Figure 1.
Cumulative distributions of the number of weak ties in five empirical networks.
For every network, the community structure is detected by the method proposed by Newman and Girvan [55]. Weak ties and bridge nodes are then identified. The number of weak ties emanating from each bridge node is recorded to give the cumulative distribution. Results are shown for students' networks for Caltech (red squares), Princeton (green circles), Georgetown (blue up triangles), Oklahoma (cyan down triangles) and North Carolina (magenta diamonds).
Table 1.
Structural Properties of the five empirical networks.
Figure 2.
Comparison of efficacy of immunization algorithms in simulated networks.
The difference in the final epidemic ratios (a) (left panel) between ACQ and BHD, and (b) (right panel)
between CBF and BHD, are shown for simulated networks with different network modularity
and immunization coverage
. The colors indicate the differences in percentages (see color codes). Results are obtained by averaging over
realizations for each pair of
and
values. The parameters associated with the SIR dynamics are
and
.
Figure 3.
Comparison of efficacy of immunization algorithms in empirical networks.
The differences in the final epidemic ratios (left panel) and
(right panel) using different immunization algorithms are shown for each of the five empirical networks as a function of the immunization coverage
. A positive value indicates that BHD is more effective than the other algorithms. Results are obtained by averaging over
realizations for each value of
.
Figure 4.
Comparison of giant components of different immunization algorithms in empirical networks.
The difference in the sizes of the giant component (left panel) and
(right panel) using different immunization algorithms are shown for each of the five empirical networks as a function of the immunization coverage
, where
is the size of the giant component before an immunization algorithm is applied. A positive value indicates that BHD is more effective in breaking up the network. Results are obtained by averaging over
realizations for each value of
.
Figure 5.
Comparison of average number of weak ties among immunized bridge hubs identified by BHD, bridge nodes identified by BHD and bridge nodes identified by CBF. BHD identifies a pair of nodes for immunization via a self-avoiding walk algorithm.
One node is a bridge hub and another a bridge node. The average number of weak ties among these two types of immunized nodes [red squares (BHD hubs) and gray diamonds (BHD nodes)] are shown for different values of immunization coverage , together with the results from immunized bridge nodes identified by CBF (blue circles). Results are obtained by averaging over
realizations for each value of
. For comparison, the results based on the method proposed by Newman and Girvan [55] are shown as the green dashed lines.
Figure 6.
Comparison of number of nodes visited before identifying immunization nodes in CBF and BHD.
The ratio is shown of the number of nodes visited by the self-avoiding walks in CBF and BHD for achieving an immunization coverage
. A large
implies a longer search for immunization nodes. The results show that BHD identifies the immunization nodes faster than CBF for
. Results are obtained by averaging over
realizations for each value of
.
Figure 7.
Robustness of BHD in networks with noise as modeled by random addition and removal of links.
The quantity is shown as a function of the number of links randomly added to or removed from an empirical network, where
and
are the final epidemic ratios when BHD is applied and not applied to the modified network, respectively. The robustness of BHD is indicated by the relatively stable values of
. Results are shown for each of the five empirical networks as labeled. The immunization coverage is
. The SIR parameters are
and
. Results are obtained by averaging over
realizations for each value of added and removed links.
Figure 8.
Schematic illustration of CBF strategy and BHD strategy.
The CBF strategy: (a) A self-avoiding walk starts at a randomly chosen node . (b)-(c) The walk visited
after two steps. The node
does not have any other links back to
and
other than the link that took the walk from
to
. The node
is a potential candidate of a bridge node. Two neighbors of
, namely
and
, are randomly picked and each is examined for connections to the visited set of nodes. As
has links with
and
, the target node
is dismissed as a bridge node and the walk continues. (d) The walk moves to
after three steps. The node
has links to previously visited nodes
and
, and thus the walk continues. (e) The walk moves to
after four steps. The node
does not have any other links back to previously visited nodes other than the link that took the walk from
to
. The node
is a potential target of a bridge node. Two neighbors of
are randomly chosen. (f) If
and
are chosen, these nodes do not connect back to the previously visited nodes and
is identified as a bridge node and immunized. The BHD strategy: (a) A self-avoiding walk starts at a randomly chosen node
. (b)-(c) The walk visited
,
and
after two steps. The set
is the union of all the neighboring nodes or friendship circles of
,
and
, as shaded in (c). (d) The walk moves to
after three steps. The friendship circle
of node
consists of
,
,
, and
. As all the nodes in
either belong to
or have a link to at least a node in
, the node
is not a potential target for immunization. (e) The union of friendship circles is updated to
as shaded. The walk continues and reaches node
after four steps. The friendship circle
of
consists of
,
,
,
and
, among them
and
do not belong to
and do not have a link to nodes in
. The node
is then identified as a bridge node for immunization. In addition, among those nodes in
that cannot be linked back to
, one node, e.g.,
, is randomly chosen and identified as a bridge hub for immunization.
Figure 9.
Visualizing the community structure and a bridge hub in the Caltech network.
The Caltech network can be divided into 13 communities based on the method of Newman and Girvan [55], as illustrated by the different colors. Also illustrated is a bridge hub that carries the largest number of weak ties and connects with other bridge nodes labeled as
to
. The number on a weak tie shows the difference in the numbers of weak ties originated from the node
and the bridge node at the other end of the weak tie.