Spectral analysis and slow spreading dynamics on complex networks

The susceptible-infected-susceptible (SIS) model is one of the simplest memoryless systems for describing information or epidemic spreading phenomena with competing creation and spontaneous annihilation reactions. The effect of quenched disorder on the dynamical behavior has recently been compared to quenched mean-field (QMF) approximations in scale-free networks. QMF can take into account topological heterogeneity and clustering effects of the activity in the steady state by spectral decomposition analysis of the adjacency matrix. Therefore, it can provide predictions on possible rare-region effects, thus on the occurrence of slow dynamics. I compare QMF results of SIS with simulations on various large dimensional graphs. In particular, I show that for Erdős-Rényi graphs this method predicts correctly the occurrence of rare-region effects. It also provides a good estimate for the epidemic threshold in case of percolating graphs. Griffiths Phases emerge if the graph is fragmented or if we apply a strong, exponentially suppressing weighting scheme on the edges. The latter model describes the connection time distributions in the face-to-face experiments. In case of a generalized Barabási-Albert type of network with aging connections, strong rare-region effects and numerical evidence for Griffiths Phase dynamics are shown. The dynamical simulation results agree well with the predictions of the spectral analysis applied for the weighted adjacency matrices.

Figure 1

Finite size scaling of QMF results on the ER model with $\langle k\rangle =4$ for $N=1000$, 2000, $4000,...,128000$. Bullets, ${\lambda }_c$; boxes, IPR; up-triangles, $a_1$; down-triangles, $a_2$; right-triangles, $a_3$. (Line) Least-squares fitting with the form $\sim 1/N$.

Figure 2

Density decay as a function of time for the SIS on ER graph with $\langle k\rangle =1$. Network size $N={10}^6$. Different curves correspond to $\lambda =$ 1.05, 1.09, 1.095, 1.1, 1.02, 1.05, 1.095, 1.115, 1.12, 1.15 (from bottom to top curves). (Inset) Effective exponents of the same data near the phase transition point.

Figure 3

Finite size scaling of QMF results for ER graph with $\langle k\rangle =0.3$$N=1000$, 2000, $4000,...128000$. Bullets, ${\lambda }_c$; boxes, IPR; up-triangles, $a_1$; down-triangles, $a_2$; right-triangles, $a_3$. (Line) Least-squares fitting with a form $1/N$. (Inset) ${\Lambda }_1\left(N\right)$ (bullets) and the form (8) (line).

Figure 4

Density decay as a function of time in the SIS defined on ER graph with $\langle k\rangle =0.3$. Network size $N={10}^6$. Different curves correspond to $\lambda =$ 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 20, 50, 100 (from bottom to top curves).

Figure 5

Finite size scaling of QMF results for the weighted ER graphs for $\langle k\rangle =4$, in the range of sizes $N=1000$, $2000...200000$. Bullets, ${\lambda }_c$; boxes, IPR; up-triangles, $a_1$; down-triangles, $a_2$; right-triangles, $a_3$. (Line) Least-squares fitting with a $1/N$. (Inset) ${\Lambda }_1\left(N\right)$ (bullets) and the form (8) (line).

Figure 6

Finite size scaling of QMF SD results of the SIS on generalized BAT with $N=1000$, 2000, 4000, 8000, $16000$, $32000$ nodes. Bullets, ${\lambda }_c$; boxes, IPR; up-triangles, $a_1$; down-triangles, $a_2$; right-triangles, $a_3$. (Line) Least-squares fitting with the form $\sim 1/N$.

Figure 7

Density decay as a function of time for the SIS on BA graph with network sizes: $N={10}^5$ (thin lines), $N={10}^6$ (thick lines). Different curves correspond to $\lambda =$ 2.4, 2.45, 2.47, 2.5, 2.55, 2.6, 2.7 (from bottom to top curves). (Left inset) Local slopes of the same data. (Right inset) Degree distribution of the aging BA graph for $N=4\times {10}^6$ nodes.