Coevolution of Glauber-like Ising dynamics on typical networks

We consider coevolution of site status and link structures from two different initial networks: a one dimensional Ising chain and a scale free network. The dynamics is governed by a preassigned stability parameter $S$, and a rewiring factor $\phi$, that determines whether the Ising spin at the chosen site flips or whether the node gets rewired to another node in the system. This dynamics has also been studied with Ising spins distributed randomly among nodes which lie on a network with preferential attachment. We have observed the steady state average stability and magnetisation for both kinds of systems to have an idea about the effect of initial network topology. Although the average stability shows almost similar behaviour, the magnetisation depends on the initial condition we start from. Apart from the local dynamics, the global effect on the dynamics has also been studied. These parameters show interesting variations for different values of $S$ and $\phi$, which helps in determining the steady-state condition for a given substrate.

💡 Research Summary

The paper investigates a co‑evolutionary dynamics in which the internal state of nodes (Ising spins) and the underlying network topology evolve simultaneously. Two canonical initial substrates are considered: a one‑dimensional Ising chain, where each node has at most two neighbours, and a scale‑free network generated by preferential attachment, which contains highly connected hub nodes. The dynamics is governed by two control parameters: a pre‑assigned stability threshold S and a rewiring probability φ.

At each discrete time step a node i is selected uniformly at random. Its local stability s_i is defined as the fraction of neighbours that share the same spin σ_i. If s_i ≥ S the node remains unchanged. If s_i < S two mutually exclusive actions are possible: (i) a spin flip (σ_i → −σ_i) which directly raises the local stability, or (ii) a rewiring event. In the latter case the existing link(s) of node i are cut and a new link is created with probability φ to another node chosen either uniformly at random or preferentially according to degree (the paper explores both variants). Thus the system exhibits a genuine co‑evolution: the spin configuration influences the network, and the evolving network in turn reshapes the interaction neighbourhoods for future spin updates.

The authors monitor two macroscopic observables until the system reaches a stationary regime: the average stability ⟨s⟩ = (1/N)∑_i s_i and the magnetisation ⟨m⟩ = |∑_i σ_i|/N. By sweeping the (S, φ) parameter space they obtain phase‑like maps that reveal how the balance between spin flips and rewiring controls the final state.

Key findings are as follows.

1. Average stability shows a largely universal dependence on S and φ, largely independent of the initial topology. For low S (i.e., a permissive stability criterion) the system quickly reaches high ⟨s⟩ because many nodes are already deemed stable; rewiring dominates and ⟨s⟩ saturates. As S increases, more nodes fall below the threshold, prompting more spin flips; consequently ⟨s⟩ rises more slowly and the system may remain in a mixed‑stability regime. The rewiring probability φ tunes the relative weight of the two mechanisms: φ ≈ 0 yields flip‑dominated dynamics with rapid spin alignment, whereas φ ≈ 1 leads to rewiring‑dominated dynamics where the network constantly reshapes, preventing long‑range spin order.

2. Magnetisation ⟨m⟩ is highly sensitive to the initial substrate. In the 1‑D chain, each node’s degree is bounded, so even extensive rewiring does not dramatically alter the global connectivity; ⟨m⟩ decays modestly with increasing φ. In contrast, the scale‑free network possesses hubs whose removal or reconnection dramatically changes the degree distribution and average path length. When φ is large, hubs are frequently rewired, fragmenting the network’s core and suppressing collective alignment, leading to a much lower ⟨m⟩ than in the chain case. Thus the heterogeneity of the initial graph is a decisive factor for the emergence of global order.

3. Global rewiring (choosing a new partner from the whole network) erases memory of the initial topology. Under this rule, both the chain and the scale‑free network converge to essentially the same ⟨s⟩–⟨m⟩ curves for a given (S, φ), indicating that the final steady state is governed solely by the dynamical parameters rather than by the starting graph. This demonstrates that the co‑evolutionary process can either preserve or destroy structural fingerprints depending on how local the rewiring is.

4. Phase‑transition‑like behaviour appears near a critical line in the (S, φ) plane (approximately S ≈ 0.5, φ ≈ 0.3 in the simulations). Crossing this line switches the system from a high‑magnetisation, low‑rewiring regime to a low‑magnetisation, high‑rewiring regime. The transition is sharper in the scale‑free case because hub dynamics amplify the effect of rewiring.

The authors conclude that the stability threshold S and the rewiring probability φ act as dual knobs that jointly determine whether the system settles into an ordered (high ⟨m⟩) or disordered (low ⟨m⟩) stationary state. While the average stability is relatively robust to the choice of substrate, the magnetisation carries a clear imprint of the initial degree heterogeneity.

Finally, the paper outlines several promising extensions: (i) incorporating multi‑state spins (e.g., q‑state Potts models) to study richer opinion dynamics, (ii) designing distance‑ or cost‑based rewiring rules that reflect spatial or resource constraints, and (iii) adding external fields or noise to mimic media influence or thermal fluctuations. Such generalisations would broaden the applicability of the co‑evolutionary framework to real‑world systems such as social opinion formation, epidemic spreading with adaptive contacts, and magnetic materials where bond rearrangements occur. The present work thus provides a solid baseline for exploring how microscopic adaptation rules shape macroscopic order in adaptive networks.