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From Network Structure to Dynamics and Back Again: Relating Dynamical Stability and Connection Topology in Biological Complex Systems

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Dynamics On and Of Complex Networks

To see a world in a grain of sand,

And a heaven in a wild flower,

Hold infinity in the palm of your hand,

And eternity in an hour.

– William Blake, Auguries of Innocence Auguries of Innocence

Like Blake, physicists look for universal principles that are valid across many different systems, often spanning several length or time scales. While the domain of physical systems has often offered examples of such widely applicable “laws,” biological phenomena tended to be, until quite recently, less fertile in terms of generating similar universalities, with the notable exception of allometric scaling relations [20]. However, this situation has changed with the study of complex networks emerging into prominence. Such systems comprise a large number of nodes (or elements) linked with each other according to specific connection topologies, and are seen to occur widely across the biological, social and technological worlds [1, 9, 16]. Examples range from the intra-cellular signaling system which consists of different kinds of molecules affecting each other via enzymatic reactions, to the internet composed of servers around the world which exchange enormous quantities of information packets regularly, and food webs which link, via trophic relations, large numbers of inter-dependent species. While the existence of complex networks in various domains had been known for some time, the recent excitement among physicists working on such systems has to do with the discovery of certain universal principles among systems which had hitherto been considered very different from each other.

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Notes

  1. 1.

    The Laplacian matrix is also referred to as the Kirchhoff matrix (e.g., see Ref. [10]).

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Acknowledgements

I would like to thank my collaborators with whom the work described here has been carried out, in particular, R. K. Pan, S. Sinha, N. Chatterjee, M. Brede, C. C. Wilmers, J. Saramäki and K. Kaski, as well as S. Vemparala, D. Kumar, K. V. S. Rao and B. Saha for helpful discussions.

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  1. The Institute of Mathematical Sciences, CIT Campus, 600113, Taramani, Chennai, India

    Sitabhra Sinha

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    Sinha, S. (2009). From Network Structure to Dynamics and Back Again: Relating Dynamical Stability and Connection Topology in Biological Complex Systems. In: Ganguly, N., Deutsch, A., Mukherjee, A. (eds) Dynamics On and Of Complex Networks. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4751-3_1

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    • DOI: https://doi.org/10.1007/978-0-8176-4751-3_1

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