Heterogeneous excitable systems exhibit Griffiths phases below hybrid phase transitions

In $d>2$ dimensional, homogeneous threshold models discontinuous transition occur, but the mean-field solution provides $1/t$ power-law activity decay and other power laws, and thus it is called mixed-order or hybrid type. It has recently been shown that the introduction of quenched disorder rounds the discontinuity and second-order phase transition and Griffiths phases appear. Here we provide numerical evidence that even in case of high graph dimensional hierarchical modular networks a Griffiths phase in the $K=2$ threshold model is present below the hybrid phase transition. This is due to the fragmentation of the activity propagation by modules, which are connected via single links. This provides a widespread mechanism in the case of the threshold type of heterogeneous systems, modeling the brain, or epidemics for the occurrence of dynamical criticality in extended Griffiths phase parameter spaces. We investigate this in synthetic modular networks with and without inhibitory links as well as in the presence of refractory states.

Figure 1

Plot of the adjacency matrix of an $N=1024$-sized sample of the HMN2d graph. Black dots denote connections between nodes $i$ and $j$. The four-level structure is clearly visible in the blocks along the diagonal; additional long-range edges are scattered points around it.

Figure 2

In-degree distribution of four randomly selected $l=5$ HMN2d graphs.

Figure 3

Avalanche size distributions at different $\lambda$ branching rates, denoted by the symbols, in the presence of excitatory links in the HMN2d with $l=5,6$ levels. From top to bottom curves: $\lambda =$ 0.33, 0.325, 0.322, 0.32 ($l=5$ cyan and $l=6$ green), 0.315, 0.31. Dashed lines show PL fits for the tails: $s>1000$ at $\lambda =$ 0.315, 0.322, 0.33.

Figure 4

Survival probability of the activity at different branching rates in the $K=2$ threshold model with excitatory links. From top to bottom curves: $\lambda =$ 0.33, 0.325, 0.322, 0.32 ($l=5$ and $l=6$), 0.315. Dashed lines show PL fits for the tails: $s>{10}^4$ at $\lambda =$ 0.315, 0.32, 0.322, 0.33.

Figure 5

Avalanche size distributions at different $\lambda$ branching rates, denoted by the symbols, in the presence of inhibitory links in HMN2d with $l=5,6$ levels. From top to bottom curves: $\lambda =0.55$, 0.54, 0.53 ($l=5$ green and $l=6$ cyan), 0.52, 0.51, 0.50 ($l=5$ triangle and $l=6$ diamond). Dashed lines show power-law fits for the tails of $\lambda =0.55,0.51$ cases, for $t>1000$. Inset: overlapping avalanches case for half-filled initial condition at $\lambda =0.51,0.515,0.52,0.525$ (bottom to top symbols).

Figure 6

Survival probability of the activity at different branching rates and $\nu =1-\lambda$ for the $K=2$ threshold model with levels: $l=5,6$ for the case with $20\%$ of inhibitory links. From bottom to top symbols: $\lambda =$ 0.5, 0.505, 0.510, 0.515, 0.520 ($l=5$ purple cross and $l=6$ blue circle), 0.525 ($l=5$ brown cross and $l=6$ brown circle). Dashed lines are PL fits for the tails of $\lambda =$ 0.505, 0.52 curves.

Figure 7

Avalanche size distributions at different $\lambda$ branching rates, denoted by the symbols, in case of the refractory model, in the presence of inhibitory links in HMN2ds with $l=5,6$ levels. From bottom to top symbols: $\lambda =$ 0.39, 0.40 ($l=5$ left triangle and $l=6$ up triangle), 0.41, 0.42, 0.43. Dashed lines are PL fits for the tails of $\lambda =0.39,0.4,0.41,0.43$ cases for $t>1000$. The inset shows the oscillatory behavior of $\rho \left(t\right)$ of a single run for $\mathrm{\Delta }t=10$.

Figure 8

Survival probability of the activity at different branching rates $\lambda$ for the levels $l=5,6$, in the case of the inhibitory-refractory model. From bottom to top symbols: $\lambda =0.40$, 0.41 ($l=5$ and $l=6$), 0.42 ($l=5$ light green and $l=6$ dark green), 0.43. Dashed lines show PL fits for $t>1000$ for the $\lambda =$ 0.4, 0.41, 0.43 cases. Inset: $\rho \left(t\right)$ at $\lambda =1$, $l=7$ averaged over ${10}^5$ realizations. Blue boxes: excitatory; red diamonds: inhibitory. Black bullets: BFS $\rho \left(r\right)$ results. Dashed lines are PL fits for the initial regions: $1\le t<10$) resulting in effective dimensions: $d_{\mathrm{eff}}=1.84\left(3\right)$ (excitatory), $d_{\mathrm{eff}}=1.19\left(1\right)$ (inhibitory), $d=4.18\left(5\right)$ (graph dimension estimated for $5<r<10$).

Figure 9

Evolution of $\rho \left(t\right)$ for different $\lambda$ in case of starting from fully active state in the excitatory model with levels: $l=5,6$. From bottom to top symbols: $\lambda =$ 0.30, 0.32, 0.321 ($l=6$), 0.322, 0.322 ($l=6$), 0.325, 0.33, 0.34 ($l=6$), 0.35, 0.4, 0.5, 0.6, 0.7.

Figure 10

Steady-state behavior for the excitatory, inhibitory, and refractory-inhibitory cases. Inset: evolution of $\rho$ in an inhibitory HMN2d with $N=4096$ for different initial activity densities: $\rho \left(0\right)=$ 0.0005, 0.001, 0.01, 0.1, 1 (bottom to top curves).

Figure 11

Evolution of $\rho \left(t\right)$ for different $\lambda$, shown by legends in the case of starting from active states in the inhibitory model. Thick, normal, and thin lines correspond to $l=7$, $l=6$, and $l=5$, accordingly. From bottom to top curves: $\lambda =$ 0.51, 0.52, 0.53, 0.54, 0.55, 0.56, 0.6, 0.78, 0.8, 0.85, 0.9, 0.999. The graph shows results with initial condition $\rho \left(0\right)=0.5$ for $\lambda \le 0.6$, except for $\lambda =0.51$, $l=7$. In all the other cases $\rho \left(0\right)=1$ is applied.