Individual plant-fungus symbioses do not exist in isolation. A single fungal mycelium can connect dozens of trees of several species, forming a common mycorrhizal network (CMN) — the so-called "wood-wide web". These networks redistribute carbon, nutrients and warning signals across the forest, with profound consequences for community ecology. To analyse them rigorously, we need the language of graph theory.
Forest as a graph
Model the forest as a graph $G = (V, E, W)$ where $V$ is a set of N tree vertices, $E$ a set of edges representing fungal hyphal connections between trees, and$W$ a weight matrix where $W_{ij} > 0$ is the conductance of the hyphal link between trees i and j. Each tree contributes a 6-variable subset of the cellular model from Module 2 (C, P_plant, P_fungus, N_plant, CCaMK, LCO are kept; SL and the JA pair are folded into background parameters). The full network ODE is then 6N-dimensional, coupled through the weighted edges.
Network topology
Empirical mycorrhizal networks show scale-free degree distributions:$P(k) \propto k^{-\gamma}$ with γ ≈ 2.0–2.5. A few "mother trees" act as hubs with hundreds of connections, while most trees have few. This was demonstrated by Beiler et al. (2010), who mapped the genets of Rhizopogon in a Douglas fir stand and showed scale-free hub structure.
We generate test networks via the Barabási–Albert (BA) model: start with a small connected graph, then add nodes one at a time, each new node forming m edges to existing nodes with probability proportional to their current degree (preferential attachment). The parameter m controls hub strength: m = 1 gives a tree, m = 2–3 gives realistic forest-like structures, m ≥ 5 gives highly redundant networks. The simulation panel lets you vary m and the number of trees N.
Network ODE — Forest Mycorrhizal Network + Villani Analysis
Generates a Barabási–Albert scale-free tree network with N trees connected through a common mycorrhizal network, integrates the 6N-variable coupled ODE, and computes the spectral gap (Fiedler λ₂), HWI inequality, and Talagrand T₂ saturation ratio.
The graph Laplacian and the Fiedler value
The graph Laplacian $\mathbf{L} = \mathbf{D} - \mathbf{W}$, where$\mathbf{D}$ is the diagonal degree matrix, encodes the network's connectivity in a single linear operator. Its eigenvalues, in increasing order, satisfy $0 = \lambda_1 \le \lambda_2 \le \dots \le \lambda_N$.
The second eigenvalue $\lambda_2$ — the Fiedler value or algebraic connectivity — characterises how well the network propagates information. It satisfies the variational principle
$$\lambda_2(\mathbf{L}) = \min_{\mathbf{x} \perp \mathbf{1},\,\|\mathbf{x}\|=1} \mathbf{x}^\top \mathbf{L}\, \mathbf{x} = \min_{\mathbf{x} \perp \mathbf{1}} \frac{\sum_{(i,j)\in E} W_{ij}(x_i - x_j)^2}{\sum_i x_i^2}.$$
A large $\lambda_2$ means nutrients and signals spread rapidly through the network (small mixing time); a small $\lambda_2$ signals a fragile network — one that can be broken into disconnected components by removing only a few central hubs.
Mixing time and the spectral gap
Consider a passive diffusion on the network: a quantity (say, fungal carbon) diffuses from a source to equilibrium. The rate of approach to equilibrium is set by the spectral gap$\Delta = \lambda_2 - \lambda_1 = \lambda_2$ (because $\lambda_1 = 0$ for a connected graph). The mixing time, defined as the time to reach within ε of equilibrium, scales as$\tau_{\text{mix}} \sim (\ln 1/\varepsilon)/\lambda_2$. For a forest with$\lambda_2 \sim 0.1$ h⁻¹ this gives mixing times of ~30 h — consistent with field measurements of carbon redistribution timescales.
Villani hypocoercivity
The full 6N-dimensional forest ODE is not a simple diffusion: it is a non-self-adjoint, structured dynamical system with conserved quantities and slow modes. Villani's hypocoercivity framework (2009) is the tool of choice for analysing convergence to equilibrium in such systems. The idea: even when the linearised operator is not coercive in the natural norm, one can construct a modified entropy that decays exponentially.
The HWI inequality ties together entropy $H$, Wasserstein distance$W$ (optimal transport cost), and Fisher information $I$:
$$H(\rho \,\|\, \rho_\infty) \;\le\; W(\rho, \rho_\infty)\,\sqrt{I(\rho \,\|\, \rho_\infty)} \;-\; \tfrac{1}{2}\,\lambda\, W(\rho, \rho_\infty)^2,$$
where $\lambda > 0$ is tied to the spectral gap $\lambda_2$. The Talagrand T2 inequality is the special case$W^2 \le 2H/\lambda$ that follows from the HWI bound. The backend simulation computes aT2 saturation ratio — how close the realised forest is to the optimal Talagrand-saturating equilibrium. A value near 1 indicates the network has hardened into an equilibrium transport plan; values closer to 0 indicate the system is still relaxing.
What the simulation reports
The forest panel returns:
fiedler_lambda2: the algebraic connectivity.spectral_gap: $\lambda_2 - \lambda_1$ of the dynamics linearised about equilibrium.mixing_time_h: mixing time in hours derived from the gap.hwi_satisfied: boolean check that the HWI bound holds on the computed trajectory.T2_saturation: Talagrand T2 saturation ratio.hub_degrees: degrees of the three highest-connected nodes (the mother trees).resonant_period_s: dominant resonance period of the forest's collective mode.
What changes with hub strength
Increasing m (more hub-like networks) raises $\lambda_2$ on average — better connectivity, faster mixing. But (as Module 5 will show) it also makes the network more hub-dependent: removing the top hub disproportionately collapses connectivity. The trade-off between equilibrium efficiency and worst-case resilience is fundamental and well-studied in network science (Cohen, Erez, Ben-Avraham & Havlin 2000–2001).