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I'm reading Dynamical Processes on Complex Networks (link), which makes frequent use of dirac delta integrals to examine evolving networks. I'm trying to get a good sense of how to evalute them and when they are appropriate to use. I am currently unable to crack the following integral in Equation 3.31:

$$ P(k, t) = \int_0^t \delta[k-k_s(t)] ds = -\bigg(\frac{\partial k_s(t)}{\delta s}\bigg)^{-1}\bigg|_{s=s(k,t)} $$

Where $k_s(t)$ is $0$ for $s<t$ and monotonically increasing otherwise, and $s(k,t)$ is the solution of the implicit equation $k = k_s(t)$.

The book just states this result and moves on and I'd love to understand what is going on here. I should say, I'm not so interested in the results so much as the machinery of the dirac delta function and how one can use such tricks/witchcraft in general.

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By substitution and composition,$$P(k,\,t)=\int_{k_0(t)}^{k_s(t)}\delta[k-k_s(t)]\frac{\partial s}{\partial k}dk=-\left(\frac{\partial s}{\partial k}\right)_{k=k_s(t)},$$where the overall $-$ sign is due to the order of the integral's limits. But of course, we can rewrite this as$$-\left(\frac{\partial k_s(t)}{\partial s}\right)^{-1}_{s=s(k,\,t)},$$as required.

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  • $\begingroup$ Thank you! And sorry if these are some obvious questions but: A) is the upper limit meant to be $k_t(t)$? B) I'm not quite sure I understand where the overall $-$ is coming from, would it possible to elaborate? C) Why is it straightforward that we can flip the last partial derivative, and where does the inverse in the solution go? (As in, in the equation I listed there's an inverse sign at the end?) $\endgroup$ – Sue Doh Nimh Nov 14 at 22:54
  • $\begingroup$ @SueDohNimh I forgot the inverse as a typo, but that's fixed now. I'll return to your other questions later. $\endgroup$ – J.G. Nov 15 at 7:15

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