Nonlinear Dynamics, Graphs, and Pattern Formation

When considering a nonlinear dynamics for a set of variables living on the nodes of a network (like a two-dimensional lattice), the variables may get stuck in arrays of dots, stripes, or other quasiperiodic patterns, or even get attracted to peridoc motion. Such dynamics seems to be used by Nature to produce patterns in biological tissue. Although various models of this type have been studied, this has mostly been done using numerical methods. My interst here lies with developing and using analytical methods to gain a more profound understanding of the mechanisms behind pattern formation and other nontrivial phenomena involving nonlinear dynamics and transport, such as disease spreading and traffic jams.

Some real-world networks are more or less ordered, with a regular, predictable structure, e.g. in the form of a lattice. In other networks such a regular structure is completely absent. There are also intermediary forms.

When a networks is formed, the formation mechanism often contains random elements, and the resulting network can be seen as a kind of random graph. My interest is focused on truly random graphs, where an underlying regular structure is absent.

A random graph really stands for a statistical ensemble of graphs, i.e. a set of graphs on which a probabilistic measure is defined, yielding a definite probability for each possible graph.

Most graphs of interest are sparse, i.e. the number of connections of each node typically stays finite even when the number of nodes becomes large.

The mother of all random graph models is the

However, the classic model does not describe most real-world networks, and workers in the field have turned to more general models of random graphs. Today a multitude of models exist, most of which are more or less specialized for describing a certain target type of networks

A problem with the existing models of random graphs, in particular for the possibility of statistical inference of models from the observed properties of real networks, is the lack of a general formalism for random graph ensembles, where more specialized models appear as special cases of a unifying model class.

The most commonly studied class of models that possess some degree of generality is

My recent research on random graphs is centered on the possibility of defining classes of random graph ensembles, general enough to contain more specific models, yet simple enough to allow for the computability of observable local and global structural properties. One possibility is to extend existing models by means of a

- Applications of network theory: from mechanisms to large-scale structure, Nordita, Stockholm, 2011.
- Discrete Probability, Institute Mittag-Leffler, Stockholm, 2009.
- Nordic Workshop on Networks, Nordita, Copenhagen, 2004.
- Science of Complex Networks, Aveiro, Portugal, 2004.
- Random Geometry, Krakow, Poland, 2003.