Last modified 3 November, 2008. |

## Research Interests

### Concise Version

&bull Discrete dynamics &bull Complex networks &bull Fundamental laws of biology

My co-workers and I use statistical mechanics to study complex systems in physics and interdisciplinary fields, including systems biology.

### Detailed Version

Discrete dynamics:

&bull Dynamics of small networks
&bull Cellular automata
&bull Self-assembly, competition

Complex networks:

&bull Weighted & directed networks
&bull Biological & social networks
&bull Inferring genetic networks

Fundamental laws of biology:

&bull Microarray expression
&bull Coding & non-coding DNA
&bull Pre-heteropolymer evolution

Discrete dynamics

We are interested in the discrete time evolution of systems in which elementary interactions give rise to non-trivial spatial or temporal patterns.

A major focus is understanding how network connectivity constrains network dynamics. Unlike much of the past work on the subject, which considers the large, disordered network limit, we are studying exact dynamics over the space of architectures for small networks (2 - 5 nodes).

We are also interested in probabilistic cellular automata (PCA): elementary cellular automata in which the update rules are random variables. We have recently solved this for the simplest category of PCA: those that are random in time and for which the mean field approximation is exact.

More recently, we have studied a simple model of competition among a collection of players. Each player has a fixed strength. Randomly selected pairs of players compete, the stronger one wins and the loser is eliminated. We show that the best indicator of future success is not the number of wins but a player's wealth: the accumulated wealth of all defeated players.

Complex networks

Complex networks are found in the brain, genetic regulation and transcription, the World Wide Web, transport and traffic and ecological food webs.

We have worked on generalizing quantities used to characterize classical networks to weighted and directed networks. Directed networks - in which every edge is an arrow, rather than an undirected link - exhibit a much more complicated connectivity than undirected networks. We show that a set of four directed clustering coefficients provide a useful space for classifying a wide range of real-world directed networks, ranging from social networks to transcription networks, language networks and food webs.

We are using these ideas to identify biological and other emergent structure in naturally occurring and designed networks.

A related interest is the inference of genetic networks from genomic expression data.

Fundamental laws of biology

'Fundamental laws of biology' is a phrase coined by DARPA to describe the application of existing and new mathematics to identify fundamental laws that govern biology across multiple time and length scales.

In most eukaryotes, a large proportion of the genome does not code for proteins. The non-coding part is observed to vary greatly in size even between closely related species. We report evidence that eukaryotes require a certain minimum amount of non-coding DNA (ncDNA), and that this minimum increases quadratically with the amount of coding DNA.

A subject which I have not begun working on, but would like to, is pre-heteropolymer evolution. A serious theory of evolution must address evolution prior to the existence of the central dogma of biology - that information flows from DNA to RNA to proteins. In other words, we need a theory of evolution in the absence of heteropolymers. Progress in this area will likely be at the intersection of information theory, thermodynamics, artificial life and the evolution of model organisms.