A comparative study of robustness measures for cancer signaling networks

Mingxing Zhou; Jing Liu; Shuai Wang; Shan He

doi:10.3934/bdia.2017011

Previous Article
BDIA Home
This Issue
Next Article
A clustering based mate selection for evolutionary optimization

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:

[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	P. Bonacich, Some unique properties of eigenvector centrality, Social Networks, 29 (2007), 555-564. doi: 10.1016/j.socnet.2007.04.002. Google Scholar
[5]	U. Brandes, On variants of shortest-path betweenness centrality and their generic computation, Social Networks, 30 (2008), 136-145. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Mathematical Journal, 23 (1973), 298-305. Google Scholar
[15]	H. Frank and I. T. Frisch, Analysis and design of survivable networks, IEEE Transactions on Communication Technology, 18 (1970), 501-519. Google Scholar
[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. Google Scholar
[17]	F. Harary, Conditional connectivity, Networks, 13 (1983), 347-357. doi: 10.1002/net.3230130303. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	A. Zeng and W. Liu, Enhancing network robustness against malicious attacks, Physical Review E, 85 (2012), 066130. doi: 10.1103/PhysRevE.85.066130. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[4]	P. Bonacich, Some unique properties of eigenvector centrality, Social Networks, 29 (2007), 555-564. doi: 10.1016/j.socnet.2007.04.002. Google Scholar
[5]	U. Brandes, On variants of shortest-path betweenness centrality and their generic computation, Social Networks, 30 (2008), 136-145. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[14]	M. Fiedler, Algebraic connectivity of graphs, Czechoslovak Mathematical Journal, 23 (1973), 298-305. Google Scholar
[15]	H. Frank and I. T. Frisch, Analysis and design of survivable networks, IEEE Transactions on Communication Technology, 18 (1970), 501-519. Google Scholar
[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. Google Scholar
[17]	F. Harary, Conditional connectivity, Networks, 13 (1983), 347-357. doi: 10.1002/net.3230130303. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[29]	A. Zeng and W. Liu, Enhancing network robustness against malicious attacks, Physical Review E, 85 (2012), 066130. doi: 10.1103/PhysRevE.85.066130. Google Scholar

Table 1. Cancer survival probabilities and network statistics for CSN-EG and CSN-CO

Cancer site	5-y survival probability	CSN-EG		CSN-GO
Cancer site	5-y survival probability	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

Table Options

Download as excel

Cancer site	5-y survival probability	CSN-EG		CSN-GO
Cancer site	5-y survival probability	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

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

Table Options

Download as excel

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

Table Options

Download as excel

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

Table Options

Download as excel

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

Download as excel

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

[1]	Jinzhi Lei, Frederic Y. M. Wan, Arthur D. Lander, Qing Nie. Robustness of signaling gradient in drosophila wing imaginal disc. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 835-866. doi: 10.3934/dcdsb.2011.16.835
[2]	Zari Dzalilov, Iradj Ouveysi, Alexander Rubinov. An extended lifetime measure for telecommunication network. Journal of Industrial & Management Optimization, 2008, 4 (2) : 329-337. doi: 10.3934/jimo.2008.4.329
[3]	A. Marigo. Robustness of square networks. Networks & Heterogeneous Media, 2009, 4 (3) : 537-575. doi: 10.3934/nhm.2009.4.537
[4]	Luis Barreira, Claudia Valls. Noninvertible cocycles: Robustness of exponential dichotomies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4111-4131. doi: 10.3934/dcds.2012.32.4111
[5]	Tiffany A. Jones, Lou Caccetta, Volker Rehbock. Optimisation modelling of cancer growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 115-123. doi: 10.3934/dcdsb.2017006
[6]	Desheng Li, P.E. Kloeden. Robustness of asymptotic stability to small time delays. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1007-1034. doi: 10.3934/dcds.2005.13.1007
[7]	Cristopher Hermosilla. Stratified discontinuous differential equations and sufficient conditions for robustness. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4415-4437. doi: 10.3934/dcds.2015.35.4415
[8]	Juan Manuel Pastor, Silvia Santamaría, Marcos Méndez, Javier Galeano. Effects of topology on robustness in ecological bipartite networks. Networks & Heterogeneous Media, 2012, 7 (3) : 429-440. doi: 10.3934/nhm.2012.7.429
[9]	Xiaobin Mao, Hua Dai. Partial eigenvalue assignment with time delay robustness. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 207-221. doi: 10.3934/naco.2013.3.207
[10]	Jin Zhang, Yonghai Wang, Chengkui Zhong. Robustness of exponentially κ-dissipative dynamical systems with perturbations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3875-3890. doi: 10.3934/dcdsb.2017198
[11]	Rubén Caballero, Alexandre N. Carvalho, Pedro Marín-Rubio, José Valero. Robustness of dynamically gradient multivalued dynamical systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1049-1077. doi: 10.3934/dcdsb.2019006
[12]	Christoph Sadée, Eugene Kashdan. A model of thermotherapy treatment for bladder cancer. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1169-1183. doi: 10.3934/mbe.2016037
[13]	Avner Friedman. A hierarchy of cancer models and their mathematical challenges. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 147-159. doi: 10.3934/dcdsb.2004.4.147
[14]	Lars Grüne, Vryan Gil Palma. Robustness of performance and stability for multistep and updated multistep MPC schemes. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4385-4414. doi: 10.3934/dcds.2015.35.4385
[15]	Jingang Zhai, Guangmao Jiang, Jianxiong Ye. Optimal dilution strategy for a microbial continuous culture based on the biological robustness. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 59-69. doi: 10.3934/naco.2015.5.59
[16]	Yuncheng You. Random attractors and robustness for stochastic reversible reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 301-333. doi: 10.3934/dcds.2014.34.301
[17]	Floriane Lignet, Vincent Calvez, Emmanuel Grenier, Benjamin Ribba. A structural model of the VEGF signalling pathway: Emergence of robustness and redundancy properties. Mathematical Biosciences & Engineering, 2013, 10 (1) : 167-184. doi: 10.3934/mbe.2013.10.167
[18]	Giovanni Scardoni, Carlo Laudanna. Identifying critical traffic jam areas with node centralities interference and robustness. Networks & Heterogeneous Media, 2012, 7 (3) : 463-471. doi: 10.3934/nhm.2012.7.463
[19]	Frederic Mazenc, Gonzalo Robledo, Michael Malisoff. Stability and robustness analysis for a multispecies chemostat model with delays in the growth rates and uncertainties. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1851-1872. doi: 10.3934/dcdsb.2018098
[20]	Daniel Peterseim. Robustness of finite element simulations in densely packed random particle composites. Networks & Heterogeneous Media, 2012, 7 (1) : 113-126. doi: 10.3934/nhm.2012.7.113

Impact Factor:

Tools

Metrics

Tools

Article outline

Figures and Tables

Impact Factor:

Tools

Figures and Tables

Impact Factor:

American Institute of Mathematical Sciences

A comparative study of robustness measures for cancer signaling networks

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

A comparative study of robustness measures for cancer signaling networks

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors