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Complex and Simple Systems

So far, the course has focused on low-complexity, low degrees of freedom systems, such as pendulums, electronic circuits, and apples falling from trees. We have also looked at low degrees of freedom yet high complexity systems, with coupled pendulums, Lorenz systems, and 3-body problems.

As we move further right on the graph of interacting elements, or degrees of freedom, we start to see models which can approximate social interactions, economic markets, ideal gases, magnets, social networks, formations of galaxies, climate dynamics, and economics in general.

These can be viewed on a graph:

Traditional View vs Network View

The traditional view of complex systems is with systems of differential equations:

\[ \frac{dx_i}{dt} = F_i(x_1(t), x_2(t), \dots, x_n(t)) \]

Here, \(x_i\) consists of the variables, elementary units, or agents that make up the complex systems, and are the individual coefficients in the differential equations. If we now consider these variables \(x_i\) as the vertices of a graph \(G\) with edges as the weighted links of \(G\), then we can model these systems differently to the differential equations as given earlier.

Applications of Networks

Recently, the structure of different networked systems, e.g., the WWW, IP networks, biochemical, social, etc. have all been captured and described. Whilst each network serves a distinct purpose to another, they all broadly share the same common properties.

Power Grids

Power grids are a type of network, with power generating stations, substations, and businesses and homes comprising the nodes of the network, with the interconnects between the two as the edges of the network graph. In this type of network, the main concerns are the vulnerability to attacks and the resiliency of the network.

Also, more and more, we are looking at the capacity of the network to handle power generation distributed much more evenly across the grid, whilst at the same time considering how electric vehicles will also affect the grid.

Airline Transportation Networks

If we can understand the flows of people, then we can begin to look at why they migrate, and the spread of epidemics globally. There is a wealth of readily available data on this type of network, as all commercial flights are tracked, and border enforcement log all entries and exits at specific borders.

Trade Networks

Let's briefly consider trading networks at a country level, where each country is a node, and each edge of the graph is the monetary flows between each of the countries. If we can understand the patterns in economic activity, then we can begin to better understand economic growth.

TBC need to watch the lecture videos as slides are hard to understand alone