Controllability and observability in complex networks – the effect of connection types

Network theory based controllability and observability analysis have become widely used techniques. We realized that most applications are not related to dynamical systems, and mainly the physical topologies of the systems are analysed without deeper considerations. Here, we draw attention to the importance of dynamics inside and between state variables by adding functional relationship defined edges to the original topology. The resulting networks differ from physical topologies of the systems and describe more accurately the dynamics of the conservation of mass, momentum and energy. We define the typical connection types and highlight how the reinterpreted topologies change the number of the necessary sensors and actuators in benchmark networks widely studied in the literature. Additionally, we offer a workflow for network science-based dynamical system analysis, and we also introduce a method for generating the minimum number of necessary actuator and sensor points in the system.


I Introduction
The Supplementary Information is organized as follows: In Section II, we introduce our "path-nding" method which was used to calculate the necessary driver and sensor nodes. In Section III, we present the studied networks. In Section IV we show the results.

II The path-nding method
The number of driver nodes can be dened as the sum of the number of unmatched nodes and the number of root strongly connected components (SCC), where all nodes are matched [1]. A root SCC, R, is a set of nodes, where there is no edge from node x j to node x i for all x j / ∈ R and for all x i ∈ R. R is a matched SCC if any node x i is matched by a node x j such that x i , x j ∈ R. Since a matched root SCC is inaccessible structurally, and uncontrolled as no unmatched node determined by maximum matching in it, we have to deal with this phenomenon separately. In the literature, one method deals with this problem using sharing input signals [1], but here, we recommend another method that can provides solution without the sharing of input signals and in some cases it grants input conguration with less driver nodes than provided by the existing method.
The signal sharing method achieves controllability by sharing the signal of an existing input on an arbitrary node from each matched root SCC. This is possible, since a matched root SCC can control itself, and only a shared signal necessary on one of its nodes. This is not true for unmatched nodes, we cannot control an unmatched nodes with a shared signal. Thus, the number of generated inputs is equal to the number of unmatched node, but the number of driver nodes is higher, it is increased by the number of matched root SCCs, since shared signals creates new driver nodes. The creation of a new approach was motivated by three reasons. The rst is that we found that in some cases we can control a system with less driver nodes, than determined by the signal sharing method. The second was the presence of SCCs where unmatched nodes were identied, so we assumed that the phenomenon was not generated by SCC. The third was the fact that sharing an input signal in some applied area is impossible, and signal sharing makes controller design more complex. With the path-nding method we want to answer these remarks.
As a result of our research, we found that matched root SCCs are results of Hamiltonian cycles. To eliminate the sharing of an input signal, the method cuts each Hamiltonian cycle and creates a Hamiltonian path so that, if it is possible, then the path continues in an unmatched node. Formally, if the matched edge set is denoted by M , and matched root SCC by R, then we nd nodes x 1 , x 2 , and x 3 such that Then by removing (x 2 , x 1 ) from M and by adding (x 2 , x 3 ) to M we create a new maximum matching, where x 1 is an unmatched node in SCC R, and x 3 is a matched node. The signal sharing method determines x 3 as a driver node, and one node from R, while the path-nding method determines only x 1 a driver node. In Figure S1 the visualization of this example can be seen. If there is no such unmatched node, then the method only removes an edge from the matching, thus creates an unmatched node for all matched root SCCs. Since our method identies Hamiltonian paths, we called this new method path-nding, and the original one, as it shares signal, the signal sharing method.
x 1 Figure S1: An illustrative example with input congurations (u) generated by the path-nding and signal sharing methods. Dashed lines show the matched edges. The unmatched node in (a) is In the case of the signal sharing method, node x 3 is controlled by u 1 and matched root SCC {x 1 , x 2 } should be also controlled by this input, which results in another driver node: x 1 . In contrast, the path-nding method modies the maximum matching by the exchange of the edge (x 2 , x 1 ) with the edge (x 2 , x 3 ). By changing the edges, the path-nding method can control the system without sharing the input signal.
Although path-nding method simplies the controlling process, the signal sharing method has its advantage as well. Since the path nding method assigns separate input for each matched root SCC, it can produce more inputs than signal sharing method, which can control all the matched root SCCs with only one input, as shown in Figure S2. The maximum matching method does not generate a unique solution, but matching of the same number of observer and controller nodes. The path nding method exhibits the same properties. The result of the path nding method is not unique, since it is possible that from a matched root SCC more unmatched nodes can be accessed. Here, the selection of Hamiltonian path determines the resultant input conguration. Nonetheless, the number of provided driver nodes is the same in each case for a given topology: the sum of the number of unmatched nodes and the number of those matched root SCCs, that do not point to any unmatched node. Here, we draw attention to a special case: if more than one matched root SCC point to an unmatched node, then only one matched root SCC can be eliminated from the sum.
In the case of the signal sharing method, the results can also be dierent, but the number of driver nodes depends on initial maximum matching (e.g. Figure S1).

III The studied networks
The networks used in the article can be seen in Table S1. Network Set I contains networks, which are used in controllability examinations and their dynamical behaviour is interpretable. In contrast, topologies in Network Set II describe processes in which dynamical behaviours are not interpretable. Network Set III contains topologies which originate from state-transition matrices of real dynamical systems.
Most of the networks originated from dierent sources, as can be seen in Table S1, but almost all of the topologies of Network Set III are analysed in the same study. We need to mention that the state-transition matrices were available on the Internet [2], and they were revealed by the authors of [3], and the topologies were not created by us. We uploaded the data to our website also [4].

IV Results
In Tables S2, S3, S4 and S5 the generated measures can be seen for the previously presented networks with Inuence, Self-inuencing inuence, Interaction and Self-inuencing interaction connection types, respectively.