Nonlinear Dynamics, Graphs, and Pattern Formation
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.
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
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
Recent and upcoming
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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.