Organizations: Institute of Risk Analysis, Prediction and Management, Southern University of Science and Technology, Shenzhen, China · Warwick Business School, University of Warwick, Coventry, UK
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 & tables
Figure 1: Temporal evolution of the event intensities λi(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 (μ1,μ2,μ3)=(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 λ1 , λ2 , and λ3 fluctuating throughout the simulation. By contrast, the first-moment Hawkes solution from Eq. ( 4a ) and the Katz centralities from Eq. ( 3 ) 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 {λi(t)}i=1M 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 μ10 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 λi(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.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
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 ni,j defined in equation ( 30 ). The left plot shows that a memory time τ of 2 is amplified to 4 for a spectral radius equal to 0.5 . The right plot shows that a memory time τ 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 ℓ . 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,ℓ) .
Link prediction models are increasingly used to recommend interactions in evolving networks, yet their impact on network structure is typically assessed from static snapshots. In particular, observed homophily conflates intrinsic interaction tendencies with amplification effects induced by network dynamics and algorithmic feedback. We propose a temporal framework based on multivariate Hawkes processes that disentangles these two sources and introduce an instantaneous bias measure derived from interaction intensities, capturing current reinforcement dynamics beyond cumulative metrics. We provide a theoretical characterization of the stability and convergence of the induced dynamics, and experiments show that the proposed measure reliably reflects algorithmic feedback effects across different link prediction strategies.
Dynamics of interacting systems in engineering, society, and nature often evolve over latent networks that govern which entities can interact. We study the problem of inferring these networks from event-based observations, which arise naturally in finance, seismology, and neuroscience. While there is substantial algorithmic work addressing this important problem, theoretical results are scarce. In this paper we ask the following fundamental question: what is the minimum time that one must observe the dynamics in order to exactly recover the underlying network, as a function of the number d of interacting entities? For a class of stationary Hawkes processes with sparse, weak interactions, we prove that an observation time of order logd is sufficient and necessary. For the upper bound we construct a two-stage estimator that uses clipped and binned event data for screening, followed by a least-squares refinement, and apply concentration bounds derived from the Poisson cluster representation. For the lower bound we combine Fano's inequality with Jacod's Girsanov formula for point processes on a suitable subclass of networks.
Jonas Linkerhägner, Michele Bortolasi, Lorenzo Baldassari +2
This paper develops a unified framework for measuring concentration in weighted systems embedded in networks of interactions. While traditional indices such as the Herfindahl-Hirschman Index capture dispersion in weights, they neglect the topology of relationships among the elements receiving those weights. To address this limitation, we introduce a family of topology-aware concentration indices that jointly account for weight distributions and network structure. At the core of the framework lies a baseline Network Concentration Index (NCI), defined as a normalized quadratic form that measures the fraction of potential weighted interconnection realized along observed network links. Building on this foundation, we construct a flexible class of extensions that modify either the interaction structure or the normalization benchmark, including weighted, density-adjusted, null-model, degree-constrained, transformed-data, and multi-layer variants. This family of indices preserves key properties such as normalization, invariance, and interpretability, while allowing concentration to be evaluated across different dimensions of dependence, including intensity, higher-order interactions, and extreme events. Theoretical results characterize the indices and establish their relationship with classical concentration and network measures. Empirical and simulation evidence demonstrate that systems with identical weight distributions may exhibit markedly different levels of structural concentration depending on network topology, highlighting the additional information captured by the proposed framework. The approach is broadly applicable to economic, financial, and complex systems in which weighted elements interact through networks.
L. Riso, M. G. Zoia
Department of Economic Policy, Universit`a Cattolica del Sacro Cuore, Milan, Italy