Theory of Asynchronous NetworksIntroduction and background
Many problems in technology, engineering and biology can be modelled by networks of
interacting subsystems or `nodes'. These networks often exhibit dynamics that
cannot be adequately reproduced using classical network
models given by smooth dynamical systems and a fixed network structure or topology.
Asynchronous networks are an approach to network dynamics
that take account of features encountered in networks from
modern technology, engineering, and biology, especially neuroscience.
For these networks, dynamics can involve a mix of distributed and decentralized control, adaptivity,
event driven dynamics, switching, varying network topology and hybrid dynamics
(a mix of continuous and discrete dynamics). Network dynamics will typically be piecewise smooth,
nodes may stop and later restart and there may be no intrinsic global time  there may be local clocks
that are not globally synchronized over the network.
Crucially, these networks often have a function: transportation networks bring people
and goods from one point to another, neural
networks may perform pattern recognition or computation, and power grid networks may be
optimizing cost and balancing load and power generation in a context of varying energy supply
(wind power and other renewables) and loads.
The key goal of the project was to
"Find conditions on asynchronous networks yielding functionality
with no deadlocks or race conditions."
Asynchronous Networks
(Large parts of this work were done in collaboration with Christian Bick.)
During the first year of the project, a transparent mathematical model was developed for asynchronous
networks that incorporated most of the features seen in contemporary network problems. Dynamically
speaking these networks have close connections with parts of the theory of nonsmooth dynamics currently
being actively investigated in the mathematics and engineering communities and originated by the
Soviet school of mathematics from the 1960s. A key advance was
formalizing the idea of a functional asynchronous network: a network with a specific function.
A major breakthrough was obtained in the second year of the project when a foundational result was obtained that
clarified the structure of functional asynchronous networks and went far beyond the original goals of the project. The result allows for a reductionist approach to
asynchronous networks. Specifically, a large class of functional asynchronous networks was identified
for which one can describe the function
of the network in terms of the function of constituent subnetworks. The result answers a question originally raised by the systems biologist Uri Alon
in connection with gene transcription networks (from page 27 of Alon's 2007 book on systems biology):
"Ideally, we would like to understand the dynamics of the entirenetwork, based on the dynamics of the individual building blocks."
The underlying premise behind Alon's comment is that
a modular, or engineering, approach to network dynamics is feasible:
identify building blocks, connect together to form networks and then describe
the dynamical properties of the resulting network in terms of the dynamics of its components.
While it is wellknown by mathematicians and physicists that in classical nonlinear or complex systems this reductionist approach
will not work, engineers do successfully couple nonlinear gadgets together to make more complex systems with specified properties.
It is natural to ask if it is possible to reconcile these viewpoints.
Our main result gives a resolution of this dichotomy and shows it is possible to describe
the function of a large class of asynchronous networks in terms of the function of constituent
subnetworks. Replacing the emphasis on dynamics by an emphasis on function enables
the use of reductionist techniques in the analysis and design of networks that figure
prominently in modern technology and science.
Articles describing these results, general theory, and applications, have been submitted for publication. We
refer to publications for details and links to preprints.
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