Network Controllability Is Determined by the Density of Low In-Degree and Out-Degree Nodes

Abstract

The problem of controllability of the dynamical state of a network is central in network theory and has wide applications ranging from network medicine to financial markets. The driver nodes of the network are the nodes that can bring the network to the desired dynamical state if an external signal is applied to them. Using the framework of structural controllability, here we show that the density of nodes with in-degree and out-degree equal to 0, 1 and 2 determines the number of driver nodes of random networks. Moreover we show that networks with minimum in-degree and out-degree greater than 2, are always fully controllable by an infinitesimal fraction of driver nodes, regardless on the other properties of the degree distribution. Finally, based on these results, we propose an algorithm to improve the controllability of networks. The controllability of a network [1-10] is a fundamental problem with wide applications ranging from medicine and drug discovery [11], to the characterization of dynamical processes in the brain [12][13][14], or the evaluation of risk in financial markets [15]. While the interplay between the structure of the network [16][17][18][19] and the dynamical processes defined on them has been an active subject of complex network research for more than ten years [20,21], only recently the rich interplay between the controllability of a network and its structure has started to be investigated. A pivotal role in this respect has been played by a paper by Liu et al. [6], in which the problem of finding the minimal set of driver nodes necessary to control a network was mapped into a maximum matching problem. Using a well established statistical mechanics approach [22][23][24][25][26][27], Liu et al. [6] characterize in detail the set of driver nodes for real networks and for ensembles of networks with given in-degree and out-degree distribution. By analyzing scale-free networks with minimum in-degree and minimum out-degree equal to 1 they have found that the smaller is the power-law exponent γ of the degree distribution, the larger is the fraction of driver nodes in the network. This result has prompted the authors of [6] to say that the higher is the heterogeneity of the degree distribution, the less controllable is the network. Later, different papers have addressed questions related to controllability of networks with similar tools [7,28].In this Letter we consider the network controllability and its mapping to the maximum matching problem, exploring the role of low in-degree and low out-degree nodes in the network. We show that by changing the fraction of nodes with in-degree and out-degree less than 3, the number of driver nodes of a network can change in a dramatic way. In particular if the minimum in-degree and the minimum out-degree of a network are both greater than 2 then any network, independently on the level of heterogeneity of the degree distribution, is fully controllable by an infinitesimal fraction of nodes. Therefore we show that the heterogeneity of the network is not the only e...

Control principles of complex systems

“…For uncorrelated directed networks, the density of nodes with k in , k out = 1 or 2 determine the size of maximum matchings (Menichetti et al, 2014). This suggests that random networks whose minimum k in and k out are greater than two typically have perfect matchings and hence can be fully controlled via a single control input (i.e.…”

“…Equation (25) indicates that as γ → 2, n D → 1, which is consistent with the result that γ c = 2 for a purely SF network. The systematic dependence of n D on k and γ prompts us to ask: How do other network characteristics, like degree correlations, clustering, modularity, or the fraction of low degree nodes, influence n D (Menichetti et al, 2014;Pósfai et al, 2013). A combination of analytical and numerical results indicate that the clustering coefficient and modularity have no discernible effect on n D .…”

Fluctuations in percolation of sparse complex networks

“…These algorithms are becoming increasingly popular in network theory and they have been used to characterize the percolation of single [26] and multilayer networks [15,[27][28][29][30][31], to predict and monitor epidemic spreading [32][33][34][35][36][37], to identify the driver nodes of a network ensuring its controllability [38,39] and to solve a number of other optimization problems on networks [40][41][42].…”

Interdisciplinary and physics challenges of network theory

“…In fact phase diagrams of critical phenomena and dynamical processes change drastically when the dynamics is defined on complex networks. Complex networks are responsible for significant changes in the critical behaviour of percolation, Ising model, random walks, epidemic spreading, synchronization, and controllability of networks [10][11][12][13].The need to characterize complex systems, to extract relevant information from them, and to understand how dynamical processes are affected by network structure, has never been more severe than in the XXI century when we are witnessing a Big Data explosion in social sciences, information and communication technologies and in biology. …”

“…In fact phase diagrams of critical phenomena and dynamical processes change drastically when the dynamics is defined on complex networks. Complex networks are responsible for significant changes in the critical behaviour of percolation, Ising model, random walks, epidemic spreading, synchronization, and controllability of networks [10][11][12][13].…”