Abstract
Understanding structural controllability of a complex network requires to identify a Minimum Input nodes Set (MIS) of the network. Finding an MIS is known to be equivalent to computing a maximum matching of the network, where the unmatched nodes constitute an MIS. However, maximum matching is often not unique for a network, and finding all possible input nodes, the union of all MISs, may provide deep insights to the controllability of the network. Here we present an efficient enumerative algorithm for the problem. The main idea is to modify a maximum matching algorithm to make it efficient for finding all possible input nodes by computing only one MIS. The algorithm can also output a set of substituting nodes for each input node in the MIS, so that any node in the set can replace the latter. We rigorously proved the correctness of the new algorithm and evaluated its performance on synthetic and large real networks. The experimental results showed that the new algorithm ran several orders of magnitude faster than an existing method on large real networks.
Highlights
Controlling complex networks[1,2,3] is of great importance in many applications, such as social, biological and technical networks
The unmatched nodes related to a maximum matching constitute a Minimum Input nodes Set, or MIS
We developed an efficient algorithm for finding all possible input nodes of a network
Summary
Consider a directed network G(V, E) over a set of nodes V and a set of edges E. The in-node nin of the bipartite graph B can be reached from an input node min related to M through an alternating path pnm. There must be an alternating path pnm related to M’ which starts with unmatched node nin and end with a matched node min. The alternating path pnm must end with an unmatched node min related to M because nin is matched by M This contradicts the fact that nin cannot be reached by any input node related to M, which completes the proof. This observation leads to a novel two-step approach to identification of all possible input nodes, i.e., we first compute an MIS and consider all of its alternating paths These two steps can be combined using a simple modification to the Hopcroft–Karp maximum matching algorithm for undirected graphs[27].
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