Complex Systems is a broad and rapidly growing area of research.
In general, its studies include:
 ·  behaviors of systems as a whole (e.g., ant colonies)
 ·  behaviors of individual actors within a system (e.g., worker ants)
 ·  and how actor-level interactions give rise to system-level properties.

It is divisible into a many subfields, six of which I introduce here:

Network Science

Posts about Network Science

Network Science includes the theory, application, and math behind graphs.
These are simplified models made up of only two types of components: nodes/vertices connected together by links/edges.

Network Analysts seek to define and calculate various metrics that may help explain a given node’s importance within the network, how robust the network is against pandemic, and so on. A few common metrics are:

  • Degree: The degree of a node is the number of nodes it is directly linked to.
  • Betweeness: The betweeness of a node is the number of shortest paths between all nodes that pass through it. *PageRank: Named after Google’s Larry Page, the PageRank of a node is based on the chances that a user will visit it by randomly “walking” from one node to another.

Destroyed Termite Mound

Emergent Behavior

Posts about Emergence

Emergence is the study of how the behavior of individuals working together in large numbers leads to system-wide effects that do not occur when the individuals work alone.
For example, a single termite is capable of building small hills out of sand, pebbles, and other materials; a colony of these pesky insects working in unison are able to produce structures of startling complexity, without any central direction at all!

We say these complex structures are an emergent behavior of the colony as a whole, a product of the combined behaviors of individual termites.

Further examples include:

  • birds flocking together without central direction
  • Spiderman’s villian Sandman arising from individual grains of sand
  • and medieval city patterns arising without central planning.

Special thanks to @BrentBrewington and @svalver for providing examples.

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Pattern Formation

Posts about Pattern Formation

Pattern Formation is an emergent behavior of a system that is both visual and statistically orderly.
These patterns have received special attention because they can be numerically modeled by partial differential equations and other methods.

These studies have produced insightful equations for such patterns as:

  • the intricate detail on seashells like Oliva Porphyria
  • irregular hexagons on the surface of cooking oil
  • and the symmetrical construction of crystals and snowflakes

Butterfly Effect

Chaos Theory

Posts about Chaos Theory

Chaos Theorists concern themselves with how slight changes in initial settings lead to drastic changes in the final product.
A common analogy is the Butterfly Effect, where a butterfly’s flapping causes tiny changes that lead to disastrous weather conditions in the long run.

Chaos is also a contrast to Emergence, where interaction between individuals have higher effect than initial settings. Like Pattern Formation, though, it is mathematically grounded; therefore, its definitions tend to have precise meaning.

A few definitions are:

  • Chaotic: A system is chaotic if approximations are not enough to make predictions, but precise measurements are.
  • Fractals: A fractal is a geometric figure made up of an infinite number of similar copies of itself. The Sierpinski Triangle is a well-known example.
  • Strange Attractors: An attractor is a point or region a system is drawn towards; for example, a pendulum is drawn towards hanging straight down. A strange attractor is one whose borders are fractals.

Tic-tac-toe-full-game-tree-x-rational

Game Theory

Posts about Game Theory

Game Theory joins Economics with Artificial Intelligence, mathematics with decision making.
A game between two or more players is typically modeled by a game or decision tree. These visuals map out each available move to each player for each turn of the game.

Once the game has been mapped into a tree, victory is just a matter of searching for “good” branches to follow.

Notable games include:

  • Chess, whose game tree contains more decisions than atoms on Earth (explaining why computers haven’t beaten us yet)
  • the El Farol Bar game, where everyone wants to go to the bar on the same night, but if too many go, it’ll be less fun than drinking alone at home
  • and the Prisoner’s Dilemma, a famous Economics problem where working together pays well, but stabbing someone in the back pays better (so long as the players don’t turn on each other).

Systems Thinking

Posts about Systems Thinking

Systems Thinking requires an understanding that all things are part of a larger system and are constantly interacting. It asks us to think outside of “mechanical thinking,” where cause and effect are neatly explained by inputs and outputs.

Thinking systemically, processes are oftentimes seen as the root “causers” of catastrophes, not just the individual actors that carrying them out. This can be uncomfortable ethically, because when a process is as fault, who do we hold accountable?

As an exercise, consider the following and ask, “Who’s to blame?”:

  • market failures, like the Great Depression
  • natural disasters, like Hurricane Katrina
  • and epidemics, like Ebola ∎