  Network controllability measures how well a networked system can be
controlled to a target state, and its robustness reflects how well the system
can maintain the controllability against malicious attacks by means of
node-removals or edge-removals. The measure of network controllability is
quantified by the number of external control inputs needed to recover or to
retain the controllability after the occurrence of an unexpected attack. The
measure of the network controllability robustness, on the other hand, is
quantified by a sequence of values that record the remaining controllability of
the network after a sequence of attacks. Traditionally, the controllability
robustness is determined by attack simulations, which is computationally time
consuming. In this paper, a method to predict the controllability robustness
based on machine learning using a convolutional neural network is proposed,
motivated by the observations that 1) there is no clear correlation between the
topological features and the controllability robustness of a general network,
2) the adjacency matrix of a network can be regarded as a gray-scale image, and
3) the convolutional neural network technique has proved successful in image
processing without human intervention. Under the new framework, a fairly large
number of training data generated by simulations are used to train a
convolutional neural network for predicting the controllability robustness
according to the input network-adjacency matrices, without performing
conventional attack simulations. Extensive experimental studies were carried
out, which demonstrate that the proposed framework for predicting
controllability robustness of different network configurations is accurate and
reliable with very low overheads.
