2.4- Weak Links
In the random graph model that we have examined so far, a node is able to infect only a few other
nodes in the graph (at least for the typical situation in which
We study the effect of infrequent sharing with a large number of other individuals by modifying the
random graph model slightly, giving a node a small but finite chance of infecting any node which is
not explicitly connected to it. In addition to the previously-defined infection rate
where
is the ratio between the total weak and strong infection rates.
Figure 6 displays the effect upon the extinction probability and the
average number of infections in equilibrium of various ratios
The simulation results of Figs. 5 and 6 suggest that the
homogeneous limit is reasonably good when the total infection rate is spread among a
sufficient number of nodes. However, when too much of the total infection rate is
concentrated into too few nodes, epidemics have a much harder time establishing
themselves than would be predicted by the homogeneous approximation. For example,
according to Fig. 6a, when the connectivity
|
).
Although it is probably true that most users share most of their programs with a small number of
other individuals, there is generally a much larger group of other individuals with whom they
exchange programs every once in a while. To what extent will these extra pathways enable viruses to
spread?
, which
we shall now refer to as the ``strong'' infection rate, we define the ``weak'' infection rate
. As before, the average total infection rate through the ``strong'' links is given by
. The average total infection rate through the ``weak'' links is
. Thus the total infection rate through all links is:
.
The results for
are simply those of Fig. 
, which is
practically impervious to infection when there are no weak links, becomes
behaviorally equivalent to a graph with
: the extinction
probability drops to nearly 1/2, and nearly half of the population is infected
in equilibrium. When
there is a finite probability
(0.40) for an epidemic to occur, and the average number of infected individuals
in equilibrium is 40. These limiting values are easily understood by completely
disregarding the strong links, in which case we have from Eq.
is 1/3 of its nominal
value, or
. Since
, the
conditions for an epidemic to occur are satisfied by the weak links alone, with
an effective value of
,
in agreement with the simulation.
for random graph with weak links. Three different values of the ratio
and
. Each point represents an average over
2500 simulation runs on 100-node graphs. As the weak links become stronger, the extinction
probability and the average number of infections approach their homogeneous limits over a wider
range of connectivities.
and there are no
weak links, the epidemic threshold is approximately
, which is five times the
classical value. Other simulations that we have not presented here indicate that, for
, the epidemic threshold is about twice the classical value. These results are
unaffected by the number of nodes N in the graph, provided that N exceeds about 100.
Weak links diminish the increase in the threshold, but do not eliminate it. For example,
when the total infection rate through the weak links is 0.2 times the total infection
rate through the strong links (
), the threshold can be five times the
classical value, but only if the graph is twice as tenuous (
) as it needs to
be when there are no weak links.