HCTCD: Temporal & Collaboration Decay Metric

• The paper introduces HCTCD, which integrates temporal and collaboration decay into harmonic closeness to predict citation outcomes.
• The methodology applies exponential decay functions on collaboration recency and frequency, computed via BFS-derived shortest paths in co-authorship graphs.
• Empirical findings demonstrate HCTCD outperforms standard centrality measures, achieving up to a 4% improvement in citation prediction accuracy.

Harmonic Closeness with Temporal and Collaboration Count Decay (HCTCD) is a centrality metric developed to assess the structural position of authors in scientific collaboration networks while explicitly accounting for both the recency and the frequency of collaborations. By extending the classical harmonic closeness framework with decay mechanisms for temporal distance and collaboration intensity, HCTCD quantifies an author's global network reach, emphasizing the dynamic, evolving topology of co-authorship graphs. The metric has demonstrated superior predictive power for citation outcomes compared to standard centrality measures, indicating its utility in explaining disparities in scientific recognition within top-tier academic venues (Jie et al., 26 Dec 2025).

1. Formal Mathematical Definition

Let $G=(V,E)$ denote the undirected co-authorship graph at a given publication year $t_p$. For every ordered author pair $(i,j)$, the components are defined as:

• $d(i,j)$: the shortest-path distance in $G$ (unweighted)
• $\Delta t_{ij} = t_p - t^*_{ij}$: “age” since the last collaboration (where $t^*_{ij}$ is the year of the most recent prior joint publication)
• $c_{ij}$: total number of publications between $i$ and $j$ before $t_p$

Two exponential decay terms are introduced:

• Temporal decay: $w^{(T)}_{ij} = \exp[-\alpha \cdot \Delta t_{ij}]$, $\alpha \in \mathbb{R}$
• Collaboration count decay: $w^{(C)}_{ij} = \exp[-\beta \cdot c_{ij}]$, $\beta \geq 0$

The effective pairwise weight is $$w_{ij} = \exp[-\alpha \cdot \Delta t_{ij} - \beta \cdot c_{ij}]$$.

HCTCD for node $i$ is:

$$C_{HCTCD}(i) = \frac{1}{N-1} \sum_{j \neq i} \frac{1}{d(i,j)} \exp \left( -\alpha (t_p - t^*_{ij}) - \beta c_{ij} \right)$$

where $N = |V|$ is the total number of authors.

2. Algorithmic Computation and Aggregation

For a given publication $p$ at $t_p$:

Inputs:

• $A_p$: Author list of paper $p$
• The full co-authorship graph constructed from all publications before $t_p$

The computation proceeds as follows:

1. Precompute all-pairs shortest paths $d(i,j)$ via BFS for the unweighted graph.
2. For each node $i$:
   • For every $j \neq i$:
     • Compute $\Delta t_{ij}$ and $c_{ij}$.
     • Calculate $$w_{ij} = \exp(-\alpha \cdot \Delta t_{ij} - \beta \cdot c_{ij})$$.
     • Accumulate $(1 / d(i,j)) \cdot w_{ij}$.
   • Normalize: $C_{HCTCD}(i) = \text{sum}_i / (N-1)$.

For team-level aggregation on an $n$-author paper:

Aggregation Type	Formula
Unweighted sum	$C_{sum} = \sum_{k=1}^n C_{HCTCD}(a_k)$
Unweighted average	$C_{ave} = \frac{1}{n} \sum_{k=1}^n C_{HCTCD}(a_k)$
Weighted sum	$$C_{w\_sum} = \sum_{k=1}^n e^{-\tau (k-1)} C_{HCTCD}(a_k)$$
Weighted average	$$C_{w\_ave} = C_{w\_sum} / \sum_{k=1}^n e^{-\tau(k-1)}$$

Here, $\tau$ modulates emphasis on lead authors ($\tau = 0.3$ implies second author weight $\approx 0.74$ of the first).

3. Parameterization and Default Values

Empirically optimized parameters are:

• Temporal decay rate $\alpha = -0.2$ (negative $\alpha$ allows older collaborations to retain partial influence)
• Collaboration decay rate $\beta = 0.45$ (diminishing marginal returns for repeat co-authorship)
• Author-order weight $\tau = 0.3$
• Time windows for dynamic centrality: 1, 2, 4, 8, 16 years pre-$t_p$; a fixed 8-year window is standard for regression analysis

A grid search over 2015–2016 data identified robust optimality ranges: $\alpha \in [-0.3, 0]$ (best at $-0.2$), $\beta \in [0.4, 0.6]$ (best at $0.45$), $\tau \approx 0.3$, after which performance plateaus.

4. Comparative Perspective

Standard harmonic closeness centrality is defined as $C_{HC}(i) = (1/(N-1)) \sum_{j} 1/d(i,j)$, neglecting both tie recency and collaboration frequency. HCTCD down-weights obsolete links and adjusts for repeated collaborations, thus capturing:

• The reduced importance of aged collaborations (via $\alpha$)
• Saturation effects from prolific dyads (via $\beta$)

Unlike metrics focusing solely on time or frequency, HCTCD integrates both axes, reflecting that aged collaborations diminish in importance while repeated collaboration with the same partner yields diminishing additional network reach. A plausible implication is that HCTCD better models real-world scholarly influence dynamics in evolving scientific communities than single-factor extensions.

5. Empirical Findings and Predictive Utility

Empirical evaluation against citation percentile statistics for 17,942 papers yields:

• Bivariate Pearson correlation values (16-year window): degree centrality ($0.275$), closeness ($0.389$), HCTCD unweighted sum ($0.382$), HCTCD weighted sum ($0.397$, highest observed)
• Beta regression (citation percentile as outcome, controlling for content and covariates): including HCTCD increases $R^2$ from $0.6045$ to $0.6083$ ($\Delta$AIC$=-42$), with estimated coefficient $\gamma \approx 1.305$, $p<0.001$
• Predictive models: XGBoost MSE decreases from $0.03645$ to $0.03502$ ($-3.9\%$), correlation rises from $0.7025$ to $0.7165$; Random Forest correlation $0.717$ (with centralities) versus $0.700$ (without)

The combination of recency and tie-strength decay, along with harmonic closeness, affords HCTCD consistent improvements in both explanatory and predictive contexts for citation outcomes.

6. Implementation Complexity and Workflow

• All-pairs shortest paths: $O(N \cdot (N + M))$ via BFS per node on an unweighted co-authorship graph
• Weight matrices (last collaboration time, collaboration count) are pre-aggregated by a single scan of the publication history
• Centrality scores are computed dynamically for each focal window; team-level HCTCD aggregation per publication uses the above author-weighted sums or averages
• Parameter optimization for $\alpha, \beta, \tau$ is performed by grid search maximizing Pearson correlation to citation percentiles in a held-out subset
• Beta regression uses statsmodels.betareg (Python); model evaluation utilizes both regression and tree-based ensemble methods

7. Interpretation and Significance

HCTCD unifies global network centrality, recency weighting, and saturation-adaptive tie strength to model how changing collaboration patterns influence future scholarly impact. Temporal decay ($\alpha$) ensures that recent collaborations exert stronger influence, while collaboration-count decay ($\beta$) prevents overemphasis on highly prolific dyads. Weighted author-order aggregation ($\tau$) captures the hierarchical prominence of lead authors. Across all tested scenarios, HCTCD outperforms classical centralities in accounting for citation disparities, offering up to a 4% reduction in predictive mean squared error and revealing the centrality-driven inequities embedded in citation dynamics. This metric substantiates arguments for network-aware evaluation frameworks in the assessment of scientific recognition (Jie et al., 26 Dec 2025).