Figure 1.
Vertex load profiles in RGGs and ER networks.
Calculated on (A) random geometric graphs and (B) Erdős-Rényi random grahs, composed of nodes with
and averaged over
network realizations. (C) Positive correlations are shown in the case of ER graphs, while these correlations seem to disappear in RGGs for degree classes higher than the average degree of the network ensemble. Data were averaged over more than
network realizations for networks of
and
. The fluctuating tail of the red curve originates from the lack of sufficient number of samples in the specific degree classes. The error bars correspond to one standard deviations. (D) A single network realization (
,
) showing the vertex loads. Note, that the node with the highest connections (blue arrows indicate the 3 highest degree nodes) does not carry the highest load in the network (loads are color coded, and node sizes are proportional to loads).
Figure 2.
Correlations of vertex loads as a function of distance between vertices and extreme value characteristics of loads.
(A) Load and distance correlations in RGGs and rewired RGGs with rewiring probability (see text). Pearson correlation coefficient as function of distance
measured between two arbitrary nodes. Data were averaged over
network realizations for networks having
nodes. (B) System size dependence of the maximum vertex load in networks with
. Data were averaged over
network realizations. The parameter
corresponds to the rewiring probability for links in the RGG.
Figure 3.
Cascades triggered by targeted and random removals.
(A) Probability that a single node removal will trigger a cascade as function of the tolerance parameter. (B) The ratio of the size of the largest surviving network component to the initial network size, as function of the tolerance parameter
, when the initial failure triggers a cascade. (C) Similar to (B), except for the case where the initial failure does not trigger a cascade. In all (A), (B) and (C) subplots the red curve corresponds to the case when the triggered node is the node with the highest load, the blue curve to the case when the triggered node is the most connected (highest degree) node in the network and the green curve shows the case when the triggered node was chosen randomly. Network parameters are
,
, while the data was averaged over
network realizations. Error bars correspond to the standard error of the mean. (D)
as a function of tolerance parameter, unconditioned on whether or not a cascade was triggered for RGGs, SF networks and ER networks. For each point, the data was averaged over
network realizations.
Figure 4.
Cascades on single network realizations.
Simulations were performed on networks of size . Fractional size of surviving giant component as a function of
for (A),(B) RGGs, (C),(D) ER networks and (E),(F) SF networks.
Figure 5.
(A) Transition in network structure from an RGG towards an ER network through the process of rewiring. Multiple rewired versions (different values of ) are shown together with the two extreme cases. (B) Average vertex load in RGG, ER and rewired versions of RGG as function of the fraction of rewired links
. Network parameters are:
,
. Data were averaged over
network realizations.
Figure 6.
Effect of rewiring on cascades in RGGs.
Cascades were triggered by the removal of the highest load. Simulations were performed on networks of size with
. As
is increased the lack of self-averaging that manifests itself in the form of non-monotonicities in the curves for
versus
, disappears.
Figure 7.
(A) Position (distance and angle) of failed nodes relative to the initially removed one, here the highest load in the network. Different colors correspond to different iterations of the cascade: blue squares (1st), red squares (2nd), green squares (3rd), light blue triangles (4th), black squares (5th), magenta circles (6th), orange circles (7th), light green squares (8th), yellow triangles (9th). Network parameters are the same as in Fig. 1, while each data point is the averaged location of nodes removed in a given stage over independent network cascades. (B) The distribution of the distance
from the cascade-triggering node for nodes that fail in the course of a cascade (See text for details).
Figure 8.
The effect of average degree upon cascading failures.
Fraction of the largest surviving network component following cascading failures triggered by the removal of a single, randomly chosen node as function of
tolerance parameter. The two curves correspond to two ensembles of random geometric graphs, one with
(maroon) and one with
(green). Data were obtained for RGGs of size
, averaged over more than
network realizations. The error bars correspond to the standard error of the mean.
Figure 9.
Preemptive node removal in RGGs.
(A) Probability that a cascade occurs after removal of the node with highest load, despite a fraction of nodes being preemptively removed immediately after the initial trigger. (B) Fractional size of the largest surviving network component
as a function of preemptively removed fraction
, when there is a cascade. (C) Fractional size of the largest surviving network component
as a function of preemptively removed fraction
for a single network instance for different values of the tolerance parameter
. (D) The ratio of the throughput (defined in text) of the surviving giant component and the throughput of the original network as a function of the altruist node fraction. The red circle corresponds to the case when there no nodes are preemptively removed. Network parameters are:
,
, and
for results shown in (A),(B) and (D).
Figure 10.
Increasing the resilience of the network by introducing altruist nodes.
(A) Probability that a cascade is triggered for an altruist/preemptively removed fraction . The orange squares indicated the probability of cascade when no nodes (other than the initial cascade-triggering node) are removed, but when the current per source is reduced by
(upper square) or
(lower square) immediately after the initial node removal. (B) The fractional size of the surviving giant component
when a cascade is triggered, as a function of the altruist/preemptively-removed node fraction. Also shown are the results when the current per source is reduced by
(upper square) or
(lower square) immediately after the initial node removal, which coincide with the
and
results respectively for altruistic node removal. (C) Similar to (B), but for the cases where a cascade is not triggered. (D) The ratio of the effective throughput (defined in text) of the surviving giant component and the throughput of the original network as a function of the altruist node fraction. The red circle corresponds to the case when there are no altruist nodes. Network parameters for all these plots are:
,
, and
for results shown in (A),(B) and (D).
Figure 11.
Characteristics of the UCTE network.
The network consists of nodes with an average degree of
. (A) Scatter of loads as a function of node degree
(black squares) and the average load (red squares) as a function of node degree
. (B) The load distribution on the intact UCTE network. (C) The degree distribution of the UCTE network. (D) A visualization of the UCTE network with loads indicated using both node size and color.
Figure 12.
(A) Cascades triggered by the removal of a single node where the node was chosen using three different criteria i.e. randomly, highest load or highest degree. (B) Cascades triggered by the removal of a single edge where the edge was either chosen randomly or was the one with the highest load. Data obtained for cascade triggered by the random removal of a single node (edge) were averaged over different scenarios.
Figure 13.
UCTE network snapshots before and after cascades.
(A) The intact network with node sizes in proportion to their respective steady-state loads. (B) The network and the loads after a highest-load-removal-triggered cascade has terminated, with the tolerance parameter . (C) The network and the loads after a highest-load-removal-triggered cascade has terminated, with the tolerance parameter
. The red nodes here indicate nodes that were removed in the cascade leading to (B), but survived in the cascade leading to (C).
Figure 14.
Cascade mitigation on the UCTE network.
Comparison between the preemptive and altruistic node removal strategies on the UCTE network with tolerance parameter (A) and (B)
.
Figure 15.
Cascades triggered by concentrated versus randomly distributed removals.
(A) Fractional surviving giant component size after a cascade as a function of number of initial nodes removed in concentrated and random removals for RGGs with and
. (B) Fractional surviving giant component size after a cascade as a function of number of initial nodes removed in concentrated and random removals for the UCTE network. Each data point is an average over more than
realizations.