Structurally robust control of complex networks

Jose C. Nacher and Tatsuya Akutsu

Phys. Rev. E 91, 012826 – Published 30 January 2015

Abstract

Robust control theory has been successfully applied to numerous real-world problems using a small set of devices called controllers. However, the real systems represented by networks contain unreliable components and modern robust control engineering has not addressed the problem of structural changes on complex networks including scale-free topologies. Here, we introduce the concept of structurally robust control of complex networks and provide a concrete example using an algorithmic framework that is widely applied in engineering. The developed analytical tools, computer simulations, and real network analyses lead herein to the discovery that robust control can be achieved in scale-free networks with exactly the same order of controllers required in a standard nonrobust configuration by adjusting only the minimum degree. The presented methodology also addresses the probabilistic failure of links in real systems, such as neural synaptic unreliability in Caenorhabditis elegans, and suggests a new direction to pursue in studies of complex networks in which control theory has a role.

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Received 3 May 2014
Revised 16 July 2014

DOI:https://doi.org/10.1103/PhysRevE.91.012826

Authors & Affiliations

Jose C. Nacher^†

Department of Information Science, Faculty of Science, Toho University, Miyama 2-2-1, Funabashi, Chiba 274-8510, Japan

Tatsuya Akutsu^‡

Bioinformatics Center, Institute for Chemical Research, Kyoto University, Uji, 611-0011, Japan

^*Corresponding authors.
^†nacher@is.sci.toho-u.ac.jp
^‡takutsu@kuicr.kyoto-u.ac.jp

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Vol. 91, Iss. 1 — January 2015

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Figure 1
The MDS model and the robust domination configuration. (a) The network is structurally controllable by selecting a minimum dominating set (MDS) because each dominated node has its own control signal. A maximum matching approach needs three driver nodes $v_{1}$ , $v_{2}$ , and $v_{4}$ , assuming a matching link from $v_{1}$ to $v_{3}$ . In contrast, the MDS only requires one node, $v_{1}$ . The labels $u_{1}^{1}$ , $u_{1}^{2}$ , and $u_{1}^{3}$ indicate control. (b) Example of a dominating set (DS) in a graph $G$ . A set of nodes $S$ (filled) in a graph $G$ is a dominating set if every node in $G$ is either an element of $S$ or adjacent to an element of $S$ . (c) The graph $G$ allows for an MDS with cover $C = 1$ composed of only three nodes. (d) Example of a robust minimum dominating set (RMDS) (i.e., an MDS with cover $C = 2$ ). In an RMDS, each node in $G$ is either an element of $S$ or adjacent to at least two elements of $S$ . (e) Each edge ( $t_{i}$ , $t_{j}$ ) has a failure probability of $P_{t_{i}, t_{j}}$ highlighted in the figure. In a probabilistic minimum dominating set (PMDS), the approach shown here, each node must be covered by multiple nodes in the MDS so the probability that at least one edge is active is at least $θ = 0.7$ . For instance, $t_{4}$ must cover itself because the link ( $t_{2}, t_{4}$ ) is unreliable with a failure probability of 0.8, which is higher than $θ = 0.7$ .
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Figure 2
The theoretical predictions for the MDS order. Theoretical analysis illustrates that the MDS size scales according to $n^{δ}$ [see Eq. (3)] in scale-free networks of size $n$ . The scaling exponent $δ$ as a function of the degree exponent $γ$ for the (a) $C = 1$ , $D = 2$ and (b) $C = 1$ , $D = 1$ configurations. (c) The degree exponent that minimises the MDS as a function of the minimum degree $D$ . (d) The dependence of the $δ$ exponent on the degree exponent $γ$ calculated for several $D$ values.
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Figure 3
Analytical results for several configurations. Comparison of upper bounds for $(C, D) = (1, 1), (1, 2)$ and lower bounds for $(C, D) = (1, 1)$ .
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Figure 4
The theoretical predictions for MDS ( $C = 1$ ) and robust domination RMDS ( $C = 2, 3$ ) with $D = C$ . (a) Lower and upper bound predictions for the fraction of nodes required to control the entire network for covers $C = 1, 2, 3$ . (b) Ratio of lower bound MDS sizes for $C = 2$ and $C = 1$ (circles) and $C = 3$ and $C = 1$ (triangles). (c) Ratio of upper bound MDS sizes for $C = 2$ and $C = 1$ (circles) and $C = 3$ and $C = 1$ (triangles).
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Figure 5
Computer simulations for MDS size for $γ < 2$ . The MDS size calculated in computer-generated scale-free networks for $γ = 1.3$ , 1.5, and 1.7 are shown in (a), (b), and (c), respectively. The configurations for minimum degree $D$ and cover $C$ are shown in the figure legend. The lines display a scaling law of $n^{δ}$ . The precise values for $δ$ are, from top to bottom, (a) $0.804 \pm 0.011$ , $0.8234 \pm 0.009$ , $0.791 \pm 0.015$ ; (b) $0.771 \pm 0.007$ , $0.781 \pm 0.006$ , $0.704 \pm 0.007$ ; and (c) $0.810 \pm 0.006$ , $0.804 \pm 0.003$ , $0.714 \pm 0.008$ . As predicted by the theory, configurations $D = 1$ , $C = 1$ (blue) and $D = 2$ , $C = 2$ (red) exhibit very similar scaling exponents. Note that $D = 2$ significantly decreases the MDS size. (d) The scaling exponent $δ$ predicted by theory compared with the scaling exponent observed in computer simulations for each $D$ and $C$ configurations. The results were averaged over 10 realizations. The error bars (s.e.m.) are shown in the figure. The correlation coefficient $r$ is above 0.999 in all cases.
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Figure 6
The MDS size calculated in computer generated scale-free networks for $γ = 2.3$ , 2.5. Configurations for minimum degree $D$ and cover $C$ are shown in figure legend. The lines show a scaling law as $n^{δ}$ . The precise values for $δ$ are from up to down (a) $0.976 \pm 0.005$ , $0.978 \pm 0.011$ , $0.951 \pm 0.010$ and (b) $0.993 \pm 0.005$ , $0.995 \pm 0.005$ , $0.986 \pm 0.010$ . All three configurations show very similar scaling exponents close to 1, as predicted by theory. Note that $D = 2$ significantly decreases the MDS size. The error bars (s.e.m.) are shown in the figure. The correlation coefficient $r$ is above 0.999 in all cases.
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Figure 7
The MDS size calculated in computer generated scale-free networks for $γ = 2.7$ , 3.0. Configurations for minimum degree $D$ and cover $C$ are shown in figure legend. The lines show a scaling law as $n^{δ}$ . The precise values for $δ$ are from up to down (a) $0.994 \pm 0.002$ , $0.999 \pm 0.004$ , $0.986 \pm 0.006$ and (b) $0.998 \pm 0.008$ , $0.992 \pm 0.011$ , $0.993 \pm 0.022$ . All three configurations show very similar scaling exponents close to 1, as predicted by theory. Note that $D = 2$ significantly decreases the MDS size. The error bars (s.e.m.) are shown in the figure. The correlation coefficient $r$ is above 0.999 in all cases.
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Figure 8
The MDS fraction in real undirected networks. The MDS fraction as a function of the degree exponent $γ$ for real undirected networks $C = 1$ in (a) and $C = 2$ in (b). The MDS fraction as a function of the network size for the same real undirected networks for $C = 1$ in (c) and $C = 2$ in (d). Classification of nodes into network classes is only displayed for (a) and (b) (see figure legend). Highlighted regions are visual guidance of the observed tendency.
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Figure 9
The MDS size scaling in real undirected and directed networks. The MDS size as a function of the network size $n$ for real (a) undirected and directed networks with (b) incoming and (c) outgoing links. The results display a scaling function of $n^{δ}$ with $δ$ close to 1, as predicted by the theoretical analysis. The precise values of the scaling exponent $δ$ for $C = 1$ (plus) and $C = 2$ (circle) are (a) $0.934 \pm 0.046$ ( $r = 0.981$ ), $0.931 \pm 0.036$ ( $r = 0.989$ ); (b) $1.094 \pm 0.053$ ( $r = 0.990$ ), $1.034 \pm 0.044$ ( $r$ =0.992); and (c) $1.134 \pm 0.071$ ( $r$ =0.984), $1.092 \pm 0.067$ ( $r = 0.985$ ), respectively. The correlation coefficient $r$ is indicated between parentheses. The ratio between the MDS sizes (MDS2:MDS1) computed with covers $C = 2$ and $C = 1$ is shown in parts (d)–(f) for the same real networks. The results show that the ratio is almost always lower than 2. See Tables 1 and 2 for real data details.
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Figure 10
Comparison between theoretical predicted bounds and MDS in real undirected networks. The overlap between the MDS fraction with $C = 1$ (MDS1) for undirected networks and the predictions of Eqs. (6) and (7) for the lower and upper bounds of networks with $γ > 2$ . The real data are always within the theoretical boundaries.
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Figure 11
The dependence of the MDS fraction on the degree distribution for real undirected networks. The MDS fraction with $C = 1$ (MDS1) (a) and $C = 2$ (MDS2) (b) as a function of the degree probability with degree 1 [ $P (1) = n (1) / n$ ] and 2 [ $P (2) = n (2) / n$ ]. The ratio of MDS2:MDS1 versus (c) $P (1)$ and (d) $P (2)$ , respectively. The ratio is below 2 in most cases. The results demonstrate that small $P (1)$ and large $P (2)$ tend to be associated with a small MDS fraction.
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Figure 12
The MDS fraction as a function of the degree exponent $γ_{out}$ for real directed networks $C = 1$ in (a) and $C = 2$ in (b). The MDS fraction as a function of the network size for the same real directed networks for $C = 1$ in (c) and $C = 2$ in (d). In all cases the MDS fraction is lower than 0.5 (dotted line). Note that four of seven food webs show power-law behavior for outgoing degrees. Each color in circles corresponds to a network type as shown in the legend.
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Figure 13
The MDS fraction as a function of the degree exponent $γ_{in}$ for real directed networks $C = 1$ in (a) and $C = 2$ in (b). The MDS fraction as a function of the network size for the same real directed networks for $C = 1$ in (c) and $C = 2$ in (d). In all cases the MDS fraction is lower than 0.5 (dotted line). Note that five of seven food webs show power-law behavior for incoming degrees. The data for the transcriptional regulatory network is absent because it does not follow a power law for incoming degrees. Each color in circles corresponds to a network type as shown in the legend.
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Figure 14
The MDS size computed on real-world bipartite networks. (a) Social networks and (b) biological networks computed for $C = 1$ and $C = 2$ as shown in the legend. Note that social networks tend to be controlled with a smaller fraction of nodes compared to biological networks. [(c)–(d)] Computation of the MDS in synthetic bipartite networks generated with the model shown in Sec. 4 C with several values of degree exponents $γ_{1}$ and $γ_{2}$ and three configurations ( $C = 1$ , $D = 1$ ), ( $C = 2$ , $D = 1$ ), and ( $C = 2$ , $D = 2$ ). Note that the minimum degree $D = 2$ almost completely compensates the increasing of the MDS size from the robust control $C = 2$ .
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Figure 15
Graphical representation of the case $D = 2$ and $w > 0$ . In order to estimate the fraction of degree-2 nodes to be added to a DS, it is enough to consider the regions (A) and (B) for the cases of $(p + w) (p - w) < 1 - θ < (p + w) (p + w)$ and $(p - w) (p - w) < 1 - θ < (p + w) (p - w)$ , respectively.
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Figure 16
Probabilistic MDS simulation results. The probabilistic MDS fraction in simulated scale-free networks with ( $a$ ) $D = 1$ and ( $b$ ) $D = 2$ for $γ = 1.5$ and $D = 2$ ( $c$ ) $D = 1$ and (d) $D = 2$ for $γ = 1.7$ . The predicted theoretical thresholds (dashed lines) that significantly changes the MDS size are in fair agreement with observed results in computer simulations. The configurations for the probability of link failure $P$ and the variability change $w$ of the failure probability [ $p - w$ , $p + w$ ] are shown in the figure legend.
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Figure 17
The probabilistic robust domination on synthetic scale-free networks. The probabilistic MDS fraction in simulated scale-free networks with (a) $D = 1$ and (b) $D = 2$ for $γ = 2.5$ and (c) $D = 1$ and (d) $D = 2$ for $γ = 3.0$ . The predicted theoretical thresholds (dashed lines) that significantly change the MDS size are in fair agreement with the observed results from the computer simulations. The configurations for the probability of link failure $p$ and the variability change $w$ of the failure probability [ $p - w$ , $p + w$ ] are shown in the figure legend.
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Figure 18
Theoretical predictions and simulations of PMDS. Comparison of the probabilistic MDS size computed in simulated scale-free networks with $D = 2$ and $γ = 3$ and the case of $D = 2$ predicted by theory. Theoretical values are scaled so these take almost the same values as the simulated ones at the beginning and ending points because, in the theoretical analysis, it is assumed that all nodes are of degree 2 and the effects of the other nodes are ignored.
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Figure 19
The neural gap junction network. Visualization of the experimental gap junction undirected network for C. elegans. A probability distribution of synapse transmission failure with a peak at $p = 0.71$ and width of $w = 0.24$ (0.47–0.95) is mapped onto the links.
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Figure 20
The probabilistic MDS size computed on the neural gap junction undirected network. The computation of the probabilistic MDS size using the real neural gap junction network for C. elegans organism where a distribution of link failure was mapped onto the network as shown in legend for three values of $w$ (circle, square, and diamond symbols). The results of synthetic scale-free networks constructed using the model shown in Sec. 3 D and calculated with the same degree exponent and the number of nodes observed in the real C. elegans network (star symbols). When simulated networks have a minimum degree $D = 2$ , the MDS size decreases as predicted by theory. Because in this simulation the synthetic network have different average degree, the results tend to be higher than those from the real network.
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Figure 21
The probabilistic robust domination on a real neural gap junction network. The computation of the probabilistic MDS size using (a) the real neural gap junction network for C. elegans where the distribution of link failures was mapped to the network as shown in the legend. (b) The results of synthetic scale-free networks constructed using the HMC model with the same number of nodes and average degree display a similar tendency. The MDS size decreases as predicted by theory when simulated networks have a minimum degree $D = 2$ .
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Figure 22
The probabilistic MDS size computed on the chemical synapse directed network. The computation of the probabilistic MDS size using the real chemical synapse directed network for C. elegans where a distribution of link failure was mapped onto the network as shown in legend for three values of $w$ . Note that the distribution follows a curve function rather than a straight line as shown in the undirected case.
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