## A certain class of non Hermitian systems with real energy

In this AIM-100 paper, we propose a method to enhance controllability by changing the direction of as few links as possible. The basic idea is to find the candidate links based on the type of their nodes, which is determined by one maximum matching in the network. For the candidate links, we develop a strategy to assign the direction of links. We apply our method to two common network models: the Erdös–Rényi (ER) model [24] and [25] and scale-free model (SF) [26] and [27]. Our method will be compared with the simple 2-matching method and the method of randomly changing link directions.

2. Method

Fig. 1. The bipartite representation of a simple directed network and its a maximum matching. (a) A simple directed network. (b) The bipartite representation of the network. (c) A maximum matching of the network.Figure optionsDownload full-size imageDownload as PowerPoint slide

As pointed out in earlier works, the maximum matching is composed of matching paths, matching cycles and isolated node. Since there does not exist unmatched node in matching cycles, matching cycles are not considered. Each matching path has only an unmatched node. Isolated nodes are always unmatched. Therefore, NDND equals the total number of matching paths and isolated nodes. Connecting a matching path or isolated node to another matching path or isolated node can make NDND decrease by one.