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Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes

In a number of natural and social systems, the response to an exogenous shock relaxes back to the average level according to a long-memory kernel $\sim 1/t^{1+θ}$ with $0 \leq θ<1$. In the presence of an epidemic-like process of triggered shocks developing in a cascade of generations at or close to criticality, this &#34;bare&#34; kernel is renormalized into an even slower decaying response function $\sim 1/t^{1-θ}$. Surprisingly, this means that the shorter the memory of the bare kernel (the larger $1+θ$), the longer the memory of the response function (the smaller $1-θ$). Here, we present a detailed investigation of this paradoxical behavior based on a generation-by-generation decomposition of the total response function, the use of Laplace transforms and of &#34;anomalous&#34; scaling arguments. The paradox is explained by the fact that the number of triggered generations grows anomalously with time at $\sim t^θ$ so that the contributions of active generations up to time $t$ more than compensate the shorter memory associated with a larger exponent $θ$. This anomalous scaling results fundamentally from the property that the expected waiting time is infinite for $0 \leq θ\leq 1$. The techniques developed here are also applied to the case $θ>1$ and we find in this case that the total renormalized response is a {\bf constant} for $t < 1/(1-n)$ followed by a cross-over to $\sim 1/t^{1+θ}$ for $t \gg 1/(1-n)$.