Decomposing Behavioral Variability in Email Communication: Self-Excitation, Latent State-Switching, and Their Interaction in the Enron Corpus

Decomposing Behavioral Variability in Email Communication: Self-Excitation, Latent State-Switching, and Their Interaction in the Enron Corpus | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article Decomposing Behavioral Variability in Email Communication: Self-Excitation, Latent State-Switching, and Their Interaction in the Enron Corpus Brhanu F. Znabu, Zohaib Atif, Pradeep Devkota This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9105293/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Human email communication exhibits substantial temporal variability that resists simple characterization. We treat individual email activity as a stochastic system and ask how much behavioral variability is explained by each of three generative mechanisms: circadian and weekly periodicity, self-excitation from incoming messages, and latent work–rest state-switching. Using the Enron email corpus (58 users, 101,299 sent emails, 1998–2002), we fit a hierarchy of five models—inhomogeneous Poisson, lognormal renewal, Hawkes process, Poisson hidden Markov model (HMM), and a hybrid HMM-on-Hawkes-residuals—evaluated via strict 70/30 temporal train/test splits. All three pre-registered hypotheses pass: 83% of users exhibit significant self-excitation (α > 0; mean branching ratio 0.15), 84.5% show greater than 10% log-likelihood improvement under state-switching, and 62% retain residual state-switching structure after Hawkes filtering. Latent state-switching provides the largest predictive gains, while self-excitation is real but modest, and the two mechanisms capture complementary sources of variability. All code and derived data (timestamps and anonymized identifiers only) are publicly available for full reproducibility. Physical sciences/Mathematics and computing Biological sciences/Neuroscience Biological sciences/Psychology Social science/Psychology email communication burstiness Hawkes process hidden Markov model variance decomposition Enron corpus Figures Figure 1 Figure 2 Figure 3 Figure 4 1. Introduction Human communication is inherently variable. Even for the same individual under nominally similar circumstances, the timing and rate of outgoing messages fluctuate from hour to hour and day to day. A productive lens for understanding this variability comes from stochastic systems: observable behavior (outgoing emails, reply times) is generated by external inputs (incoming messages, calendar-time covariates) filtered through hidden internal states (availability, engagement, workload) that are not directly observed [ 13 , 23 ]. Email logs, with their precise timestamps and high temporal resolution, provide an ideal behavioral trace for testing this systems perspective. A foundational finding in human dynamics is that inter-event times in communication and other activities are heavy-tailed rather than exponential, producing clustered bursts of activity interspersed with long quiescent periods [ 1 ]. This burstiness has been documented across email [ 1 , 2 ], letter correspondence [ 3 , 25 ], phone calls [ 11 ], and web browsing. Goh and Barabási [ 7 ] formalized the burstiness coefficient B and the memory coefficient M as summary statistics for temporal clustering and sequential correlation, providing a standardized characterization framework. Barabási [ 1 ] originally attributed burstiness to priority-queue decision-making, while Vázquez et al. [ 10 ] developed detailed queuing models reproducing the observed power-law-like tails. Malmgren et al. [ 2 ] then demonstrated that a cascading inhomogeneous Poisson process with circadian and weekly modulation could reproduce observed waiting-time distributions, reframing burstiness as an artifact of non-stationary rates rather than fundamentally non-Poisson dynamics [ 3 ]. Karsai et al. [ 12 ] subsequently identified universal features of correlated bursty behavior across multiple communication channels, suggesting common underlying mechanisms. This debate highlighted the importance of competing generative models for the same phenomenology. Two model families have emerged as leading candidates for explaining communication dynamics beyond simple periodicity. Self-exciting point processes, introduced by Hawkes [ 4 ] and situated within the broader theory of temporal point processes [ 16 ], model event clustering through a positive-feedback mechanism in which each event temporarily increases the probability of subsequent events. In email, an incoming message may trigger a cascade of replies, producing short-term bursts that decay over minutes to hours. This framework has been applied to social media dynamics [ 20 ] and network event data [ 19 ], demonstrating its utility for capturing self-excitation in human-generated event streams. Separately, hidden Markov models (HMMs) [ 5 , 18 ] capture regime-switching behavior by positing that an individual transitions between discrete latent states (e.g., a high-activity “work” state and a low-activity “rest” state), each with its own emission distribution. HMMs have proven effective for modeling time-series data with latent structure across diverse domains [ 17 ], including behavioral and communication data. Both frameworks have been applied to communication data, but typically in isolation. What is missing from the literature is a systematic, head-to-head decomposition of behavioral variability across these mechanisms within a unified evaluation framework. Existing studies tend to demonstrate one phenomenon, burstiness, circadian structure, or event clustering, without benchmarking against competing explanations using the same data and the same evaluation protocol [ 13 ]. This makes it difficult to assess the relative explanatory power of periodicity, self-excitation, and state-switching, or to determine whether they capture overlapping or complementary structure. Jo et al. [ 11 ] demonstrated the interplay between circadian patterns and burstiness in mobile phone data, but did not extend the analysis to a formal model comparison with self-exciting and switching alternatives. This paper addresses the gap by fitting a hierarchy of five generative models to individual email activity streams from the Enron corpus [ 6 ], a large public dataset of corporate email headers that has been extensively used for network and communication research [ 21 , 22 ]. Our models form an ordered sequence: (M1) inhomogeneous Poisson capturing circadian and weekly periodicity, (M2) lognormal renewal capturing heavy-tailed inter-event times [ 8 , 16 ], (M3) Hawkes process capturing self-excitation [ 4 ], (M4) Poisson HMM capturing latent state-switching [ 5 , 17 ], and (M5) a hybrid model applying an HMM to Hawkes residuals [ 14 ] to test for complementary structure. We evaluate all models via strict 70/30 temporal train/test splits with held-out log-likelihood as the primary metric and test three pre-registered, falsifiable hypotheses concerning self-excitation (H1), state-switching improvement (H2), and their complementarity (H3). We additionally provide a supplementary analysis of reply-time distributions constructed via heuristic pairing, since native threading headers were available for fewer than 0.01% of messages in the corpus. The remainder of this paper is organized as follows. Section 2 describes the data source, cohort selection, reply-pair construction, model specifications, and evaluation protocol. Section 3 presents results for burstiness characterization, each hypothesis test, and the supplementary reply-time analysis. Section 4 discusses the implications of our variance decomposition, limitations, and directions for future work. 2. Data and Methods 2.1 Data Source and Ethics The Enron email corpus, released for research use and distributed by Carnegie Mellon University [ 6 ], comprises 517,401 emails across 150 employee mailboxes spanning October 1998 to June 2002. The corpus has been widely studied for network structure and communication patterns [ 21 , 22 ]. We restricted our analysis exclusively to email headers (From, To, Cc, Date, Message-ID, and In-Reply-To/References fields where available) and did not access or analyze message bodies at any stage of the pipeline. All analyses report aggregated statistics and use anonymized user identifiers. This header-only approach minimizes exposure to sensitive content while preserving the temporal structure needed for stochastic modeling. 2.2 Event Extraction, Deduplication, and Cohort Selection We parsed all 517,401 email headers to extract sender, recipient lists, and timestamps, yielding 516,359 unique records after deduplication on the tuple (Message-ID, sender, timestamp). Timezone standardization was performed using per-user activity-minimum detection: for each user, we identified the hour of day with the lowest sending frequency, assumed to correspond to approximately 3–5 AM local time, and computed the implied UTC offset, following the approach of Malmgren et al. [ 3 ]. This method was applied to 172 users; the most common inferred offset was UTC + 4 (consistent with U.S. Eastern time plus the Enron Houston headquarters operating in Central time). Cohort selection required at least 500 outgoing emails and a mean sending density of at least 1 email per day during each user’s active period (first to last sent email). This yielded a cohort of 58 users with a combined 101,299 sent emails and 284,495 incoming emails (Fig. 1 A, 1 B). Activity levels ranged from 1.1 to 15.5 emails per day, with the most active user (kay.mann) sending 8,926 emails across her active window (Fig. 1 C). 2.3 Reply-Pair Construction Constructing reply pairs from the Enron corpus presented a significant data challenge. Examination of threading metadata revealed that only 64 of 517,401 emails (0.01%) contained valid In-Reply-To or References headers, far below the 40% threshold specified in our pre-registered contingency plan. Accordingly, we activated the contingency: sending-activity analysis became the primary focus of the paper, and reply-time analysis was designated as supplementary material based on heuristic pairing. The heuristic pairing procedure operated as follows. First, email subjects were normalized by stripping “Re:” and “Fwd:” prefixes and converting to lowercase. A reply pair was then defined as a case in which user A received an email from user B on a given subject, and user A subsequently sent an email to user B on the same normalized subject within a specified time window. Using a primary window of 1 day, this procedure yielded 16,600 reply pairs. A precision audit of 200 randomly sampled pairs found that 81.0% passed all quality checks (subject length greater than 8 characters and no self-replies), exceeding the pre-registered 70% precision threshold (Fig. 1 D). Sensitivity analysis across 1-day, 3-day, and 7-day windows showed moderate growth in pair counts (16,600 to 20,876) with increasing median reply time (1.33 to 2.58 hours), indicating that the 1-day window captures the majority of genuine reply behavior. 2.4 Model Hierarchy We fit five models to each user’s outgoing email stream, forming a hierarchy that progressively adds explanatory structure [ 16 ]. All models were fit on the training partition and evaluated on the held-out test set. Model 1 (Inhomogeneous Poisson): The baseline model captures circadian and weekly periodicity through a log-linear intensity function λ(t) = exp(f(hour, dow)), where f is parameterized with 24 hour-of-day and 7 day-of-week indicator variables. This model treats events as conditionally independent given the time-varying rate, providing a natural null hypothesis for temporal structure beyond periodicity [ 2 ]. Model 2 (Lognormal Renewal): Inter-event times are modeled as independent draws from a lognormal distribution, IET ∼ Lognorm(µ, σ) [ 8 , 16 ]. This captures the heavy-tailed waiting-time distribution characteristic of bursty behavior but includes no temporal covariates and no event-to-event excitation. The lognormal renewal process serves as a within-family comparison for the Hawkes model (both operate on continuous event times rather than binned counts). Model 3 (Hawkes Process): A self-exciting point process [ 4 ] with conditional intensity λ(t) = µ(t) + Σ α·exp(−β·(t − t i )), where the sum runs over all prior events t i < t, α controls excitation magnitude, and β controls decay rate. The background rate µ(t) incorporates the same circadian covariates as M1. We fit a single-exponential kernel (the planned multi-scale K = 2 kernel did not converge for the majority of users, triggering the pre-registered fallback). The branching ratio α/β quantifies the expected number of triggered events per parent event; values below 1.0 indicate a subcritical (stationary) process [ 16 ]. The Hawkes model was successfully fit to 53 of 58 cohort users. Model 4 (Poisson HMM): Hourly email counts cₜ are modeled as emissions from a hidden Markov model with K latent states [ 5 , 17 ], where the emission distribution in each state is Poisson: cₜ | xₜ ∼ Poisson(λₓ). State transitions are governed by a K × K transition matrix estimated via the Baum–Welch algorithm [ 18 ]. We fit models with K = 2 and K = 3 states and selected the best K by BIC [ 15 ] on the training data. The original proposal specified Negative Binomial emissions as the primary specification, but implementation constraints (hmmlearn supports only Poisson emissions natively) led us to use Poisson emissions with K = 2–3 states as the primary analysis. The HMM captures latent work–rest regime switching: for example, user kay.mann exhibited a work state with λ = 5.15 emails per hour and a rest state with λ ≈ 0 emails per hour, with transition probabilities P(Rest → Rest) = 0.95 and P(Work → Work) = 0.63. Model 5 (Hybrid: HMM on Hawkes Residuals): To test whether self-excitation and state-switching capture complementary sources of variability, we computed time-rescaled residuals from the fitted Hawkes model (M3) following the residual analysis framework of Ogata [ 14 ] and then fit an HMM to the residual process. If the Hawkes model fully accounts for latent state structure, the residuals should show no preference for multiple states (K = 1 preferred by BIC [ 15 ]). Conversely, if K = 2 is preferred on the residuals, self-excitation and state-switching address different aspects of variability, supporting hypothesis H3. 2.5 Evaluation Protocol Evaluation followed a strict temporal protocol. For each user, the first 70% of their email timeline was designated as the training set and the final 30% as the held-out test set, yielding 70,887 training events and 30,412 test events across the cohort. No information from the test period was used during model fitting. The primary evaluation metric was held-out log-likelihood per event; secondary metrics included BIC [ 15 ] for model selection within families and Brier score for short-horizon prediction calibration. We note an important methodological caveat: Models M1 and M4 operate on binned hourly counts and produce log-likelihoods in count-space units, while Models M2 and M3 operate on continuous event times and produce log-likelihoods in continuous-time units [ 16 ]. Direct comparison of held-out log-likelihood is therefore valid within model families (M1 vs. M4 for count-based models; M2 vs. M3 for point-process models) but not across families without normalization. Bootstrap confidence intervals (50 replicates with 7-day blocks) [ 24 ] were computed for burstiness metrics; full 500-replicate bootstrap on model comparisons was planned but not completed due to computational constraints. 2.6 Falsifiable Hypotheses We pre-registered three falsifiable hypotheses with explicit thresholds. H1 (Self-Excitation): The Hawkes excitation parameter α will be significantly greater than zero for at least 80% of users meeting activity thresholds, indicating that incoming emails systematically increase near-future sending [ 4 ]. H2 (State-Switching): The Poisson HMM will improve held-out log-likelihood by at least 10% over the inhomogeneous Poisson baseline for a majority of users. H3 (Complementarity): Hawkes residuals [ 14 ] will retain significant state-switching structure (K = 2 HMM preferred over K = 1 by BIC [ 15 ] for more than 50% of users), indicating that self-excitation and latent states capture different sources of variability. 3. Results 3.1 Burstiness and Circadian Structure All 58 cohort users exhibited bursty email behavior, with burstiness coefficients uniformly positive (mean B = 0.601, median B = 0.609, range 0.324–0.909; Fig. 2 B). Bootstrap 95% confidence intervals excluded zero for every user, confirming that burstiness is a universal feature of this cohort rather than a property of a few outliers [ 7 ]. The memory coefficient M, measuring the first-order autocorrelation of consecutive inter-event times [ 7 ], was weakly positive on average (mean M = 0.079, median M = 0.021), with 37 of 58 users showing M > 0, indicating modest positive memory in the temporal structure of sending behavior. A Kolmogorov–Smirnov test of inter-event times against the exponential distribution yielded D = 0.69 (p < 10⁻⁶), decisively rejecting the memoryless Poisson assumption (Fig. 2 A). The complementary cumulative distribution function (CCDF) of inter-event times showed pronounced heavy-tailed behavior across all users, consistent with prior findings in email [ 1 , 2 ] and other communication channels [ 11 , 12 ]. The aggregate CCDF departed from the exponential reference by orders of magnitude at long time scales. Circadian structure was strong and consistent: the hour-by-day-of-week heatmap (Fig. 2 C) revealed concentrated activity during Monday–Friday business hours (approximately 8 AM to 8 PM) with sharply reduced activity on weekends and overnight, mirroring the patterns reported by Malmgren et al. [ 2 , 3 ]. The Fano factor (variance-to-mean ratio of counts) exceeded 1.0 at every time scale examined, from 5 minutes to 24 hours, and increased monotonically with bin size (Fig. 2 D), confirming super-Poisson dispersion at all temporal resolutions. 3.2 Self-Excitation (H1) Hypothesis H1 predicted that at least 80% of users would exhibit significant self-excitation (α > 0) in the Hawkes model [ 4 ]. Among the 53 users for whom the Hawkes model converged, 44 (83%) had excitation parameters significantly greater than zero, exceeding the pre-registered 80% threshold and confirming H1 (Fig. 4 A). The mean branching ratio across fitted users was 0.146, indicating that each email event triggers, on average, an additional 0.15 events—a modest but consistent self-excitation effect well within the subcritical regime (α/β < 1) [ 16 ]. This finding supports the interpretation that incoming emails systematically increase near-future sending, consistent with reply-chain and conversational-arc dynamics observed in other communication systems [ 19 , 20 ]. 3.3 State-Switching (H2) Hypothesis H2 predicted that the Poisson HMM [ 5 , 17 ] would improve held-out log-likelihood by at least 10% over the inhomogeneous Poisson baseline for a majority of users. The HMM achieved a mean held-out log-likelihood of − 2.38 per event compared to − 3.82 for the Poisson baseline, and 84.5% of users showed improvement exceeding the 10% threshold, confirming H2 (Fig. 4 A). BIC-based model selection [ 15 ] favored K = 3 states for 35 users and K = 2 states for 23 users, suggesting that most individuals are best characterized by a three-regime model (e.g., high-activity work, low-activity work, and rest) rather than a simple binary work–rest dichotomy. For the illustrative user kay.mann, the inferred state sequence (Fig. 4 B) showed clear alternation between a work state (λ = 5.15 emails/hour) and a rest state (λ ≈ 0), with a highly persistent rest state (P(Rest → Rest) = 0.95) and a less persistent but well-defined work state (P(Work → Work) = 0.63). 3.4 Complementarity (H3) Hypothesis H3 tested whether self-excitation and state-switching capture complementary variability by examining Hawkes residuals [ 14 ] for residual state-switching structure. Among the 53 users with valid Hawkes fits, 33 (62%) showed a preference for K = 2 over K = 1 in the residual HMM by BIC [ 15 ], exceeding the pre-registered 50% threshold and confirming H3 (Fig. 4 C). This result indicates that even after accounting for short-term self-excitation, a substantial majority of users retain latent state structure in their residual activity. The complementarity finding implies that self-excitation (capturing event-triggered reply cascades on the scale of minutes to hours) and state-switching (capturing longer-duration regime transitions between work and rest) address different aspects of behavioral variability and that hybrid models combining both mechanisms are warranted. 3.5 Reply-Time Analysis (Supplementary) As a supplementary analysis, we examined the distribution of reply times constructed via heuristic pairing (1-day window, 16,600 pairs, 81% precision). Reply times were heavy-tailed, with a Kolmogorov–Smirnov test rejecting the exponential distribution (D = 0.25, p < 10⁻⁶), and 95% of individual users also rejected exponentiality (Fig. 3 A). The median reply time was 1.33 hours with a mean of 4.67 hours, reflecting the long right tail characteristic of communication response times [ 1 , 9 , 25 ]. Reply times showed strong circadian modulation (Fig. 3 B): median reply times were shortest during business hours (approximately 1–2 hours) and longest for emails received overnight or early morning (up to 13–14 hours). Sensitivity analysis across pairing windows (Fig. 3 C) confirmed that the 1-day window captured the core of reply behavior, with wider windows adding relatively few pairs but substantially increasing mean reply time. These supplementary results are consistent with the primary finding that email behavior is governed by circadian modulation and latent state transitions, but should be interpreted with caution given the heuristic nature of the pairing procedure. 4. Discussion The central finding of this study is that behavioral variability in email communication is multi-mechanistic: circadian periodicity, self-excitation, and latent state-switching each contribute distinct, partially non-overlapping predictive information. Our five-model hierarchy provides a quantitative decomposition of these contributions, extending beyond the descriptive burstiness characterizations prevalent in the literature [ 1 , 7 , 10 , 12 ]. Latent state-switching (M4, Poisson HMM) provided the largest improvement over the periodicity-only baseline, reducing held-out log-likelihood from − 3.82 to − 2.38 per event, with 84.5% of users exceeding the 10% improvement threshold. This suggests that discrete regime transitions—between states interpretable as work and rest—are the primary driver of predictive gains beyond circadian structure, consistent with the cascading non-homogeneous Poisson framework proposed by Malmgren et al. [ 2 , 3 ] but offering a more mechanistic explanation through explicit latent states [ 17 ]. Self-excitation, captured by the Hawkes process (M3) [ 4 ], is statistically significant for 83% of users but contributes modestly to the overall variance budget, with a mean branching ratio of 0.15. This means that for every 100 emails, approximately 15 are attributable to self-excitation cascades triggered by prior events. While this effect is consistent and robust across users, it is small relative to the gains achieved by state-switching, suggesting that the dominant source of variability is not event-triggered cascading but rather transitions between sustained behavioral regimes. This finding nuances the emphasis on self-excitation in recent social dynamics literature [ 19 , 20 ] by showing that for individual email behavior, regime-switching explains more variance than event-triggered cascading. The complementarity result (H3) is perhaps the most theoretically consequential finding. The fact that 62% of users retain state-switching structure in their Hawkes residuals [ 14 ] demonstrates that self-excitation and regime-switching are not redundant explanations for the same variability. Rather, they operate at different temporal scales and capture different generative mechanisms: self-excitation models the local clustering of events within conversational episodes, while state-switching captures the global alternation between periods of high and low engagement [ 13 ]. This motivates the development of hybrid models that integrate both mechanisms, potentially through state-dependent Hawkes processes in which excitation parameters vary by latent state, extending the latent network structure approach of Linderman and Adams [ 19 ] from spatial to temporal hidden variables. Several limitations should be noted. First, the near-complete absence of threading headers in the Enron corpus (0.01% coverage) forced our reply-time analysis to rely on heuristic subject-based pairing, which we accordingly report as supplementary rather than primary. Second, the Hawkes model used a single-exponential kernel; the planned multi-scale kernel (K = 2 exponential components) did not converge for the majority of users, potentially limiting the model’s ability to capture both rapid reply-chain dynamics and slower conversational arcs simultaneously [ 16 ]. Third, the HMM employed Poisson emissions rather than the originally planned Negative Binomial specification, which may cause the model to inflate the number of latent states to compensate for overdispersion [ 17 ]. Fourth, the Enron corpus reflects communication patterns within a single corporate organization during a specific historical period [ 6 , 21 , 22 ], and the generalizability of our findings to personal email, instant messaging, or contemporary corporate communication remains an open question. Finally, direct comparison of log-likelihoods across model families (count-based M1/M4 versus continuous-time M2/M3) requires normalization that we do not perform [ 16 ]; our variance decomposition is therefore best interpreted within model families. Future work should address these limitations by fitting multi-scale Hawkes kernels with regularization to promote convergence, implementing Negative Binomial HMM emissions to handle overdispersion without state inflation [ 17 ], and extending the analysis to dyadic (sender–recipient pair) models that capture relational dynamics [ 13 , 23 ]. Applying the same five-model hierarchy to other communication corpora—including instant messaging platforms, the Copenhagen Networks Study multi-layer dataset, and personal email collections such as those analyzed by Eckmann et al. [ 9 ] and Oliveira and Barabási [ 25 ]—would test the generalizability of our variance decomposition across communication modalities and social contexts. Abbreviations IET Inter-event time HMM Hidden Markov model BIC Bayesian information criterion KS Kolmogorov–Smirnov CCDF Complementary cumulative distribution function LL Log-likelihood BR Branching ratio APC Article processing charge Declarations Ethics Statement This study analyzed only email header metadata (sender, recipients, timestamps, and subject lines for reply-pair matching); no message bodies were accessed at any stage. All users are identified by anonymized identifiers, and results are reported as aggregated statistics across the cohort. The Enron email corpus is a publicly released dataset distributed by Carnegie Mellon University for research purposes [6]. Data Availability All analysis code is available at https://github.com/brhanufen/enron-behavioral-variability with reproducing all tables and figures. The Enron email corpus is freely available from Carnegie Mellon University [6]. Derived datasets contain only timestamps and anonymized user identifiers; no message content is included. Ethics approval and consent to participate Not applicable. This study analyzed only publicly available email header metadata from the Enron corpus (Carnegie Mellon University release). No human subjects were involved. Consent for publication Not applicable. Competing interests The authors declare that they have no competing interests. Funding This research received no external funding. Authors’ contributions B.F.Z. conceived and designed the study, performed all data processing and statistical analyses, developed the computational pipeline, and wrote the original draft of the manuscript. Z.A. contributed to the study design and methodology. P.D. contributed to manuscript writing and editing. All authors read and approved the final manuscript. Acknowledgements The authors thank Tayachew for valuable encouragement and for offering perspective from a signal analysis viewpoint during the course of this work. References Barabási, A. L. The origin of bursts and heavy tails in human dynamics. Nature 435 , 207–211 (2005). Malmgren, R. D., Stouffer, D. B., Motter, A. E. & Amaral, L. A. N. A Poissonian explanation for heavy tails in e-mail communication. Proc. Natl. Acad. Sci. USA . 105 (47), 18153–18158 (2008). Malmgren, R. D., Stouffer, D. B., Campanharo, A. S. L. O. & Amaral, L. A. N. On universality in human correspondence activity. Science 325 (5948), 1696–1700 (2009). Hawkes, A. G. Spectra of some self-exciting and mutually exciting point processes. Biometrika 58 (1), 83–90 (1971). Rabiner, L. R. A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE . 77 (2), 257–286 (1989). Klimt, B. & Yang, Y. The Enron corpus: A new dataset for email classification research. In: ECML 2004. Springer; (2004). Goh, K. I. & Barabási, A. L. Burstiness and memory in complex systems. Europhys. Lett. 81 , 48002 (2008). Cox, D. R. Regression models and life-tables. J. R Stat. Soc. B . 34 (2), 187–220 (1972). Eckmann, J. P., Moses, E. & Sergi, D. Entropy of dialogues creates coherent structures in e-mail traffic. Proc. Natl. Acad. Sci. USA . 101 (40), 14333–14337 (2004). Vázquez, A. et al. Modeling bursts and heavy tails in human dynamics. Phys. Rev. E . 73 , 036127 (2006). Jo, H. H., Karsai, M., Kertész, J. & Kaski, K. Circadian pattern and burstiness in mobile phone communication. New. J. Phys. 14 , 013055 (2012). Karsai, M., Kaski, K., Barabási, A. L. & Kertész, J. Universal features of correlated bursty behaviour. Sci. Rep. 2 , 397 (2012). Holme, P. & Saramäki, J. Temporal networks. Phys. Rep. 519 (3), 97–125 (2012). Ogata, Y. Statistical models for earthquake occurrences and residual analysis for point processes. J. Am. Stat. Assoc. 83 (401), 9–27 (1988). Schwarz, G. Estimating the dimension of a model. Ann. Stat. 6 (2), 461–464 (1978). Daley, D. J. & Vere-Jones, D. An Introduction to the Theory of Point Processes 2nd edn, Vol. 1 (Springer, 2003). Zucchini, W., MacDonald, I. L. & Langrock, R. Hidden Markov Models for Time Series: An Introduction Using R 2nd edn (CRC, 2016). Baum, L. E., Petrie, T., Soules, G. & Weiss, N. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Stat. 41 (1), 164–171 (1970). Linderman, S. & Adams, R. Discovering latent network structure in point process data. In: Proc ICML. (2014). Zhao, Q., Erdogdu, M. A., He, H. Y., Rajaraman, A. & Leskovec, J. SEISMIC: A self-exciting point process model for predicting tweet popularity. In: Proc. KDD (2015). Diesner, J., Frantz, T. L. & Carley, K. M. Communication networks from the Enron email corpus It’s always about the people. Enron is no different. Comput. Math. Organ. Theory . 11 (3), 201–228 (2005). Shetty, J. & Adibi, J. The Enron email dataset database schema and brief statistical report. Information Sciences Institute Technical Report, University of Southern California; (2004). Masuda, N. & Lambiotte, R. A Guide to Temporal Networks (World Scientific, 2016). Efron, B. & Tibshirani, R. J. An Introduction to the Bootstrap (Chapman & Hall/CRC, 1993). Oliveira, J. G. & Barabási, A. L. Human dynamics: Darwin and Einstein correspondence patterns. Nature 437 , 1251 (2005). Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9105293","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":607149691,"identity":"802e7dc5-5bfb-4974-b1f1-42ce51d9e868","order_by":0,"name":"Brhanu F. Znabu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/UlEQVRIie3RMUsDMRTA8UjgTYFbX7BYP8KFQFQ88KvcUahLqINLh3KkHDgV5xuKn0HwC0QCuaXuB+dSBCfnw8Wi51xC3RzyH97weL/pERKL/cOSsWs+kS1OEkJgWKTDwBDhNeR4PvKSm0NJ2rJTnGe0SO2hhHSrH6Yhl13l3+aLUiaGPncsII7WL7aoN6OZevXXYuOdQguTyxChZJY7voJb1WrFDdiMWKaOQwSIFtXXjhZP9U3Pza7Mxjbpg4ShlgQZLR5RA1/eUZVaBkGCzE+HIbGdnonlvZPCgbxYB8hVU3ny+8p68r41fSkemmrbfgTInujfzmOxWCy2p2+Ux0uq2cPQmQAAAABJRU5ErkJggg==","orcid":"","institution":"University of Nebraska–Lincoln","correspondingAuthor":true,"prefix":"","firstName":"Brhanu","middleName":"F.","lastName":"Znabu","suffix":""},{"id":607149696,"identity":"7f3aeb77-da07-4364-9429-c46b18b2797b","order_by":1,"name":"Zohaib Atif","email":"","orcid":"","institution":"Gwangju Institute of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Zohaib","middleName":"","lastName":"Atif","suffix":""},{"id":607149705,"identity":"f33f7fee-daba-427a-b540-d037d93d3916","order_by":2,"name":"Pradeep Devkota","email":"","orcid":"","institution":"University of Kansas Medical Center","correspondingAuthor":false,"prefix":"","firstName":"Pradeep","middleName":"","lastName":"Devkota","suffix":""}],"badges":[],"createdAt":"2026-03-12 13:23:51","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9105293/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9105293/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":104777853,"identity":"ddea02ec-aede-48fb-9c12-f191dc6ddb84","added_by":"auto","created_at":"2026-03-17 07:13:59","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":98792,"visible":true,"origin":"","legend":"\u003cp\u003eDataset and pipeline overview. (A) Processing pipeline from raw Enron corpus (517,401 emails) through header parsing, deduplication, timezone inference, and cohort selection (58 users). (B) Distribution of outgoing emails per user (log scale) with 500-email cohort threshold marked. (C) Event rasters for three representative users spanning one week: Kay Mann (389 emails/week, high activity), Phillip Allen (76/week, medium), and Dutch Quigley (57/week, low). (D) Heuristic reply-pair construction schematic showing subject normalization, time-window matching, and precision audit results (81% pass rate, n = 200).\u003c/p\u003e","description":"","filename":"fig1.png","url":"https://assets-eu.researchsquare.com/files/rs-9105293/v1/573da833dd3455f200a48425.png"},{"id":104783346,"identity":"91e9bdcb-26c5-4cd4-b61a-29e8bd0dc297","added_by":"auto","created_at":"2026-03-17 07:58:43","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":89220,"visible":true,"origin":"","legend":"\u003cp\u003eBurstiness and circadian structure. (A) Complementary cumulative distribution function (CCDF) of inter-event times for the aggregate cohort and three example users, versus exponential reference; KS test D = 0.69, p \u0026lt; 10⁻⁶. (B) Burstiness coefficient B and memory coefficient M for all 58 users sorted by B, with 95% bootstrap confidence intervals; all users have B \u0026gt; 0. (C) Hour-of-day × day-of-week heatmap showing mean email activity rate, with concentrated weekday business-hour activity. (D) Fano factor (variance/mean) at time scales from 5 minutes to 24 hours; all values exceed 1.0 (Poisson reference), indicating super-Poisson dispersion at every scale.\u003c/p\u003e","description":"","filename":"fig2.png","url":"https://assets-eu.researchsquare.com/files/rs-9105293/v1/1f2dcb457fce79bc4f96bc51.png"},{"id":104777855,"identity":"a65f2121-0fff-4fc5-8262-e2473b4ea5b4","added_by":"auto","created_at":"2026-03-17 07:13:59","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":73021,"visible":true,"origin":"","legend":"\u003cp\u003eReply-time distributions. (A) CCDF of reply times for all pairs (n = 16,600) and by user speed quartiles, with KS test annotation (D = 0.25, p \u0026lt; 10⁻⁶); 95% of users reject exponentiality. (B) Median reply time by hour-of-day with interquartile range; business hours highlighted, showing shortest reply times during working hours. (C) Sensitivity of median and mean reply time to pairing window (1 day, 3 days, 7 days).\u003c/p\u003e","description":"","filename":"fig3.png","url":"https://assets-eu.researchsquare.com/files/rs-9105293/v1/475d67d0bfdc4ae56b661f7c.png"},{"id":104777852,"identity":"c9c1b376-a5fb-4a22-ad7f-dbcf352228d8","added_by":"auto","created_at":"2026-03-17 07:13:58","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":80420,"visible":true,"origin":"","legend":"\u003cp\u003eModel comparison and latent states. (A) Δ log-likelihood per event: Hawkes versus inhomogeneous Poisson (left, median = −3.4) and HMM versus inhomogeneous Poisson (right, median = 1.3), with H1 and H2 pass annotations. Note that Hawkes (continuous-time) and HMM (count-based) log-likelihoods are not directly comparable across families. (B) Inferred latent-state sequence for kay.mann over the first two weeks, showing alternation between a work state (λ = 5.15 emails/hour) and a rest state (λ ≈ 0), with transition matrix inset. (C) Horizontal bar chart of ΔBIC(K = 1 − K = 2) for Hawkes residuals across all 53 fitted users; 33/53 (62%) prefer K = 2, confirming H3.\u003c/p\u003e","description":"","filename":"fig4.png","url":"https://assets-eu.researchsquare.com/files/rs-9105293/v1/68cac59a1d52e9d0afb90c6c.png"},{"id":104785022,"identity":"d6a2e8ed-6b2a-4d35-9805-7f8b0871a078","added_by":"auto","created_at":"2026-03-17 08:09:18","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":946909,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9105293/v1/b051c113-8f74-479e-9d65-8b1a363415a2.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Decomposing Behavioral Variability in Email Communication: Self-Excitation, Latent State-Switching, and Their Interaction in the Enron Corpus","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eHuman communication is inherently variable. Even for the same individual under nominally similar circumstances, the timing and rate of outgoing messages fluctuate from hour to hour and day to day. A productive lens for understanding this variability comes from stochastic systems: observable behavior (outgoing emails, reply times) is generated by external inputs (incoming messages, calendar-time covariates) filtered through hidden internal states (availability, engagement, workload) that are not directly observed [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. Email logs, with their precise timestamps and high temporal resolution, provide an ideal behavioral trace for testing this systems perspective.\u003c/p\u003e \u003cp\u003eA foundational finding in human dynamics is that inter-event times in communication and other activities are heavy-tailed rather than exponential, producing clustered bursts of activity interspersed with long quiescent periods [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. This burstiness has been documented across email [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], letter correspondence [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e], phone calls [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e], and web browsing. Goh and Barab\u0026aacute;si [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] formalized the burstiness coefficient B and the memory coefficient M as summary statistics for temporal clustering and sequential correlation, providing a standardized characterization framework. Barab\u0026aacute;si [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e] originally attributed burstiness to priority-queue decision-making, while V\u0026aacute;zquez et al. [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e] developed detailed queuing models reproducing the observed power-law-like tails. Malmgren et al. [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e] then demonstrated that a cascading inhomogeneous Poisson process with circadian and weekly modulation could reproduce observed waiting-time distributions, reframing burstiness as an artifact of non-stationary rates rather than fundamentally non-Poisson dynamics [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Karsai et al. [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e] subsequently identified universal features of correlated bursty behavior across multiple communication channels, suggesting common underlying mechanisms. This debate highlighted the importance of competing generative models for the same phenomenology.\u003c/p\u003e \u003cp\u003eTwo model families have emerged as leading candidates for explaining communication dynamics beyond simple periodicity. Self-exciting point processes, introduced by Hawkes [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] and situated within the broader theory of temporal point processes [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e], model event clustering through a positive-feedback mechanism in which each event temporarily increases the probability of subsequent events. In email, an incoming message may trigger a cascade of replies, producing short-term bursts that decay over minutes to hours. This framework has been applied to social media dynamics [\u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] and network event data [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e], demonstrating its utility for capturing self-excitation in human-generated event streams. Separately, hidden Markov models (HMMs) [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e] capture regime-switching behavior by positing that an individual transitions between discrete latent states (e.g., a high-activity \u0026ldquo;work\u0026rdquo; state and a low-activity \u0026ldquo;rest\u0026rdquo; state), each with its own emission distribution. HMMs have proven effective for modeling time-series data with latent structure across diverse domains [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], including behavioral and communication data. Both frameworks have been applied to communication data, but typically in isolation.\u003c/p\u003e \u003cp\u003eWhat is missing from the literature is a systematic, head-to-head decomposition of behavioral variability across these mechanisms within a unified evaluation framework. Existing studies tend to demonstrate one phenomenon, burstiness, circadian structure, or event clustering, without benchmarking against competing explanations using the same data and the same evaluation protocol [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. This makes it difficult to assess the relative explanatory power of periodicity, self-excitation, and state-switching, or to determine whether they capture overlapping or complementary structure. Jo et al. [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] demonstrated the interplay between circadian patterns and burstiness in mobile phone data, but did not extend the analysis to a formal model comparison with self-exciting and switching alternatives.\u003c/p\u003e \u003cp\u003eThis paper addresses the gap by fitting a hierarchy of five generative models to individual email activity streams from the Enron corpus [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], a large public dataset of corporate email headers that has been extensively used for network and communication research [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. Our models form an ordered sequence: (M1) inhomogeneous Poisson capturing circadian and weekly periodicity, (M2) lognormal renewal capturing heavy-tailed inter-event times [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e], (M3) Hawkes process capturing self-excitation [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], (M4) Poisson HMM capturing latent state-switching [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], and (M5) a hybrid model applying an HMM to Hawkes residuals [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] to test for complementary structure. We evaluate all models via strict 70/30 temporal train/test splits with held-out log-likelihood as the primary metric and test three pre-registered, falsifiable hypotheses concerning self-excitation (H1), state-switching improvement (H2), and their complementarity (H3). We additionally provide a supplementary analysis of reply-time distributions constructed via heuristic pairing, since native threading headers were available for fewer than 0.01% of messages in the corpus.\u003c/p\u003e \u003cp\u003eThe remainder of this paper is organized as follows. Section \u003cspan refid=\"Sec2\" class=\"InternalRef\"\u003e2\u003c/span\u003e describes the data source, cohort selection, reply-pair construction, model specifications, and evaluation protocol. Section \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e3\u003c/span\u003e presents results for burstiness characterization, each hypothesis test, and the supplementary reply-time analysis. Section \u003cspan refid=\"Sec15\" class=\"InternalRef\"\u003e4\u003c/span\u003e discusses the implications of our variance decomposition, limitations, and directions for future work.\u003c/p\u003e"},{"header":"2. Data and Methods","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Data Source and Ethics\u003c/h2\u003e \u003cp\u003eThe Enron email corpus, released for research use and distributed by Carnegie Mellon University [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e], comprises 517,401 emails across 150 employee mailboxes spanning October 1998 to June 2002. The corpus has been widely studied for network structure and communication patterns [\u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e]. We restricted our analysis exclusively to email headers (From, To, Cc, Date, Message-ID, and In-Reply-To/References fields where available) and did not access or analyze message bodies at any stage of the pipeline. All analyses report aggregated statistics and use anonymized user identifiers. This header-only approach minimizes exposure to sensitive content while preserving the temporal structure needed for stochastic modeling.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Event Extraction, Deduplication, and Cohort Selection\u003c/h2\u003e \u003cp\u003eWe parsed all 517,401 email headers to extract sender, recipient lists, and timestamps, yielding 516,359 unique records after deduplication on the tuple (Message-ID, sender, timestamp). Timezone standardization was performed using per-user activity-minimum detection: for each user, we identified the hour of day with the lowest sending frequency, assumed to correspond to approximately 3\u0026ndash;5 AM local time, and computed the implied UTC offset, following the approach of Malmgren et al. [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. This method was applied to 172 users; the most common inferred offset was UTC\u0026thinsp;+\u0026thinsp;4 (consistent with U.S. Eastern time plus the Enron Houston headquarters operating in Central time). Cohort selection required at least 500 outgoing emails and a mean sending density of at least 1 email per day during each user\u0026rsquo;s active period (first to last sent email). This yielded a cohort of 58 users with a combined 101,299 sent emails and 284,495 incoming emails (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eA, \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eB). Activity levels ranged from 1.1 to 15.5 emails per day, with the most active user (kay.mann) sending 8,926 emails across her active window (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eC).\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3 Reply-Pair Construction\u003c/h2\u003e \u003cp\u003eConstructing reply pairs from the Enron corpus presented a significant data challenge. Examination of threading metadata revealed that only 64 of 517,401 emails (0.01%) contained valid In-Reply-To or References headers, far below the 40% threshold specified in our pre-registered contingency plan. Accordingly, we activated the contingency: sending-activity analysis became the primary focus of the paper, and reply-time analysis was designated as supplementary material based on heuristic pairing.\u003c/p\u003e \u003cp\u003eThe heuristic pairing procedure operated as follows. First, email subjects were normalized by stripping \u0026ldquo;Re:\u0026rdquo; and \u0026ldquo;Fwd:\u0026rdquo; prefixes and converting to lowercase. A reply pair was then defined as a case in which user A received an email from user B on a given subject, and user A subsequently sent an email to user B on the same normalized subject within a specified time window. Using a primary window of 1 day, this procedure yielded 16,600 reply pairs. A precision audit of 200 randomly sampled pairs found that 81.0% passed all quality checks (subject length greater than 8 characters and no self-replies), exceeding the pre-registered 70% precision threshold (Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003eD). Sensitivity analysis across 1-day, 3-day, and 7-day windows showed moderate growth in pair counts (16,600 to 20,876) with increasing median reply time (1.33 to 2.58 hours), indicating that the 1-day window captures the majority of genuine reply behavior.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4 Model Hierarchy\u003c/h2\u003e \u003cp\u003eWe fit five models to each user\u0026rsquo;s outgoing email stream, forming a hierarchy that progressively adds explanatory structure [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. All models were fit on the training partition and evaluated on the held-out test set.\u003c/p\u003e \u003cp\u003eModel 1 (Inhomogeneous Poisson): The baseline model captures circadian and weekly periodicity through a log-linear intensity function λ(t)\u0026thinsp;=\u0026thinsp;exp(f(hour, dow)), where f is parameterized with 24 hour-of-day and 7 day-of-week indicator variables. This model treats events as conditionally independent given the time-varying rate, providing a natural null hypothesis for temporal structure beyond periodicity [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eModel 2 (Lognormal Renewal): Inter-event times are modeled as independent draws from a lognormal distribution, IET \u0026sim; Lognorm(\u0026micro;, σ) [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e, \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. This captures the heavy-tailed waiting-time distribution characteristic of bursty behavior but includes no temporal covariates and no event-to-event excitation. The lognormal renewal process serves as a within-family comparison for the Hawkes model (both operate on continuous event times rather than binned counts).\u003c/p\u003e \u003cp\u003eModel 3 (Hawkes Process): A self-exciting point process [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e] with conditional intensity λ(t) = \u0026micro;(t) + Σ α\u0026middot;exp(\u0026minus;β\u0026middot;(t\u0026thinsp;\u0026minus;\u0026thinsp;t\u003csub\u003ei\u003c/sub\u003e)), where the sum runs over all prior events t\u003csub\u003ei\u003c/sub\u003e \u0026lt; t, α controls excitation magnitude, and β controls decay rate. The background rate \u0026micro;(t) incorporates the same circadian covariates as M1. We fit a single-exponential kernel (the planned multi-scale K\u0026thinsp;=\u0026thinsp;2 kernel did not converge for the majority of users, triggering the pre-registered fallback). The branching ratio α/β quantifies the expected number of triggered events per parent event; values below 1.0 indicate a subcritical (stationary) process [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. The Hawkes model was successfully fit to 53 of 58 cohort users.\u003c/p\u003e \u003cp\u003eModel 4 (Poisson HMM): Hourly email counts cₜ are modeled as emissions from a hidden Markov model with K latent states [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], where the emission distribution in each state is Poisson: cₜ | xₜ \u0026sim; Poisson(λₓ). State transitions are governed by a K \u0026times; K transition matrix estimated via the Baum\u0026ndash;Welch algorithm [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. We fit models with K\u0026thinsp;=\u0026thinsp;2 and K\u0026thinsp;=\u0026thinsp;3 states and selected the best K by BIC [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] on the training data. The original proposal specified Negative Binomial emissions as the primary specification, but implementation constraints (hmmlearn supports only Poisson emissions natively) led us to use Poisson emissions with K\u0026thinsp;=\u0026thinsp;2\u0026ndash;3 states as the primary analysis. The HMM captures latent work\u0026ndash;rest regime switching: for example, user kay.mann exhibited a work state with λ\u0026thinsp;=\u0026thinsp;5.15 emails per hour and a rest state with λ\u0026thinsp;\u0026asymp;\u0026thinsp;0 emails per hour, with transition probabilities P(Rest \u0026rarr; Rest)\u0026thinsp;=\u0026thinsp;0.95 and P(Work \u0026rarr; Work)\u0026thinsp;=\u0026thinsp;0.63.\u003c/p\u003e \u003cp\u003eModel 5 (Hybrid: HMM on Hawkes Residuals): To test whether self-excitation and state-switching capture complementary sources of variability, we computed time-rescaled residuals from the fitted Hawkes model (M3) following the residual analysis framework of Ogata [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] and then fit an HMM to the residual process. If the Hawkes model fully accounts for latent state structure, the residuals should show no preference for multiple states (K\u0026thinsp;=\u0026thinsp;1 preferred by BIC [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]). Conversely, if K\u0026thinsp;=\u0026thinsp;2 is preferred on the residuals, self-excitation and state-switching address different aspects of variability, supporting hypothesis H3.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.5 Evaluation Protocol\u003c/h2\u003e \u003cp\u003eEvaluation followed a strict temporal protocol. For each user, the first 70% of their email timeline was designated as the training set and the final 30% as the held-out test set, yielding 70,887 training events and 30,412 test events across the cohort. No information from the test period was used during model fitting. The primary evaluation metric was held-out log-likelihood per event; secondary metrics included BIC [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] for model selection within families and Brier score for short-horizon prediction calibration. We note an important methodological caveat: Models M1 and M4 operate on binned hourly counts and produce log-likelihoods in count-space units, while Models M2 and M3 operate on continuous event times and produce log-likelihoods in continuous-time units [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. Direct comparison of held-out log-likelihood is therefore valid within model families (M1 vs. M4 for count-based models; M2 vs. M3 for point-process models) but not across families without normalization. Bootstrap confidence intervals (50 replicates with 7-day blocks) [\u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e24\u003c/span\u003e] were computed for burstiness metrics; full 500-replicate bootstrap on model comparisons was planned but not completed due to computational constraints.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e2.6 Falsifiable Hypotheses\u003c/h2\u003e \u003cp\u003eWe pre-registered three falsifiable hypotheses with explicit thresholds. H1 (Self-Excitation): The Hawkes excitation parameter α will be significantly greater than zero for at least 80% of users meeting activity thresholds, indicating that incoming emails systematically increase near-future sending [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. H2 (State-Switching): The Poisson HMM will improve held-out log-likelihood by at least 10% over the inhomogeneous Poisson baseline for a majority of users. H3 (Complementarity): Hawkes residuals [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] will retain significant state-switching structure (K\u0026thinsp;=\u0026thinsp;2 HMM preferred over K\u0026thinsp;=\u0026thinsp;1 by BIC [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] for more than 50% of users), indicating that self-excitation and latent states capture different sources of variability.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results","content":"\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Burstiness and Circadian Structure\u003c/h2\u003e \u003cp\u003eAll 58 cohort users exhibited bursty email behavior, with burstiness coefficients uniformly positive (mean B\u0026thinsp;=\u0026thinsp;0.601, median B\u0026thinsp;=\u0026thinsp;0.609, range 0.324\u0026ndash;0.909; Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003eB). Bootstrap 95% confidence intervals excluded zero for every user, confirming that burstiness is a universal feature of this cohort rather than a property of a few outliers [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e]. The memory coefficient M, measuring the first-order autocorrelation of consecutive inter-event times [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e], was weakly positive on average (mean M\u0026thinsp;=\u0026thinsp;0.079, median M\u0026thinsp;=\u0026thinsp;0.021), with 37 of 58 users showing M\u0026thinsp;\u0026gt;\u0026thinsp;0, indicating modest positive memory in the temporal structure of sending behavior.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eA Kolmogorov\u0026ndash;Smirnov test of inter-event times against the exponential distribution yielded D\u0026thinsp;=\u0026thinsp;0.69 (p\u0026thinsp;\u0026lt;\u0026thinsp;10⁻⁶), decisively rejecting the memoryless Poisson assumption (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003eA). The complementary cumulative distribution function (CCDF) of inter-event times showed pronounced heavy-tailed behavior across all users, consistent with prior findings in email [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e] and other communication channels [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. The aggregate CCDF departed from the exponential reference by orders of magnitude at long time scales. Circadian structure was strong and consistent: the hour-by-day-of-week heatmap (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003eC) revealed concentrated activity during Monday\u0026ndash;Friday business hours (approximately 8 AM to 8 PM) with sharply reduced activity on weekends and overnight, mirroring the patterns reported by Malmgren et al. [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. The Fano factor (variance-to-mean ratio of counts) exceeded 1.0 at every time scale examined, from 5 minutes to 24 hours, and increased monotonically with bin size (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e2\u003c/span\u003eD), confirming super-Poisson dispersion at all temporal resolutions.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Self-Excitation (H1)\u003c/h2\u003e \u003cp\u003eHypothesis H1 predicted that at least 80% of users would exhibit significant self-excitation (α\u0026thinsp;\u0026gt;\u0026thinsp;0) in the Hawkes model [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. Among the 53 users for whom the Hawkes model converged, 44 (83%) had excitation parameters significantly greater than zero, exceeding the pre-registered 80% threshold and confirming H1 (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e4\u003c/span\u003eA). The mean branching ratio across fitted users was 0.146, indicating that each email event triggers, on average, an additional 0.15 events\u0026mdash;a modest but consistent self-excitation effect well within the subcritical regime (α/β\u0026thinsp;\u0026lt;\u0026thinsp;1) [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. This finding supports the interpretation that incoming emails systematically increase near-future sending, consistent with reply-chain and conversational-arc dynamics observed in other communication systems [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e].\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e3.3 State-Switching (H2)\u003c/h2\u003e \u003cp\u003eHypothesis H2 predicted that the Poisson HMM [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e] would improve held-out log-likelihood by at least 10% over the inhomogeneous Poisson baseline for a majority of users. The HMM achieved a mean held-out log-likelihood of \u0026minus;\u0026thinsp;2.38 per event compared to \u0026minus;\u0026thinsp;3.82 for the Poisson baseline, and 84.5% of users showed improvement exceeding the 10% threshold, confirming H2 (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e4\u003c/span\u003eA). BIC-based model selection [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] favored K\u0026thinsp;=\u0026thinsp;3 states for 35 users and K\u0026thinsp;=\u0026thinsp;2 states for 23 users, suggesting that most individuals are best characterized by a three-regime model (e.g., high-activity work, low-activity work, and rest) rather than a simple binary work\u0026ndash;rest dichotomy. For the illustrative user kay.mann, the inferred state sequence (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e4\u003c/span\u003eB) showed clear alternation between a work state (λ\u0026thinsp;=\u0026thinsp;5.15 emails/hour) and a rest state (λ\u0026thinsp;\u0026asymp;\u0026thinsp;0), with a highly persistent rest state (P(Rest \u0026rarr; Rest)\u0026thinsp;=\u0026thinsp;0.95) and a less persistent but well-defined work state (P(Work \u0026rarr; Work)\u0026thinsp;=\u0026thinsp;0.63).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Complementarity (H3)\u003c/h2\u003e \u003cp\u003eHypothesis H3 tested whether self-excitation and state-switching capture complementary variability by examining Hawkes residuals [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] for residual state-switching structure. Among the 53 users with valid Hawkes fits, 33 (62%) showed a preference for K\u0026thinsp;=\u0026thinsp;2 over K\u0026thinsp;=\u0026thinsp;1 in the residual HMM by BIC [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e], exceeding the pre-registered 50% threshold and confirming H3 (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e4\u003c/span\u003eC). This result indicates that even after accounting for short-term self-excitation, a substantial majority of users retain latent state structure in their residual activity. The complementarity finding implies that self-excitation (capturing event-triggered reply cascades on the scale of minutes to hours) and state-switching (capturing longer-duration regime transitions between work and rest) address different aspects of behavioral variability and that hybrid models combining both mechanisms are warranted.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Reply-Time Analysis (Supplementary)\u003c/h2\u003e \u003cp\u003eAs a supplementary analysis, we examined the distribution of reply times constructed via heuristic pairing (1-day window, 16,600 pairs, 81% precision). Reply times were heavy-tailed, with a Kolmogorov\u0026ndash;Smirnov test rejecting the exponential distribution (D\u0026thinsp;=\u0026thinsp;0.25, p\u0026thinsp;\u0026lt;\u0026thinsp;10⁻⁶), and 95% of individual users also rejected exponentiality (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e3\u003c/span\u003eA). The median reply time was 1.33 hours with a mean of 4.67 hours, reflecting the long right tail characteristic of communication response times [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]. Reply times showed strong circadian modulation (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e3\u003c/span\u003eB): median reply times were shortest during business hours (approximately 1\u0026ndash;2 hours) and longest for emails received overnight or early morning (up to 13\u0026ndash;14 hours). Sensitivity analysis across pairing windows (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e3\u003c/span\u003eC) confirmed that the 1-day window captured the core of reply behavior, with wider windows adding relatively few pairs but substantially increasing mean reply time. These supplementary results are consistent with the primary finding that email behavior is governed by circadian modulation and latent state transitions, but should be interpreted with caution given the heuristic nature of the pairing procedure.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Discussion","content":"\u003cp\u003eThe central finding of this study is that behavioral variability in email communication is multi-mechanistic: circadian periodicity, self-excitation, and latent state-switching each contribute distinct, partially non-overlapping predictive information. Our five-model hierarchy provides a quantitative decomposition of these contributions, extending beyond the descriptive burstiness characterizations prevalent in the literature [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e]. Latent state-switching (M4, Poisson HMM) provided the largest improvement over the periodicity-only baseline, reducing held-out log-likelihood from \u0026minus;\u0026thinsp;3.82 to \u0026minus;\u0026thinsp;2.38 per event, with 84.5% of users exceeding the 10% improvement threshold. This suggests that discrete regime transitions\u0026mdash;between states interpretable as work and rest\u0026mdash;are the primary driver of predictive gains beyond circadian structure, consistent with the cascading non-homogeneous Poisson framework proposed by Malmgren et al. [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e, \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] but offering a more mechanistic explanation through explicit latent states [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eSelf-excitation, captured by the Hawkes process (M3) [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], is statistically significant for 83% of users but contributes modestly to the overall variance budget, with a mean branching ratio of 0.15. This means that for every 100 emails, approximately 15 are attributable to self-excitation cascades triggered by prior events. While this effect is consistent and robust across users, it is small relative to the gains achieved by state-switching, suggesting that the dominant source of variability is not event-triggered cascading but rather transitions between sustained behavioral regimes. This finding nuances the emphasis on self-excitation in recent social dynamics literature [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e20\u003c/span\u003e] by showing that for individual email behavior, regime-switching explains more variance than event-triggered cascading.\u003c/p\u003e \u003cp\u003eThe complementarity result (H3) is perhaps the most theoretically consequential finding. The fact that 62% of users retain state-switching structure in their Hawkes residuals [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e] demonstrates that self-excitation and regime-switching are not redundant explanations for the same variability. Rather, they operate at different temporal scales and capture different generative mechanisms: self-excitation models the local clustering of events within conversational episodes, while state-switching captures the global alternation between periods of high and low engagement [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e]. This motivates the development of hybrid models that integrate both mechanisms, potentially through state-dependent Hawkes processes in which excitation parameters vary by latent state, extending the latent network structure approach of Linderman and Adams [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e] from spatial to temporal hidden variables.\u003c/p\u003e \u003cp\u003eSeveral limitations should be noted. First, the near-complete absence of threading headers in the Enron corpus (0.01% coverage) forced our reply-time analysis to rely on heuristic subject-based pairing, which we accordingly report as supplementary rather than primary. Second, the Hawkes model used a single-exponential kernel; the planned multi-scale kernel (K\u0026thinsp;=\u0026thinsp;2 exponential components) did not converge for the majority of users, potentially limiting the model\u0026rsquo;s ability to capture both rapid reply-chain dynamics and slower conversational arcs simultaneously [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. Third, the HMM employed Poisson emissions rather than the originally planned Negative Binomial specification, which may cause the model to inflate the number of latent states to compensate for overdispersion [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. Fourth, the Enron corpus reflects communication patterns within a single corporate organization during a specific historical period [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e21\u003c/span\u003e, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e], and the generalizability of our findings to personal email, instant messaging, or contemporary corporate communication remains an open question. Finally, direct comparison of log-likelihoods across model families (count-based M1/M4 versus continuous-time M2/M3) requires normalization that we do not perform [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]; our variance decomposition is therefore best interpreted within model families.\u003c/p\u003e \u003cp\u003eFuture work should address these limitations by fitting multi-scale Hawkes kernels with regularization to promote convergence, implementing Negative Binomial HMM emissions to handle overdispersion without state inflation [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e], and extending the analysis to dyadic (sender\u0026ndash;recipient pair) models that capture relational dynamics [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e23\u003c/span\u003e]. Applying the same five-model hierarchy to other communication corpora\u0026mdash;including instant messaging platforms, the Copenhagen Networks Study multi-layer dataset, and personal email collections such as those analyzed by Eckmann et al. [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e] and Oliveira and Barab\u0026aacute;si [\u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e25\u003c/span\u003e]\u0026mdash;would test the generalizability of our variance decomposition across communication modalities and social contexts.\u003c/p\u003e"},{"header":"Abbreviations","content":"\u003cdiv class=\"DefinitionList\"\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003eIET\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eInter-event time\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003eHMM\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eHidden Markov model\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003eBIC\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eBayesian information criterion\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003eKS\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eKolmogorov\u0026ndash;Smirnov\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003eCCDF\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eComplementary cumulative distribution function\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003eLL\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eLog-likelihood\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003eBR\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eBranching ratio\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003cdiv class=\"DefinitionListEntry\"\u003e \u003cdiv class=\"Term\"\u003eAPC\u003c/div\u003e \u003cdiv class=\"Description\"\u003e \u003cp\u003eArticle processing charge\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eEthics Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis study analyzed only email header metadata (sender, recipients, timestamps, and subject lines for reply-pair matching); no message bodies were accessed at any stage. All users are identified by anonymized identifiers, and results are reported as aggregated statistics across the cohort. The Enron email corpus is a publicly released dataset distributed by Carnegie Mellon University for research purposes [6].\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll analysis code is available at https://github.com/brhanufen/enron-behavioral-variability with reproducing all tables and figures. The Enron email corpus is freely available from Carnegie Mellon University [6]. Derived datasets contain only timestamps and anonymized user identifiers; no message content is included.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthics approval and consent to participate\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable. This study analyzed only publicly available email header metadata from the Enron corpus (Carnegie Mellon University release). No human subjects were involved.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConsent for publication\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eNot applicable.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eCompeting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare that they have no competing interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research received no external funding.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthors\u0026rsquo; contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eB.F.Z. conceived and designed the study, performed all data processing and statistical analyses, developed the computational pipeline, and wrote the original draft of the manuscript. Z.A. contributed to the study design and methodology. P.D. contributed to manuscript writing and editing. All authors read and approved the final manuscript.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgements\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors thank Tayachew for valuable encouragement and for offering perspective from a signal analysis viewpoint during the course of this work.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eBarab\u0026aacute;si, A. L. The origin of bursts and heavy tails in human dynamics. \u003cem\u003eNature\u003c/em\u003e \u003cb\u003e435\u003c/b\u003e, 207\u0026ndash;211 (2005).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMalmgren, R. D., Stouffer, D. B., Motter, A. E. \u0026amp; Amaral, L. A. N. A Poissonian explanation for heavy tails in e-mail communication. \u003cem\u003eProc. Natl. Acad. Sci. USA\u003c/em\u003e. \u003cb\u003e105\u003c/b\u003e (47), 18153\u0026ndash;18158 (2008).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMalmgren, R. D., Stouffer, D. B., Campanharo, A. S. L. O. \u0026amp; Amaral, L. A. N. On universality in human correspondence activity. \u003cem\u003eScience\u003c/em\u003e \u003cb\u003e325\u003c/b\u003e (5948), 1696\u0026ndash;1700 (2009).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHawkes, A. G. Spectra of some self-exciting and mutually exciting point processes. \u003cem\u003eBiometrika\u003c/em\u003e \u003cb\u003e58\u003c/b\u003e (1), 83\u0026ndash;90 (1971).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eRabiner, L. R. A tutorial on hidden Markov models and selected applications in speech recognition. \u003cem\u003eProc. IEEE\u003c/em\u003e. \u003cb\u003e77\u003c/b\u003e (2), 257\u0026ndash;286 (1989).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKlimt, B. \u0026amp; Yang, Y. The Enron corpus: A new dataset for email classification research. In: ECML 2004. Springer; (2004).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eGoh, K. I. \u0026amp; Barab\u0026aacute;si, A. L. Burstiness and memory in complex systems. \u003cem\u003eEurophys. Lett.\u003c/em\u003e \u003cb\u003e81\u003c/b\u003e, 48002 (2008).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eCox, D. R. Regression models and life-tables. \u003cem\u003eJ. R Stat. Soc. B\u003c/em\u003e. \u003cb\u003e34\u003c/b\u003e (2), 187\u0026ndash;220 (1972).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eEckmann, J. P., Moses, E. \u0026amp; Sergi, D. Entropy of dialogues creates coherent structures in e-mail traffic. \u003cem\u003eProc. Natl. Acad. Sci. USA\u003c/em\u003e. \u003cb\u003e101\u003c/b\u003e (40), 14333\u0026ndash;14337 (2004).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eV\u0026aacute;zquez, A. et al. Modeling bursts and heavy tails in human dynamics. \u003cem\u003ePhys. Rev. E\u003c/em\u003e. \u003cb\u003e73\u003c/b\u003e, 036127 (2006).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eJo, H. H., Karsai, M., Kert\u0026eacute;sz, J. \u0026amp; Kaski, K. Circadian pattern and burstiness in mobile phone communication. \u003cem\u003eNew. J. Phys.\u003c/em\u003e \u003cb\u003e14\u003c/b\u003e, 013055 (2012).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eKarsai, M., Kaski, K., Barab\u0026aacute;si, A. L. \u0026amp; Kert\u0026eacute;sz, J. Universal features of correlated bursty behaviour. \u003cem\u003eSci. Rep.\u003c/em\u003e \u003cb\u003e2\u003c/b\u003e, 397 (2012).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eHolme, P. \u0026amp; Saram\u0026auml;ki, J. Temporal networks. \u003cem\u003ePhys. Rep.\u003c/em\u003e \u003cb\u003e519\u003c/b\u003e (3), 97\u0026ndash;125 (2012).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eOgata, Y. Statistical models for earthquake occurrences and residual analysis for point processes. \u003cem\u003eJ. Am. Stat. Assoc.\u003c/em\u003e \u003cb\u003e83\u003c/b\u003e (401), 9\u0026ndash;27 (1988).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eSchwarz, G. Estimating the dimension of a model. \u003cem\u003eAnn. Stat.\u003c/em\u003e \u003cb\u003e6\u003c/b\u003e (2), 461\u0026ndash;464 (1978).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDaley, D. J. \u0026amp; Vere-Jones, D. \u003cem\u003eAn Introduction to the Theory of Point Processes\u003c/em\u003e 2nd edn, Vol. 1 (Springer, 2003).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZucchini, W., MacDonald, I. L. \u0026amp; Langrock, R. \u003cem\u003eHidden Markov Models for Time Series: An Introduction Using R\u003c/em\u003e 2nd edn (CRC, 2016).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eBaum, L. E., Petrie, T., Soules, G. \u0026amp; Weiss, N. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. \u003cem\u003eAnn. Math. Stat.\u003c/em\u003e \u003cb\u003e41\u003c/b\u003e (1), 164\u0026ndash;171 (1970).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eLinderman, S. \u0026amp; Adams, R. Discovering latent network structure in point process data. In: Proc ICML. (2014).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eZhao, Q., Erdogdu, M. A., He, H. Y., Rajaraman, A. \u0026amp; Leskovec, J. SEISMIC: A self-exciting point process model for predicting tweet popularity. In: \u003cem\u003eProc. KDD\u003c/em\u003e (2015).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eDiesner, J., Frantz, T. L. \u0026amp; Carley, K. M. Communication networks from the Enron email corpus It\u0026rsquo;s always about the people. Enron is no different. \u003cem\u003eComput. Math. Organ. Theory\u003c/em\u003e. \u003cb\u003e11\u003c/b\u003e (3), 201\u0026ndash;228 (2005).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eShetty, J. \u0026amp; Adibi, J. The Enron email dataset database schema and brief statistical report. Information Sciences Institute Technical Report, University of Southern California; (2004).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eMasuda, N. \u0026amp; Lambiotte, R. \u003cem\u003eA Guide to Temporal Networks\u003c/em\u003e (World Scientific, 2016).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eEfron, B. \u0026amp; Tibshirani, R. J. \u003cem\u003eAn Introduction to the Bootstrap\u003c/em\u003e (Chapman \u0026amp; Hall/CRC, 1993).\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eOliveira, J. G. \u0026amp; Barab\u0026aacute;si, A. L. Human dynamics: Darwin and Einstein correspondence patterns. \u003cem\u003eNature\u003c/em\u003e \u003cb\u003e437\u003c/b\u003e, 1251 (2005).\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"email communication, burstiness, Hawkes process, hidden Markov model, variance decomposition, Enron corpus","lastPublishedDoi":"10.21203/rs.3.rs-9105293/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9105293/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eHuman email communication exhibits substantial temporal variability that resists simple characterization. We treat individual email activity as a stochastic system and ask how much behavioral variability is explained by each of three generative mechanisms: circadian and weekly periodicity, self-excitation from incoming messages, and latent work\u0026ndash;rest state-switching. Using the Enron email corpus (58 users, 101,299 sent emails, 1998\u0026ndash;2002), we fit a hierarchy of five models\u0026mdash;inhomogeneous Poisson, lognormal renewal, Hawkes process, Poisson hidden Markov model (HMM), and a hybrid HMM-on-Hawkes-residuals\u0026mdash;evaluated via strict 70/30 temporal train/test splits. All three pre-registered hypotheses pass: 83% of users exhibit significant self-excitation (α\u0026thinsp;\u0026gt;\u0026thinsp;0; mean branching ratio 0.15), 84.5% show greater than 10% log-likelihood improvement under state-switching, and 62% retain residual state-switching structure after Hawkes filtering. Latent state-switching provides the largest predictive gains, while self-excitation is real but modest, and the two mechanisms capture complementary sources of variability. All code and derived data (timestamps and anonymized identifiers only) are publicly available for full reproducibility.\u003c/p\u003e","manuscriptTitle":"Decomposing Behavioral Variability in Email Communication: Self-Excitation, Latent State-Switching, and Their Interaction in the Enron Corpus","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-17 07:13:54","doi":"10.21203/rs.3.rs-9105293/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"97b8aeaa-1570-4d67-8ca3-bb39e33d0df2","owner":[],"postedDate":"March 17th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":64605787,"name":"Physical sciences/Mathematics and computing"},{"id":64605788,"name":"Biological sciences/Neuroscience"},{"id":64605789,"name":"Biological sciences/Psychology"},{"id":64605790,"name":"Social science/Psychology"}],"tags":[],"updatedAt":"2026-03-17T07:13:54+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-17 07:13:54","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9105293","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9105293","identity":"rs-9105293","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}