Fig. 6

Hubs and asymmetric saturation time. A ER graphs have less inter-hub links, hence less shortcuts inside the network. B Due to assortative mixing (hubs are linked), in BA graphs the hubs are more connected among themselves. This creates many shortcuts for the walkers within different parts of the network that provide a stable saturation time for dynamical processes. C In ER graph the number of hubs (determined by the condition \(deg(v)^2 > \sum _{j \in v.neighbors}{deg(j)}\)) negatively correlate with the saturation time, meaning that having more hubs tend to decrease the saturation time. But, having more hubs increases the instability as it positively correlates with the skewness of the saturation time, hence having asymmetric behavior. D In contrast, in BA graphs we observe having more hubs is correlated with more stability in the saturation time (negative correlation with skewness), at the same time having more hubs increases the saturation time as it takes walkers more time to find nodes with very high degree. The correlation coefficients in (C, D) are obtained by generating 100 different ER and BA networks with 1000 nodes and 6 average degrees. Then, we ran 100 simulations (number of initial selected nodes is 4 and diffusion rate is 0.4) over each network thus, for each network, we found the skewness of the duration time distribution of these 100 simulations