Fig. 2.

Representative dynamics of the proposed model. Each column shows a population of $n=8$ nodes simulated for 2,000 time steps using the SpringRank score function with $m\hspace{0.17em}=\hspace{0.17em}1$ update per time step, varying the preference parameters ${\beta }_1$ and ${\beta }_2$. A, C, E, and G show the simulated rank vector $\mathit{\gamma }$ over time; different colors track the ranks of different nodes. B, D, F, and H show the adjacency matrix $\mathbf{A}$ at time step t = 2,000 for the corresponding parameter combinations. See SI Appendix, Fig. S1 for additional examples with SpringRank; SI Appendix, Fig. S2 for examples with PageRank; and SI Appendix, Fig. S3 for examples with Root-Degree. See SI Appendix, Fig. S4 for the dependence of rank variance on ${\beta }_1$ and ${\beta }_2$ jointly. Parameters: $\lambda \hspace{0.17em}=\hspace{0.17em}0.995$, ${\alpha }_s\hspace{0.17em}=\hspace{0.17em}10^{-8}$.