Fig. 3.

Bifurcations in models with Root-Degree (A), PageRank (B), and SpringRank (C and D) score functions with ${\beta }_2=0$ and $m=1$ update per time step. Points give the value of the rank vector $\mathit{\gamma }$ averaged over the final 500 time steps of a $5\times 10^4$-step simulation with $n=8$ nodes, memory parameter $\lambda =0.9995$, and varying ${\beta }_1$ specified by the horizontal axis. Solid curves show stationary points of the long-memory dynamics obtained by numerically solving the equation $\mathbf{f}\left(\mathbf{s},\mathbf{A}\right)=\mathbf{0}$, subject to the restriction that nodes separate into two groups with identical ranks in each. Black curves are linearly stable, while gray curves are unstable. Stability was determined by studying the spectrum of the Jacobian matrix of $\mathbf{f}$. Vertical lines give the critical value ${\beta }_1^c$, at which the egalitarian solution becomes linearly unstable, according to Theorem 1. Parameters: ${\alpha }_p=0.85$, ${\alpha }_s=10^{-8}$.