Effect of degree correlation on edge controllability of real networks

doi:10.1088/1674-1056/ab99ab

Abstract

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.

Keywords: complex network edge controllability degree correlation controllability limit

Received: 05 April 2020
Revised: 24 May 2020
Accepted manuscript online: 05 June 2020

PACS:	02.30.Yy	(Control theory)
	89.75.Fb	(Structures and organization in complex systems)

Corresponding Authors: ^†Corresponding author. E-mail: shaopengpang@qlu.edu.cn

About author:

†Corresponding author. E-mail: shaopengpang@qlu.edu.cn

* 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 $ {n}_{{\rm{D}}}^{{\rm{real}}} $ obtained directly and $ {n}_{{\rm{D}}}^{{\rm{rand}}} $ of real networks with no degree correlation. (b) The fraction of driven edges $ {m}_{{\rm{D}}}^{{\rm{real}}} $ obtained directly and $ {m}_{{\rm{D}}}^{{\rm{rand}}} $ of real networks with no degree correlation. (c) and (d) The differences $ {\varDelta }_{n}={n}_{{\rm{D}}}^{{\rm{rand}}}-{n}_{{\rm{D}}}^{{\rm{real}}} $ and $ {\varDelta }_{m}={m}_{{\rm{D}}}^{{\rm{rand}}}-{m}_{{\rm{D}}}^{{\rm{real}}} $ 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 K_in = {0,1,2,3} and out-degree sequence K_out = {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 $ {k}_{i}^{-}\lt {k}_{j}^{+} $ . Its generated network in (e) has $ {N}_{{\rm{D}}}^{{\rm{U}}}=2 $ . (b) In the bipartite graph $ \bar{H} $ , the node i from in-degree sequence and the node j from out-degree sequence will be connected if $ {k}_{i}^{-}\ge {k}_{j}^{+} $ . Its generated network in (f) has $ {N}_{{\rm{D}}}^{{\rm{L}}}=1 $ . (c) The weighted bipartite graph H^* has the same topological structure as H, and the weight of each edge is $ {k}_{j}^{+}-{k}_{i}^{-} $ . Its generated network in (g) has $ {M}_{{\rm{D}}}^{{\rm{U}}}=3 $ . (d) The weighted bipartite graph $ \bar{{H}^{* }} $ is generated by connecting arbitrary two nodes, and assigning the weight $ {k}_{i}^{-}-{k}_{j}^{+} $ to the edges satisfying $ {k}_{i}^{-}\lt {k}_{j}^{+} $ , and 0 for other edges. Its generated network in (h) has $ {M}_{{\rm{D}}}^{{\rm{L}}}=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.

Fig. 3.

Controllability limit of real networks. (a) The fraction of driver nodes $ {n}_{{\rm{D}}}^{{\rm{real}}} $ obtained directly and the controllability limit ($ {n}_{{\rm{D}}}^{{\rm{U}}} $ and $ {n}_{{\rm{D}}}^{{\rm{L}}} $ ) of real networks. (b) The fraction of driven edges $ {m}_{{\rm{D}}}^{{\rm{real}}} $ obtained directly and the controllability limit ($ {m}_{{\rm{D}}}^{{\rm{U}}} $ and $ {m}_{{\rm{D}}}^{{\rm{L}}} $ ) of real networks. (c) and (d) The differences $ {n}_{{\rm{D}}}^{{\rm{U}}}-{n}_{{\rm{D}}}^{{\rm{L}}} $ and $ {m}_{{\rm{D}}}^{{\rm{U}}}-{m}_{{\rm{D}}}^{{\rm{L}}} $ 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 Δ_n of real networks. (b) The Pearson correlation coefficient P and the differences Δ_m 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	$ {n}_{{\rm{D}}}^{{\rm{real}}} $	$ {m}_{{\rm{D}}}^{{\rm{real}}} $	$ {n}_{{\rm{D}}}^{{\rm{rand}}} $	$ {m}_{{\rm{D}}}^{{\rm{rand}}} $	$ {n}_{{\rm{D}}}^{{\rm{U}}} $	$ {n}_{{\rm{D}}}^{{\rm{L}}} $	$ {m}_{{\rm{D}}}^{{\rm{U}}} $	$ {m}_{{\rm{D}}}^{{\rm{L}}} $
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 ($ {n}_{{\rm{D}}}^{{\rm{real}}} $ and $ {m}_{{\rm{D}}}^{{\rm{real}}} $ ), after randomization ($ {n}_{{\rm{D}}}^{{\rm{rand}}} $ and $ {m}_{{\rm{D}}}^{{\rm{rand}}} $ ), and the controllability limits ($ {n}_{{\rm{D}}}^{{\rm{U}}} $ , $ {n}_{{\rm{D}}}^{{\rm{L}}} $ , $ {m}_{{\rm{D}}}^{{\rm{U}}} $ and $ {m}_{{\rm{D}}}^{{\rm{L}}} $ ).

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 $ {P}_{{\rm{ND}}}^{{\rm{U}}} $ (red) and $ {P}_{{\rm{ND}}}^{{\rm{L}}} $ (blue) in [(a), (c)] model networks and (e) real networks. The Pearson correlation coefficient $ {P}_{{\rm{MD}}}^{{\rm{U}}} $ (green) and $ {P}_{{\rm{MD}}}^{{\rm{L}}} $ (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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