HawkesRank: Dynamic Event-Driven Ranking

• HawkesRank is a dynamic, event-driven framework that computes node importance as instantaneous event intensities, integrating both exogenous inputs and network amplification.
• It leverages an exponential kernel to weight recent events more heavily, enabling real-time updates and adaptive rankings that respond to temporal shifts in activity.
• Its formulation bridges traditional centrality measures like Katz and PageRank, revealing how static metrics emerge as mean-field limits of a time-resolved process.

HawkesRank is a fully dynamic, event-driven framework for importance ranking based on multivariate Hawkes point processes. It defines each node’s score as the instantaneous rate at which events occur on that node, with that rate driven jointly by exogenous inputs, interpreted as intrinsic or external background activity, and endogenous amplification through self- and cross-excitation. In this formulation, importance is time-dependent, empirically calibrated, and adaptive rather than a property of a static graph; classical indices such as Katz centrality and PageRank arise as mean-field limits, while the operative score remains the instantaneous intensity itself (Sornette et al., 12 Mar 2026).

1. Formal specification

The framework considers $M$ event types, or nodes, indexed by $i=1,\dots,M$, with $N_i(t)$ denoting the counting process of events of type $i$. For each node, HawkesRank specifies the intensity

$$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$

Here, $\mu_i(t)$ is the exogenous or background rate of type-$i$ events, capturing intrinsic appeal or external shocks. The coefficient $n_{j,i}$ is the fertility or branching ratio from node $j$ to $i$, defined as the expected number of direct $i=1,\dots,M$0-events triggered by each $i=1,\dots,M$1-event. The collection $i=1,\dots,M$2 generalizes the adjacency matrix. The kernel is a normalized exponential,

$i=1,\dots,M$3

with memory time-scale $i=1,\dots,M$4 and $i=1,\dots,M$5 (Sornette et al., 12 Mar 2026).

In vector form, with $i=1,\dots,M$6, $i=1,\dots,M$7, and $i=1,\dots,M$8, the model is written as

$i=1,\dots,M$9

The semantic interpretation follows directly from point-process theory: because $N_i(t)$0, the intensity $N_i(t)$1 is itself the instantaneous importance or centrality score of node $N_i(t)$2 at time $N_i(t)$3. This gives HawkesRank an explicit connection to observable activity rather than to an abstract graph statistic.

2. Time-dependent centrality and event-level semantics

Under HawkesRank, the score of node $N_i(t)$4 at time $N_i(t)$5 is

$N_i(t)$6

Each past event $N_i(t)$7 of type $N_i(t)$8 contributes $N_i(t)$9 to the current score of node $i$0. Nodes with high current exogenous input, or with many recent and strongly connected triggers, therefore rank highly at that instant. Because the score is defined at the event level and updated continuously in time, the framework yields fully adaptive, finely resolved rankings (Sornette et al., 12 Mar 2026).

A common misconception is to treat HawkesRank as a static summary applied to temporal data. The defining quantity is not an aggregate count or a time average, but the instantaneous event intensity. This is the central distinction from centrality measures that rely on static representations, heuristic network constructions, or purely endogenous notions of importance. The framework is therefore dynamic in definition rather than merely in application.

The same formulation also makes the dependence on memory explicit. The exponential kernel weights recent events more strongly than older ones, with the characteristic time-scale $i$1 determining how quickly past influence decays. This gives the ranking a direct temporal semantics: a node can rise or fall in importance as activity arrives, decays, or is amplified through the network.

3. Mean-field limits and the relation to Katz centrality and PageRank

Classical centralities emerge from the first moment of the process. Taking expectation and using $i$2 gives

$i$3

In the infinitely short-memory limit $i$4,

$i$5

so that

$i$6

If one sets $i$7 and $i$8, this becomes Katz’s equation,

$i$9

In the stationary long-time limit, with $$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$0 and $$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$1,

$$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$2

which again recovers Katz centrality with arbitrary $$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$3. PageRank follows by choosing $$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$4, where $$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$5 is the row-normalized adjacency and $$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$6 is the damping parameter, together with

$$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$7

yielding

$$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$8

the usual PageRank solution (Sornette et al., 12 Mar 2026).

These derivations locate Katz centrality and PageRank as mean-field limits of a more general event-driven model. The framework thereby clarifies both the validity and the limitations of classical indices. Their validity appears when the system can be summarized by stationary or short-memory expectations; their limitations appear when instantaneous rankings, shocks, or time-varying exogenous inputs materially affect importance.

4. Real-time computation and statistical estimation

Because the intensities satisfy the Hawkes recursion, they can be updated event by event in $$\lambda_i(t) = \mu_i(t) + \sum_{j=1}^M \int_{-\infty}^{t} n_{j,i}\,\phi(t-s)\,dN_j(s).$$9 time and tracked between events by simple exponential decay. If the last event occurred at time $\mu_i(t)$0 and no further events occur in $\mu_i(t)$1, then

$\mu_i(t)$2

When a new event of type $\mu_i(t)$3 arrives at time $\mu_i(t)$4, the intensity undergoes the jump

$\mu_i(t)$5

The per-event update therefore requires adding $\mu_i(t)$6 to the intensity vector, with cost proportional to the number of nonzero $\mu_i(t)$7. Between events, the intensities decay toward $\mu_i(t)$8, which may be evaluated pointwise or assumed piecewise constant. This gives a direct route to real-time ranking from streaming event data (Sornette et al., 12 Mar 2026).

Parameter estimation for $\mu_i(t)$9, $i$0, and $i$1 proceeds by maximum likelihood on the observed event streams $i$2. The log-likelihood is

$i$3

For fixed $i$4, this objective is concave in $i$5 and can be optimized by convex solvers or by an EM-type approach. Scalability to large $i$6 can be achieved by sparsity constraints on $i$7.

A common misconception is that a dynamic centrality of this kind must be computationally impractical in online settings. The update equations show the opposite: with exponential memory, the required state is low-dimensional, event-local, and directly maintainable.

5. Endogenous–exogenous decomposition, shocks, and empirical behavior

At every time $i$8, the total intensity decomposes exactly into an exogenous term and an endogenous term,

$i$9

with

$n_{j,i}$0

One may define the endogenous share

$n_{j,i}$1

so that $n_{j,i}$2 measures the fraction of node $n_{j,i}$3’s activity due to network-based amplification, while $n_{j,i}$4 measures purely exogenous or intrinsic influence. The paper characterizes this decomposition as exact, instantaneous, and fully interpretable (Sornette et al., 12 Mar 2026).

This decomposition is central to the framework’s treatment of shocks. Because $n_{j,i}$5 can vary arbitrarily, HawkesRank immediately adapts to sudden injections of exogenous activity. In simulation, a ten-fold shock to the smallest $n_{j,i}$6 at time $n_{j,i}$7 sharply alters the true ranking $n_{j,i}$8. Static benchmarks—Katz, eigenvector, and PageRank—show a drop in Spearman correlation with the ground-truth ordering, while HawkesRank’s dynamic $n_{j,i}$9 continues to track the new ordering perfectly.

The empirical illustration is a YouTube live-chat emotion analysis in which HawkesRank is fitted by maximum likelihood to the six basic emotions. The estimated branching matrix $j$0, which includes self-excitation plus strong anger↔disgust links, differs markedly from a correlation-based network. The resulting $j$1 trajectories exhibit multiple crossings that static centralities would miss. The exo–endo ratio $j$2 reveals when viewers’ emotional bursts are driven by the video, corresponding to high exogenous share, versus by contagion among participants, corresponding to high endogenous share.

These examples underscore the distinction between an intensity-based ranking and a static centrality computed on an average network. In HawkesRank, ranking is not only descriptive but also predictive, because it is directly tied to event intensities that govern future arrivals.

6. Related Hawkes-based ranking models

A related line of work uses network Hawkes process models to infer latent hierarchy in social animal interactions rather than to define node importance directly as instantaneous intensity. In that setting, each individual $j$3 is associated with a latent dominance score $j$4, and directed interaction processes $j$5 are modeled so that pair-specific Hawkes parameters are deterministic functions of $j$6 and a small number of global parameters (Ward et al., 2020).

That framework develops three models of increasing complexity: the Cohort Hawkes Process (C-HP), the Cohort Degree-Corrected Hawkes Process (C-DCHP), and the Cohort Markov-Modulated Hawkes Process (C-MMHP). In C-MMHP, each directed pair $j$7 carries an independent two-state continuous-time Markov chain $j$8, with $j$9 representing an inactive or sporadic phase and $i$0 an active or bursty phase. Baselines, Hawkes amplitudes, decay rate, and CTMC rates are tied to $i$1 through functions involving exponentials and logistic structure, so that the dominance ordering is built into the point-process dynamics.

Inference there is fully Bayesian and implemented via Hamiltonian Monte Carlo in Stan, with the discrete CTMC states analytically marginalized rather than sampled. Priors are specified as $i$2, $i$3, and $i$4. Diagnostics include the time-rescaling theorem with Kolmogorov–Smirnov tests and QQ-plots, Pearson residual heatmaps, simulation recovery summarized by root-mean-square error and Spearman $i$5, posterior predictive checks for event counts using Mean Absolute Error, and posterior $i$6 against an external Glicko ranking.

This suggests a broader family of Hawkes-based ranking methodologies. One formulation, exemplified by HawkesRank proper, identifies importance with instantaneous event intensities on nodes. Another formulation embeds latent ranks into pairwise Hawkes intensities and uses posterior inference to recover hierarchy. The two approaches share event-driven Hawkes structure, but they operationalize “ranking” differently: one as a time-resolved centrality score, the other as a latent ordering variable inferred from interaction dynamics.