Emergent behavior is the process in which larger, complex patterns arise from smaller, simpler agents. An emergent system should be unpredictable from a low level perspective.  Conway's Game of Life is a classic example of emergent behavior. Four rules govern the actions of each cell:
- Any live cell with fewer than two live neighbours dies, as if caused by under-population.
- Any live cell with two or three live neighbours lives on to the next generation.
- Any live cell with more than three live neighbours dies, as if by over-population.
- Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.
From these rules appear patterns ranging from the simple blocks and beehives, to the more complex oscillators, to the very complex "spaceships".
It is the interactions that are important, not the individuals. Although these rules apply to each cell, a cell's state is dependent on the state of its neighbors. Like the Game of Life, the focus of emergent behavior is on how the interactions create the whole. In short, emergent behavior can be described as a complex pattern created from simple rules.
Self-organization, swarm intelligence, and stigmergy are all group-behavioral ideas that fall under emergent behavior. While the theory of emergent behavior encompasses much more than these 3 examples, they provide an introduction to the subject.
Self-organization is the process where signs of organized behavior forms from local interactions in a chaotic system. Self-organization relies on four basic operations: positive feedback, negative feedback, a balance of exploitation and exploration, and multiple interactions.  The key component in self-organization like in the other group intelligence ideas is communication. An agent needs to be able to communicate its discoveries to other agents to start the positive or negative feedback. Findings that are rewarding send a positive feedback to the group, increasing the incidents of such behavior which leads to increased system stability. Findings that are undesirable are thus communicated as such. Others will seek such behavior less often which still results in system stability. Eventually, the system will have identified certain actions that are favored and others that are not so, creating an equilibrium where the probability distribution of potential actions is stable.
The idea of self-organization has been around longer than the actual term. Descartes mentions this idea in the fifth part of his Discourse on Method, where he mentions God's laws and how they effect greater truths.
And I discovered certain laws that God has established in nature; he has implanted notions of these laws in our minds, in such a way that after adequate reflection we can’t doubt that the laws are exactly observed in everything that exists or occurs in the world. Moreover, by considering what follows from these laws I have discovered (it seems to me) many truths that are more useful and important than anything I had previously learned or even hoped to learn.
Although the phrase "self-organization" never appears in the text, his statement sums up the crux of the idea, that a set of rules can effect a greater pattern or in his case, truths.
According to Google's Ngram viewer, the term "self-organization" does not see use until the 1800s, well after the date of Descartes' works. Emergent behavior follows a similar trend to self-organization in that the idea has existed for some time now whether it is in self-organization, swarm intelligence, or stigmergy.
In early August 2014, Michael Brown was shot and killed in Ferguson, Missouri. His death prompted peaceful protests which later escalated and turned violent with police intervention.  Braha claims that social instability can be explained as a spatial epidemics phenomenon. Like an epidemic, civil unrest can be spread seemingly randomly but actually with a set of simple rules. The rules govern the individual, not the whole, but nonetheless it spreads. 
Self-organization in Society and Technology
Social media provides a case of self-organized behavior. Facebook, for example, as a system where people can “like” a comment or post. A post with a high amount of likes can be viewed as being rewarding. A post with little to no likes can be seen as unrewarding. When others view these well-received posts, they are encouraged to post something similar. In this example, the number of likes is the reward function of the system. Knowing that certain posts will be well-received creates a systematic movement towards certain areas. Each person on an individual level is merely working to maximize their post popularity, not working towards a collective goal or ideal. However, on a high level one can see that certain ideas are encouraged while others fade.
Self-organization requires communication in order for agents to provide feedback. In 2014, Open Garden released a product called FireChat. It is a messaging app that does not require internet or mobile connectivity. The app directly communicates with other phones in the area to create a peer-to-peer messenger. The app grew in popularity with its use in Iraq when the government restricted internet use.  The app also saw popular usage in Hong Kong during student protests against Beijing's political decisions. 
Swarm intelligence is the complex behavior of decentralized, self-organized systems. Individuals in the swarm follow very simple rules and are not aware of resulting large scale behaviors. Swarm intelligent systems are often found in nature. Such systems are often used as models for systems created by humans because they are more resilient than centralized systems.
Ant colonies function without central coordination. Each colony has a queen, but her purpose is to reproduce, not organize the colony. Using only simple interactions, ant colonies are able to organize and coordinate the processes necessary to maintain the health of the colony. Ants interact using pheromones sensed by their antennae. Pheromones identify other ants and indicate which task an ant is working on. An ant’s behavior is influenced primarily by the frequency of interactions with other ants and the pheromones it senses during each interaction.
The colony must forage efficiently to avoid wasting resources. The colony uses feedback, based off the frequency of ants returning with food, to increase, or decrease, the number of ants actively foraging. As the number of actively foraging ants returning with food increases, more ants take on the foraging role and search for food. Conversely, as ants with food return less frequently, fewer ants take on the foraging role. To optimize the path to the food source, ants from the colony travel in the direction of ants returning with food. Each ant is able to remember the direction back to the colony, so the ants incrementally form direct path to and from the food source.
Certain birds exhibit swarm intelligence while flying, known as flocking. Flocking behavior results from individual birds following three simple concepts: separation, alignment, cohesion. By attempting to keep a constant distance from and similar alignment with neighboring individuals, complex, unpredictable flight patterns can emerge.
Emergent behavior can involve individual organisms working together in the form of a single force or entity. In these situations, there needs to be a means to organize the group, especially when direct communication is not possible. One way of countering this problem is to utilize stigmergy. Stigmergy is a method of communication in which the individual parts in the system communicate with one another by modifying and sensing their local environment.  .
This term was originally coined by French entomologist Pierre-Paul Grassé in the late 1950’s in reference to the ability of social insects such as termites, ants, and bees to coordinate their efforts.  It allowed Grassé to explain how these insect species were able to carry out complex projects without direct interaction, while also having limited intelligence.  Termite mounds, anthills, and beehives are just a few examples of stigmergy structures built by these insects. Pheromone trails used by ant colonies when searching for food sources are another example of stigmergy use by social insect species.
The term stigmergy was used only to explain insects’ communication behaviors until around 1990, when Jean-Louis Deneubourg, who was a member of the Brussels School of Complex Systems, proposed that stigmergy was a broader example of “spontaneous organization”.  From that point forward, stigmergy was increasingly used to explain to many other types of emergent behaviors, from swarm intelligence to computer network communications.
Stigmergy in Biology
While it is apparent that stigmergy is prevalent in insects, the phenomenon is also well documented within plants and animals. The internal functions of cells are organized with the help of their cytoskeletons, which are stigmergy structures.  On a slightly larger scale, the entire bodies of plants and animals are stigmergy structures. Cells within these organisms deposit cues in the form of messenger molecules throughout the body. Examples of where these cues could be placed include tissues, bones, bark, and many more. Another key stigmergy feature in multicellular organisms is the brain. Neurons and synapses undergo physical changes when long-term memories are created.  These neurons and synapses are used to form behaviors, which affects memory, and so on. These patterns are made possible with the principle of stigmergy.
Stigmergy in Society and Technology
This broader form of stigmergy can be witnessed in a continuously growing number of applications. In fact, stigmergy is so common in social settings that it may be harder to find examples to which stigmergy does not apply.  It is easiest to see stigmergy in technology, most notably within computers. Digital memory, such as RAM, ROM, databases, or even the entire Internet all rely on stigmergy when data is saved and later used to perform tasks.  Another example includes the virtual environment of a wiki, such as Wikipedia or Lentis. By adding one user’s idea to an article (such as this emergent behavior article), other users encounter this and either modify or add to it . This eventually leads to a complicated system of ideas.
An Example of Emergent Behavior in Humans - Landscaping
Emergent behavior can be witnessed in all walks of life. One example is how a pathway can be created during a construction project. Pictured to the right is two, parallel sidewalks on McCormick Road at the University of Virginia. The sidewalk on the right is the original sidewalk, while the one on the left is newly developed. During the fall semester semester of 2015, a construction project blocked off the original sidewalk on the right. Students walking through this area either could walk on the road (far right) to get around the project, or they could walk through the grass. Students choosing to take the safer route through the grass began to form a worn down trail, and as a result the construction crew decided to add the newly developed sidewalk to provide a new path around the construction. This is why students walking down McCormick Road will come across this odd pair of parallel sidewalks.
This is an example of emergent behavior in humans and landscaping. The students who walked around the construction on the worn down trail did not necessarily expect a sidewalk to be built there. Each student was acting on a simple idea: he or she wanted to reach the other side of the construction safely. Over time, they created a trail, similar to a trail created by ants foraging for food. This example shows how emergent behavior can be used in landscaping. Instead of having a project manager planning out where to build a sidewalk at the beginning of a construction project, there may be times when he will wait and see where individuals within the area create trails. These trails, which are formed by individuals simply trying to quickly get from one point to another, are used to decide where to pave the final sidewalks on a construction site.
While self-organization, swarm behavior, and stigmergy are all major categories within the broader topic of emergent behavior, there are many more topics worthy of research. Topics could include the history of emergent behavior, emergent behaviors within cities or traffic, and other emergent behaviors in humans ranging from the stock market to consciousness.
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- Descartes, R. (199). Discourse on the method of rightly conducting the reason, and seeking truth in the sciences. Raleigh, N.C.: Alex Catalogue.
- Follman, M. (2014, August 27). Michael Brown's Mom Laid Flowers Where He Was Shot—and Police Crushed Them. MotherJones. Retrieved December 13, 2015, from http://www.motherjones.com/politics/2014/08/ferguson-st-louis-police-tactics-dogs-michael-brown
- Associated Press. (2014, August 13). Ferguson protests erupt in violence as people lob Molotov cocktails, police use tear gas. Cleveland. Retrieved December 13, 2015, from http://www.cleveland.com/nation/index.ssf/2014/08/ferguson_protests_erupt_in_vio.html
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- Francis Heylighen (2012). Stigmergy as a generic mechanism for coordination: definition, varieties, and aspects. Evolution, Complexity, and Cognition group. Vrije Universiteit Brussel. http://pespmc1.vub.ac.be/papers/Stigmergy-WorkingPaper.pdf