Emergence

 

A flock of starlings
The shapes and patterns are an example of emergence.

 

Generally, complexity science is the study of emergence, and complex systems are systems which undergo or demonstrate emergence. Complex networks are networks which undergo emergence.

 

Emergence refers to the unveiling of macroscopic states or observables which do not exist at the microscopic level. At the molecular level, water does not have 'wetness'. Snowflake crystallization patterns are also emergent, in that they are macroscopic, as a. result of H2O molecule properties, temperature, and the large number of water molecules. There are a myriad examples if you look around.

 

The macroscopic states also have to do with the combinatorics of the system. For example, there are a combinatorially huge number of ways to arrange water molecules, but many of these would not macroscopically appear to be snowflakes. So the number of arrangements which we accept as snowflakes is very small compared to the total number of arrangements. Each arrangement is called a 'microstate', and the set of arrangements corresponding to some large scale state is called a a 'macrostate'. Ice, liquid water, and gas are three macrostates of water.

 

You can see this with coin-flipping. With a fair coin, there are 2^100 ways to flip it 100 times. These are all the microstates. But how many ways are there to flip 50 heads and 50 tails? We can count (100 choose 50) ways to get 50 heads out of 100 flips. So the probability of 50 heads is (100 choose 50) / (2^100), which is 0.0796. If you do the same calculation for 49 heads and 51 tails, and so on, you will see that the probabilty is less.

 

So the macrostate is dominated by the 50-50 result. This is an emergent property, resulting from the fair coin and flipping it 100 times. Just as an individual water molecule doesn't have wetness, a single coin flip does not have 50 heads and 50 tails. The tendency to 50 heads and 50 tails is revealed (emerges) by scaling up the number of flips.  Also notice that predicting the macroscopic result by seeing the microscopic system elements is not always obvious or easy. This is often true in systems where the scaling up is on many orders of magnitude.

 

So, 'more is different', as Philip Anderson famously said in 1972.

 

Formation of the
Giant Connected Component (GCC)
at center, in a random graph.

Some graphs/networks undergo emergence, the classic example being network percolation in random
(Erdos-Renyi) graphs, where the size of the giant connected component (the largest connected set of nodes) suddenly explodes when the network connectivity increases to a critical density.

 

This may not seem interesting at first, but it is actually very deep, since it means that any time I start having random interactions between any entities (people, atoms, grains of sand), if the density of those interactions reaches a critical point, I will very suddenly start to have system-wide interactions (interactions which scale proportional to the size of the system itself). Most real interactive systems are far from random, so this gives us a worst case situation.

 

Network Science approximately is the study of networks, which tend to be naturally-occurring networks, vs. graph theory the study of graphs, which are mathematical objects. There is a lot of overlap between these two, since they often look very similar or have similar properties, so they are used more or less interchangeably (e.g. above).

 

So, we can study networks but not focus on the emergent aspects of the system. But of course in networks, there are so many possible ways to have various kinds of emergence, that it becomes implicit that network science is about complexity.

As an exercise, calculate the most probable sum of rolling a 6-sided die 100 times.

 

For more on emergence, which is a fascinating topic, check the book Complexity Science by Henrik Jensen.