HawkesRank: Event-Driven Centrality for Real-Time Importance Ranking

Abstract

Quantifying influence in networks is important across science, economics, and public health, yet widely used centrality measures remain limited: they rely on static representations, heuristic network constructions, and purely endogenous notions of importance, while offering little semantic connection to observable activity. We introduce HawkesRank, a dynamic framework grounded in multivariate Hawkes point processes that models exogenous drivers (intrinsic contributions) and endogenous amplification (self- and cross-excitation). This yields a principled, empirically calibrated, and adaptive importance measure. Classical indices such as Katz centrality and PageRank emerge as mean-field limits of the framework, clarifying both their validity and their limitations. Unlike static averages, HawkesRank measures importance through instantaneous event intensities, enabling prediction, transparent endo-exo decomposition, and adaptability to shocks. Using both simulations and empirical analysis of emotion dynamics in online communication platforms, we show that HawkesRank closely tracks system activity and consistently outperforms static centrality metrics.

Figures (5)

• Figure 1: Temporal evolution of the event intensities $\lambda_i(t)$ for $i = 1,2,3$ in a three-dimensional Hawkes process. The branching ratio matrix $N$ is illustrated in the top-right corner as a weighted network, where edge thickness indicates the strength of excitation between nodes. The intensities evolve dynamically in response to exogenous inputs with baseline values $(\mu_1, \mu_2, \mu_3) = (0.05,\,0.08,\,0.04)$, generating endogenous bursts of activity through self- and cross-excitation. As a result, the relative ranking of the intensities changes over time, with the ordering of $\lambda_1$, $\lambda_2$, and $\lambda_3$ fluctuating throughout the simulation. By contrast, the first-moment Hawkes solution from Eq. \ref{['eq:first_moment_general0']} and the Katz centralities from Eq. \ref{['eq:katz1']} remain constant over time.
• Figure 2: For each of four static centrality measures, we compute the Spearman rank correlation between the corresponding static ranking and the ground-truth ranking induced by the instantaneous intensities $\{\lambda_i(t)\}_{i=1}^M$ at each time $t$. Higher correlation values indicate better agreement between the static and dynamic rankings. The vertical dashed line at $t=150$ marks the onset of an exogenous shock, implemented by increasing the smallest baseline intensity $\mu_{10}$ by a factor of $10$. The correlation curves are smoothed using a centered moving average over $50$ time steps to reduce high-frequency stochastic fluctuations in the event-driven rankings.
• Figure 3: Emotion dynamics in a YouTube live-chat video. Top: Network adjacency matrix (left) and estimated branching ratio matrix $N$ (right), where entry $(i,j)$ quantifies the influence of source emotion $j$ on target emotion $i$. Middle: Hawkes intensity trajectories $\lambda^{i}(t)$ for five basic emotions across the video. The sixth emotion (joy), dominates across emotions and is therefore omitted from plots for visual clarity. The intensities are smoothed using a centered 2-time-step moving average. Bottom: Time-varying ratio of endogenous activity, defined as the proportion of self- and cross-excitation relative to total intensity. The ratios are smoothed using a centered 10-time-step moving average.
• Figure 4: Intensity as a function of observation time of a simulated Hawkes process with $M=3$ event types, for two different values of the spectral radius of the matrix ${N}$ of branching ratios $n_{i,j}$ defined in equation (\ref{['dhtyh3wybn2']}). The left plot shows that a memory time $\tau$ of $2$ is amplified to $4$ for a spectral radius equal to $0.5$. The right plot shows that a memory time $\tau$ of $2$ is amplified to $20$ for a spectral radius equal to $0.9$.
• Figure 5: Adjacency matrices of correlation-based emotion networks constructed using different combinations of bin size $b$ and lag parameter $\ell$. The bin size determines the temporal aggregation window used to construct emotion activity time series, while the lag parameter specifies the number of bins separating the two observations when computing lead–lag correlations. Each panel is computed from the same YouTube live-chat data using different parameter values $(b,\ell)$.