# American Institue of Mathematical Sciences

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

## A comparative study of robustness measures for cancer signaling networks

 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, United Kingdom

* 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.

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:
 [1] R. Albert, H. Jeong, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, I. T. Frisch, Analysis and design of survivable networks, IEEE Transactions on Communication Technology, 18 (1970), 501-519. [16] P. Hage, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, M. Vidal, Drug-target network, Nature Biotechnology, 25 (2007), 1119-1126. [29] A. Zeng, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, I. T. Frisch, Analysis and design of survivable networks, IEEE Transactions on Communication Technology, 18 (1970), 501-519. [16] P. Hage, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, M. Vidal, Drug-target network, Nature Biotechnology, 25 (2007), 1119-1126. [29] A. Zeng, W. Liu, Enhancing network robustness against malicious attacks, Physical Review E, 85 (2012), 066130. doi: 10.1103/PhysRevE.85.066130.
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 leukemia 23.6% 57 152 32 39 Basal cell carcinoma 91.4% 47 304 13 11 Bladder cancer 78.1% 29 46 21 19 Chronic myeloid leukemia 55.2% 73 185 44 47 Colorectal cancer 63.6% 49 104 34 33 Endometrial cancer 68.6% 46 88 24 23 Glioma 33.4% 69 209 55 61 Melanoma 91.2% 70 282 22 23 Nonsmall-cell lung cancer 18.0% 73 183 36 43 Pancreatic cancer 5.5% 67 134 43 43 Prostate cancer 99.4% 99 333 40 45 Renal cell carcinoma 69.5% 57 104 36 33 Small cell lung cancer 6.2% 86 238 31 37 Thyroid cancer 97.2% 28 49 18 14
 Cancer site 5-y survival probability CSN-EG CSN-GO Nodes Edges Nodes Edges Acute myeloid leukemia 23.6% 57 152 32 39 Basal cell carcinoma 91.4% 47 304 13 11 Bladder cancer 78.1% 29 46 21 19 Chronic myeloid leukemia 55.2% 73 185 44 47 Colorectal cancer 63.6% 49 104 34 33 Endometrial cancer 68.6% 46 88 24 23 Glioma 33.4% 69 209 55 61 Melanoma 91.2% 70 282 22 23 Nonsmall-cell lung cancer 18.0% 73 183 36 43 Pancreatic cancer 5.5% 67 134 43 43 Prostate cancer 99.4% 99 333 40 45 Renal cell carcinoma 69.5% 57 104 36 33 Small cell lung cancer 6.2% 86 238 31 37 Thyroid cancer 97.2% 28 49 18 14
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
 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
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
 Network parameters H Q λ γ -0.62 -0.21 -0.60
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
 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
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
 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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