Laplace transformation to solve a differential equation

 

Laplace transform to solve a differential equation

Laplace transform

 

https://en.wikipedia.org/wiki/Laplace_transform

"In particular, it transforms differential equations into algebraic equations and convolution into multiplication.^[1][2][3] "

https://mathvault.ca/laplace-transform/

In fact, it takes a time-domain function, where $t$ t is the variable, and outputs a frequency-domain function, where s$s$ is the variable

It seems to me that Fourier transformation can be viewed as a special form of Laplace transformation when the alpha is zero. 

Laplace transformation is not Lapalce operator

https://en.wikipedia.org/wiki/Laplace_operator

linear dynamic systems, laplace transform

 

https://see.stanford.edu/Course/EE263 

https://see.stanford.edu/materials/lsoeldsee263/13-lin-sys.pdf

advantages of temporal network, continued

 Li, ..., Barabasi, Science, 2017,

temporal network advantages.

Energy needed from state vector x0 to final state xf
  E(x0, xf) = 1/2 d^T x W^01_eff  x d

where Weff encode the energy structure of the network.

I did not follow S1.1 method. 

After reading Laplace transformation, and watched YouTube movies on solving differential equations with Laplace transformation, it seems the Li17 presented a general solution. 

Li17 cited a yeast dynamic protein network: 

27. J. Wang, X. Peng, M. Li, Y. Pan, Construction and application of dynamic protein interaction

network based on time course gene expression data. Proteomics 13, 301–312 (2013).

doi:10.1002/pmic.201200277 Medline

In Figure S6, a digram is presented to describe the construct of a yeast dynamic PPI. An interaction is consider as active when both protein are active at that time point. based on  X. Tang, J. Wang, B. Liu, M. Li, G. Chen, Y. Pan, A comparison of the functional modules identified from time course and static PPI network data. BMC Bioinformatics 12, 339

(2011). doi:10.1186/1471-2105-12-339 Medline

https://bmcbioinformatics.biomedcentral.com/articles/10.1186/1471-2105-12-339#Sec17

In Fig 2. Protein networks used contain only 84, 74, and 85 nodes. So, it seems Li17 only used a small subset of PPI. 

Li17 fixed the number of driver nodes to 20% of the nodes. Hong did not find the criteria on the reasons behind this choice. 

Laplace transformation of graphic function

https://www.youtube.com/watch?v=f1mZArY0lLE

https://www.youtube.com/watch?v=ZGPtPkTft8g