Abstract
A dynamical system is controllable if by imposing appropriate external signals on a subset of its nodes, it can be driven from any initial state to any desired state in finite time. Here we study the impact of various network characteristics on the minimal number of driver nodes required to control a network. We find that clustering and modularity have no discernible impact, but the symmetries of the underlying matching problem can produce linear, quadratic or no dependence on degree correlation coefficients, depending on the nature of the underlying correlations. The results are supported by numerical simulations and help narrow the observed gap between the predicted and the observed number of driver nodes in real networks.
Introduction
While during the past decade significant efforts have been devoted to understanding the structure, evolution and dynamics of complex networks^{1,2,3,4,5,6}, only recently has attention turned to an equally important problem: our ability to control them. Given the problem's importance, recent work has extended the concept of pinning control^{7,8,9} and structural controllability^{10,11,12,13} to complex networks. Here we focus on the latter approach. A networked system is considered controllable if by imposing appropriate external signals on a subset of its components, called driver nodes, the system can be driven from any initial state to any final state in finite time^{14,15,16,17}. As the control of a system requires a quantitative description of the governing dynamical rules, progress in this area was limited to small engineered systems. Yet, recently Liu et al.^{10} showed that the identification of the minimal number of driver nodes required to control a network, N_{D}, can be derived from the network topology by mapping controllability^{16} to the maximum matching in directed networks^{18}. The mapping indicated that N_{D} is mainly determined by the degree distribution P(k_{in}, k_{out}). We know, however, that a series of characteristics, from degree correlations^{19,20,21} to local clustering^{22} and communities^{23,24,25,26}, cannot be accounted for by P(k_{in}, k_{out}) alone, prompting us to ask: which network characteristics affect the system's controllability?
The three most commonly studied deviations from the random network configuration are (i) clustering, manifested as a higher clustering coefficient C than expected based on the degree distribution^{27}; (ii) community structure, representing the agglomeration of nodes into distinct communities, captured by the modularity parameter Q^{25}; (iii) degree correlations^{28}. First, we motivate our work by showing that network characteristics other than the degree distribution also affect network control. Next, we use numerical simulations to identify the network characteristics that affect controllability, finding that only degree correlations have a discernible effect. We then analytically derive n_{D} = N_{D}/N for random networks with a given degree distribution and correlation profile. More detailed calculations are provided in the Supplementary Information Sec. III. Finally, we test our predictions on real networks.
Results
Prediction based on the degree distribution
To motivate our study we compared the observed N_{D} to the prediction based on the degree sequence for several real networks. For this we randomize each network preserving its degree sequence and we calculate , the number of driver nodes for the randomized network. Plotting N_{D} versus on loglog scale indicates that the degree sequence correctly predicts the order of magnitude of N_{D} despite known correlations^{19,20} (Fig. 1a). However, by plotting n_{D} = N_{D}/N versus we observe clear deviations from the degree based prediction (Fig. 1b). Our goal is to understand the origin of these deviations and the degree to which network correlations can explain the observed n_{D}.
Numerical simulations
We start from a directed network with Poisson^{29,30} or scalefree degree distribution^{31,32}. The scalefree network is generated by the static model described in the Methods section. We use simulated annealing to add various network characteristics by link rewiring, while leaving the in and outdegrees unchanged, tuning each measure to a desired value, for details see the Methods section. We computed n_{D} using the HopcroftKarp algorithm^{33}.
Clustering
We use the global clustering coefficient^{27} defined for directed networks as
The simulations indicate that changes in C only slightly alter n_{D} and that the effect is not systematic (Fig. 2a). Hence we conclude that C plays a negligible role in determining n_{D}.
Modularity
We quantify the community structure using^{25,26}:
where A_{vw} is the adjacency matrix, c_{v} and c_{w} are the communities the v and w nodes belong to, respectively. Specifying Q still leaves a great amount of freedom in the number and size of the communities. We therefore choose to randomly divide the nodes into N_{C} equally sized groups and increase the edge density within these groups, elevating Q to the desired value.
The simulations indicate that this community structure has no effect on n_{D} (Fig. 2b). While adding communities to networks can be achieved in many different ways and the effect of modularity can be explored in more detail (e.g. hierarchical organization of communities^{23,34,35}, overlapping community structure^{24,36}, etc), we have failed to detect systematic, modularity induced changes in n_{D}, prompting us to conclude that Q does not play a leading role in n_{D}.
Degree correlations
In directed networks each node has an indegree (k_{i}) and an outdegree (k_{o}), thus we can define four correlation coefficients: correlations between the source node's in and outdegree and the target node's in and outdegree (Figs. 3, 4)^{28}. We use the Pearson coefficient to quantify each correlation with a single parameter:
where · sums over all edges, α, β ∈ {in, out} is the degree type, k^{(α)} is the degree of the source node, j^{(β)} is the degree of the target node. And is the average degree of the nodes at the beginning of each link, is the variance; and σ^{(β)} are defined similarly.
Simulations shown in Figs. 3 and 4 indicate that degree correlations systematically affect n_{D}. We observe three distinct types of behavior:

i
n_{D} depends monotonically on r^{(outin)}, so that low (negative) correlations increase n_{D} and high (positive) correlations lower n_{D} (Figs. 3c, 4c);

ii
Both r^{(inin)} and r^{(outout)} increase n_{D}, independent of the sign of the correlations (Figs. 3a, 3d, 4a, 4d);
 iii
The behavior is qualitatively the same for ErdősRényi (Fig. 3) and scalefree (Fig. 4) networks.
The diversity of these numerical results require a deeper explanation. Therefore in the remaining of the paper we focus on understanding analytically the role of degree correlations, which, by systematically altering n_{D}, affect the system's controllability.
Analytical framework
The task of identifying the driver nodes can be mapped to the problem of finding a maximum matching of the network^{10}. A matching is a subset of links that do not share start or end points. We call a node matched if a link in the matching points at it and we gain full control over a network if we control the unmatched nodes. The cavity method has been successfully used to calculate the size of the maximum matching for undirected^{37} and directed^{10} network ensembles with given degree distribution. Here we study network ensembles with a given degree correlation profile.
We calculate n_{D} analytically for a given P(k_{in}, k_{out}) and selected degreedegree correlation e(j_{in}, j_{out}; k_{in}, k_{out}), representing the probability of a directed link pointing from a node with degrees j_{in} and j_{out} to a node with degrees k_{in} and k_{out}. In the absence of degree correlations (neutral case)
where , and 〈k〉 is the average degree. To ensure analytical tractability we chose^{21}
By fixing m^{(α–β)}(j, k) (α, β ∈ {in, out}) we obtain a one parameter network ensemble characterized by r^{(α–β)}, where m^{(α–β)}(j, k) satisfies the constraints
and all elements of e^{(α–β)}(j, k) are between 0 and 1.
Our goal is to understand the relation between n_{D} and the degree correlation coefficient r^{(α–β)}. Assuming that r^{(α–β)} is small we treat the correlations as perturbations to the neutral case, discussing the impact of the four r^{(α–β)} correlations separately.
Outin correlations
Using equation (5c) and keeping the first nonzero correction we obtain (Supplementary Information Sec. III.):
where is the fraction of driver nodes of the uncorrelated network; w_{i} and only depend on P(k_{in}, k_{out})^{10} and
Equation (8) predicts that depends linearly on r^{(outin)}, a prediction supported by simulations for small r^{(outin)} (Figs. 3c and 4c). This behavior is also revealed by the equivalent problem of finding the maximum matching of graphs^{10}. For a node A with outdegree k_{0}, by definition only one edge can be in the matching. If the remainder k_{0} – 1 edges point to nodes with degree 1 (disassortative case), A inhibits them from being matched, so we have to control each of them individually, increasing n_{D}. If the remainder k_{0} – 1 edges point to hubs (assortative case), these hubs are likely to be matched through another incoming edge, decreasing n_{D}.
Outout correlations
The cavity method indicates that for outout correlations the first nonzero correction is of order (r^{(outout)})^{2}:
where (α ∈ {in, out}) only depends on P(k_{in}, k_{out}) and
Equation (10) predicts that does not depend on the outout correlation of the directly connected nodes, but only on the correlation between the second neighbors, hence its dependence is quadratic in r^{(outout)}, a prediction supported by numerical simulations (Figs. 3d and 4d). Indeed, positive (negative) r^{(outout)} correlation between the immediate neighbors means that if node A has high outdegree, then node B is expected to have high (low) outdegree and therefore C is likely to have high outdegree (Fig. 5). That is, both positive and negative onestep outout correlations induce positive twostep correlations, accounting for the symmetry of the effect observed in simulations (Figs. 3d and 4d).
Inin correlations
Switching the direction of each link does not change the matching, but turns outout correlations into inin correlations. So can be obtained by exchanging P^{(in)}(k_{in}) and P^{(out)}(k_{out}) in equation (10), predicting again a quadratic dependence on r^{(inin)}, supported by the numerical simulations (Figs. 3a and 4a).
Inout correlations
The equations for do not depend on the indegree of the source and the outdegree of the target of a link, hence we predict that r^{(inout)} does not play a role in network controllability, a prediction supported by the simulations (see Figs. 3b and 4b).
Taken together, we predict that the functional dependence of on degree correlations defines three classes of behaviors, depending on the matching problem's underlying symmetries: has no dependence on r^{(inout)}, linear dependence on r^{(outin)} and quadratic dependence on r^{(inin)} and r^{(outout)}. These predictions are fully supported by numerical simulations Figs. 3 and( 4): for small r we see no dependence on r^{(inout)}, an asymmetric, monotonic dependence on r^{(outin)} and a symmetric on r^{(inin)} and r^{(outout)}.
To directly compare the analytical predictions to simulations we need to know the complete e(j_{i}, j_{o}; k_{i}, k_{o}) distribution, which is not explicitly set in our simulations. So to test the results we use a rewiring method that sets the e(j_{i}, j_{o}; k_{i}, k_{o}) distribution, not only the r correlation coefficient^{21}. This method is not as robust as our original algorithm and the range of accessible r values is more restricted. However, since our results are based on perturbation scheme we only expect them to be correct for small r values. Indeed, we find that the predictions quantitatively reproduce the numerical results in a fair interval of r^{(α–β)} (Fig. 6).
Real networks
We test the predictions provided by the developed analytical and numerical tools on a set of publicly available network datasets. When complex systems are mapped to networks, the links connecting the nodes represent interactions between them. In this context selfloops represent selfinteractions, with a strong, well understood impact on controllability^{10,38}. While in some systems selfloops are obviously present (e.g. neural networks), in others they are manifestly absent (e.g. electric circuits^{39}). Our purpose here is to test the effect of correlations, hence we rely on datasets that capture the wiring diagram of various complex systems with different correlation properties. Therefore, even if in a few of these maps selfloops are missing, it is beyond the scope of this work to complete these networks. However, when studying controllability of a particular system, careful thought has to be put into whether selfloops are present or not. We present a systematic study on the effect of selfloops in the Supplementary Information Sec. II.B.
To test the impact of our predictions on real networks we calculate
where represent the number of driver nodes for the degreepreserved randomized version of the original network. Hence if Δ = 0 then P(k_{in}, k_{out}) accurately determines N_{D}; if Δ ≠ 0 then the structural properties not captured by the degree sequence influence its controllability. We measure the correlations in several real networks and based on our numerical and analytical results we predict the sign of Δ (Fig. 7). We grouped the networks according to our predictions. We provide the details of each network dataset in the Supplementary Information Table SI.
Group A
The networks of p2p Internet (Gnutella filesharing clients) do not have strong correlations, therefore we expect n_{D} to be correctly approximated by the prediction based on P(k_{in}, k_{out}) (i.e. Δ ≈ 0), in line with the empirical observations.
Group B
As in most networks the three relevant correlations coexist to some degree (Fig. 7), it is impossible to isolate their individual role. Yet, the networks in this group (electric circuits, metabolic networks, neural networks, power grids and food webs with exception of the Seagrass network) all have negative outin and nonzero inin and outout correlations, each of which individually increase n_{D} as we showed above. Therefore we predict Δ > 0, in line with the empirical observations.
Group C
Only the prison socialtrust and the cell phone network feature significant positive outin correlations. These networks also display nonzero inin and outout correlation, leading to the coexistence of two competing effects: outin correlations decrease n_{D} and the outout and inin correlations increase n_{D}. Since the outin correlation is a first order effect (equation (8)), while outout and inin correlations are only of second order (equation (10)), we expect a decrease in n_{D} (i.e. Δ < 0), consistent with the empirical results.
Group D
The Seagrass food web and citation networks do not feature significant outin correlations, only the secondary inin and outout correlations, hence we expect n_{D} to increase (Δ > 0), consistent with the observations.
Group E
Only the transcriptional regulatory networks are somewhat puzzling in that they show degree correlations, yet the degree sequence still correctly gives n_{D}. However, the simulations indicated that the effect of correlations is negligible for high n_{D}. And our analytical results showed that the value of the correction depends on details of e(j_{i}, j_{o}; k_{i}, k_{o}), not captured by the Pearson coefficient r. These observations highlight that even though in most cases our qualitative predictions based on r are valid, in some cases further investigation is required.
Discussion
The goal of our paper was to clarify the higher order network characteristics that influence controllability. We studied the effect of three topological characteristics: clustering, modularity and degree correlations. We used numerical simulations to identify the role of the relevant characteristics, finding that changes in the clustering coefficient and the community structure have no systematic effect on the the minimum number of driver nodes n_{D}. In contrast degree correlations showed a robust effect, whose magnitude and direction depends on the type of correlation. Using the cavity method we derived n_{D} for networks with given degree distribution and correlation profiles, finding results that are consistent with our numerical simulations. For real networks these numerical and analytic results enabled us to qualitatively explain the deviation of the observed n_{D} from the prediction based only on P(k_{in}, k_{out}).
Our results not only offer a new perspective on the role of topological properties on network controllability, but also raise several questions. Future research directions include determining the optimal network structure to minimize the number of necessary driver nodes and studying how different network characteristics influence the robustness of the control configuration.
Methods
Generating a scalefree network
We use the static model to generate directed scalefree networks^{40}. We start from N disconnected nodes and assign a weight w_{i} = (i + i_{0})^{–α} to each node i (i = 1 … N). We randomly select two nodes i and j with probability proportional to w_{i} and w_{j} respectively and if they are yet not connected, we connect them. We allow selfloops, but avoid multiedges. We repeat the process until L links have been placed. The resulting network has average degree 〈k〉 = 2L/N and P^{(in/out)}(k) ~ k^{–γ} for large k, where and maximum degree .
To systematically study correlations, the starting network has to be uncorrelated. However, the presence of hubs may induce unwanted degree correlations^{41} and may also considerably limit the maximum and minimum correlations accessible via rewiring^{42}. We overcome these difficulties by introducing a structural cutoff in the degrees, choosing i_{0} to ensure k_{max} < (〈k〉N)^{1/2} ^{43}. Note, that in the static model of Goh et al. i_{0} = 0^{40}.
As both in and outdegree of node i is proportional to w_{i}, the above procedure results in correlations between the in and outdegrees of node i. To eliminate the correlations, we randomize the indegree sequence while keeping the outdegree sequence unchanged.
Rewiring algorithm
We use degree preserving rewiring^{20} to add each network characteristic. Suppose that the chosen network characteristic is quantified by a metric X. To set its value to X*, we define the E(X) = X − X* energy, so E(X*) is a global minimum. We minimize this energy by simulated annealing^{44}: (1) choose two links at random with uniform probability; (2) rewire the two links and calculate the energy E(X) of the resulted network; (3) accept the new configuration with probability
where the β parameter is the inverse temperature; (4) repeat from step one and gradually increase β. Stop if E(X) − E(X*) is smaller than a predefined value.
Note, that keeping the degree sequence bounds the possible values of X that can be reached by rewiring. In all cases we study the full interval of accessible X values.
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Acknowledgements
This work was supported by the Network Science Collaborative Technology Alliance sponsored by the US Army Research Laboratory under Agreement Number W911NF0920053; the Defense Advanced Research Projects Agency under Agreement Number 11645021; the Defense Threat Reduction Agency award WMD BRBAA07J20035; FET IP project MULTIPLEX (3A532) and the generous support of Lockheed Martin. M. Pósfai has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under grant agreement No. 270833.
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All authors designed and did the research. M.P. analysed the empirical data and did the analytical and numerical calculations. A.L.B. was the lead writer.
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Pósfai, M., Liu, YY., Slotine, JJ. et al. Effect of correlations on network controllability. Sci Rep 3, 1067 (2013). https://doi.org/10.1038/srep01067
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