Biology Reference
In-Depth Information
represent cities, counties and in general municipalities or
urban aggregations coupled by connections that correspond
to the commuting flows of individuals. The analysis of these
networks uncovered rather homogeneous topologies
molecules or fluid elements do not care about the multi-
scale nature of turbulent fluid. However, the collective
dynamic behavior and our ability to conduct mathematical/
computational analyses of techno-social systems are con-
strained by the heterogeneous and multi-scale character-
istic of the system, and we must develop appropriate
formalisms and techniques, as have done researchers
studying multi-scale physical systems
[30]
(fluids, solids,
distribution of masses in the universe, etc.). In the context
of networks and techno-social systems, the multi-scale
challenge is making its appearance now because of the
availability of large-scale datasets. To achieve analytical
understanding of techno-social systems and approach them
computationally, we must find different strategies to deal
with dynamical behavior/equations that work at very
different characteristic scales but still influence each other.
e
mainly due to strong spatial constraints
associated with
very large fluctuations in individuals' travel flows
[28]
.
Finally, the most global scale is characterized by the air
connections infrastructure, composed of airports (nodes)
and direct flights among them (links). Data representing the
travel flow of passengers defines the weight to each
connection
[27]
. This transportation network displays
several strong levels of heterogeneity. The distribution of
degrees (i.e., of the number of connections of an airport) is
scale free and the traffic is very broadly distributed, varying
over several orders of magnitude
[27,29]
. This points to
a structure composed of airports having large fluctuations
in their number of connections to other airports and,
moreover, to the number of passengers traveling on a given
route, ranging from a few to millions of individuals in
a given period of time.
Figure 27.2
shows two networks for human mobility at
different scales ranging from the airline transportation
network at the distance of hundreds and thousands of
kilometers and several days to the commuting network that
connects neighboring cities within the span of a few hours.
We see that the scale and complexity extend at all granu-
larities, creating a huge multi-scale network where the time
and spatial separations range from a few hundred meters and
a few minutes to thousands of kilometers and several days.
The challenge in providing a holistic description of
multi-scale networks is the necessity of dealing with
multiple time and length scales simultaneously. The final
system's dynamical behavior at any scale is the product of
the events taking place on all scales. The single agent
spreading a disease or single node is apparently not affected
by the multi-scale nature of the network, just as single
e
CONTAGION PHENOMENA IN COMPLEX
SOCIAL NETWORKS
Contagion processes are usually seen as a transmission
process, either as a pathogen that spreads from host to host,
or a piece of information/knowledge that is transmitted
during social interactions. Let us consider the simple sus-
ceptible
recovered (SIR) epidemic model. In
this model, infected individuals (labeled with the state I)
can propagate the contagion to susceptible neighbors
(labeled with the state S) with rate
l
, while infected indi-
viduals recover with rate
m
and become removed from the
population. This is the prototypical model for infectious
disease spreading, where individuals recover and are
immune to disease after a typical time that on average
can be expressed as the inverse of the recovery
rate. A classic variation of this model is the susceptible
infected
e
e
e
infected
susceptible (SIS) model, in which individuals go
e
FIGURE 27.2
Multiscale structure of human mobility. (Left) The flight network (top) involves typical periods of several days, whereas the
commuting flows (below) occur within a single day but are typically much more intense. (Right) The weight distribution of the flight and commuting
networks.
