Growing Unlabeled Trees

• Growing unlabeled trees are network models where node identities are suppressed, emphasizing automorphism orbits and symmetry-driven growth dynamics.
• They employ attachment kernels and leaf-based statistics to capture degree heterogeneity and predict scaling behaviors distinct from labeled models.
• This label-free framework offers broad implications for understanding network evolution in areas like quantum gravity, plant architecture, and communication systems.

Models of growing unlabeled trees concern the evolution of tree structures when node labels are suppressed, so that only the combinatorial structure—encoded in isomorphism types and automorphism symmetries—matters. In contrast to classical labeled growth models (such as uniform or preferential attachment), unlabeled growth operates over orbits of structurally indistinguishable positions in a tree. This label-free perspective exposes pronounced effects caused by symmetries, especially among leaves, and yields notably altered growth dynamics—most prominently, enhanced degree heterogeneity and suppressed attachment probabilities for symmetric positions. Analytical tools such as leaf-based statistics and attachment kernels formalize these processes, enabling direct comparison of unlabeled and labeled outcomes and scaling behaviors.

1. Unlabeled Growth Formalism: Orbits and Attachment Kernels

Growth in the unlabeled tree setting is defined not over explicit vertices but over automorphism orbits—collections of positions indistinguishable by symmetry. At every step, one attaches a new leaf to an orbit $u \in D(G)$ of the current tree $G$, not to a specific labeled node. The general rule for transition probabilities is formulated as

$$P_n(G'|G) = \frac{f(G,G')}{\sum_{G'' \in \mathcal{G}(G)} f(G,G'')}$$

where $f$ is an attachment kernel encoding characteristics (e.g., degree or leaf-count) invariant under automorphisms. For degree-based kernels,

$$f(G,G') = \varphi(G, v(G,G')), \quad \varphi(G,u)=k_{u}+\delta$$

with $k_u$ the degree of the nucleus node underlying orbit $u$ and $\delta$ an additive shift. Under uniform attachment (UUA), $\varphi(G,u)=1$ for all $u$; under preferential attachment (UPA), $\varphi(G,u)=k_u$.

This approach ensures the growth process respects the network’s structural symmetries—multiple symmetric positions collapse into single growth possibilities, fundamentally altering the effective sample space and transition probabilities.

2. Leaf-Based Analytical Statistics

Because nontrivial symmetries in trees almost always involve leaves (nodes of degree 1), a leaf-based formalism efficiently captures the essential evolution of unlabeled trees. Key variables are:

• $L$: Total number of leaves;
• $M_\ell$: Number of nuclei with exactly $\ell$ leaf neighbors.

These obey the relations: $$L + \sum_{\ell=0}^\infty M_\ell = n,\quad \sum_{\ell=0}^\infty \ell M_\ell = L$$ where $n$ is the number of tree nodes. Each growth event (attaching a leaf to an orbit of leaf-degree $\ell$) updates the $M_\ell$ and $L$ variables via explicit recurrence: $$M_\ell \rightarrow M_\ell-1,\quad M_{\ell-1} \rightarrow M_{\ell-1}+1,\quad M_1 \rightarrow M_1+1$$ Tracking these variables allows prediction of the limiting leaf fraction and degree distribution via recursive ensemble-averaged equations.

3. Role of Symmetry in the Collapse of Attachment Events

The presence of automorphism symmetries causes a structural collapse in possibility space: in the labeled process, each node’s arrival time distinguishes it; in the unlabeled process, symmetric nodes merge into a single combinatorial position. Thus, the probability to attach a new leaf to any member of an orbit is not multiplied by the number of symmetric nodes—attachment probabilities for leaves are suppressed relative to the labeled case.

For example, in $n$-node trees, several leaves sharing a neighbor may form a large orbit; the growth rule only counts this as one possibility rather than several. This phenomenon fundamentally differentiates unlabeled from labeled models and is responsible for qualitative differences in structural outcomes.

4. Degree Heterogeneity: Enhancement and Dynamics

Unlabeled growth rules increase degree heterogeneity compared to classical labeled models, with the specific enhancement contingent on the growth kernel:

• Unlabeled Uniform Attachment (UUA): The tail of the degree distribution for nuclei decays geometrically with base $c\approx 0.57$ (versus $1/2$ in labeled RRT), i.e., $p(k)\sim c^k$. This means high-degree nodes are more common in the unlabeled model.
• Unlabeled Preferential Attachment (UPA): Yields “leaf-proliferation”—almost all nodes eventually become leaves, and nuclei amass extreme degrees. The degree sequence satisfies a power-law with anomalous exponent $\gamma_0\approx1.84$ (compared to $\gamma=3$ in classical BA trees).

With an additive shift (UPA($\delta$)), the power-law exponent $\gamma(\delta)$ varies, with anomalously heavy-tailed distributions for small $\delta$ ($\gamma(\delta)\leq2$), and extensive scaling for larger $\delta$.

5. Comparative Analysis: Labeled vs Unlabeled Tree Growth

The collapse of symmetric possibilities in the unlabeled setting causes distinct differences from labeled models:

• The sample space is reduced to isomorphism classes, fundamentally altering local attachment probabilities and global statistical averages.
• Growth steps from small trees may yield equiprobable outcomes in the unlabeled model (e.g., line and star shapes with four nodes), whereas labeled models may favor one outcome due to multiplicity of labeled attachment events.

This leads to enhanced structural diversity—especially among high-degree nodes—and altered scaling behaviors for key network parameters.

6. Extensions and Generalizations

The leaf-based analytical formalism extends naturally (albeit with additional technical challenges) to network classes beyond trees, such as growing unlabeled hypergraphs, simplicial complexes, or forests. Symmetry-induced collapse must then be accounted for in more complex orbit structures.

This framework also enables paper of non-degree parameters (maximal depth, entropy, motif counts), and can inform inference tasks (e.g., parameter estimation) in biological or technological networks where node identity is inherently irrelevant.

7. Broader Implications

Recognition that network symmetries fundamentally impact growth dynamics has implications for disciplines where intrinsic structure matters—such as causal sets in quantum gravity (where Lorentz invariance implies unlabeled dynamics), physiological modeling of plant architectures, or the analysis of transportation or communication networks. Enhanced heterogeneity and altered scaling laws derived here may inform the modeling and statistical analysis of such systems.

Researchers in statistical physics, complex networks, and mathematical biology may find the formalism and explicit degree scaling exponents directly relevant to the characterization of evolving networks under symmetry-constrained growth.

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In summary, models of growing unlabeled trees, analyzed via leaf-based statistics and orbit-based growth kernels, uncover enhanced degree heterogeneity and profound differences from labeled growth models, stemming primarily from network symmetries. These insights carry both specific mathematical consequences—such as anomalous power-law exponents—and broad conceptual implications for network science and the modeling of intrinsically unlabeled structures (Hartle et al., 21 Sep 2025).