Triple-Dyad Ratio Estimator

• The paper introduces a triple-dyad ratio estimator that leverages algebraic identities from dyad triplets to infer global and nodal parameters in directed graphs.
• It employs explicit ratio formulas and bias correction techniques, ensuring computational tractability and asymptotic consistency with normality.
• The method offers practical testing procedures and scalable implementations, outperforming traditional MLE in large network applications.

The triple-dyad ratio estimator is a parametric estimation method for the $p_1$ model of directed graphs, enabling statistically principled inference of global and nodal parameters including density and reciprocity. This estimator leverages algebraic identities satisfied by triple patterns of dyad indicators, yielding explicit, scalable estimators with guaranteed asymptotic consistency and normality. Its computational tractability, combined with rigorous bias correction and testing methods, makes it a foundational tool for the analysis of large directed networks (Feng et al., 10 Jan 2026).

1. Mathematical Foundation and Model Context

The $p_1$ model concerns a directed graph $G_n$ with $n$ labeled vertices. The adjacency matrix $X=(X_{ij})$ encodes directed edges with $X_{ij}=1$ if an edge exists from $i$ to $j$ and $X_{ii}=0$. For each unordered node pair $\{i, j\}$, the dyad is $D_{ij}=(X_{ij}, X_{ji}) \in \{0,1\}^2$, leading to four dyad-types: $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$, with binary indicators $I_{ij}^{ab} = \mathbf{1}\{D_{ij} = (a, b)\}$.

The $p_1$ model posits independent dyads with probabilities parameterized as

$$\mathbb{P}(D_{ij} = (a, b)) = p_{ij}^{ab} = \frac{\exp\{a(\theta + \alpha_i + \beta_j) + b(\theta + \alpha_j + \beta_i) + ab\rho\}}{k_{ij}}$$

where $k_{ij}$ normalizes over all $(a,b)$, $\theta$ governs density, $\rho$ governs reciprocity, and $\alpha_i, \beta_j$ are nodal “expansiveness” and “popularity” parameters subject to $\sum_{i}\alpha_i=0$, $\sum_{j}\beta_j=0$ (Feng et al., 10 Jan 2026).

2. Triple-Dyad Indicator Construction

For estimation, the triple-dyad indicator $$T_{itj}^{(a_1b_1),(a_2b_2),(a_3b_3)} = I_{it}^{a_1b_1} I_{tj}^{a_2b_2} I_{ji}^{a_3b_3}$$ is defined over triplets $(i, t, j)$ of distinct nodes and their dyad configurations. The expectation factorizes: $$\mathbb{E}T_{itj}^{(a_1b_1),(a_2b_2),(a_3b_3)} = p_{it}^{a_1b_1} p_{tj}^{a_2b_2} p_{ji}^{a_3b_3}$$ Strategic selection of dyad patterns and ratios enables isolation of target parameter combinations through empirical counts.

3. Explicit Estimator Formulas

Density Estimation

For fixed $t$, define \begin{align*} A_t &= \sum_{i \neq t} \sum_{j \neq t, i} I_{it}^{01} I_{tj}^{01} I_{ij}^{00} \ B_t &= \sum_{i \neq t} \sum_{j \neq t, i} I_{it}^{00} I_{tj}^{00} I_{ij}^{01} \end{align*} The estimator for $\theta + \alpha_t + \beta_t$ is $\log(A_t / B_t)$. Averaging over $t$ and exploiting identifiability yields

$$\hat{\theta} = \frac{1}{n}\sum_{t=1}^n \log\left(\frac{A_t}{B_t}\right)$$

Reciprocity Estimation

For another triple pattern, isolating $\rho + \theta + \alpha_t + \beta_t$ and subtracting $\hat{\theta}$ gives

$$\hat{\rho} = \frac{1}{n}\sum_{t=1}^n \log\left(\frac{\sum_{i,j \neq t} I_{it}^{11} I_{tj}^{11} I_{ij}^{10}}{\sum_{i,j \neq t} I_{it}^{10} I_{tj}^{10} I_{ij}^{11}}\right) - \hat{\theta}$$

Nodal Parameters

For each $i$ \begin{align*} \hat{\alpha}i &= \frac{1}{n}\sum{t \neq i} \log\left(\frac{\sum_{k,l \neq i, t} I_{ki}^{01} I_{kl}^{00} I_{tl}^{01}}{\sum_{k,l \neq i, t} I_{ki}^{00} I_{kl}^{01} I_{tl}^{00}}\right) - \hat{\theta} \ \hat{\beta}j &= \frac{1}{n}\sum{t \neq j} \log\left(\frac{\sum_{k,l \neq j, t} I_{kt}^{01} I_{kl}^{00} I_{jl}^{01}}{\sum_{k,l \neq j, t} I_{kt}^{00} I_{kl}^{01} I_{jl}^{00}}\right) - \hat{\theta} \end{align*} Collectively, $$\widehat{\Theta} = (\hat{\rho}, \hat{\theta}, \{ \hat{\alpha}_i \}, \{ \hat{\beta}_j \})$$ defines the triple-dyad ratio estimator (Feng et al., 10 Jan 2026).

4. Statistical Guarantees and Bias Correction

Consistency and Asymptotics

Under bounded parameter norms and non-sparsity,

$$\|\widehat{\Theta} - \Theta\|_\infty = O\left(\sqrt{\frac{\log n}{n}}\right)$$

Sharper rates are available in regimes with specified dyad probability bounds.

Central Limit Theorem

For large $n$,

$$\frac{\hat{\theta} - \theta - \theta^*}{\sigma_\theta} \xrightarrow{d} \mathcal{N}(0,1), \quad \frac{\hat{\rho} - \rho - \rho^*}{\sigma_\rho} \xrightarrow{d} \mathcal{N}(0,1)$$

where the explicit bias terms $\theta^*, \rho^*$ are $O(1/n)$ and variances are analytic sums over probabilities and mixing weights. Nodal parameters admit a joint asymptotic normality with explicit block-diagonal covariance formulas.

Bias Correction

Plug-in empirical versions of all dyad probabilities yield $\hat{\theta}^*, \hat{\rho}^*$; bias-corrected estimators are

$$\hat{\theta}_{\mathrm{BC}} = \hat{\theta} - \hat{\theta}^*, \quad \hat{\rho}_{\mathrm{BC}} = \hat{\rho} - \hat{\rho}^*$$

5. Testing Procedures and Inferential Extensions

To test for reciprocity ($H_0: \rho = 0$), the asymptotic normality and bias-correction yield the statistic

$$T = \frac{\hat{\rho} - \hat{\rho}^*}{\hat{\sigma}_\rho}$$

$H_0$ is rejected at significance $\alpha$ if $|T| > z_{1-\alpha/2}$. Confidence intervals and two-sample tests comparing graphs are constructed via analogous formulas, with coverage frequencies empirically verified to be near nominal values.

6. Computational Complexity and Implementation

Naïve evaluation of all node triples is prohibitive at $O(n^3)$ time. The estimator is efficiently computed via matrix-product chains. Configuration matrices $A^{ab}$ encode dyad indicators; products such as $A^{01}A^{00}A^{01}$ allow extraction of all necessary triple sums for every $t$ via diagonal entries, computable in $O(n^\omega)$ time ($\omega \approx 2.373$), or $O(n^2)$ for sparse graphs. For $n$ up to 5,000, the triple-dyad estimator executes in under 1.5 seconds, compared to 8–90 seconds for MLE fitting.

7. Empirical Performance and Real-World Applications

Simulation studies across multiple density regimes and node counts show that the triple-dyad ratio estimator matches the accuracy of the MLE for global parameters and approaches it for nodal parameters as the network size grows. In a real-world example on the Sina Weibo following network ($n=4077$), the estimator recovered strong reciprocity ($\hat{\rho} \approx 7.56$) and heterogeneous node degrees, yielding statistically significant reciprocity test results. The estimator’s speed and scalability are confirmed in extensive numerical experiments (Feng et al., 10 Jan 2026).