A model for biological dynamic networks

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December 2011, 6(4): 647-663. doi: 10.3934/nhm.2011.6.647

A model for biological dynamic networks

Alessia Marigo ^1, and Benedetto Piccoli ^2,

1.	Department of Mathematical Sciences, Rutgers University - Camden, 311 N 5th Street, Camden, NJ 08102, United States
2.	Department of Mathematical Sciences and Center for Computational and Integrative Biology, Rutgers University - Camden, 311 N 5th Street, Camden, NJ 08102, United States

Received March 2011 Revised September 2011 Published December 2011

Abstract
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The main aim of this paper is to introduce a mathematical framework to study stochastically evolving networks. More precisely, we provide a common language and suitable tools to study systematically the probability distribution of topological characteristics, which, in turn, play a key role in applications, especially for biological networks. The latter is possible via suitable definition of a random network process and new results for graph isomorphism, which, under suitable generic assumptions, can be stated in terms of the graph walk matrix and computed in polynomial time.

Keywords: Dynamic networks, Biological networks, Isomorphic graphs..

Mathematics Subject Classification: Primary: 05C60, 92C42; Secondary: 05C8.

Citation: Alessia Marigo, Benedetto Piccoli. A model for biological dynamic networks. Networks and Heterogeneous Media, 2011, 6 (4) : 647-663. doi: 10.3934/nhm.2011.6.647

References:

[1]	U. Alon, "An Introduction to Systems Biology: Design Principles of Biological Circuits," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.
[2]	A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509.
[3]	A.-L. Barabási and R. E. Crandall, Linked: The new science of networks, Am. J. Phys., 71 (2003), 409-410. doi: 10.1119/1.1538577.
[4]	B. Bollobás, C. Borgs, J. Chayes and O. Riordan, Directed scale-free graphs, in "Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms" (Baltimore, MD, 2003), 132-139, ACM, New York, 2003.
[5]	M. Chaves and E. D. Sontag, State-estimation for chemical reaction networks of Feinberg-Horn-Jackson zero deficiency type, Europ. J. of Control, 8 (2002), 343-359. doi: 10.3166/ejc.8.343-359.
[6]	C. Cooper and A. Frieze, A general model of web graphs, Random Struct. Alg., 22 (2003), 311-335. doi: 10.1002/rsa.10084.
[7]	D. M. Cvetković, M. Doob and H. Sachs, "Spectra of Graphs: Theory and Applications," Third edition, Johann Ambrosius Barth, Heidelberg, 1995.
[8]	D. Del Vecchio, A. J. Ninfa and E. D. Sontag, Modular cell biology: Retroactivity and insulation, Mol. Syst. Biology, 4 (2008), Article number 161. doi: 10.1038/msb4100204.
[9]	R. Durrett, "Random Graph Dynamics," Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2007.
[10]	P. Erdős and A. Renyi, On random graphs, Publ. Math. Debrecen, 6 (1959), 290-297.
[11]	M. Farina, R. Findeisen, E. Bullinger, S. Bittanti, F. Allgower and P. Wellstead, Results towards identifiability properties of biochemical reaction networks, in "Proceedings of the 45^th IEEE Conference on Decision & Control," San Diego, CA, USA, December 13-15, (2006), 2104-2109.
[12]	E. M. Hagos, Some results on graph spectra, Linear Algebra Appl., 356 (2002), 103-111. doi: 10.1016/S0024-3795(02)00324-5.
[13]	S. Mangan and U. Alon, Structure and function of the feed-forward loop network motif, PNAS, 100 (2003), 11980-11985. doi: 10.1073/pnas.2133841100.
[14]	M. E. J. Newman, The structure and functions of complex networks, SIAM Review, 45 (2003), 167-256. doi: 10.1137/S003614450342480.
[15]	B. O. Palsson, "Systems Biology-Properties of Reconstructed Networks," Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511790515.
[16]	E. D. Sontag, Molecular systems biology and control, Europ. J. of Control, 11 (2005), 396-435.
[17]	D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. doi: 10.1038/30918.

show all references

References:

[1]	U. Alon, "An Introduction to Systems Biology: Design Principles of Biological Circuits," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.
[2]	A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509.
[3]	A.-L. Barabási and R. E. Crandall, Linked: The new science of networks, Am. J. Phys., 71 (2003), 409-410. doi: 10.1119/1.1538577.
[4]	B. Bollobás, C. Borgs, J. Chayes and O. Riordan, Directed scale-free graphs, in "Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms" (Baltimore, MD, 2003), 132-139, ACM, New York, 2003.
[5]	M. Chaves and E. D. Sontag, State-estimation for chemical reaction networks of Feinberg-Horn-Jackson zero deficiency type, Europ. J. of Control, 8 (2002), 343-359. doi: 10.3166/ejc.8.343-359.
[6]	C. Cooper and A. Frieze, A general model of web graphs, Random Struct. Alg., 22 (2003), 311-335. doi: 10.1002/rsa.10084.
[7]	D. M. Cvetković, M. Doob and H. Sachs, "Spectra of Graphs: Theory and Applications," Third edition, Johann Ambrosius Barth, Heidelberg, 1995.
[8]	D. Del Vecchio, A. J. Ninfa and E. D. Sontag, Modular cell biology: Retroactivity and insulation, Mol. Syst. Biology, 4 (2008), Article number 161. doi: 10.1038/msb4100204.
[9]	R. Durrett, "Random Graph Dynamics," Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2007.
[10]	P. Erdős and A. Renyi, On random graphs, Publ. Math. Debrecen, 6 (1959), 290-297.
[11]	M. Farina, R. Findeisen, E. Bullinger, S. Bittanti, F. Allgower and P. Wellstead, Results towards identifiability properties of biochemical reaction networks, in "Proceedings of the 45^th IEEE Conference on Decision & Control," San Diego, CA, USA, December 13-15, (2006), 2104-2109.
[12]	E. M. Hagos, Some results on graph spectra, Linear Algebra Appl., 356 (2002), 103-111. doi: 10.1016/S0024-3795(02)00324-5.
[13]	S. Mangan and U. Alon, Structure and function of the feed-forward loop network motif, PNAS, 100 (2003), 11980-11985. doi: 10.1073/pnas.2133841100.
[14]	M. E. J. Newman, The structure and functions of complex networks, SIAM Review, 45 (2003), 167-256. doi: 10.1137/S003614450342480.
[15]	B. O. Palsson, "Systems Biology-Properties of Reconstructed Networks," Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511790515.
[16]	E. D. Sontag, Molecular systems biology and control, Europ. J. of Control, 11 (2005), 396-435.
[17]	D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. doi: 10.1038/30918.

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