heat kernel
the multi-scale smoothing operator in the tri-kernel
$$H_\tau = \exp(-\tau L)$$
where $L$ is the graph Laplacian and $\tau$ is the temperature parameter. one of three components of the tru computation alongside diffusion and springs
the heat equation on graphs
the operator arises from the graph heat equation:
$$\frac{\partial H}{\partial \tau} = -L H, \quad H_0 = I$$
the solution is the matrix exponential $H_\tau = \exp(-\tau L)$. applied to a signal $\phi$ on the cybergraph, $H_\tau \phi$ returns a smoothed version where each particle's value blends with its neighborhood — Gaussian-like decay over graph distance
temperature as scale
$\tau$ controls the spatial reach of smoothing:
| regime | behavior | analogy |
|---|---|---|
| $\tau \to 0$ | identity — each particle sees only itself | crystallization, commitment |
| small $\tau$ | local smoothing — nearby particles blend | fine-grained context |
| large $\tau$ | global smoothing — distant particles contribute | annealing, exploration |
| $\tau \to \infty$ | uniform distribution — all structure erased | maximum entropy |
high $\tau$ explores. low $\tau$ commits. the temperature parameter is the thermostat of collective focus — controlling how much the system adapts versus how much it preserves local structure
spectral decomposition
in the eigenbasis of $L$ with eigenvalues $0 = \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n$:
$$H_\tau = \sum_{k=1}^{n} e^{-\tau \lambda_k} \, v_k v_k^\top$$
low-frequency eigenvectors (small $\lambda_k$) persist. high-frequency eigenvectors (large $\lambda_k$) are exponentially suppressed. this is why the operator acts as a multi-scale smoother: it preserves global community structure while erasing local noise
the spectral gap $\lambda_2$ (the Fiedler eigenvalue) governs the contraction rate:
$$\|H_\tau \phi - H_\tau \psi\|_2 \leq e^{-\tau \lambda_2} \|\phi - \psi\|_2$$
properties
positivity-preserving: if $\phi \geq 0$ then $H_\tau \phi \geq 0$. the operator never creates negative values from positive inputs
semigroup: $H_{\tau_1} H_{\tau_2} = H_{\tau_1 + \tau_2}$. smoothing at scale $\tau_1$ then at scale $\tau_2$ equals smoothing at scale $\tau_1 + \tau_2$
locality: Gaussian tail decay means the effect at distance $d$ falls as $\exp(-d^2 / 4\tau)$. for precision $\varepsilon$, only $h = O(\log(1/\varepsilon))$ hops are needed. Chebyshev polynomial approximation computes $H_\tau \phi$ locally with bounded error
contraction: the collective focus theorem proves $H_\tau$ contracts with rate $e^{-\tau \lambda_2} < 1$, contributing to the composite contraction of the full tri-kernel
role in the tri-kernel
the tri-kernel blends three operators:
$$\phi^{(t+1)} = \text{norm}\big[\lambda_d \cdot D(\phi^t) + \lambda_s \cdot S(\phi^t) + \lambda_h \cdot H_\tau(\phi^t)\big]$$
diffusion explores via random walks. springs enforce structural consistency via the screened Laplacian. the heat kernel provides adaptive context — what the graph looks like at scale $\tau$. together the three forces converge to focus: the fixed point that is cyberank
see cyber/heat for the full specification. see tri-kernel for the composite operator. see collective focus theorem for the convergence proof. see Fan Chung for the foundational work on heat kernels and PageRank. see spectral gap for the eigenvalue that governs contraction rate