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.
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.
Rule chosen: 101