Fig. 5: Sustaining neuronal and spin dynamics.
From: Sustaining a network by controlling a fraction of nodes
a Neuronal dynamics based on Wilson–Cowan model52,53. b Phase diagram for dynamics of the forced system via fraction ρ = 0.02 and holding value Δ = 15 according to Eq. (8). The network is Erdős-Rényi (ER) with N = 104 and κ = 40. The forced curve is shifted left relative to the free system curve (thin and light lines), yielding a window of sustaining (blue shade) (c) (κ, λ)-space shows a sustaining phase diagram containing five phases. Our theory, Eqs. (8) and (11), predicts for high degrees well the transition between unsustainable and sustainable by fraction ρ = 0.02 and Δ = 15 (the white line). The light blue area is the sustainable phase for controlling a fraction ρ = 0.02, and the dark blue phase is the sustainable region when controlling a single node. d (β, ρ) phase diagram for κ = 20, 60, 100. e Model for spin dynamics based on Ising-Glauber model54. f While the free system (light and thin lines) shows a diagram with a zero regime and bi-stable symmetric regime, the forced system (thick lines) exhibits two regions: for a dense network (large β) coexistence of x0 and x1, and for a sparse network (small β) only x1 appears. Consequently, there is a range of sustaining (blue shade). Here ρ = 0.1 and Δ = 1. The network is ER with N = 104 and κ = 40. g The (κ, λ) phase diagram for ρ = 0.1 and Δ = 1. Here there is no sustainable phase when holding a single node since controlling a single node does not change the global system states in this dynamics. The simulations were averaged over 10 realizations of ER networks with N = 104. The white line represents the theory of Eq. (S4.41) in Supplementary Note 4 with Eq. (11). h The (β, ρ) phase diagram for fixed Δ = 1. Color represents simulations on ER with κ = 20, 60, 100 and N = 104. The results were averaged over ten realizations. The black line stands for the theory of Eq. (S4.41) in Supplementary Note 4 with Eq. (11).
