Survival probability of random networks

Abstract

In this work we study in detail all phases of the time evolution of a delta-like excitation in Erdös-Renyi (ER) random networks by means of the survival probability (SP): The initial decay of the SP (both, the fast decay followed by the power-law decay), the correlation hole regime (the regime between the minimum value of the SP and its saturation value), and the saturation of the SP. Specifically, we found that (i) the power-law decay of the SP and the time-averaged SP are proportional to $t^{-D_{2}}$ and $t^{-\widetilde{D}_{2}}$, respectively (where $D_2$ and $\widetilde{D}_2$ are the correlation dimension of the eigenstates of the randomly weighted adjacency matrices of the ER random networks and the correlation dimension associated with the initial state, respectively) and (ii) the relative depth of the correlation hole of the SP scales with the average degree $\langle k\rangle\approx np$ (here, $n$ and $p$ are the size and the connection probability of the ER random networks). In addition, we show that the eigenstates of the randomly weighted adjacency matrices of ER networks display clear multifractal properties.

Figures (9)

• Figure 1: Histograms of the local density of states (LDOS) at the center of the band of ER random networks. Several combinations of sizes $n$ (different rows) and connection probabilities $p$ (different columns) are considered. Specifically, (a,d,g) $p=0.25p_c$, (b,e,h) $p=p_c$, and (c,f,i) $p=8p_c$ with $p_{c}=n^{2/3}/(n-1)$. Each histogram was constructed from a single random network realization. Black lines are the semicircles of Eq. (\ref{['eq:Semicircle']}) with $\sigma_{ini}$ given by Eq. (\ref{['eq:sigma']}).
• Figure 2: Survival probability $SP(t)$ of ER random networks of size $n=1000$ and several values of the connection probability $p$. Insets in panels (b,c) are enlargements of the corresponding dashed squares. Horizontal dashed black lines indicate the saturation value of the curves. The connection probability $p$ increases from top to bottom.
• Figure 3: (a-h) Survival probability $SP(t)$ of ER random networks for several values of the average degree $\langle k\rangle$, as indicated in the panels. Four graph sizes are reported in each panel ($n=250$, 500, 1000, 2000); they increase from top to bottom. Magenta dashed lines correspond to $t^{-D_{2}}$ with (a) $D_2=0.0446$, (b) $D_2=0.1316$, (c) $D_2=0.234$, (d) $D_2=0.3509$, (e) $D_2=0.4885$, (f) $D_2=0.6868$, (g) $D_2=0.8615$, and (h) $D_2=0.9477$. Cyan solid lines correspond to $t^{-\widetilde{D}_{2}}$ with (a) $\widetilde{D}_2=0.0269$, (b) $\widetilde{D}_2=0.1467$, (c) $\widetilde{D}_2=0.2647$, (d) $\widetilde{D}_2=0.4062$, (e) $\widetilde{D}_2=0.5589$, (f) $\widetilde{D}_2=0.7786$, (g) $\widetilde{D}_2=0.8398$, and (h) $\widetilde{D}_2=0.8873$. Green dashed lines correspond to $1-\langle k\rangle t^{2}$; i.e. the decay at very short times. (g) Standard deviation of the energy distribution of the initial state $\sigma_{ini}$ (Eq. (\ref{['eq:sigma']})) as a function of the connection probability $p$ for ER random networks of size $n=1000$. The red dashed line is a fit of the function $\sigma_{ini}=Ap^{B}$ to the data with $A=31.682$ and $B=0.501$.
• Figure 4: (a-h) Time-averaged survival probability $C(t)$ of ER random networks for several values of the average degree $\langle k\rangle$, as indicated in the panels. Four graph sizes are reported in each panel ($n=250$, 500, 1000, 2000); they increase from top to bottom. Red dashed lines correspond to $t^{\widetilde{D_{2}}}$. In panel (a) the blue dashed line corresponds to $t^{D_{2}}$ and is plotted for comparison purposes. (g) Generalized dimension $\widetilde{D}_{2}$ of the initial state $\ket{\phi_{ini}}$ as a function of the average degree $\langle k\rangle$. The blue dashed line indicates $\widetilde{D_{2}}=1$.
• Figure 5: Thouless time $t_{Th}$ of ER random networks of size $n$ as a function of the connection probability $p$. Dashed lines correspond to fittings of Eq. (\ref{['eq:ThoulessDecay']}) to the data with fitting parameters reported in Table \ref{['tab:Thouless']}.