## Complex network

The scientific study of networks is an interdisciplinary field that combines ideas from mathematics, physics, biology, computer science, the social sciences, and many other areas. In online databases one can find examples of computer networks, social networks, and biological networks (e.g. the structure of primitive brains).

A project on any level could include studying the properties of one of these networks, comparing two different networks, or generating a new type of network and compare it to existing ones. Is it a small-world network? Is it more clustered that one would expect? Does it contain communities? Here, one can either adopt a very mathematical approach based on linear algebra, or develop computer algorithms to explore the networks.

*Suitable for:* First year project, Bachelor project, Master’s project

*Main disciplines: *Complex systems, network science, programming (any language), linear algebra

*Related work:*

## The Ising model

Perhaps the most beautiful and elegant model of all times. Atoms of spin up and down are arranged on a 2D-lattice, interacting only with their nearest neighbours. At low temperatures all spins are aligned. At high temperatures, spins are randomly distributed. At a certain critical temperature, a phase transition occurs, which captures all the essential features of a real magnetic system. The model has been studied and build upon for almost a century and keeps fascinate and surprise scientists world-wide. There is plenty to be done and much to be learned!

*Suitable for:* First year project, Bachelor’s project

*Main disciplines: *Programming (any language), complex systems, phase transitions

*Related work:*

## Fractals, diffusion limited aggregation, and viscous fingering

For decades the beautiful and complex properties of fractals have continued to fascinate scientists and the public alike. Fractals have an non-integer dimension, they are self-similar, and they can be found many places in Nature. A first year project about fractals can include a mathematical description of different dimension-measures, investigation of some intuitive geometric tools like the Poincaré map, examples of strange attractors like the Lorentz attractor, and perhaps computer simulations illustrating some fractal properties of diffusion limited aggregation. It is also possible to create beautiful fractals experimentally through viscous fingering.

*Suitable for:* First year project

*Main disciplines: *Complex systems, mathematical physics, programming (any language), experiments

*Related work:*

## Sandpile models on networks

Sandpile models were the first models found to exhibit self-organized criticality. Grains of sand are added on a lattice, one at a time, until the amount of sand on a site exceeds a specific threshold value at which time that site collapses transferring sand into the adjacent sites. If this causes the amount of sand there to exceed the threshold, an avalanche of sand may start.

Sand leaves the model through the boundaries of the lattice. On a network you have no boundary, so you can either add ‘boundary nodes’, let sand dissipate gradually over time, during avalanches, or at the end of avalanches. How does this change the model?

*Suitable for:* First year project, Bachelor’s project, Master’s project

*Main disciplines: *Programming (any language), self-organization, complex systems, networks

*Related work: *

## Multiple disease spread model

In my master’s thesis I studied a multiple disease spread model, which self-organize to a state where the disease clusters get fractal shapes. Much additional work can be done on this, or similar, models. What will happen if diseases can be lethal? If parts of the network is vaccinated? If the network is scale free? Or if some diseases are so similar, that cross-immunization becomes relevant?

*Suitable for:* Bachelor project, Master’s thesis

*Main disciplines: *Programming (Java), percolation, epidemiology, self-organization.

*Related work:*

Locally self-organized quasi-critical percolation in a multiple disease model

A minimal model for multiple epidemics and immunity spreading

Introduction to percolation theory