Griffiths phase on hierarchical modular networks with small-world edges

The Griffiths phase has been proposed to induce a stretched critical regime that facilitates self-organizing of brain networks for optimal function. This phase stems from the intrinsic structural heterogeneity of brain networks, i.e., the hierarchical modular structure. In this work, the Griffiths phase is studied in modified hierarchical networks with small-world connections based on the 3-regular Hanoi network. Through extensive simulations, the hierarchical level-dependent inter-module wiring probabilities are identified to determine the emergence of the Griffiths phase. Numerical results and the complementary spectral analysis of the relevant networks can be helpful for a deeper understanding of the essential structural characteristics of finite-dimensional networks to support the Griffiths phase.

Figure 1

Depiction of the Hanoi network of generation $g=6$. The network features a regular geometric structure, in the form of a one-dimensional backbone, and a distinct set of recursive small-world links. The node degree is uniformly 3.

Figure 2

(a) IPR vs $m$ for HMN1 network configurations with different maximum generation $g$. Red squares show values of IPR for $g=10$; black circles, for $g=11$; blue diamonds, for $g=12$. Each data point averages values of IPR over 100 independent realizations of the HMN1 configuration. (b) Localized eigenvectors corresponding to the five largest eigenvalues of the adjacency matrix of one graph realization of the HMN1 configuration with $g=11,m=16$.

Figure 3

(a) IPR vs $m$ for HMN2 network configurations with different $g$ values. Red squares show values of the IPR with $g=10$; black circles, for $g=11$. The hierarchical level-dependent intermodule probability is $p_i=4^{-\left(i+1\right)}$. Compared to them, blue diamonds are IPRs for $g=10$ with a level-dependent probability $p_i=4^{-i}$. Each data point averages IPRs over 100 independent realizations of the HMN2 configuration. (b) Localized eigenvectors corresponding to the five largest eigenvalues of the adjacency matrix of one graph realization of the HMN2 configuration with $g=14,m=2,p_i=4^{-\left(i+1\right)}$. (c) Localized eigenvectors corresponding to the five largest eigenvalues of the adjacency matrix of one graph realization of HMN2 with $g=14,m=3,p_i=4^{-\left(i+1\right)}$.

Figure 4

Lifshitz tails for HMN1 and HMN2. Red circles show the tail distribution for the HMN1 configuration with $g=11$ and $m=8$; black squares, the distribution for the HMN2 configuration with $g=13, m=2$, and $p_i=4^{-\left(i+1\right)}$, fitted with a power law $P\left(\mathrm{\Lambda }\right)\sim {\mathrm{\Lambda }}^{0.6828\cdots }$; and blue diamondsfor the tail distribution for the HMN2 configuration with $g=12, m=3$, and $p_i=4^{-\left(i+1\right)}$, fitted with $P\left(\mathrm{\Lambda }\right)\sim {\mathrm{\Lambda }}^{0.8226\cdots }$.

Figure 5

(a) $\rho$ vs $t$ for the HMN1 network configuration with $g=11, m=8$. Lines from bottom to top are for $\lambda =0.1650$, 0.1651, 0.1652, and 0.1653. (b) $ln\left(\rho \right)$ versus $ln\left(t\right)$. The straight black line is the fitted curve with $\rho \sim t^{-0.2849...}$. The critical propagation rate is ${\lambda }_c\approx 0.1652$.

Figure 6

(a) $\rho$ vs $t$ for the HMN1 network configuration with $g=11,m=16$. Lines from bottom to top are for $\lambda =0.07475,0.07480,0.07485,0.0749$. (b) $ln\left(\rho \right)$ versus $ln\left(t\right)$. The straight black line shows the fitted curve with $\rho \sim t^{-0.3127...}$. The critical propagation rate is ${\lambda }_c\approx 0.07485$.

Figure 7

(a) $\rho$ vs $t$ for HMN2 network configurations with $g=13,m=2$ and with $g=14,m=2$. Lines from bottom to top are for $\lambda =0.46$, 0.47, 0.48, 0.49, 0.50, 0.51, 0.52, and 0.53. (b) $ln\left(\rho \right)$ versus $ln\left(t\right)$, the black straight lines are the fitted curves with $\rho \sim t^{-0.9094...}, \rho \sim t^{-0.6989...}, \rho \sim t^{-0.5356...}, \rho \sim t^{-0.3962...}$, and $\rho \sim t^{-0.3054...}$ from bottom to top for $\lambda =0.49$, 0.50, 0.51, 0.52, and 0.53.

Figure 8

(a) $\rho$ vs $t$ for HMN2 network configurations with $g=13,m=3$ and with $g=12,m=3$. Lines from bottom to top are for propagation rates $\lambda =0.258$, 0.259, 0.260, 0.261, and 0.262. (b) $ln\left(\rho \right)$ versus $ln\left(t\right)$. The straight black lines from bottom to top are the fitted curves with $\rho \sim t^{-0.5339...}, \rho \sim t^{-0.4325...}$, and $\rho \sim t^{-0.3605...}$ for $\lambda =0.260$, 0.261, and 0.262.

Figure 9

Dynamic susceptibility $\mathrm{\Sigma }$, measured for the HMN2 network configurations with $g=11,m=3$ (red diamonds), $g=12,m=3$ (blue circles), and $g=13,m=3$ (black squares) with $p_i=4^{-\left(i+1\right)}$. The overall response increases significantly near and in the Griffiths phase (from $\lambda =0.260$ to $\lambda =0.262$) and decreases away from the critical region.