In a <a href="https://www.nature.com/articles/359826a0">1992 Nature article</a>, Nowak &amp; May published surprising findings in their spatial model of the <a href="https://en.wikipedia.org/wiki/Prisoner%27s_dilemma">Prisoner&#8217;s Dilemma</a>. Although the Nash equilibrium of the standard Prisoner&#8217;s Dilemma is for neither player to cooperate, in Nowak &amp; May&#8217;s model, cooperation and defection dynamically coexist, spreading and receding across the 2-dimensional lattice of &#8216;players.&#8217; Each player considers the payoffs that each of its neighbors receives by playing their current strategy (cooperate or defect), and then changes their strategy to the highest-scoring strategy among their neighbors. Defecting (colored in red, with first-time defectors yellow) spreads from a single initial defector at the center, but cooperating (blue, with first-time cooperators green) is always able to remain a significant strategy.
A year after the publishing of Nowak &amp; May&#8217;s model, <a href="https://www.pnas.org/content/90/16/7716">Huberman &amp; Glance responded in PNAS</a>, proving that the hypnotizing dynamic patterns created by the Nowak &amp; May model were an artifact of synchronous updating across all players — if players update asynchronously, defecting will become the dominant equilibrium strategy.

Nowak and May's Prisoner's Dilemma

In this model, designed and presented at the <a href="https://vermontcomplexsystems.org/events/cnww/">Complex Networks Winter Workshop 2021</a> by Herzog, Johnson, Stone, Gao, &amp; Donaldson, a contagion is released on a network. Agents who are infected might infect others, but they might also self-isolate, reducing their likelihood of infecting others to 0. If enough agents become infected, a global &#8216;lockdown&#8217; is imposed, and all agents may choose to self-isolate (with a higher likelihood if one is infected or has a neighbor who is infected).
In a typical simulation run, we can see dynamics similar to the spread of COVID-19 in the United States and other countries in 2020. Harsh measures are imposed, and the contagion spread dies down. However, once lockdowns are lifted and typical behavior resumes, the rate of spread begins increasing again, and more become infected. Since there is no immunity in this model, these cyclical dynamics could continue indefinitely.

Edge Weight Adaptation in Response to Contagion

In network science, a preferential attachment network is a type of network structure where just a few nodes have a large number of connections to others, while most nodes have a small number of connections. This results in highly unbalanced networks with large disparities of degree, although degree is predictable based on the size of the network and the number of connections used in generating it. An example of a preferential attachment network is Twitter, where a few accounts have huge number of followers, where most accounts have relatively few.
This model shows how to generate a preferential attachment network from a target size and degree for new nodes. When agents are added to the network, they connect to a certain number of other agents based on how many connections existing agents in the network have. This results in those that were added earliest having the highest number of connections, on average.

Preferential Attachment Network

In this remix of the <a href="/models/flocking">original Flocking model</a>, the distribution of <code>Agent</code>s in space, rather than the individual <code>Agent</code>s themselves, is visualized using a <a href="/docs/heatmap"><code>Heatmap</code></a>. While it is more difficult to distinguish the behavior of individuals, group behavior becomes evident as flocks, emerging and dispersing over time, are presented as cells on the <code>Heatmap</code>, similar to emergent lifeforms in the <a href="/models/gameoflife">Game of Life</a>.

Flocking (Heatmap)

<a href="https://arxiv.org/pdf/1909.06483.pdf">This model</a> (Krause, Weyhausen-Brinkmann &amp; Bornholdt, 2019) shows public opinion on an issue on which agents can be for, against, or neutral. With each tick, agents holding non-neutral opinions converse (or debate) with another agent.
<ul>
<li>If the second has a neutral opinion, there is a baseline 50% chance that the first will convince it of their opinion. However, the first may also <code>repel</code> the second, in which case the second now has the opposite opinion.</li>
<li>If the two have opposite opinions, then there is a chance that the second will <code>doubt</code> its opinion and begin holding a neutral opinion (as if persuaded by a strong argument).</li>
<li>If the two have the same opinion, there is a (pretty slim) chance that the second will <code>doubt</code> its opinion and begin holding a neutral opinion (as if realizing flaws in the supporting argument).</li>
</ul>
With starting parameters of <code>0.5</code> for <code>repel</code> and <code>0.1</code> for <code>doubt</code>, public opinion always reaches a 50-50 stalemate, even though initially many more agents are for versus against or neutral. By adjusting the parameters, the dynamics of public opinion can sway, from relatively uncontroversial (one view dominates), to volatile (large swings in which one is the majority opinion), to the stalemate shown below.

Controversial Debate and Public Opinion

<a href="https://ndg.asc.upenn.edu/wp-content/uploads/2016/04/Centola-et-al-2005-AJS.pdf">The Emperor&#8217;s Dilemma</a> (Centola, Willer &amp; Macy, 2005) models how norms spread among populations despite low popularity. Similar to the <a href="https://en.wikipedia.org/wiki/Salem_witch_trials">Salem witch trials</a> of the 17th century, the phenomenon is caused &#8220;not by an outbreak of deviance, but of enforcement.&#8221; A few agents who are &#8216;true believers&#8217; in the norm make it costly for nonbelievers to not take a stance — despite their indifference or even opposition, they become enforcers of the norm and cause it to spread across the population. This results in an environment where few actually believe in the dominant norm, but nearly all enforce it.
In this simulation, agents are connected in a small world network (most agents connected to those close to them, with a few &#8216;random&#8217; connections to far-flung others). Red agents represent those complying with the norm, and the few yellow agents are those &#8216;true believers&#8217; who always enforce the norm. Agents who do not believe in the norm will comply and enforce it if they see enough of their neighbors enforcing it. By adjusting the <code>cost</code>, percentage of <code>believers</code>, <code>strength</code> (maximum conviction against the norm), and <code>rewire</code> parameters, the spread of the norm will take different trajectories. What dynamics lead to it spreading more quickly, slowing, or halting?

The Emperor's Dilemma

In this model, translated from the <a href="https://www.complexity-explorables.org/explorables/particularly-stuck/">Complexity Explorables model &#8220;Particularly Stuck&#8221;</a> (Janina Schöneberger &amp; Dirk Brockmann), particle-agents move noisily around in space, slowly spiraling toward the center. One particle is already stationary at the center, considered the &#8216;aggregation,&#8217; and when other particles collide with it, they become part of a growing, emergent structure. Although the byproduct of random movement, the structure appears to have a fractal logic, branching outward like coral, neurons, or other natural objects.

Diffusion Limited Aggregation

This model, based on Robert Axelrod&#8217;s <a href="http://www-personal.umich.edu/~axe/research/Building.pdf">Building New Political Actors</a>, attempts to show how state-like actors can emerge, gain power over other entities, and dissipate over time. In 20th century history, the Soviet Union provides an example of this lifecycle, emerging with the Russian Revolution of 1917, building its sphere of influence post-World War II, and dissolving in the early 1990s.
In this model, ten entities are situated on a linear landscape. Each turn, three are randomly chosen to demand tribute from others, who in turn may choose to either pay or to fight. By paying tribute, commitments are formed between more powerful and less powerful entities, and entities committed to each other may fight together against their enemies (fighting leads to decreased commitment). The functions of tribute and bidirectional commitments can lead to a dynamic political landscape, with actors gaining power and influence and sometimes losing it as others emerge over time.

Tribute Model

<a href="https://mathworld.wolfram.com/LangtonsAnt.html">Langton&#8217;s Ant</a> (designed by artificial life researcher <a href="https://en.wikipedia.org/wiki/Christopher_Langton">Christopher Langton</a>) is a cellular automaton with a single agent (the &#8216;ant&#8217;) moving across a two-dimensional grid. When the ant moves one step forward, it changes the color of the cell it occupies (from black to white and vice-versa). It also rotates clockwise or counter-clockwise depending on if it is on a black or white cell.
Although at the beginning of the simulation the ant appears to traverse a chaotic path, leaving behind random black and white cells in its wake, at about 10,000 steps in, it discovers a pattern, and begins tracing a wide path toward the southeast. On an infinite grid, it would continue in this direction forever, but on a toroidal grid (as shown here), it eventually collides with cells it previously traced, and returns to chaos.

Langton's Ant

This model visualizes the <a href="https://en.wikipedia.org/wiki/Mandelbrot_set">Mandelbrot set</a>, a mathematical construct. The set consists of a relatively simple formula that results in complex fractal patterns when visualized with a <code>Terrain</code>. Each coordinate represents a complex number, with the x-value corresponding to the real component and y the imaginary component. 
Use the buttons and drag the canvas to explore how the Mandelbrot set exhibits self-similarity (fractals) at different scales and at different boundary areas.

Mandelbrot Set Explorer

This implementation of <a href="https://www.complexity-explorables.org/explorables/cycledelic/">Dirk Brockmann&#8217;s &#8220;Cycledelic&#8221; model</a> simulates three &#8216;species&#8217; interacting in space. When A encounters B, B diminishes and A moves into its territory. In turn, B defeats C, and C defeats A. One way of conceptualizing this model is of simultaneous games of rock-paper-scissors playing out in two-dimensional space. The abstractly cyclic rule generates physical cycles (spirals and eddies) when visualized on a <code>Terrain</code>. 
By adjusting the parameters for predation (how aggressively a species moves into another&#8217;s space), competition (how difficult it is for a species to defeat another), and diffusion (absent of competition, how much species move in space), a variety of complex patterns will emerge.

Cyclic Mobility

<a href="https://workthatreconnects.org/resource/the-systems-game/">The Systems Game</a>, by Joanna Macy, is played in an open space by a group of people. Each person selects two others in the group and tries to keep an equal distance between them and each of the other two. Physically playing The Systems Game tangibly illustrates many of the dynamics of complex systems — self-regulation through activity, emergent layers of activity (clockwise or counterclockwise group motion), and interdependence of parts. 
<a href="https://www.youtube.com/watch?v=o_XRwN7UvIE">See The Systems Game played by a group of Carnegie Mellon University students.</a>
In this computational model of The Systems Game, agents randomly select two others from which to find an equidistant position. By clicking &#8220;Show Constraints&#8221; and &#8220;Next Agent,&#8221; you can see the intent behind an agent&#8217;s movement. If the system reaches a static configuration, clicking &#8220;Remove&#8221; will randomly remove five agents, and start the system in motion again.

The Systems Game

This is a model of the <a href="https://en.wikipedia.org/wiki/Iowa_caucuses" target="_blank" rel="noopener">Iowa caucuses</a>, a political event that combines voting and debate/discussion. In this model, voters and candidates are represented on a political compass, with each point in space representing a unique ideological viewpoint, all drawn from the same distribution function. Voters are also connected to each other, representing social relationships. They are connected to their closest neighbors ideologically, with random connections drawn to voters on other parts of the political spectrum. 
The caucus proceeds like this: In the first round, each voter casts their vote for the candidate whose ideological location is &#8216;closest&#8217; to theirs. Then, in following rounds, candidates receiving less than 15% of the total vote are marked not &#8216;viable,&#8217; and voters who chose a nonviable candidate vote again by randomly picking one of the candidates chosen by their connections (including the same candidate). This continues until all candidates receiving votes have over 15% of the total. 
By situating the landscape of voters on a social network, it is possible for candidates receiving less than 15% of the vote in the first round to become viable candidates through their voters influencing others in subsequent rounds.

Iowa Caucuses

Starting with no common language, a group of agents learn to communicate with each other through a guessing game. Each turn, an agent points to a unique object in a group of three objects. A second agent guesses the &#8216;meaning&#8217; behind the first agent&#8217;s act of pointing — a distinguishing feature, like &#8220;blue,&#8221; &#8220;circle,&#8221; or &#8220;1st&#8221; — and produces a word for that meaning. If it matches the word the first agent had in mind, the association between the word and the meaning are strengthened (however, in ambiguous contexts, it&#8217;s possible for the second agent to match the first agent&#8217;s word but for a different meaning). At the start, since the agents have no vocabulary, they generate new words and use them randomly, but over time, a shared language emerges that allows them to successfully communicate about the features of objects. 
Despite the simplicity of this model, a number of complex linguistic phenomena emerge, including synonyms (multiple words with the same meaning), differing usage/&#8217;slang&#8217; among sub-populations, and context-specificity (ex. a word that is associated with &#8220;blue square,&#8221; but not squares in general). This model is based on Luc Steels and Frédéric Kramer&#8217;s <a href="https://ai.vub.ac.be/sites/default/files/steels-ground1999.pdf">Bootstrapping Grounded Word Semantics</a> (1999).

Language Evolution

This model, based on observations of fireflies in Thailand, shows how a system of distributed entities with periodic behavior can eventually become nearly perfectly in sync. Each &#8216;firefly&#8217; agent has an internal clock, and when it flashes, nearby fireflies accelerate their clocks forward slightly. As the fireflies randomly navigate the environment, they encounter more and more flashing fireflies and grow more in sync with them. After a relatively short amount of time, all the fireflies are flashing in unison, with no top-down or centralized behavior organizing them. Like in the <a href="https://www.flocc.network/model/flocking">flocking model</a>, this behavior emerges out of the interactions of many individual agents. 
The implementation of this model is indebted to Nicky Case&#8217;s interactive explorable <a href="https://ncase.me/fireflies/">&#8220;Fireflies&#8221;</a>, and the phenomenon is described in the opening chapter of Steven Strogatz&#8217;s book <a href="https://www.stevenstrogatz.com/books/sync-the-emerging-science-of-spontaneous-order">Sync: How Order Emerges from Chaos in the Universe, Nature, and Daily Life</a>.

Fireflies Syncing

The <a href="https://en.wikipedia.org/wiki/Lorenz_system">Lorenz system</a> is a set of three interrelated differential equations that, with certain starting parameters, can result in locally chaotic but globally predictable values. In this model, five agents start at random positions (between 0 and 1 for x, y, and z values) and follow the update rules of the Lorenz system. Although their paths are initially similar, they soon diverge and one agent&#8217;s position is totally unrelated to the next. However, the five paths trace the form of a Lorenz attractor — a set of solutions to the system within which any given agent&#8217;s position will lie. Since there are no feedback loops between agents interacting with one another, this is an example of a complex physical system rather than a complex adaptive system. 
The visualization uses Three.js for rendering.

Lorenz Attractor

This model, outlined in Miller &amp; Page&#8217;s <a href="https://press.princeton.edu/books/paperback/9780691127026/complex-adaptive-systems">Complex Adaptive Systems</a>, represents an organization that makes &#8220;decisions&#8221; — binary outputs of either &#8220;0&#8221; or &#8220;1&#8221; — based on four-bit binary input &#8220;problems&#8221; (for example, &#8220;0110&#8221; or &#8220;1101&#8221;). The organization is structured with three agents, who each make decisions based on two-bit binary inputs (&#8220;00,&#8221; &#8220;01,&#8221; &#8220;10,&#8221; or &#8220;11&#8221;). They are all initialized with a &#8220;decision table&#8221; that determines what output they produce for each input. Two of the agents take input directly from the four-bit problem (the first two and the last two digits, respectively), and the third agent takes input from these agents&#8217; outputs, outputting the organization&#8217;s decision. 
Since each problem has a &#8220;correct&#8221; solution, and there are 16 unique problem-solution pairs, it&#8217;s unlikely that the organization will correctly decide every problem. The organization&#8217;s baseline (percent of problems it will correctly decide) is displayed at the top, and the line chart updates to show a running tally of the mean percent of problems decided correctly.

Organizational Decision Making

This simple model reflects a phenomenon of two-party politics where both candidates shift their positions to garner more votes. Voters are <a href="https://en.wikipedia.org/wiki/Left%E2%80%93right_political_spectrum">represented along a single axis</a>, and two candidates are initially placed at random positions in the left half and the right half (blue and red, respectively, as in U.S. politics). A voter casts their ballot for the candidate whose position is closer to theirs.
After each round of voting, the candidate with fewer votes shifts their position (there is no incentive for an incumbent to adopt a new position). With LEARNING_RATE set to 1, they will inevitably move toward the center, with the opponent doing the same if and when they are the minority candidate. With a lower LEARNING_RATE, the process still moves toward the center, but resembles a random walk, where candidates try out new positions that may be less optimal and find equilibrium more slowly.

Race to the Center

In the Predator-Prey model, there are two agent types: Sheep and wolves. Sheep eat grass, and their population is limited by both their food source — above a certain carrying capacity there aren&#8217;t enough resources to go around — and by the wolves who prey on nearby sheep. The nonlinear dynamics of this model are extremely sensitive to the starting conditions and parameters of reproductive and metabolic rates.

Predator-Prey

In the minority game (based on the NetLogo model of the same name), agents make a guess of either 0 or 1 with each tick of the simulation. After each tick, the goal for each agent is to be in the minority (guessing the same as less than half of all other agents). 
Agents are initialized with a set of strategies — binary strings, i.e. 010111011000&#8230; — and the history of recent winning (minority) guesses are stored as another binary string. A strategy &#8216;chooses&#8217; the agent&#8217;s guess by converting the history to a decimal number (i.e. 1101 → 13) and selecting that digit from the strategy (which is always either 0 or 1). After each tick, an agent updates the scores of all its strategies and selects the one with the highest score, using it for subsequent guesses. Although this behavior is somewhat opaque, it does result in unpredictable, chaotic behavior as agents attempt to guess at and outperform the market.

Minority Game

The Game of Life shows how remarkable complexity and emergent patterns can result from simple rules. 
From one tick of the game to the next, a live cell with fewer than two or more than three live neighbors dies. A live cell with two or three neighbors lives on, and a dead cell with exactly three live neighbors becomes a live cell.

Game of Life

Each agent in this model begins with a random direction, but is constantly shifting its direction to be closer to the average of its closest neighbors. Eventually, all agents will be moving in roughly the same direction.

Flocking

This model extends the 2d flocking model into 3 dimensions, using Three.js instead of Flocc renderers to visualize the emergence of flocking behavior. 
Like many other models, the environment is bounded such that agents that go outside of the cube will emerge on the other side.

Flocking (3d)

The Schelling Segregation model (1971) demonstrates how group-level patterns can emerge that are disconnected from the intentions of individuals. In this model, agents are represented by individual red or blue pixels on a <code>Terrain</code>. An agent belonging to each group wants to be surrounded by more of its own type than the other, and will move to an empty cell in the environment if it is in the minority. 
These agents do not exhibit aversion to other types of agents per se, and only want to find a position in a non-minority group, but the resulting movements deterministically result in a segregated environment — an illustration of how systems-level effects (i.e. systemic bias) can arise unintuitively from lower-level behavior.

Segregation

If you&#8217;ve ever been on a freeway at rush hour, you know the stop-and-start nature of heavy traffic. In this model, an agent with an empty &#8216;road&#8217; in front of it will attempt to accelerate until it gets close to another agent, at which point it decelerates (i.e. slamming on the brakes). Watch how this behavior sends &#8216;waves&#8217; backwards to the agents behind it, causing &#8216;traffic jams.&#8217; Under the right conditions, agents are able to separate such that traffic becomes smooth, but other environments are plagued by disruptive, halting motion.

Traffic

The fire model simulates how fire spreads through a forest. It starts with a wave of trees simultaneously catching fire, and with every tick of the simulation, a tree with at least one neighboring tree (to the north, south, east, or west) on fire will also catch fire. 
The PERCENT_FULL parameter determines how crowded the forest is. Set to 1.0 (100% full), every cell in the grid will be occupied by a tree, and the fire will deterministically sweep across the entire forest. However, around 0.59 (59% full), there is a threshold, and value below 0.59 will rarely result in the entire forest being burned, while above 0.59 the entire forest will almost always catch fire. 
Read more at https://en.wikipedia.org/wiki/Forest-fire_model

Forest Fire

Cellular automata (CA) are discrete models that usually take place on a grid (or lattice), with cells that take action based on their own state (for example, &#8220;on&#8221; or &#8220;off&#8221;) and the state of nearby cells. The <a href="/model/gameoflife">Game of Life</a> is an example of a cellular automata. Elementary cellular automata are a special type of CA that take place on a 1-dimensional grid (a single row). 
Reading from top to bottom, each visualized row represents the state of the cells over time. Each rule from 0 to 255 represents the action a cell will take given its own state and that of its immediate neighbors. For example, in rule 0, every cell will turn off no matter what. In Rule 184, the &#8220;traffic rule,&#8221; the number of cells in each state always remains the same, but sends &#8216;waves&#8217; throughout the lattice. With random starting states, rules 30, 90, 110 exhibit unpredictable, chaotic behavior. Try other rules below!

Elementary Cellular Automata

This intentionally abstract simulation depicts the dynamics of a network tending toward the mean. 
Each agent is connected to its closest geographic neighbors, and possibly a few other random agents in the network, depending on the value of REWIRING_PROBABILITY. With each tick of the simulation, an agent that is larger than its neighbor will transfer some of its size to the neighbor, shrinking the agent and growing the neighbor. With enough time, the network reaches stasis. 
Try adjusting the parameters to see how it affects the network dynamics.

Network Diffusion

This implementation of Joshua Epstein&#8217;s <a href="https://www.pnas.org/content/99/suppl_3/7243">Civil Violence Model</a> simulates outbreaks of unrest against a central authority by agents questioning its legitimacy. 
In this visualization, cops are represented as blue pixels. Civilians are represented by gray, red, or white pixels. Every turn, some civilians will go from inactive (gray) to active (red), depending on their own threshold and the exogenous legitimacy of the government. However, a cop who is next to an active civilian will arrest them (turning them white and unable to move) for a fixed number of turns set by the jail term. By updating parameters for legitimacy, jail term, % full, and % Cops, different dynamics will arise, from a peaceful population to periodic outbursts to near-constant anarchy.

Civil Violence

A Voronoi diagram divides a plane into a set of non-overlapping regions, where each region is the set of points closest to a given point in that region. 
One method of deriving a Voronoi diagram from a set of points on a plane is through the emergent movement of (many more) random vectors on the plane. In this example, 25 random points are placed on the plane to divide it into regions. Then, 1,000 random vectors (points with a random directional velocity) are placed and begin moving until they are the same distance from the two nearest points. Then, they stop. The resulting accumulation of stationary vectors describes the boundaries of the Voronoi diagram.

Emergent Voronoi

Based on the model described by Rudy Rucker in &#8220;Artificial Life Lab&#8221; and implemented in NetLogo, this model simulates a voting process and the emergence of polarized opinions. Each agent samples its neighbors within RADIUS and, if there&#8217;s a clear super-majority with a different vote from the agent, it will change its vote (i.e. bowing to social pressure). 
By expanding the RADIUS parameter, regions of agents with the same opinion become larger and more defined.

Flocc

Examples