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January  2017, 2(1): 87-96. doi: 10.3934/bdia.2017011

A comparative study of robustness measures for cancer signaling networks

Mingxing Zhou 1, , Jing Liu 1,, , Shuai Wang 1, and  Shan He 2,

1. 

Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, Xidian University, Xi'an 710071, China
2. 

School of Computer Science, University of Birmingham, Birmingham, U.K

* Corresponding author: Jing Liu

Published  September 2017

Fund Project: The corresponding author is supported by NSF grant 61522311,61773300,61528205 and 2017JZ017.

Network robustness stands for the capability of networks in resisting failures or attacks. Many robustness measures have been proposed to evaluate the robustness of various types of networks, such as small-world and scale-free networks. However, the robustness of biological networks is different for their special structures related to the unique functionality. Cancer signaling networks which show the information transformation of cancers in molecular level always appear with robust complex structures which mean information exchange in the networks do not depend on skimp pathways in which resulting the low rate of cure, high rate of recurrence and especially, the short time in survivability caused by constantly destruction of cancer. So a network metric that shows significant changes when one node is removed, and further to correlate that metric with survival probabilities for patients who underwent cancer chemotherapy is meaningful. Therefore, in this paper, the relationship between 14 typical cancer signaling networks robustness and those cancers patient survivability are studied. Several widely used robustness measures are included, and we find that the natural connectivity, in which the redundant circles are satisfied with the need of information exchange of cancer signaling networks, is negatively correlated to cancer patient survivability. Furthermore, the top three affected nodes measured by natural connectivity are obtained and the analysis on these nodes degree, closeness centrality and betweenness centrality are followed. The result shows that the node found are important so we believe that natural connectivity will be a great help to cancer treatment.

Keywords: Network robustness, robustness measure, cancer signaling network, cancer patient survivability.
Mathematics Subject Classification: Primary: 37F20, 68U05; Secondary: 65Y20.
Citation: Mingxing Zhou, Jing Liu, Shuai Wang, Shan He. A comparative study of robustness measures for cancer signaling networks. Big Data & Information Analytics, 2017, 2 (1) : 87-96. doi: 10.3934/bdia.2017011
References:
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R. Albert, H. Jeong and A. -L. Barabási, Error and attack tolerance of complex networks, Nature, 406 (2000), 378-382.  doi: 10.1038/35019019.

[2]

M. Andrec, B. N. Kholodenko, R. M. Levy and E. Sontag, Inference of signalingand gene regulatory networks by steady-state perturbation experiments: Structure and accuracy, Journal of Theoretical Biology, 232 (2005), 427-441.  doi: 10.1016/j.jtbi.2004.08.022.

[3]

D. Bauer, F. Boesch, C. Suffel and R. Tindell, Connectivity extremal problems and the design of reliable probabilistic networks, The Theory and Application of Graphs, Wiley, New York, (1981), 45–54.

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P. Bonacich, Some unique properties of eigenvector centrality, Social Networks, 29 (2007), 555-564.  doi: 10.1016/j.socnet.2007.04.002.

[5]

U. Brandes, On variants of shortest-path betweenness centrality and their generic computation, Social Networks, 30 (2008), 136-145. 

[6]

D. Breitkreutz, L. Hlatky, E. Rietman and J. A. Tuszynski, Molecular signalingnetwork complexity is correlated with cancer patient survivability, Proceedings of the National Academy of Sciences, 109 (2012), 9209-9212. 

[7]

D. S. Callaway, M. E. J. Newman, S. H. Strogatz and D. J. Watts, Network robustness and fragility: Percolation on random graphs, Physical Review L, 85 (2000), 5468-5471.  doi: 10.1103/PhysRevLett.85.5468.

[8]

R. Cohen, K. Erez, D. Ben-Avraham and S. Havlin, Resilience of the Internet to random breakdowns, Physical Review L, 85 (2000), 4626-4628.  doi: 10.1103/PhysRevLett.85.4626.

[9]

R. Cohen, K. Erez, D. Ben-Avraham and S. Havlin, Breakdown of the Internet under intentional attack, Physical Review L, 86 (2001), 3682-3685.  doi: 10.1103/PhysRevLett.86.3682.

[10]

C. Dong and K. Hemminki, Multiple primary cancers of the colon, breast and skin (melanoma) as models for polygenic cancers, International Journal of Cancer, 92 (2001), 883-887.  doi: 10.1002/ijc.1261.

[11]

E. J. Edelman, J. Guinney, J.-T. Chi, P. G. Febbo and S. Mukherjee, Modeling cancer progression via pathway dependencies, PLoS Comput Biol, 4 (2008), e28.  doi: 10.1371/journal.pcbi.0040028.

[12]

E. Estrada, D. J. Higham and N. Hatano, Communicability betweenness in 315 complex networks, Physica A: Statistical Mechanics and its Applications, 388 (2009), 764-774. 

[13]

J. D. Feala, J. Cortes, P. M. Duxbury, C. Piermarocchi, A. D. McCulloch and G. Paternostro, Systems approaches and algorithms for discovery of combinatorial therapies, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 2 (2010), 181-193.  doi: 10.1002/wsbm.51.

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M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Mathematical Journal, 23 (1973), 298-305. 

[15]

H. Frank and I. T. Frisch, Analysis and design of survivable networks, IEEE Transactions on Communication Technology, 18 (1970), 501-519. 

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P. Hage and F. Harary, Eccentricity and centrality in networks, Social Networks, 17 (1995), 57-63.  doi: 10.1016/0378-8733(94)00248-9.

[17]

F. Harary, Conditional connectivity, Networks, 13 (1983), 347-357.  doi: 10.1002/net.3230130303.

[18]

V. H. Louzada, F. Daolio, H. J. Herrmann and M. Tomassini, Generating robust and efficient networks under targeted attacks, Propagation Phenomena in Real World Networks, 85 (2015), 215-244.  doi: 10.1007/978-3-319-15916-4_9.

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J. C. Nacher and J.-M. Schwartz, A global view of drug-therapy interactions, BMC pharmacology, 8 (2008), 5pp.  doi: 10.1186/1471-2210-8-5.

[21]

G. Paul, S. Sreenivasan and H. E. Stanley, Resilience of complex networks to random breakdown, Physical Review E, 72 (2005), 056130, 6pp.  doi: 10.1103/PhysRevE.72.056130.

[22]

C. A. Penfold, V. Buchanan-Wollaston, K. J. Denby and D. L. Wild, Nonparametric bayesian inference for perturbed and orthologous gene regulatory networks, Bioinformatics, 28 (2012), i233-i241.  doi: 10.1093/bioinformatics/bts222.

[23]

C. M. Schneider, A. A. Moreira, J. S. Andrade, S. Havlin and H. J. Herrmann, Mitigation of malicious attacks on networks, Proc. Natl. Acad. Sci. U.S.A., 108 (2011), 3838-3841.  doi: 10.1073/pnas.1009440108.

[24]

B. Shargel, H. Sayama, I. R. Epstein and Y. Bar-Yam, Optimization of robustness and connectivity in complex networks, Physical Review L, 90 (2003), 068701.  doi: 10.1103/PhysRevLett.90.068701.

[25]

C. Sonnenschein and A. M. Soto, Theories of carcinogenesis: an emerging per-275 spective, in: Seminars in cancer biology, Seminars in Cancer Biology, 18 (2008), 372-377. 

[26]

K. Takemoto and K. Kihara, Modular organization of cancer signaling networks is associated with patient survivability, Biosystems, 113 (2013), 149-154.  doi: 10.1016/j.biosystems.2013.06.003.

[27]

J. Wu, M. Barahona, Y.-J. Tan and H.-Z. Deng, Spectral measure of structural robustness in complex networks, IEEE Transactions on Syst. Man Cybern. A Syst. and Humans, 41 (2015), 1244-1252.  doi: 10.1109/TSMCA.2011.2116117.

[28]

M. A. Yildirim, K.-I. Goh, M. E. Cusick, A.-L. Barabási and M. Vidal, Drug-target network, Nature Biotechnology, 25 (2007), 1119-1126. 

[29]

A. Zeng and W. Liu, Enhancing network robustness against malicious attacks, Physical Review E, 85 (2012), 066130.  doi: 10.1103/PhysRevE.85.066130.

show all references

References:
[1]

R. Albert, H. Jeong and A. -L. Barabási, Error and attack tolerance of complex networks, Nature, 406 (2000), 378-382.  doi: 10.1038/35019019.

[2]

M. Andrec, B. N. Kholodenko, R. M. Levy and E. Sontag, Inference of signalingand gene regulatory networks by steady-state perturbation experiments: Structure and accuracy, Journal of Theoretical Biology, 232 (2005), 427-441.  doi: 10.1016/j.jtbi.2004.08.022.

[3]

D. Bauer, F. Boesch, C. Suffel and R. Tindell, Connectivity extremal problems and the design of reliable probabilistic networks, The Theory and Application of Graphs, Wiley, New York, (1981), 45–54.

[4]

P. Bonacich, Some unique properties of eigenvector centrality, Social Networks, 29 (2007), 555-564.  doi: 10.1016/j.socnet.2007.04.002.

[5]

U. Brandes, On variants of shortest-path betweenness centrality and their generic computation, Social Networks, 30 (2008), 136-145. 

[6]

D. Breitkreutz, L. Hlatky, E. Rietman and J. A. Tuszynski, Molecular signalingnetwork complexity is correlated with cancer patient survivability, Proceedings of the National Academy of Sciences, 109 (2012), 9209-9212. 

[7]

D. S. Callaway, M. E. J. Newman, S. H. Strogatz and D. J. Watts, Network robustness and fragility: Percolation on random graphs, Physical Review L, 85 (2000), 5468-5471.  doi: 10.1103/PhysRevLett.85.5468.

[8]

R. Cohen, K. Erez, D. Ben-Avraham and S. Havlin, Resilience of the Internet to random breakdowns, Physical Review L, 85 (2000), 4626-4628.  doi: 10.1103/PhysRevLett.85.4626.

[9]

R. Cohen, K. Erez, D. Ben-Avraham and S. Havlin, Breakdown of the Internet under intentional attack, Physical Review L, 86 (2001), 3682-3685.  doi: 10.1103/PhysRevLett.86.3682.

[10]

C. Dong and K. Hemminki, Multiple primary cancers of the colon, breast and skin (melanoma) as models for polygenic cancers, International Journal of Cancer, 92 (2001), 883-887.  doi: 10.1002/ijc.1261.

[11]

E. J. Edelman, J. Guinney, J.-T. Chi, P. G. Febbo and S. Mukherjee, Modeling cancer progression via pathway dependencies, PLoS Comput Biol, 4 (2008), e28.  doi: 10.1371/journal.pcbi.0040028.

[12]

E. Estrada, D. J. Higham and N. Hatano, Communicability betweenness in 315 complex networks, Physica A: Statistical Mechanics and its Applications, 388 (2009), 764-774. 

[13]

J. D. Feala, J. Cortes, P. M. Duxbury, C. Piermarocchi, A. D. McCulloch and G. Paternostro, Systems approaches and algorithms for discovery of combinatorial therapies, Wiley Interdisciplinary Reviews: Systems Biology and Medicine, 2 (2010), 181-193.  doi: 10.1002/wsbm.51.

[14]

M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Mathematical Journal, 23 (1973), 298-305. 

[15]

H. Frank and I. T. Frisch, Analysis and design of survivable networks, IEEE Transactions on Communication Technology, 18 (1970), 501-519. 

[16]

P. Hage and F. Harary, Eccentricity and centrality in networks, Social Networks, 17 (1995), 57-63.  doi: 10.1016/0378-8733(94)00248-9.

[17]

F. Harary, Conditional connectivity, Networks, 13 (1983), 347-357.  doi: 10.1002/net.3230130303.

[18]

V. H. Louzada, F. Daolio, H. J. Herrmann and M. Tomassini, Generating robust and efficient networks under targeted attacks, Propagation Phenomena in Real World Networks, 85 (2015), 215-244.  doi: 10.1007/978-3-319-15916-4_9.

[19]

R. Merris, Laplacian matrices of graphs: A survey, Linear Algebra and its Applications, 197 (1994), 143-176.  doi: 10.1016/0024-3795(94)90486-3.

[20]

J. C. Nacher and J.-M. Schwartz, A global view of drug-therapy interactions, BMC pharmacology, 8 (2008), 5pp.  doi: 10.1186/1471-2210-8-5.

[21]

G. Paul, S. Sreenivasan and H. E. Stanley, Resilience of complex networks to random breakdown, Physical Review E, 72 (2005), 056130, 6pp.  doi: 10.1103/PhysRevE.72.056130.

[22]

C. A. Penfold, V. Buchanan-Wollaston, K. J. Denby and D. L. Wild, Nonparametric bayesian inference for perturbed and orthologous gene regulatory networks, Bioinformatics, 28 (2012), i233-i241.  doi: 10.1093/bioinformatics/bts222.

[23]

C. M. Schneider, A. A. Moreira, J. S. Andrade, S. Havlin and H. J. Herrmann, Mitigation of malicious attacks on networks, Proc. Natl. Acad. Sci. U.S.A., 108 (2011), 3838-3841.  doi: 10.1073/pnas.1009440108.

[24]

B. Shargel, H. Sayama, I. R. Epstein and Y. Bar-Yam, Optimization of robustness and connectivity in complex networks, Physical Review L, 90 (2003), 068701.  doi: 10.1103/PhysRevLett.90.068701.

[25]

C. Sonnenschein and A. M. Soto, Theories of carcinogenesis: an emerging per-275 spective, in: Seminars in cancer biology, Seminars in Cancer Biology, 18 (2008), 372-377. 

[26]

K. Takemoto and K. Kihara, Modular organization of cancer signaling networks is associated with patient survivability, Biosystems, 113 (2013), 149-154.  doi: 10.1016/j.biosystems.2013.06.003.

[27]

J. Wu, M. Barahona, Y.-J. Tan and H.-Z. Deng, Spectral measure of structural robustness in complex networks, IEEE Transactions on Syst. Man Cybern. A Syst. and Humans, 41 (2015), 1244-1252.  doi: 10.1109/TSMCA.2011.2116117.

[28]

M. A. Yildirim, K.-I. Goh, M. E. Cusick, A.-L. Barabási and M. Vidal, Drug-target network, Nature Biotechnology, 25 (2007), 1119-1126. 

[29]

A. Zeng and W. Liu, Enhancing network robustness against malicious attacks, Physical Review E, 85 (2012), 066130.  doi: 10.1103/PhysRevE.85.066130.

Table 1.  Cancer survival probabilities and network statistics for CSN-EG and CSN-CO
Cancer site 5-y survival probability CSN-EG CSN-GO
Nodes Edges Nodes Edges
Acute myeloid leukemia23.6%571523239
Basal cell carcinoma91.4%473041311
Bladder cancer78.1%29462119
Chronic myeloid leukemia55.2%731854447
Colorectal cancer63.6%491043433
Endometrial cancer68.6%46882423
Glioma33.4%692095561
Melanoma91.2%702822223
Nonsmall-cell lung cancer18.0%731833643
Pancreatic cancer5.5%671344343
Prostate cancer99.4%993334045
Renal cell carcinoma69.5%571043633
Small cell lung cancer6.2%862383137
Thyroid cancer97.2%28491814
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Cancer site 5-y survival probability CSN-EG CSN-GO
Nodes Edges Nodes Edges
Acute myeloid leukemia23.6%571523239
Basal cell carcinoma91.4%473041311
Bladder cancer78.1%29462119
Chronic myeloid leukemia55.2%731854447
Colorectal cancer63.6%491043433
Endometrial cancer68.6%46882423
Glioma33.4%692095561
Melanoma91.2%702822223
Nonsmall-cell lung cancer18.0%731833643
Pancreatic cancer5.5%671344343
Prostate cancer99.4%993334045
Renal cell carcinoma69.5%571043633
Small cell lung cancer6.2%862383137
Thyroid cancer97.2%28491814
Table 2.  Pearsons correlation coefficient between CSN robustness and 5-year survival rate are showed
CSN $\setminus\gamma$ $R$ $R_l$ $p_c^r$ $p_c^t$ $a(G)$ $\bar\lambda$
CSN-EG 0.31 0.28 -0.11 0.34 0.049 0.24
CSN-GO 0.18 0.17 -0.56 0.36 0.21 -0.60
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CSN $\setminus\gamma$ $R$ $R_l$ $p_c^r$ $p_c^t$ $a(G)$ $\bar\lambda$
CSN-EG 0.31 0.28 -0.11 0.34 0.049 0.24
CSN-GO 0.18 0.17 -0.56 0.36 0.21 -0.60
Table 3.  Pearsons correlation coefficient between the network parameters of CSN-GO and 5-year survival rate is showed
Network parameters H Q λ
γ -0.62 -0.21 -0.60
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Network parameters H Q λ
γ -0.62 -0.21 -0.60
Table 4.  Degree, closeness centrality and betweenness centrality of important nodes measured by natural connectivity for each cancer site part 1
Cancer site Degree Closeness Betweenness
Acute myeloid leukemia Top 1 6.00 14.92 545.00
Top 2 7.00 12.55 269.83
Top 3 5.00 14.45 500.17
Average 2.43 10.17 98.00
Basal cell carcinoma Top 1 5.00 6.45 60.00
Top 2 2.00 5.28 50.00
Top 3 2.00 4.92 48.00
Average 1.82 4.46 22.73
Bladder cancer Top 1 3.00 5.17 38.00
Top 2 3.00 4.83 26.00
Top 3 3.00 4.67 26.00
Average 1.78 3.99 13.33
Chronic myeloid leukemia Top 1 7.00 12.38 217.00
Top 2 5.00 12.09 303.00
Top 3 5.00 11.67 318.00
Average 2.15 8.44 78.85
Colorectal cancer Top 1 5.00 8.59 188.00
Top 2 4.00 7.71 147.00
Top 3 4.00 8.74 274.00
Average 2.09 6.73 91.39
Endometrial cancer Top 1 5.00 7.30 85.00
Top 2 3.00 6.88 121.00
Top 3 3.00 6.88 142.00
Average 2.00 5.62 57.65
Glioma Top 1 5.00 9.70 78.00
Top 2 4.00 10.25 141.17
Top 3 4.00 9.15 36.17
Average 2.42 7.61 36.74
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Cancer site Degree Closeness Betweenness
Acute myeloid leukemia Top 1 6.00 14.92 545.00
Top 2 7.00 12.55 269.83
Top 3 5.00 14.45 500.17
Average 2.43 10.17 98.00
Basal cell carcinoma Top 1 5.00 6.45 60.00
Top 2 2.00 5.28 50.00
Top 3 2.00 4.92 48.00
Average 1.82 4.46 22.73
Bladder cancer Top 1 3.00 5.17 38.00
Top 2 3.00 4.83 26.00
Top 3 3.00 4.67 26.00
Average 1.78 3.99 13.33
Chronic myeloid leukemia Top 1 7.00 12.38 217.00
Top 2 5.00 12.09 303.00
Top 3 5.00 11.67 318.00
Average 2.15 8.44 78.85
Colorectal cancer Top 1 5.00 8.59 188.00
Top 2 4.00 7.71 147.00
Top 3 4.00 8.74 274.00
Average 2.09 6.73 91.39
Endometrial cancer Top 1 5.00 7.30 85.00
Top 2 3.00 6.88 121.00
Top 3 3.00 6.88 142.00
Average 2.00 5.62 57.65
Glioma Top 1 5.00 9.70 78.00
Top 2 4.00 10.25 141.17
Top 3 4.00 9.15 36.17
Average 2.42 7.61 36.74
Table 5.  Degree, closeness centrality and betweenness centrality of important nodes measured by natural connectivity for each cancer site part 2
Cancer site Degree Closeness Betweenness
Melanoma Top 1 3.00 6.58 73.00
Top 2 3.00 6.50 58.00
Top 3 3.00 6.06 43.00
Average 2.00 5.15 26.77
Nonsmall-cell lung cancer Top 1 5.00 12.45 346.73
Top 2 5.00 11.42 91.67
Top 3 5.00 11.42 91.67
Average 2.45 9.25 110.19
Pancreatic cancer Top 1 5.00 6.95 45.00
Top 2 4.00 6.78 61.00
Top 2 3.00 5.95 7.00
Average 2.33 5.24 22.17
Prostate cancer Top 1 12.00 18.07 850.67
Top 2 3.00 13.16 546.00
Top 3 4.00 8.33 93.00
Average 2.29 9.92 156.29
Renal cell carcinoma Top 1 5.00 9.08 160.00
Top 2 3.00 7.20 28.00
Top 3 3.00 7.20 28.00
Average 2.00 6.06 34.38
Small cell lung cancer Top 1 7.00 8.75 126.00
Top 2 3.00 5.78 67.00
Top 3 2.00 6.65 98.00
Average 2.00 5.65 38.80
Thyroid cancer Top 1 3.00 4.17 18.00
Top 2 3.00 4.17 18.00
Top 3 2.00 4.00 18.00
Average 1.71 3.38 7.71
Table Options

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Cancer site Degree Closeness Betweenness
Melanoma Top 1 3.00 6.58 73.00
Top 2 3.00 6.50 58.00
Top 3 3.00 6.06 43.00
Average 2.00 5.15 26.77
Nonsmall-cell lung cancer Top 1 5.00 12.45 346.73
Top 2 5.00 11.42 91.67
Top 3 5.00 11.42 91.67
Average 2.45 9.25 110.19
Pancreatic cancer Top 1 5.00 6.95 45.00
Top 2 4.00 6.78 61.00
Top 2 3.00 5.95 7.00
Average 2.33 5.24 22.17
Prostate cancer Top 1 12.00 18.07 850.67
Top 2 3.00 13.16 546.00
Top 3 4.00 8.33 93.00
Average 2.29 9.92 156.29
Renal cell carcinoma Top 1 5.00 9.08 160.00
Top 2 3.00 7.20 28.00
Top 3 3.00 7.20 28.00
Average 2.00 6.06 34.38
Small cell lung cancer Top 1 7.00 8.75 126.00
Top 2 3.00 5.78 67.00
Top 3 2.00 6.65 98.00
Average 2.00 5.65 38.80
Thyroid cancer Top 1 3.00 4.17 18.00
Top 2 3.00 4.17 18.00
Top 3 2.00 4.00 18.00
Average 1.71 3.38 7.71
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