Chinese Physics B, Volume. 29, Issue 10, (2020)

Effect of degree correlation on edge controllability of real networks

Shu-Lin Liu and Shao-Peng Pang
Author Affiliations
  • School of Electrical Engineering and Automation, Qilu University of Technology (Shandong Academy of Science), Jinan 250353, China
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    Figures & Tables(6)
    Effect of degree correlation on the edge controllability of real networks. (a) The fraction of driver nodes nDreal obtained directly and nDrand of real networks with no degree correlation. (b) The fraction of driven edges mDreal obtained directly and mDrand of real networks with no degree correlation. (c) and (d) The differences Δn=nDrand−nDreal and Δm=mDrand−mDreal as the function of the Pearson correlation coefficient P of real networks. All the numerical results are obtained by averaging over 50 independent networks realizations. See Table 1 for details.
    Controllability limit theory. The method of calculating the controllability limits of a network with in-degree sequence Kin = {0,1,2,3} and out-degree sequence Kout = {1,1,2,2}. (a) In the bipartite graph H, the node i from in-degree sequence and the node j from out-degree sequence are connected if ki−<kj+. Its generated network in (e) has NDU=2. (b) In the bipartite graph H¯, the node i from in-degree sequence and the node j from out-degree sequence will be connected if ki−≥kj+. Its generated network in (f) has NDL=1. (c) The weighted bipartite graph H* has the same topological structure as H, and the weight of each edge is kj+−ki−. Its generated network in (g) has MDU=3. (d) The weighted bipartite graph H*¯ is generated by connecting arbitrary two nodes, and assigning the weight ki−−kj+ to the edges satisfying ki−<kj+, and 0 for other edges. Its generated network in (h) has MDL=1. Note that the matching nodes of the generated networks are from the matched edges in the maximum matching, and other nodes of the generated networks are combined randomly.
    Controllability limit of real networks. (a) The fraction of driver nodes nDreal obtained directly and the controllability limit (nDU and nDL) of real networks. (b) The fraction of driven edges mDreal obtained directly and the controllability limit (mDU and mDL) of real networks. (c) and (d) The differences nDU−nDL and mDU−mDL versus the average degree 〈k〉 of real networks. The numbers in (a) and (b) refer to the network indices in Table 1.
    Anomaly in edge controllability of real networks. (a) The Pearson correlation coefficient P and the differences Δn of real networks. (b) The Pearson correlation coefficient P and the differences Δm of real networks. All the numerical results are obtained by averaging over 50 independent networks in realizations. The numbers refer to the network indices in Table 1.
    Anomaly in edge controllability. The range of Pearson correlation coefficient P in (a)–(d) model networks and (e)–(f) real networks. The Pearson correlation coefficient PNDU (red) and PNDL (blue) in [(a), (c)] model networks and (e) real networks. The Pearson correlation coefficient PMDU (green) and PMDL (orange) in [(b), (d)] model networks and (f) real networks. The model network is generated by given degree distribution, where in-degree follows exponent distribution and out-degree follows Poisson distribution in (a) and (b), and in-degree follows Poisson distribution and out-degree follows exponent distribution in (c) and (d). See Appendix for how to construct a model network. All the numerical results are obtained by averaging over 50 independent networks in realizations. The numbers in (e)–(f) refer to the network indices in Table 1.
    • Table 1. Simulation results of real networks. For each real network, we show its type, name, nodes’ number N, edges’ number M, the Pearson correlation coefficient P, the fraction of driver nodes and driven edges calculated in the real network (nDreal and mDreal), after randomization (nDrand and mDrand), and the controllability limits (nDU, nDL, mDU and mDL).

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      Table 1. Simulation results of real networks. For each real network, we show its type, name, nodes’ number N, edges’ number M, the Pearson correlation coefficient P, the fraction of driver nodes and driven edges calculated in the real network (nDreal and mDreal), after randomization (nDrand and mDrand), and the controllability limits (nDU, nDL, mDU and mDL).

      TypeNo.NameNMPnDrealmDrealnDrandmDrandnDUnDLmDUmDL
      Regulatory1Ownership-USCorp[28]84976726−0.0310.1360.9240.0860.8480.1590.0281.0000.738
      2TRN-EC-2[29]423578−0.0820.2200.8290.1660.7620.2740.0710.8790.545
      3TRN-Yeast-1[30]4684154510.0440.0520.9470.0490.9470.0640.0250.9840.803
      4TRN-Yeast-2[29]6881079–0.2360.1770.9520.1380.8410.1900.0630.9680.610
      Trust5Prison inmate[31]671820.2010.4030.3190.4500.3590.7610.1790.5110.110
      6Wiki Vote[32]71151036890.3180.2810.6530.2790.8340.3350.0660.9870.192
      Food web7St.Marks[33]45224−0.2920.5330.5630.4790.4830.7110.1560.7010.143
      8Seagrass[34]49226−0.1920.4490.5180.4410.460.7140.1020.6550.097
      9Grassland[35]88137−0.1790.3180.6060.3020.5590.3410.1480.6200.314
      10Ythan[35]1356010.1680.3040.5970.3330.6370.4740.0520.8440.195
      11Silwood[36]1543700.0140.1880.7970.1740.8060.2140.0840.8970.508
      12Little Rock[37]1832494−0.1380.6390.6030.6540.6010.8310.4970.8180.299
      Electronic13S208a[29]122189−0.1770.4510.3440.4300.3260.5490.3110.4130.201
      circuits14s420a[29]252399−0.1540.4560.3480.4390.3270.5600.3250.4160.206
      15s838a[29]512819−0.1460.4590.3500.4410.3270.5640.3320.4180.208
      Neuronal16C. elegans[38]29723590.2910.5490.3740.4940.4770.9230.0810.6390.069
      Citation17Small World[39]2331988−0.0940.2100.7290.2060.7350.3090.0470.8690.469
      18SciMet[39]2729104160.0680.3600.6230.3520.6380.6130.0370.8300.153
      19Kohonen[40]3772127310.0440.2300.7150.2150.7240.3810.0290.8760.436
      Internet20Political blogs[41]1224190900.3790.6190.5250.5530.7100.8700.1650.9080.162
      21p2p-1[42]10876399940.1450.3340.5910.3440.6470.3810.2550.8700.325
      22p2p-2[42]8846318390.1010.3440.6280.3440.6590.3870.2650.8780.352
      23p2p-3[42]8717315250.1070.3430.6250.3440.6580.3830.2640.8780.347
      Organizational24Freeman-1[43]346950.6420.3530.1110.4540.1990.7350.1180.2850.047
      25Consulting[44]468790.4820.5220.1500.4970.2660.8480.1090.3690.078
      Language26English words[31]7381462810.8570.1580.2100.3260.7550.4790.0030.8620.087
      27French words[31]8325242950.9050.1570.2160.2540.6760.3330.0090.7360.092
      Transportation28USair97[45]33221260.6080.4370.4000.4400.6890.7620.0300.8610.045
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    Shu-Lin Liu, Shao-Peng Pang. Effect of degree correlation on edge controllability of real networks[J]. Chinese Physics B, 2020, 29(10):

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    Paper Information

    Received: Apr. 5, 2020

    Accepted: --

    Published Online: Apr. 21, 2021

    The Author Email: Pang Shao-Peng (shaopengpang@qlu.edu.cn)

    DOI:10.1088/1674-1056/ab99ab

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