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In this paper we study a version of (non-Markovian) first passage percolationon graphs, where the transmission time between two connected vertices isnon-iid, but increases by a penalty factor polynomial in their expecteddegrees. Based on the exponent of the penalty-polynomial, this makes itincreasingly harder to transmit to and from high-degree vertices. This choiceis motivated by awareness or time-limitations. For the iid part of thetransmission times we allow any nonnegative distribution with regularly varyingbehaviour at $0$. For the underlying graph models we choose spatial randomgraphs that have power-law degree distributions, so that the effect of thepenalisation becomes visible: (finite and infinite) Geometric InhomogeneousRandom Graphs, and Scale-Free Percolation. In these spatial models, theconnection probability between two vertices depends on their spatial distanceand on their expected degrees. We prove that upon increasing the penaltyexponent, the transmission time between two far away vertices $x,y$ sweepsthrough four universal phases even for a single underlying graph: explosive(tight transmission times), polylogarithmic, polynomial but sublinear($\Theta(|x-y|^{\eta_0})$ for an explicit $\eta_0<1$), and linear($\Theta(|x-y|)$) in their Euclidean distance. Further, none of these phasesare restricted to phase boundaries, and those are non-trivial in the main modelparameters: the tail of the degree-distribution, a long-range parameter, andthe exponent of regular variation of the iid part of the transmission times. Inthis paper we present proofs of lower bounds for the latter two phases and theupper bound for the linear phase. These complement the matching upper boundsfor the polynomial regime in our companion paper.

Four universal growth regimes in degree-dependent first passage percolation on spatial random graphs II