, or social. We can agree that these systems keep changing; adapt, and evolve over time. For instance, social systems change as a function of people's preferences, the discovery of resources, implementation of rules, or the emergence of cultural artifacts. Quite differently, natural systems change as a function of the climate, number of species, and, among many other variables, include interaction with social systems.

Systems, like a village of human animals interacting and exchanging goods or a rainforest in Amazonia, will change over time. We can intuitively induce great differences between these systems. However, according to empirical studies, these systems can display strong resemblances. For instance, if one looks at wealth distribution in our civilization and biomass distribution in the Amazon forest, both systems display similar properties: around 1% of our population has 50% of total wealth, and around 1% of the tree species has 50% of the biomass. More interestingly, Scheffer et al. shows that similar dynamics led to these properties in both systems (figure 1).

Even though the fundamental mechanisms leading to resource inequality are very likely to be different from one system to another, two very different systems may exhibit profound resemblance in the aggregate. The field of complexity science attempts to identify common properties that apply to various dynamic systems.

Figure 1: Inequality in nature and society (Scheffer et al. 2017)

Figure 1: Inequality in nature and society (Scheffer et al. 2017)

Solving problems

From a problem-solving perspective, one can go about dealing with systems in many different ways. Like one can take different approaches to solve a Rubik's cube. What can one do?

One can take a rather stupid approach and just use the same technique over and over on all problems. Same methods for all systems. This is an illustration of stupidity given that, for a Rubik’s cube, it is impossible to solve it if one turns the same axis on and on. More importantly, solving a Rubik’s cube would not be possible if one looks at it always in the same way.

One could apply a random strategy by switching between different techniques, which already is much better than the stupid one because it adds variety. It can take ages but at some point, the Rubik’s cube will be solved.

Or one can try a systematic strategy and attempt to understand the underpinnings of the problem, to learn and eventually identify the best moves, and solve the problem more easily. Being systematic opens the door to many methods and that’s the realm of science.

Map & territory

A useful way to think about the scientific approach is the distinction of the map and the territory. A map is a simplified version of reality that use to navigate reality - the territory - better. All maps are wrong in certain ways because their purpose is to be useful. Were they an accurate representation of the territory, then we could just as well simply look at reality.

We all have maps in our head. The problem is that, as humans, we tend to confuse our map with the territory, so much so that the distinction between our understanding and reality is lost. This confusion prevents us from understanding reality better.

If we presume that (1) systems are dynamic and yet may resemble each other, (2) we should be systematic when exploring them, and (3) we must not confuse the map and territory to preserve our epistemic integrity, then we can turn to the hard task: which systematic method is useful to address complexity?


In the 1980s, a group of scientists (many of them from the Los Alamos National Laboratory) created the Santa Fe Institute. Its mission is to foster the development of the science of complex systems, namely complexity science. The goal of the founders was to respond to the prevailing and increasing specialization and reductionism in science.

So, what is 'complexity'?

According to John Holland, complexity still remains loosely defined, which is not a problem because it does not prevent one to have a systematic approach to the subject matter. Yet, a working definition helps to grok what complexity is: "complexity characterizes something with many evolving parts that interact with each other in various ways, displaying nonlinear patterns in the aggregate, which, often, is not additive. This something usually is called a ‘complex system’.” Put differently, complex systems satisfy the following characteristics:

  • Cardinality: many parts (from particles to agents) constitute the system

  • Diversity: parts are different from one another

  • Dimensionality: parts differ from one another in multiple ways, i.e. along several dimensions

  • Connectivity: parts act, interact, and adapt through networks

  • Nonlinearity: the relationship between variables is often nonlinear and the aggregate of the parts does not equal the sum of their characteristics or actions.

That said, complexity science mostly attempts to understand micro-level relationships to understand how macro-level patterns emerge. The idea is to identify common properties across complex systems and apply them to real-world phenomena.

Complex systems

A chess game is a fitting analogy to explain how common complex systems might be. Systems are characterized by their states and rules.

Figure 2: Chess game

Figure 2: Chess game

A state is how the system looks like at a given point in time. For example, figure 2 shows the state of a chess game after three white and two black moves presumably.Rules are the mechanisms that will change the states of the systems. In chess, the rules are clearly defined: there is an 8x8 field of squares, on which 32 pieces start out that have 6 different kinds of moves. There are additional rules like castling and the pawns’ double jump that define the actions players can take.

In complex physical systems, the rules are called laws (e.g. laws of thermodynamics) and states are called states (gas, liquid, solid). In complex adaptive systems (CAS), the elements following the rules (e.g. chess pieces) are called adaptive agents (e.g. animals, bacteria, artificial agents). The dynamics of CAS can serve to approximate social systems.

A key property of a complex system is the notion of perpetual novelty. In chess, the rules remain the same across countries and for centuries. But almost every chess game is different. Experts estimate the number of possible chess games to be 10120, known as the Shannon number, while the number of atoms in the observable universe is estimated to be 1080. This is because a variety of moves are possible and because the system state at time t gives almost no information about where the game can go. Therefore, the evolution of the game is stochastic.

But even though a chess game (a complex system) displays perpetual novelty, it is possible to identify particular patterns that are recurrent over time. In chess, it is possible to predict openings; the way a horse forks a king and a queen; and how end-of-the-game checkmates happen. Those patterns are the main subject of study of complexity science.

If we could predict every state, then the system would be simple. If there was no pattern recurring over time, then the system would be chaotic. System theory argues that there are enough complex systems out there that it is worth trying to look for recurring patterns across them.

Complex adaptive systems

The idea underlying complexity science is to identify properties that are valid across systems. The goal of complexity science is to offer a set of flexible frameworks that are applicable to almost all complex systems. Complex social systems often are conceptualized as complex adaptive systems (CAS) because of the adaptive nature of an agent’s behavior in interaction with their environment.

Given the focus on exploring micro-level interactions to understand the emergence of macro-level structures, the framework of complex adaptive systems (figure 3) provides helpful vocabulary to formalize the description of complex dynamics. Each component is described after another.

Figure 3: Complex adaptive system

Figure 3: Complex adaptive system

Micro-level dynamics

Most studies that apply the methodology of complexity science attempt to specify the micro-level relationships, to then describe macro-level behavior.

1. The system's micro-level is composed of heterogeneous agents that differ in terms of idiosyncratic characteristics:

  • attributes that characterize agents (e.g. job, culture, resources); and

  • rules of behavior including cognitive rules (e.g. beliefs and biases), strategies (actions agents can take like support, oppose, buy, or sell), learning (conscious or unconscious learning techniques such as trial and error or imitation), and updating (a threshold that makes the agent flip to another state).

2. Heterogeneous agents interact based on their topology of interaction, i.e. network structure which can differ in

  • size (number of agents);

  • morphology (shape);

  • complexity (amount of heterogeneity);

  • centrality (presence of hubs); and

  • density (number of links or proximity between agents).

3. Agents interact in a changing environment with, i.e. :

  • systemic factors;

  • shocks; or

  • stressors.

In chess, the systemic factor is the board’s 8x8 squares which, in this case, does not evolve. A stressor would occur if, for example, a stone would be placed in the middle of the board, constraining where pieces can move. A shock could be a player shaking the board or throwing pieces on the ground.These events can happen theoretically, though they rarely do. In social phenomena, a systemic factor can be democracy, a stressor may be Trump in power, and a shock could be a harmful pandemic spreading around the globe.

4. The system exhibits feedback loops between agents and the environment through which information travels and leads to small changes, nurturing learning capacity, and continuous adaptation.

These four points alone are a detailed micro-level perspective of a system. It remains easy to understand and manipulate to study different complex adaptive systems. The idea of complexity is that the whole forms a collective behavior that adapts over time.

The approach of complex adaptive systems can be applied to small complex systems of two or three agents to very large systems such as cities, the national economy, the internet, etc.

Macro-level dynamics

From the micro-level, collective behavior generates structures or dynamics in the aggregate, i.e. on the macro-level.


A typical observation is the self-organisation in patterns. A famous model, called the Schelling model of segregation dynamics (animation 1 and 2), simulates two kinds of agents that have one parameter: the extent to which they want to live close to an agent of the same type. And, in case they are not satisfied, they would move to empty areas.

In the model, one can vary the parameter of neighbor preference. In the example illustrations below, people actually prefer to live close to other people, respectively 70% and 55% of the time. However, one nevertheless observes clear segregation in both cases even though people were quite tolerant.

Such evolution from micro-level preferences to macro-level structures is called self-organisation. Sometimes, such as in this example, self-organisation can lead to counter-intuitive dynamics. Studies that apply complexity science to study social phenomena rely on the generation to understand self-organisation patterns. The assumption is that agents are, beyond their characteristics, relatively independent from one another and adapt over time.

Animation 1 and 2: Segregation  
“I want to live next to different individuals 70% of the time”

“I want to live next to different individuals 70% of the time”

“I want to live next to different individuals 55% of the time”

“I want to live next to different individuals 55% of the time”

Here, both initial states start with a population of two agents (green and red) that are randomly distributed spatially. The population does not cover the whole space to leave empty areas for migration (adaptation).


Another common property of complex systems is the nonlinear relationship between the local system components and the aggregate. Below are two forests that are almost identical. The one on the left has a density of 58% while the other one's is 59% (with trees randomly distributed on a spatial grid). We can see how this small difference, invisible to our eyes, can lead to drastic changes in case of a forest fire.

Animation 3 and 4: Forest fire
Density: 58%  Burn: 10.2%

Density: 58%

Burn: 10.2%

Density: 59%  Burn: 70.9%

Density: 59%

Burn: 70.9%

One important lesson beyond the nonlinear effect is that in some instances a tiny error in the measurement of the micro-level structure will lead to huge errors in predictions on the macro-level. This is why the field of complexity science does not advocate for pin-point prediction (as it is close to impossible) but for the understanding and prediction of patterns, and, more specifically, of tipping points (figure 4).

Figure 4: Tipping points

Figure 4: Tipping points

Power laws

A third macro-level property of complex systems is the occurrence of power laws - notably, in networks of agents with preferential attachment or reinforcement loops leading to centralized networks or powerful agents. Animation 5 shows the evolution of the network structure with additional agents joining the network. On the right, the upper graph shows the heavy-tailed degree distribution and the log-log graph below depicts a power law emerging over time.

Animation 5: Network evolution with preferential attachment

Animation 5: Network evolution with preferential attachment

Nonlinear effects and power laws lead to extreme events (positive and negative). They can take the form of wealth distributions (e.g. figure 1) where reinforcement loops drive up inequality; financial crises leading to a cascade of crashes; or other big events like a global pandemic spreading through networks.

Linking micro and macro

In complexity science, the concept of ‘emergence’ is used to describe how macro-level properties emerge from local dynamics. It refers to the non-additive properties of macro-level structures, i.e. they are not the sum of local interactions.

Complexity science scrutinizes how a well defined 'starting state' (e.g. local interactions like a chess game) evolves into 'mature states' (e.g. macro-level structures like a chess game after twenty moves). The animations 6, 7, and 8 below show 3 iterations of a model of 10 ants that have the same behavioral rule. The 3 iterations start with the same starting state and speed and last for the same duration. In these three isolated environments, the starting state and the rules governing it are exactly the same. However, what we observe in terms of the outcome is completely different.

This example yields the following insight: the starting state provides almost no predictive power about the mature state. Thereby supporting the argument that the behavior of non-sophisticated complex systems may sometimes be impossible to predict. These are 10 ants that have one behavioral rule, interacting in a completely flat environment and are already depicting stochastic processes.

Animations 6, 7, and 8: Ant colonies
ant gif 4.gif

It is important to note that in a more sophisticated setting with more variables that balance one another, more order may be observed, allowing for more accurate prediction. For example, the Financial Crisis Observatory at ETH Zurich does make somewhat accurate predictions of financial markets, while they are known to be sophisticated complex systems. Therefore, prediction is possible yet difficult, and should be assumed impossible in most cases. Already understanding the core dynamics between starting and mature states is a step forward that complexity science can provide.    

While models enable in silico experiments to re-run the same starting states multiple times, and, potentially, observe different or the same dynamics emerging every time, such observation cannot be made in reality. In social science and biology, there is often only one sequence of events leading to a phenomenon, with no practical possibility of iteration to analyse different scenarios. If we were able to re-run the course of evolution, would we end up with a different social world than today? Would this apply to political debates, public events, or wars?

It is very likely that we would end up in different future scenarios if we restarted history at any given point. That is a lesson complexity science teaches: the plurality and heterogeneity of the biological and social world lead to many various combinations of events that, in iterated aggregates, can depict completely different trajectories.

In the light of this insight, complexity science focuses on pattern prediction or pattern understanding to identify recurrent patterns over time. If patterns recur, complexity science attempts to understand the underpinning characteristics that lead to them. In case of multiple trajectory patterns, complexity science focuses on a certain point in time or variabilities that could make patterns bifurcate (figure 5).

Figure 5: Graph of multiple trajectory patterns

Figure 5: Graph of multiple trajectory patterns

In a nutshell, studying the properties of complex systems and stochastic or recurrent patterns is the very subject and raison d’être of complexity science. Many other disciplines explore complex systems and acknowledge complexity, but without the use of such insights.

Generative science

Complexity theory provides a set of concepts and properties to approach and conceptualize complex systems. Yet, in Epstein’s terms, “if you haven’t grown your society, you haven’t understood it”. the animations above yielded insights only because of dynamism that allowed for adaptation. Generative science refers to the use of computational models to generate macro-level dynamics from local interactions.

More specifically, creating models of complex systems serves eight purposes:

  1. data-driven models enable to predict patterns and corroborate results with real-world scenarios;

  2. models allow to understand how a well-specified starting state evolves and leads to expected or counter-intuitive mature states;

  3. through the identification of recurrent patterns, models permit to identify the key mechanisms that generate certain parameters and when they create tipping or bifurcation points;

  4. complex, multifaceted phenomena are difficult to study and monitor. Models can help to guide data collection, or remedy the lack of data in case of hypothetical scenarios;

  5. models of prevailing theory can challenge a theory’s robustness through perturbation tests or serve as evaluation method to determine causal relationships;   

  6. models enable to emphasize uncertainties;

  7. the understanding of social phenomena is often dependent on subjective understand and subject of passionate discussion. Models require and promote a scientific habit of mind, educate the general public, and discipline the policy dialogue.

The following section covers the different components inherent to building models in complexity science: conceptualization, formalization, implementation, and simulation. Generally, modelling can fall into two research realms: exploratory theoretical studies and evaluative data-driven studies.

From conceptual to simulated complexity

The components of complex systems (c.f. micro-level dynamics) serve as a starting point for everyone wanting to conceptualize a complex system. Generally, one or multiple causal flow diagram(s) are created based on explicitly stated assumptions and anchored in theoretical or empirical studies.

The shape of the conceptual model will depend on its type, whether it is an agent-based, network, system dynamics, or game theoretic model. Agent-based models (ABMs) capture the key characteristics of the other models such as interaction networks, system dynamics, and behavior adaptation. ABMs serve as an archetypical type of model used in generative social science as it models bottom-up dynamics resulting from the interaction between agents, very much like in the social world.

Figure 5 depicts a general, conceptual agent-based model. Causal flow diagrams are also accompanied by a model algorithm diagram presenting the sequence of steps the model runs through.

Figure 6: Agent-based model

Figure 6: Agent-based model

In short, an agent-based model specifies the micro-level components of a complex system with agents, attributes,  rules of behavior, interactions between agents, feedback loops, and a changing environment.

The conceptualization is the first step. The value of a model increases when it is formalized. As variables and relationships have to be well defined, vagueness shrinks. Translating text and ideas into mathematics also enables to quantify variables which then permits to seed the model with empirical data. Different types of mathematics may be used depending on one's needs. For instance, differential equations are useful to treat agents and their adaptation, and phase transition and critical phenomena enable to formalize system evolution and nonlinear dynamics.

Following the formalization, a model becomes computational once it is implemented. Causal flow diagrams and mathematical equations can be implemented in different languages such as Python, NetLogo, MatLab, and JavaScript, and can be sequenced according to the model’s algorithm. Today’s computational power allows to iterate such an algorithm hundreds or thousands of times until patterns emerge from local dynamics.

The process of running iterations of micro-level interactions to observe macro-level patterns is called simulation. In other words, simulations are real-world scenarios developed in silico. Simulation essentially permits to shed light on non-additive, nonlinear patterns that are impossible to deduce from the micro-level.

Moreover, a model can be tweaked to run what-if scenarios and run computer experiments. For instance, they may complement randomized controlled trials (on the ground) by running the simulation twice: once for the treatment group, once for the control group (Figure 6). This combination may provide additional information to strengthen the value of experiments and increase external validity. For example by testing the same treatment in different scenarios.

Figure 7: ABM for randomized controlled trials

Figure 7: ABM for randomized controlled trials

Exploratory versus evaluative modelling

As previously mentioned, complexity science is divided into two schools of thought: exploratory and evaluative research. Both have their advantages and downsides.

Exploratory research refers to the modelling of real-world phenomena without anchorage in empirical data, thus being based on theory or assumptions only. Its principal goal is to inform ‘static’ theory with more dynamism or shed light on a complex system’s properties. In fact, exploratory research has made it possible to gain insights on common properties across complex systems and was thus the main driver behind the creation of complexity theory.

In addition to theoretical work, running exploratory simulations is also helpful to understand hypothetical scenarios or phenomena that we have not yet been able to observe. For instance, disastrous events threatening civilization such as solar flares, asteroid impact, extreme climate change, global pandemics, or bio-attacks (figure 7). As we can see, exploratory research may seem very detached from reality or abstract but it can provide insights on policy-related questions.

Figure 8: Results of an exploratory agent-based model of bio-attacks

Figure 8: Results of an exploratory agent-based model of bio-attacks

Evaluative research refers to the data-driven modelling of real phenomena. Its principal goal is to provide scientific evidence on a specific topic in a given time frame. Empirical data are used to seed, calibrate, and validate the model. The emerging field of evidence-driven modelling started playing a role in e.g. conflict forecasting, and diseases modelling (figure 8).

Figure 9: Data-driven modelling of diseases

Figure 9: Data-driven modelling of diseases


Even though I argue that combining the insights of complex systems and computational modelling is a powerful method, there remain fundamental limitations.

First, models of any kind are inaccurate. Complexity science is about building better maps of the territory. It is worth emphasizing that maps are inaccurate because they have to be useful, i.e. possible to understand and to use to draw conclusions and make decisions. Should a model attempt to be completely right, then it would not be a model anymore, it would be an audacious attempt to create a carbon copy of reality.

Second, models can only disprove a mechanism or show that a given mechanism is sufficient to generate a given output. However, computational models cannot prove mechanisms, meaning that one cannot claim that a simulated mechanism, even if complex, is the sole explanation of an output.

Third, in silico simulations cannot replace real-world experiments. They can only complement them, guide their design, or attempt to remedy the impossibility to run experiments (e.g. risks modelling). For example, models can be useful to generate macro-level dynamics that are observable only after iterations, but they cannot have the causal power of real-world, controlled experiments which, explicitly or implicitly, do capture more variables. The qualitative information collected through experiments is also not provided by models.

Fourth, the replication of models and their results is needed. Otherwise, researchers may claim more explanatory power than possible by employing these seemingly powerful methods and manipulating the presentation of their results. Moreover, given the likelihood of stochasticity, replication makes sure that published research took stochasticity into account (if relevant) instead of cherry-picking ‘suitable’ scenarios.

Fifth, data-driven models do not currently pass tests of external validity. If simulations are tailored to a specific context, then the generalizability of results will be weak. Iterations of the same model on multiple cases may remedy this problem but have so far not proven entirely sufficient.

Sixth, agent-based models are not computer games. The idea, related to models being inaccurate by definition, is not about representing the world in such a fine-grained manner. Modelling attempts to specify complex micro-level interactions as good as possible, but will only analyse a selected set of emerging phenomena. Again, it is not about representing the real world. It is about answering hypotheses.


Our ability to understand the gears of the social world, to anticipate harmful events, or to build recovery programs that will foster adaptation later on is the most valuable asset we can rely on to act well. Yet, building this knowledge while accounting for complexity and preventing us from drawing too simplistic assumptions is difficult.

The above content serves as a first introduction to complexity science. It provides some first perspectives to grasp what complex systems are, how they evolve, and how one would go about studying them.

First, the structure of micro-level dynamics and emergent macro-level properties were explained to shed light on self-organisation, non-linearity, power laws, stochastics, and the difficulty of prediction. Second, the field of generative science was presented with an explanation of computational modelling and the two broad domains of exploratory and evaluative research. Finally, the limitations of models were discussed.

The main point one should keep in mind is the following: it is possible to granularize analyses by unpacking complex systems with applicable concepts, and precise vocabulary to then re-aggregate content in a bottom-up manner thanks to computational models. This enables to scrutinize social phenomena down to, for instance, actor cognition and behavior, and up to social structures and collective behavior.

More importantly, generative science enables to dynamize starting states into mature states without simplistic assumptions. While disciplines tend to have a micro versus macro focus, such as behavioral public policy versus political science; or microeconomics versus macroeconomics, complexity science offers a way to capture both: (1) formalize micro-interactions, (2) simulate their adaptation, and (3) observe macro patterns.

I believe this is a powerful approach to understand reality. On one hand, it does complicate the task of researchers and practitioners through the honest emphasis of, for instance, non-linearity and stochasticity. On the other hand, it provides a way to appreciate complexity with a vocabulary that captures the peculiarities of complex systems.

By shedding light on how complex social phenomena are, complexity science at least increases our understanding of how uncertain we should be.

Further Reading & Learning

The most complete and most recent introduction:

  • Thurner, Stefan, Rudolf Hanel, and Peter Klimek. Introduction to the theory of complex systems. Oxford University Press, 2018.

Longer introductions to complexity science for social sciences:

  • Epstein, Joshua M. Generative Social Science: Studies in Agent-Based Computational Modeling. Princeton University Press, 2006.

  • Miller, John H, and Scott E Page. Complex Adaptive Systems: An Introduction to Computational Models of Social Life. Vol. 17. Princeton university press, 2009.

General introduction to complexity science:

  • Holland, John H. Complexity: A Very Short Introduction. OUP Oxford, 2014.

  • Mitchell, Melanie. Complexity: A Guided Tour. Oxford University Press, 2009.

Advanced introduction to complexity science:

  • Holland, John H. Signals and Boundaries: Building Blocks for Complex Adaptive Systems. Mit Press, 2012.

Applications of complexity science and computational modelling to practical problems:

  • Helbing, Dirk. ‘Globally Networked Risks and How to Respond’. Nature 497, no. 7447 (2013): 51.

  • Helbing, Dirk, Dirk Brockmann, Thomas Chadefaux, Karsten Donnay, Ulf Blanke, Olivia Woolley-Meza, Mehdi Moussaid, et al. ‘Saving Human Lives: What Complexity Science and Information Systems Can Contribute’. Journal of Statistical Physics 158, no. 3 (2015): 735–81.

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