Network Dynamics
Channel: Systems Innovation
Date: 2015-04-29

Take the full course: https://bit.ly/SiCourse
Download booklet: https://bit.ly/SiBooklets

Twitter: http://bit.ly/2JuNmXX
LinkedIn: http://bit.ly/2YCP2U6
Almost all real networks are dynamic in nature and how they have evolved and change over time is a defining feature to their topology and properties. As network theory is a very new subject much of it is still focused on trying to explore the basics of static graphs, as the study of their dynamics results in the additional of a whole new sets of parameters to our models and takes us into a new level of complexity, much of which remains unexplored, and is the subject of active research.



Transcription excerpt:
So lets start by talking about growing a random network to see what it looks like, when we say growing a network we might mean adding more nodes to it, but also, more interestingly adding links to it, that increases the overall connectivity. In our random model links were just placed between nodes at random with some given probability, growing the network here just meant increasing this probability so as to have more links develop over time. One interesting thing we find when we do this is that there are thresholds and phase transitions during the network's development, by thresholds we simply mean that, by gradually increasing our link probability parameter, some property to the network suddenly appearing when we pass a critical value. 

So for example our first threshold is when the average degree goes above 1 over the total number of nodes in the network, as at this threshold we start to get our first connection. At degree one that is when every node has on average one connection the network stars to appear connected, we see one giant component  emerging within the network, that is one dominant cluster and we start to have cycles, which means there are feedback loops in the network. Another threshold occurs when nodes have an average degree of log(n) at this point everything starts to be connected meaning there is typically a path to all other nodes in the network. So this is what we see in random network but as we know most real world networks are not random as they are subject to some resource constraints and they have preferential attachment giving them clusters that we do not see in these random graphs.

One way of thinking about how real world networks form is through the lens of percolation theory, percolation theory looks at how something filters all percolates through something else like a liquid filtering through some mash structure in a material or we might think about some water running down the side of the hill, as it does the water will find the path of least resistance creating channels and furrows in the side of the hill. This network formation is then the product of the resource constraints that its environment placed upon it, but the constraints are unevenly distributed and the networks topology is then reflecting this as it follows the paths of least resistance as it avoids toughest material.

In order to demonstrate the general relevance of this we will take some other examples, if say we put on a cheap flights from one city to another then people will start using that transportation link because of financial constraints. Or because of phenomena of Homophily within social networks we will get the same peculation dynamic where it will be easier for people to make links with people who are similar to themselves than with others, again creating a particular structure based on the social constraints within the system.