Figure 1.
(a) A simple network of nodes. (b) The controlled network is represented by a directed graph
with an input node
connecting to a state node
. The stem-cycle disjoint subgraph
(shown in red) contains six edges, which is the largest number of edges among all possible stem-cycle disjoint subgraphs of the directed graph
and corresponds to the generic dimension of controllable subspace by controlling node
. The control centrality of node
is thus
. (c) The control centrality of the central hub in a directed star is always 2 for any network size
. (d) The control centrality of a node in a directed acyclic graph (DAG) equals its layer index. In applying Hosoe’s theorem, if not all state nodes are accessible, we just need to consider the accessible part (highlighted in green) of the input node(s).
Figure 2.
Distribution of normalized control centrality of several real-world networks (blue) and their randomized counterparts: rand-ER (red), rand-Degree (green), plotted in log-log scale.
(a) Intra-organizational network of a manufacturing company [49]. (b) Hyperlinks between weblogs on US politics [50]. (c) Email network in a university [51]. (d) Ownership network of US corporations [52]. In- and out-degree distributions for each network are shown in the insets. See Table 1 for other network characteristics.
Table 1.
Real networks analyzed in the paper.
Figure 3.
The impact of different attack strategies on network controllability with respective to the random attack.
with
represents the generic dimension of controllable subspace after removing a
fraction of nodes using strategy-
. The nodes are removed according to six different strategies. (Strategy-0) Random attack: randomly remove
fraction of nodes. (Strategy-1 or 2) Random upstream (or downstream) attack: randomly choose
fraction of nodes, randomly remove one of their upstream neighbors (or downstream neighbors). The results are averaged over 10 random choices of
fraction of nodes with error bars defined as s.e.m. Lines are only a guide to the eye. (Strategy-3,4, or 5) Targeted attacks: remove the top
fraction of nodes according to their control centralities (or in-degrees or out-degrees).