Liu et al.: Controllability of complex networks

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1 complex networks Sandbox slides. Peter Sheridan Dodds Department of Mathematics & Statistics Center for Complex Systems Vermont Advanced Computing Center University of Vermont of 2

2 Outline 2 of 2

3 RESEARCH From [] a x 4 ARTICLE u t u b 2 t x a 4 a b 2 2 a 3 x Initial state? Desired final state x 2 A = b b a 2 b b 2 a b 2 2 ; B = ; C = a 3 a 34 a 3 b a 34 a 4 b a 4 a 4 b N = 4, M = 2, rank(c) = N a 34 x 2 x 4 x 3 x 3 b e h Network u 3 Controlled network c Matched node Unmatched node Input signal Matching link u x x 2 x 3 f u u 2 x u 3 x 2 x 3 x 4 u u i 2 u x 4 3 x x 2 x 4 x 5 x6 x 4 Link category Critical link Redundant link Ordinary link d g j Figure Controlling a simple network. a, The small network can be controlled by an input vector u 5 (u(t), u2(t)) T (left), allowing us to move it from its initial state to some desired final state in the state space (right). Equation (2) provides the controllability matrix (C), which in this case has full rank, indicating that the system is controllable. b, Simple model network: a directed path. c, Maximum matching of the directed path. Matching edges are shown in purple, matched nodes are green and unmatched nodes are white. The unique maximum matching includes all links, as none of them share a common starting or ending node. Only the top node is unmatched, so controlling it yields full control of the directed path (ND 5 ). d, In the directed path shown in b, all links are critical, that is, their removal eliminates our ability to control the network. e, Small model network: the directed star. f, Maximum matchingsof the directed star. Only one link can be partof the maximum matching, which yields three unmatched nodes (ND 5 3). The three different maximum matchings indicate that three distinct node configurations can exert full control. g, In a directed star, all links are ordinary, that is, their removal can eliminate some control configurations but the network could be controlled in their absence with the same number of driver nodes N D. h, Small example network.i, Only two links can be part of a maximum matching for the network in h, yielding four unmatched nodes (N D 5 4). There are all together four different maximum matchings for this network. j, The network has one critical link, oneredundantlink (whichcanberemoved withoutaffectinganycontrol configuration) and four ordinary links. networks are either unknown (for example regulatory networks) or problem maps into an equivalent geometrical problem on a network: are known only approximately and are time dependent (for example we can gain full control over a directed network if and only if we Paper site: Internet traffic). Even if all weights are known, a brute-force search directly control each unmatched node and there are directed paths requires us to compute the rank of C for 2 N 2 distinct combina- from the input signals to all matched nodes 29. The possibility of 3 of 2

4 tion procedure (rand-er) that turns the network into a directed Erdős Rényi random network with N and L unchanged. For several Table The characteristics of the real networks analysed in the paper perfectly with ND rand-degree (and hence is in good agreement with the exact value, ND real ), offering an effective analytical tool to study Type Name N L nd real nd rand-degree nd rand-er Regulatory TRN-Yeast- 4,44 2, TRN-Yeast-2 688, TRN-EC-,55 3, TRN-EC Ownership-USCorp 7,253 6, Trust College student Prison inmate Slashdot 82,68 948, WikiVote 7,5 3, Epinions 75,888 58, Food web Ythan Little Rock 83 2, Grassland Seagrass Power grid Texas 4,889 5, Metabolic Escherichia coli 2,275 5, Saccharomyces cerevisiae,5 3, Caenorhabditis elegans,73 2, Electronic circuits s s s Neuronal Caenorhabditis elegans 297 2, Citation ArXiv-HepTh 27,77 352, ArXiv-HepPh 34,546 42, World Wide Web nd.edu 325,729,497, stanford.edu 28,93 2,32, Political blogs,224 9, Internet p2p-,876 39, p2p-2 8,846 3, p2p-3 8,77 3, Social communication UCIonline,899 2, epoch 3,88 39, Cellphone 36,595 9, Intra-organizational Freemans Freemans Manufacturing 77 2, Consulting For each network, we show its type and name; number of nodes (N)andedges(L); and density of driver nodes calculated in the real network (nd real ), after degree-preserved randomization (nd rand-degree )andafter full randomization (nd rand-er ). For data sources and references, see Supplementary Information, section VI. 2 Macmillan Publishers Limited. All rights reserved 2 MAY 2 VOL 473 NATURE 69 4 of 2

5 RESEARCH ARTICLE a b Erdos Rényi c d 6 5 f D k D N D rand-er Low-k Medium-k High-k Low-k Medium-k High-k k Nreal D N D rand-degree e Nreal D Regulatory Trust Food web Power grid Metabolic Electronic circuits Neuronal Citation World Wide Web Internet Social communication Intra-organizational Erdos Rényi γ = 2.5 γ = 3. γ = 4. N D rand-degree f Nanalytic D Figure 2 Characterizing and predicting the driver nodes (ND). a, b, Role of the hubs in model networks. The bars show the fractions of driver nodes, fd, among the low-, medium- and high-degree nodes in two network models, Erdős Rényi (a) and scale-free (b), with N 5 4 and Ækæ 5 3(c 5 3), indicating that the driver nodes tend to avoid the hubs. Both the Erdős Rényi and the scale-free networks are generated from the static model 38 and the results are averaged over realizations. The error bars (s.e.m.), shown in the figure, are smaller than the symbols. c, Mean degree of driver nodes compared with the mean degree of all nodes in real and model networks, indicating that in real systems the hubs are avoided by the driver nodes. d, Number of driver nodes, ND rand-er, obtained for the fully randomized version of the networks listed in Table, compared with the exact value, ND real. e, Number of driver nodes, ND rand-degree, obtained for the degree-preserving randomized version of the networks shown in Table, compared with N D real. f, The analytically predicated N D analytic calculated using the cavity method, compared with N D rand-degree.in d f, data points and error bars (s.e.m.) were determined from, realizations of the randomized networks. the impact of various network parameters on ND. Although the cavity method does not offer a closed-form solution, we can derive the dependence of nd on key network parameters in the thermodynamic limit (N R ). We find, for example, that for a directed Erdős Rényi network nd decays as agreement with the numerical results for c. 3 (Fig. 3d, e). Near c 5 2, however, nd as predicted by the cavity method deviates from the exact nd value owing to degree correlations that are prominent at cc 5 2 and can be eliminated by imposing a degree cut-off in constructing the scale-free networks 39,46 (Supplementary Information, section IV.B). 5 of 2

6 a b c d n D f Random regular P(k) P(k) log[p(k)] k k log(k) ER k N = oo N = 5 SF!= 2.2 SF = 2.5 SF = 3. SF = 4. N = oo N = 5 SF!= 2.2 SF!= 2.5 SF!= 3. SF!= 4. ER Erdos Rényi n D e g All nodes must be controlled SF k = 2 SF k = 4 SF k = 8 SF k = 6 ER N = oo N = 5 SF k = 2 SF k = 4 SF k = 8 SF k = 6 ER N = oo N = 5 Figure 3 The impact of network structure on the number of driver nodes. a c, Characteristics of the explored model networks. A random-regular digraph (a), shown here for Ækæ 5 4, is the most degree-homogeneous network as kin 5 kout 5 Ækæ/2 for all nodes. Erdős Rényi networks (b) have Poisson degree distributions and their degree heterogeneities are determined by Ækæ. networks (c) have power-law degree distributions, yielding large degree heterogeneities. d, Driver node density, n D, as a function of Ækæ for Erdős Rényi (ER) and scale-free (SF) networks with different values of c. Both the Erdős Rényi and the scale-free networks are generated from the static model 38 with N 5 5. Lines are analytical results calculated by the cavity method using the expected degree distribution in the N R limit. Symbols are calculated for the constructed discrete network: open circles indicate exact results calculated from the maximum matching algorithm, and plus symbols indicate the analytical results of the cavity method using the exact degree sequence of the constructed network. For large Ækæ, nd approaches its lower bound, N 2, that is, a single driver node (ND5) inanetworkofsizen. e, nd asafunction of c for scale-free networks with fixed Ækæ. For infinite scale-free networks, n D R asc R c c 5 2, that is, it is necessary to control almost all nodes to control the network fully. For finite scale-free networks, n D reaches its maximum as c approaches c c (Supplementary Information). f, n D as a function of degree heterogeneity, H, for Erdős Rényi and scale-free networks with fixed c and variable Ækæ. g, nd asa functionofhforerdős Rényiandscalefree networks for fixed Ækæ and variable c. Asc increases, the curves converge to the Erdős Rényi result (black) at the corresponding Ækæ value. n D n D H Furthermore, the larger are the differences between node degrees, the H To understand the factors that determine l, l and l, in Fig. 5a, c, f 6 of 2

7 RESEARCH ARTICLE l r l o l c Consulting Manufacturing Freemans-2 Freemans- Cellphone -epoch UCIonline p2p-3 p2p-2 p2p- Political blogs stanford.edu nd.edu ArXiv-HepPh ArXiv-HepTh C. elegans (neuronal) s28 s42 s838 C. elegans (metabolic) S. cerevisiae E. coli Texas Seagrass Grassland Littlerock Ythan Epinions WikiVote Slashdot Prison inmate College student Ownership-USCorp TRN-EC-2 TRN-EC- TRN-Yeast-2 TRN-Yeast- Figure 4 Link categories for robust control. The fractions of critical (red, lc), redundant (green, lr) and ordinary (grey, lo) links for the real networks named in Table. To make controllability robust to link failures, it is sufficient to double only the critical links, formally making each of these links redundant and therefore ensuring that there are no critical links in the system. Erdos Rényi matchings increases exponentially (Supplementary Information, section IV.C) and, as a result, the chance that a link does not participate in any control configuration decreases. For scale-free networks, we observe the same behaviour, with the caveat that Ækæc decreases with c (Fig. 5c, d). Discussion and conclusions Control is a central issue in most complex systems, but because a general theory toexplore it in aquantitative fashion has been lacking, little is known about how we can control a weighted, directed network the configuration most often encountered in real systems. Indeed, applying Kalman s controllability rank condition (equation (3)) to large networks is computationally prohibitive, limiting previous work to a few dozen nodes at most 7 9. Here we have developed the tools to address controllability for arbitrary network topologies and sizes.our key finding, that ND isdetermined mainly bythe degree l c a c SF!= 2.6 SF!= 2.8 SF!= 3. SF!= 4. ER l c l r l o d n core b Core Leaves e e 7 of 2

8 l c a Erdos Rényi c SF!= 2.6 SF SF SF ER!= 2.8!= 3.!= 4. l c l r l o d n core b Core Leaves e k e e k = 4 k = 5 k = 7 Core percolation k Leaf node Core node f Link category Critical link Redundant link Ordinary link Figure 5 Control robustness. a, Dependence on Ækæ of the fraction of critical (red, lc), redundant (green, lr) and ordinary (grey, lo) links for an Erdős Rényi network: lr peaks at Ækæ 5 Ækæc 5 2e and the derivative of lc is discontinuous at Ækæ 5 Ækæc. b, Core percolation for Erdős Rényi network occurs at k 5 Ækæc 5 2e, whichexplainsthelr peak. c, d, Sameas inaandbbutforscale-freenetworks. The Erdős Rényi and scale-free networks 38 have N 5 4 and the results are averaged over ten realizations with error bars defined as s.e.m. Dotted lines are only a guide to the eye. e, The core (red) and leaves (green) for small Erdős Rényi networks (N 5 3) at different Ækæ values (Ækæ 5 4, 5, 7). Node sizes are proportional to node degrees. f, The critical (red), redundant (green) and ordinary (grey) links for the above Erdős Rényi networks at the corresponding Ækæ values. 8 of 2

9 How number of emotion-assessed words decays with stopword gap size. 9 of 2

10 Time series comparisons with stop words h avg (5 h avg, 5 + h avg ) (lower values are better). Difference is RMS. Uses raw time series (no subtraction of mean). Transition at h avg =.6 is because no drops out. of 2

11 Time series comparisons with stop words h avg (5 h avg, 5 + h avg ) (higher values are better). Difference is cosine of angle between time series vectors. Uses time series with mean subtracted. Suggests h avg = or.7 is first of a stable range. 5?? of 2

12 I [] Y.-Y. Liu, J.-J. Slotine, and A.-L. Barabási.. Nature, 473:67 73, 2. 2 of 2

Supplementary Figures

Supplementary Figures Supplementary Figure S: Identifying all minimum sets of driver nodes in a directed network associated with structural matrix. (a) A directed network with structural matrix A, matrix