Figure 1.
Example of healing after single link failure.
Notice that failure of a single node can be modelled as the failure of all its links; hence, multiple links failure are the more general event to be considered. (Left Panel) In the initial state, the source node (filled square, upper left corner) is able to serve all nodes through the links of the active tree. The
dashed lines (green online) represent dormant backup links that can be activated upon failure. The redundancy of the system is
as only
of the
possible backup links are present. The link marked with an X is the one that is going to fail. (Central panel) A single link failure disconnects all the nodes of a sub-tree; in the example, a sub-tree of
nodes (red online) is left isolated from the source – i.e., the system has a damage
). (Right Panel) By activating a single dormant backup link, the self-healing protocol has been able to recover connectivity for the whole system, in this case bringing back the number of served nodes at its maximum value
. The link that has recovered the connectivity is marked with an R.
Figure 2.
Self-healing results for networks of size 104.
Panel (a): distribution networks based on square grids. The average fraction of nodes that the self-healing protocol is able to restore decreases with the number of faults
with no relevant dependency on the redundancy; results are shown for a
nodes network. Panel (b): distribution networks based on scale-free networks generated according to Barabasi-Albert[16]. The average fraction of nodes
of served nodes is plotted against the number of failures
. Even for a low
redundancy (
), the system can almost totally heal after sustaining
failures; as a comparison, for the same number of failures square grids loose
of the nodes. Panel (c): distribution grids based on small-sworld networks obtained by rewiring a fraction
of links according to Watts-Strogatz [17]. The average fraction of nodes
of served nodes is plotted against the number of failures
. At difference with square grids and scale-free networks, the restored fraction of service
shows a marked dependency upon the redundancy parameter
. Similar results are obtained for
and
.
Figure 3.
Comparison among different network structures.
Here we show the performances of our self-healing algorithm with respect to the quality of service for increasing number of removed links with the redundancy fixed to
; for
networks, the rewiring probability is
. The average fraction of nodes
of served nodes is plotted against the number of failures
.
Figure 4.
Upper panels, from left to right: planar square grid (), small-world network (
) generated according to Watts-Strogatz [17] and scale-free network (
) generated accordint to Barabasi-Albert [16]. Lower panels: random spanning trees associated with the related underlying topologies in the upper panels.