Largest connected component in duplication-divergence growing graphs with symmetric coupled divergence

Abstract

The largest connected component in duplication-divergence growing graphs with symmetric coupled divergence is studied. Finite-size scaling reveals a phase transition occurring at a divergence rate $δ_c$. The $δ_c$ found stands near the locus of zero in Euler characteristic for finite-size graphs, known to be indicative of the largest connected component transition. The role of non-interacting vertices in shaping this transition, with their presence ($d=0$) and absence ($d=1$) in duplication is also discussed, suggesting a particular transformation of the time variable considered yielding a singularity locus in the natural logarithm of Euler characteristic of finite-size graphs close to that obtained with $d=1$ but from the model with $d=0$. The findings may suggest implications for bond percolation in these growing graph models.

Figures (9)

• Figure 1: Growth of $E(\delta,t)$ versus $N(\delta,t)$ with symmetric coupled divergence ($\sigma=1/2$), for $d=0$ (points) and $d=1$ (solid curves), for various $\delta$: 0.25 ($\circ$), 0.5 ($\Diamond$), 0.75 ($\triangle$). Points and curves shown result from averaging over $10^2$ simulations ending with a total number of vertices $t=10^3$.
• Figure 2: Plots of $E(\delta,t)$ and $t$ with $t=1024$, respectively with solid and dashed lines in left panels: in (a) with $d=0$, and in (c) with $d=1$. In (b), $|t - E(\delta,t)|$ (solid curve on right panels) for the model with $d=0$ showing a locus of singularity at $\delta_{c,\xi} \approx 0.442$, while in (d) Euler entropy with $d=1$, with $\delta_{c,\xi} \approx 1 - e^{-1}$. Solid curves in (a),(b) are obtained through Eq. \ref{['eq_et']}, while in (c),(d) through averaging over $10^3$ simulations.
• Figure 3: Finite-size scaling for $t'(\delta,t)$ showing scaling collapse for $\delta \in [0.55,0.75]$. The linear behavior in log-linear plot suggests the exponential form (shown for visual reference), $ae^{-b(\delta -\delta_c)t^{1/\varphi}}$ with $a \approx 0.95$, $b \approx 1.05$ which, when unscaled, describes a subset of points of $t'(\delta,t)$ versus $\delta$ (see inset).
• Figure 4: In (a), $E(\delta,t)$ (solid line), $t'(\delta,t)$ (dashed line) as in inset of Fig. \ref{['fig3']}, simulations (points marked with $\circ$). In (b), Euler entropy curve assuming $t'(\delta,1024)$ in the model with $d=0$, with $\delta \in [0.55,0.75]$ from Fig. \ref{['fig3']} extended for visual reference to $\delta \in [0,1]$. The singularity has a locus near $\delta_{c,\xi} \approx 1-e^{-1}$ as the one for $d=1$ in Fig. \ref{['fig2']}(d).
• Figure 5: Scaling collapse of $P(\delta,t)$ and $\chi(\delta,t)$ (inset), respectively from Eq. \ref{['eq_sclP']} and Eq. \ref{['eq_sclChi']}. Each point is obtained by averaging over $3 \cdot10^3$ simulations ending at a different total number of vertices $t$ (in legend).