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### Research area: Nonlinear Dynamics, Graphs, and Pattern Formation

#### Nonlinear Dynamics

Linear systems are typically predictable and unable to produce nontrivial behavior. For this a nonlinear dynamics is required. Even such a simple system as a monotonous map of a circle upon itself can produce most interesting behavior, in the form of an intricate pattern of mode-lockings as a function of one parameter.

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.

#### Random Graphs

Networks appear in many areas of science, technology and real life. A network can be modelled by a mathematical graph, consisting of nodes (points, vertices) representing elementary units, and links (lines, bonds, edges) representing relations, connections or interactions between pairs of units.

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 Classic Model of Erdös and Rényi, where independently for each pair of distinct nodes a link exists with a fixed probability c/N, where c is an adjustable parameter governing the average degree of nodes, i.e. the average number of connections. This model yields a Poissonian degree distribution in the limit of a large network.

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 Random Graphs with a given Degree Distribution, where an arbitrary degree distribution is chosen, and a random graph of given size is defined by drawing the degree for each of the N nodes from the given distribution, and then links are defined by pairing their connections randomly. While there is no limitation on the degree distribution in this class of models, it suffers from a certain lack of correlations between the edges in the resulting graphs.

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 hidden coloring, i.e. to allow for the elementary building blocks of a graph to come in different types - or colors - that are allowed to affect the edge probabilities. The coloring can be thought to reflect some observable properties of a network, but I prefer to view it as a hidden set of variables with the sole purpose of enabling a richer and more general statistical structure.