From the Director
Networks

Doug Arnold, 
IMA Director 
The concept of a network is among the most frequently recurring in
recent IMA programming. Communication networks, power networks,
transportation networks, sensor networks, social networks, gene
regulatory networks, and protein interaction networks are but a few of
the examples that have been explored in recent IMA programs.
This should have become clear to me as soon as I arrived at the IMA
in 2001. The first two workshops during my tenure here were
Hot Topics workshops on Mathematical
Opportunities in LargeScale Network Dynamics and
Wireless Networks.
And the first technological problem I dealt with as director here was
the establishment of the IMA wireless network, which is now a widely
used part of the IMA information infrastructure. Just last week
it was extended beyond the IMA facilities to the Radisson Hotel
used by most of our visitors.
Mathematically a network is graph or digrapha collection of nodes and
connections between them by (possibly directed) edgesoften endowed
with additional information associated to the edges or nodes, such as a
bandwidth capacity linking two routers on the Internet. This simple
mathematical abstraction, which harks back at least to Euler, underlies
all the examples above. Take for example their role in modeling the
spread of disease, the topic of a workshop entitled Networks and the Population Dynamics of
Disease Transmission, which took place from November 1721, 2003 at
the IMA. Mathematical epidemiology, a subject that began with Daniel
Bernoulli's study of smallpox spread in 1760, was dominated in the 20th
century by the SIR (susceptible/infected/recovered) model of Kermack
and McKendrick, a system of ODEs that describes the population only in
terms of a small number of discrete groups of individuals, assuming
"perfect mixing" of the population. Actually the details of the
contact networks of individuals play a massive role in disease
transmission (perfect mixing is not a very attractive model
for sexually transmitted diseases!). The relevant network has as nodes the individuals in the
population and edges which describe relevant contacts between them, and understanding
this network and incorporating its characteristics into the
epidemiological model is crucial to predicting and controlling the
spread of disease. There are many impediments to obtaining such
understanding. These are big networks with the relevant number of
nodes and edges in the thousands, millions, or billions (how large is
the largest connected component of the sexual contact network?), and
are frequently highly timedependent. By contrast Euler's Konigsberg
bridge graph was static and had four nodes and seven edges. The large
dynamic networks in today's applications can rarely be exactly known
and statistical and probabilistic methods are necessary. As was clear
at the November workshop, which brought together statisticians, graph
theorists, epidemiologist, sociologists, and many others, mathematics
has many toolsrandom graph models, measures of degree and degree
distribution, of locality, transitivity, correlation, centrality, and
similar properties, simulation methods such as Markov Chain Monte
Carlo, Bayesian inference methods, etc., etc.which can help elucidate
the different network properties that contribute to disease spread.
The wonderful thing about taking a mathematical point of view is the
commonality it brings. The same ideas which can be used to model the
spread of disease can be applied to the spread of rumors, or
technology, or terroristic ideologies. But even more generally, many
network modeling, analysis, and simulation ideas apply equally
well to systems as diverse as the World Wide Web, which can be modeled
as a digraph whose nodes represent web pages and whose directed edges
represent hyperlinks, and the protein interaction network of a cell
whose nodes represent proteins in the cell and whose directed edges
describe various types of chemical interactions between them. The
poster image for this years program on Probabilistic and Statistical
Methods in Complex Systems, a detail of which appears here (click
on it for more), is a reminder of this. While at first
glance it seems to be organic and alive, it is actually a visualization of
roundtrip time measurements on the internet.
The power of mathematical abstraction and its ability to bridge
disciplines is at the heart of the workshop Robustness of Complex Systems, which
will take place at the IMA February 913, 2004. This workshop will
bring together Internet researchers, scientists working on a range of
different technological networks, biologists and biophysicists,
mathematicians, and computer scientists and provide them with the
opportunity to compare notes and share insights from engineering theory
and practice that can shed new light on biological complexity or from
biology that can illuminate existing mysteries associated with the
complexity of largescale engineered networks.
Networks are sure to be a central theme at the IMA for many years to
come. Just as recent activity has focussed on merging network theory
with mathematical epidemiology, many other mergers cry out for the
attention of mathematical scientists. For example, consider the
interplay of optimization and networks as in last years workshop on Network Management and Design,
or control theory and networks as in the upcoming workshop on Control and Pricing in Communication and
Power Networks, or game theory and networks (how does one manage
the internet to maximize throughput or fairness or load balance if
there are smart hackers willing to tweak their networking hardware and
software to gain an advantage?).
And, of course, another kind of network which is at the
heart of the IMA's mission: the collaboration network. Best known to
mathematicians by the whimsical measure of centrality known as the
Erdös number, networks that measure collaboration among scientists
are the subject of serious study (e.g., by recent IMA visitor Mark Newman, who
published "The structure
of scientific collaboration networks" in PNAS a couple of years ago). Just as air
transportation has led to connections of geographically distant
populations in the networks
of disease transmission and forever changed the ways diseases spread
across the planet (think SARS), the IMA is fundamentally in the business of
fostering the connections of scientifically distant researchers to bring
about changes in the advancement of science.