We use the controllability limit theory to study impact of correlation between in- and out-degrees (degree correlation) on edge controllability of real networks. Simulation results and analytic calculations show that the degree correlation plays an important role in the edge controllability of real networks, especially dense real networks. The upper and lower controllability limits hold for all kinds of real networks. Any edge controllability in between the limits is achievable by properly adjusting the degree correlation. In addition, we find that the edge dynamics in some real networks with positive degree correlation may be difficult to control, and explain the rationality of this anomaly based on the controllability limit theory.
* Project supported by the National Natural Science Foundation of China (Grant No. 61903208).
Cite this article:
Shu-Lin Liu(刘树林) and Shao-Peng Pang(庞少鹏)† Effect of degree correlation on edge controllability of real networks 2020 Chin. Phys. B 29 100202
Fig. 1.
Effect of degree correlation on the edge controllability of real networks. (a) The fraction of driver nodes obtained directly and of real networks with no degree correlation. (b) The fraction of driven edges obtained directly and of real networks with no degree correlation. (c) and (d) The differences and 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.
Fig. 2.
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 . Its generated network in (e) has . (b) In the bipartite graph , the node i from in-degree sequence and the node j from out-degree sequence will be connected if . Its generated network in (f) has . (c) The weighted bipartite graph H* has the same topological structure as H, and the weight of each edge is . Its generated network in (g) has . (d) The weighted bipartite graph is generated by connecting arbitrary two nodes, and assigning the weight to the edges satisfying , and 0 for other edges. Its generated network in (h) has . 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.
Fig. 3.
Controllability limit of real networks. (a) The fraction of driver nodes obtained directly and the controllability limit ( and ) of real networks. (b) The fraction of driven edges obtained directly and the controllability limit ( and ) of real networks. (c) and (d) The differences and versus the average degree 〈k〉 of real networks. The numbers in (a) and (b) refer to the network indices in Table 1.
Fig. 4.
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.
Type
No.
Name
N
M
P
Regulatory
1
Ownership-USCorp[28]
8497
6726
−0.031
0.136
0.924
0.086
0.848
0.159
0.028
1.000
0.738
2
TRN-EC-2[29]
423
578
−0.082
0.220
0.829
0.166
0.762
0.274
0.071
0.879
0.545
3
TRN-Yeast-1[30]
4684
15451
0.044
0.052
0.947
0.049
0.947
0.064
0.025
0.984
0.803
4
TRN-Yeast-2[29]
688
1079
–0.236
0.177
0.952
0.138
0.841
0.190
0.063
0.968
0.610
Trust
5
Prison inmate[31]
67
182
0.201
0.403
0.319
0.450
0.359
0.761
0.179
0.511
0.110
6
Wiki Vote[32]
7115
103689
0.318
0.281
0.653
0.279
0.834
0.335
0.066
0.987
0.192
Food web
7
St.Marks[33]
45
224
−0.292
0.533
0.563
0.479
0.483
0.711
0.156
0.701
0.143
8
Seagrass[34]
49
226
−0.192
0.449
0.518
0.441
0.46
0.714
0.102
0.655
0.097
9
Grassland[35]
88
137
−0.179
0.318
0.606
0.302
0.559
0.341
0.148
0.620
0.314
10
Ythan[35]
135
601
0.168
0.304
0.597
0.333
0.637
0.474
0.052
0.844
0.195
11
Silwood[36]
154
370
0.014
0.188
0.797
0.174
0.806
0.214
0.084
0.897
0.508
12
Little Rock[37]
183
2494
−0.138
0.639
0.603
0.654
0.601
0.831
0.497
0.818
0.299
Electronic
13
S208a[29]
122
189
−0.177
0.451
0.344
0.430
0.326
0.549
0.311
0.413
0.201
circuits
14
s420a[29]
252
399
−0.154
0.456
0.348
0.439
0.327
0.560
0.325
0.416
0.206
15
s838a[29]
512
819
−0.146
0.459
0.350
0.441
0.327
0.564
0.332
0.418
0.208
Neuronal
16
C. elegans[38]
297
2359
0.291
0.549
0.374
0.494
0.477
0.923
0.081
0.639
0.069
Citation
17
Small World[39]
233
1988
−0.094
0.210
0.729
0.206
0.735
0.309
0.047
0.869
0.469
18
SciMet[39]
2729
10416
0.068
0.360
0.623
0.352
0.638
0.613
0.037
0.830
0.153
19
Kohonen[40]
3772
12731
0.044
0.230
0.715
0.215
0.724
0.381
0.029
0.876
0.436
Internet
20
Political blogs[41]
1224
19090
0.379
0.619
0.525
0.553
0.710
0.870
0.165
0.908
0.162
21
p2p-1[42]
10876
39994
0.145
0.334
0.591
0.344
0.647
0.381
0.255
0.870
0.325
22
p2p-2[42]
8846
31839
0.101
0.344
0.628
0.344
0.659
0.387
0.265
0.878
0.352
23
p2p-3[42]
8717
31525
0.107
0.343
0.625
0.344
0.658
0.383
0.264
0.878
0.347
Organizational
24
Freeman-1[43]
34
695
0.642
0.353
0.111
0.454
0.199
0.735
0.118
0.285
0.047
25
Consulting[44]
46
879
0.482
0.522
0.150
0.497
0.266
0.848
0.109
0.369
0.078
Language
26
English words[31]
7381
46281
0.857
0.158
0.210
0.326
0.755
0.479
0.003
0.862
0.087
27
French words[31]
8325
24295
0.905
0.157
0.216
0.254
0.676
0.333
0.009
0.736
0.092
Transportation
28
USair97[45]
332
2126
0.608
0.437
0.400
0.440
0.689
0.762
0.030
0.861
0.045
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 ( and ), after randomization ( and ), and the controllability limits ( , , and ).
Fig. 5.
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 (red) and (blue) in [(a), (c)] model networks and (e) real networks. The Pearson correlation coefficient (green) and (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.
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