Dynamical Advantages of Scale-Free Networks

A dynamical analysis of common network topologies is given and it is reported that a scale-free structure has two vital and distinctive features. First, complex but nevertheless reproducible states exist and, second, single-site induced state switching reminiscent of gene-expression control exists also. This indicates that scale-free networks have key dynamical advantages over other network topologies that could have contributed to their evolutionary success and thus may provide another reason for their prevalence in nature.

Figure 1

Network topologies whose dynamics are compared: (a) diffusive, (b) hierarchical, (c) small-world, (d) Watts-Strogatz, (e) scale-free, and (f) ER-random.

Figure 2

Attractor diversity and reproducibility for various network topologies: (a) diffusive, (b) hierarchical, (c) small-world, (d) Watts-Strogatz, (e) scale-free, (f) ER-random. The system size was set to 50 nodes throughout except for the hierarchical network where it was set to 64. The number of simulations per part of a run was $M=250$ and the transient time was ${10}^5$ time steps. The blue circles denote the area of interest.

Figure 3

(a) Histogram depicting attractor reproducibility of all the network topologies shown in Fig. 1 with the system size increased to $N_N=100$. For the topologies of Fig. 1a, 1b none of the data falls within the plotted bin range. (b) Histogram of the attractor reproducibility of the scale-free and ER-random networks for the parameters in (a) that yield the highest reproducibility, $\alpha =1.7$, $ϵ=0.9$, with the frequency determined from 1000 runs per topology. (c) Same as (b) with small bins in the high reproducibility region. (d) same as (b) but with the system size increased to $N_N=200$ and the frequency determined from 1500 runs per topology.

Figure 4

Large reproducibility in the scale-free network versus $\alpha$ and $ϵ$. The yellow squares indicate a reproducibility greater than 0.99 while the blue squares indicate a reproducibility smaller or equal than 0.99. The number of nodes is $N_N=100$.

Figure 5

Single-site switchable attractors. The top and bottom attractors can be switched by providing an input of $-1$ for a few times over a short period of time at the sites indicated by the arrows. The parameters were set to $\alpha =1.7$ and $\epsilon =0.7$, and the system size was $N_N=50$.