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Wolfram Summer School


Gerhard Werner

Summer School

Class of 2008


Gerhard Werner is a physician with specialization in psychiatry and neurology. His main activity during academic work is neuroscience research. He is presently a faculty member in the Department of Biomedical Engineering of the University of Texas at Austin, TX.

Project: Network growth with a population of heterogeneous vertices

The long-standing tradition of graph theory, concerned with the study of the mathematical properties of nodes (vertices) connected by edges (links), has in recent years spawned intense research activity on the statistical mechanics of networks. This field of research seeks to determine the factors that govern the growth of networks, resulting from the stepwise accretion of new nodes to existing nets. Much of this work was motivated by the need to account for the patterns of growth of the internet.

Several basic rules were established, including that creation of new links by the added nodes can follow three distinct patterns: 1) random attachment, 2) preferential attachment on a statistical basis to nodes that already have many links (“the rich getting richer”), and 3) depending on what is designated “fitness” (i.e. an inherent property of nodes) in combination with preferential attachment. In case 2, nets that have grown in this way show a characteristic distribution of links that follows a power law; thus they are called “scale free.” In case 3, Bianconi and Barabási ascertained that the statistical distribution of the edges obeys the Bose-Einstein statistics that had been characterized for the energy distribution in a particular class of microphysical particles (the bosons). In microphysics, bosons have the capacity under extremely low temperatures to condense to large assemblies, attaining macrophysical properties. In this capacity they support superfluidity and superconductivity (the Bose-Einstein condensates). Bianconi also showed that networks with Bose-Einstein connectivity undergo phase transitions, in analogy to the Bose-Einstein condensates in microphysics.

In a separate line of research, Watts and Strogatz discovered a different kind of network, consisting of predominantly short links between next or next-to-next nodes. This class of networks has very distinct properties in terms of facilitating rapid information transfer among network constituents. Networks of this class have gained prominence in neuroscience since it is now established that connection patterns among brain centers fall into this class

The question of this project is this: are small-world networks capable of forming connection patterns with Bose-Einstein statistics, and are they thus also candidates for state transitions associated with it?

If it were possible to characterize Bose-Einstein condensation in these networks, it would be evidence in support of the notion that the architecture of brain pathways is suited for enabling phase transitions in brain networks, as has been suggested by Gerhard Verner and others as underlying neural processes supporting cognition.


Bianconi, G., and Barabási, A.-L. “Bose-Einstein Condensation in Complex Networks.” Physical Review Letters 86, no. 24 (2001): 5632-5635.

Watts, D. J., and Strogatz S. H. “Collective Dynamics of ‘Small-World’ Networks.” Nature 6884, no. 393 (1998): 440-442.

Favorite Radius 3/2 Rule

Rule 101