Controllability of flow-conservation networks

The ultimate goal of exploring complex networks is to control them. As such, controllability of complex networks has been intensively investigated. Despite recent advances in studying the impact of a network's topology on its controllability, a comprehensive understanding of the synergistic impact of network topology and dynamics on controllability is still lacking. Here, we explore the controllability of flow-conservation networks, trying to identify the minimal number of driver nodes that can guide the network to any desirable state. We develop a method to analyze the controllability on flow-conservation networks based on exact controllability theory, transforming the original analysis on adjacency matrix to Laplacian matrix. With this framework, we systematically investigate the impact of some key factors of networks, including link density, link directionality, and link polarity, on the controllability of these networks. We also obtain the analytical equations by investigating the network's structural properties approximatively and design the efficient tools. Finally, we consider some real networks with flow dynamics, finding that their controllability is significantly different from that predicted by only considering the topology. These findings deepen our understanding of network controllability with flow-conservation dynamics and provide a general framework to incorporate real dynamics in the analysis of network controllability.

Figure 1

Illustration of flow-conservation networked systems. (a) A node $i$ and its neighbors in a flow-conservation network. (b) The total outflow from node $i$ to its neighbors. (c) The total inflow from $i$'s neighbors to $i$. The amount of flow between two nodes depends on the weight of the corresponding link. The total flow in the network is viewed as a constant.

Figure 2

Controllability measure $n_{\text{D}}$ in undirected and directed networks with different network size $N$ for FC-ER networks (a),(c) and FC-SF networks (b),(d). $n_{\text{D}}$ as a function of $\langle k\rangle$ is shown in the undirected situation (a),(b) and the directed situation (c),(d), with three different network sizes, $N=1000,N=2000$, and $N=5000$, respectively. MMT denotes the results obtained from the exact controllability theory (ECT). The results of the efficient tool (ET) are obtained from the fast algorithm, $n_{\text{D}}^{\text{ET}}=max\{1/N,\text{rank}\left(W\right)/N\}$, for sparse networks according to ECT in directed FC-ER networks (a) and undirected FC-ER networks (c). ET of undirected FC-SF networks (b) and directed FC-SF networks (d) could be estimated according to Eqs. (10) and (12), respectively. The theory of FC-ER networks and FC-SF networks corresponds to the analytical results of Eqs. (6) and (9) for undirected situations, and Eqs. (7) and (11) for directed situations. The results of MMT and ET are averaged over 100 independent realizations for $N=1000$, over 50 independent realizations for $N=2000$, and over 10 independent realizations for $N=5000$. We use the original key parameters controlling $\langle k\rangle$ instead of $\langle k\rangle$ itself, $p=\langle k\rangle /N$ in ER random networks (a),(c) and $m=\langle k\rangle /2$ in SF networks (b),(d), to show the relativity between the average degree and $n_{\text{D}}$, respectively.

Figure 3

(a)–(d) $n_{\text{D}}$ as a function of the link directionality ($\tau$) for directed ER (a),(c) and directed SF (b),(d) networks. $n_{\text{D}}$ is shown as a function of $\tau$ for ER and SF networks without the flow-conservation dynamics (a),(b) and for FC-ER and FC-SF networks (c),(d). The results of the networks with different average degree are shown. ET of networks without the flow-conservation dynamics (a),(b) could be estimated by $n_{\text{D}}^{\text{ET}}=max\{1/N,\text{rank}\left(A\right)/N\}$, and ET for the flow-conservation networks (c),(d) is computed by Eq. (14). Theory denotes the results obtained from the graphical approach [13]. The sizes of networks are all 2000 in (a)–(d). The results from MMT and ET are averaged over 20 independent realizations, and those from theory are averaged over 100 independent realizations. The notations in this figure are the same as in Fig. 2.

Figure 4

The impact of the negative link ratio on $n_{\text{D}}$ in sparse situations (a) and dense situations ( b) of directed FC-ER networks, with different average degrees $\langle k\rangle$ ($\langle k\rangle =Np$). ET of networks could be estimated by the origin equations in ECT for sparse networks (a) and Eq. (16) for dense networks (b). Theory denotes the results obtained from the analytical Eq. (7) for the sparse ( a) and Eq. (15) for the dense (b). The simulated networks' sizes are all 2000. The results of MMT and ET are averaged over 50 independent realizations.