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SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Populations

Marian-Andrei Rizoiu, Swapnil Mishra, Quyu Kong, Mark Carman, Lexing Xie

arXiv:1711.01679v3cs.SIphysics.soc-ph

TL;DR

The paper asks how epidemic and Hawkes models can describe online diffusion and its uncertain popularity. It links their event rates, proposes a finite-population Hawkes model, and derives a future cascade-size distribution that tends to be bimodal.

  • Problem

    Epidemic and Hawkes models are both used for online diffusion, but their relationship and the distribution of final cascade sizes remain open questions.

    Source: §1 · p. 1

  • Method

    The paper links Hawkes events to SIR infections after marginalizing recovery events, introduces HawkesN for finite populations, and applies an SIR-derived Markov-chain method to cascades fitted with HawkesN.

    Source: §1 · pp. 1–2

  • Results

    The extended Hawkes event rate equals the SIR new-infection rate after marginalizing recoveries, and predicted cascade-size distributions tend to have peaks near zero and at large sizes.

    Source: §1 · pp. 1–2

  • Takeaways & Limitations

    The connection permits tools developed for SIR and Hawkes to be applied across the two approaches, while prefix observations update cascade-size probabilities.

    Source: §1 · pp. 1–2

    The model assumes a fixed population in which users neither enter nor exit.

    Source: §8 · p. 9

Abstract

from arXiv · show

Among the statistical tools for online information diffusion modeling, both epidemic models and Hawkes point processes are popular choices. The former originate from epidemiology, and consider information as a viral contagion which spreads into a population of online users. The latter have roots in geophysics and finance, view individual actions as discrete events in continuous time, and modulate the rate of events according to the self-exciting nature of event sequences. Here, we establish a novel connection between these two frameworks. Namely, the rate of events in an extended Hawkes model is identical to the rate of new infections in the Susceptible-Infected-Recovered (SIR) model after marginalizing out recovery events -- which are unobserved in a Hawkes process. This result paves the way to apply tools developed for SIR to Hawkes, and vice versa. It also leads to HawkesN, a generalization of the Hawkes model which accounts for a finite population size. Finally, we derive the distribution of cascade sizes for HawkesN, inspired by methods in stochastic SIR. Such distributions provide nuanced explanations to the general unpredictability of popularity: the distribution for diffusion cascade sizes tends to have two modes, one corresponding to large cascade sizes and another one around zero.

1 INTRODUCTION

The paper asks how epidemic and Hawkes models relate, how to represent diffusion in a finite population, and how to characterize uncertain cascade sizes. It links SIR and Hawkes processes, proposes HawkesN, and uses it to estimate a distribution over future cascade size.

1 INTRODUCTION

  • The work studies word-of-mouth diffusion to connect individual actions with collective measures such as popularity or fame.It frames questions about links between epidemic and Hawkes models, finite-population diffusion models, and cascade-size distributions.Source: §1 · p. 1
  • Marginalizing unobserved recovery events makes the extended Hawkes event rate identical to the SIR new-infection rate, linking the two frameworks.The connection allows tools developed for either approach to be applied to the other.Source: §1 · p. 1
  • HawkesN modulates event rates by the remaining finite population, addresses Hawkes’ unbounded event count, and better explains longer event sequences.The paper also estimates N from data and constructs a lower-bound statistic to detect when N has no valid solution.Source: §1 · p. 2
  • The predicted distribution of future cascade size has two peaks: one near zero for quickly extinguishing cascades and a smaller one at large sizes.At the beginning, the two outcomes cannot be distinguished; observing a cascade prefix updates their posterior probabilities.Source: §1 · p. 2

2.2 The SIR Model

  • The contributions include ActiveRT, a publicly available Twitter-cascade benchmark, with HawkesN simulation and fitting code.The paper also reports validating the model connection theoretically and empirically on three public retweet-cascade datasets.Source: §1 · p. 1, §2.2 · p. 2

2 PREREQUISITES

The prerequisites review point processes and SIR, then describe stochastic SIR as infection and recovery events with time-varying rates. The bivariate representation supplies the link to Hawkes processes.

2 PREREQUISITES

  • The section reviews Poisson and Hawkes point processes, then the SIR model and its bivariate process formulation.A point process records event times; for a homogeneous Poisson process, inter-arrival times are independent exponential random variables.Source: §2 · p. 2, §2.1 · p. 2

2.1 Poisson and Hawkes processes

  • SIR divides individuals into susceptible, infected, and recovered compartments; infection occurs at rate β and recovery at constant rate γ.The model assumes a homogeneous population, constant rates, and no births or deaths, so N remains fixed.Source: §2.1 · p. 2
  • The stochastic SIR formulation represents infection and recovery as two event types, with each infected individual later recovering.The infectious period is exponentially distributed with parameter γ and averages 1/γ time units.Source: §2.1 · p. 3
  • In the bivariate process, the infection rate depends on susceptible and infectious counts, while recovery events reduce the infectious count.As susceptible individuals are depleted, the infection rate falls to zero; the recovery rate also reaches zero when no infected individuals remain.Source: §2.1 · p. 3
  • The SIR bivariate point-process formulation provides the link to Hawkes point processes.Figure 1 illustrates infection and recovery events, infectious-population size, and their rates for N = 5, β = 2, and γ = 0.5.Source: §2.1 · p. 3, Figure 1 · p. 3

3 LINKING EPIDEMIC MODELS AND HAWKES PROCESSES

The section links HawkesN event rates to expected SIR infection rates after marginalizing over recovery times, and extends Hawkes to account for a finite population. With an exponential kernel, Hawkes is recovered as the infinite-population case of HawkesN.

3.1 HawkesN: a process in finite population

  • HawkesN scales event intensity by the remaining population, reducing it to zero when the population’s available event capacity is exhausted.The finite-population extension models cascades whose users can act only within a bounded population.Source: §3.1 · p. 3
  • As N → ∞, HawkesN simplifies to the Hawkes process, making Hawkes its infinite-population special case.Source: §3.1 · p. 4
  • The analysis uses an exponential kernel; applying HawkesN with non-exponential kernels is left for future work.The paper names power-law and Rayleigh kernels as alternatives used with Hawkes.Source: §3.1 · p. 4

3.2 Linking HawkesN and SIR

  • After averaging over SIR recovery times, its new-infection intensity matches HawkesN’s conditional intensity when event and infection counts align and parameters are equivalent.The expectation removes randomness from recovery times, not from event or infection times.Source: §3.2 · pp. 4–5

4 DIFFUSION SIZE DISTRIBUTION

The section adapts the SIR final-size Markov-chain method to estimate cascade-size distributions from partially observed cascades fitted with HawkesN. For partial observations, it initializes the calculation using the observed event count and an expected infected count over recovery times.

4 DIFFUSION SIZE DISTRIBUTION

  • A Markov-chain technique developed for SIR computes the final-size distribution of a partially observed diffusion cascade fitted with HawkesN.Source: §4 · p. 5

4.1 Epidemic size distributions in SIR

  • In the SIR chain, each state records susceptible and infected counts; infection and recovery are the two possible transitions, and states with no infected individuals absorb the process.Source: §4.1 · p. 5
  • The state probabilities converge after at most 2N − 1 transitions; probabilities of absorbing states yield the final-size distribution N′ = N − s.Source: §4.1 · p. 5

4.2 Cascade size distribution in HawkesN

  • For partial observations, the posterior calculation starts at {N − l, E_T[˜i]}, with expected infected count over recovery times rounded to the nearest integer.The a priori distribution instead starts at {N − 1, 1} and depends only on model parameters; the posterior also depends on observed event times.Source: §4.2 · p. 6

5 FITTING HAWKESN TO DATA

This section fits HawkesN parameters by maximizing the event-time likelihood, then examines whether population size N can be estimated from partial cascades. HawkesN is harder than SIR to estimate early because it observes infection events but not recovery events.

5.1 The likelihood of HawkesN

  • For diffusion cascades, the likelihood fit assumes µ = 0 after the first event and estimates κ, θ, and N by maximizing the HawkesN point-process log-likelihood.The optimization constrains θ > 0, κ > 0, and N ≥ n, and uses AMPL with Ipopt.Source: §5.1 · p. 6

5.2 Estimating population size N

  • When N is unknown, the section studies it as the only unknown parameter and uses a zero derivative of the log-likelihood as a necessary condition for a local maximum.The derivative-based analysis motivates the lower-bound statistic for detecting when no valid N solution exists.Source: §5.2 · p. 6
  • The statistic S guarantees that no valid N exists when S > 0, because the log-likelihood then increases monotonically for N ≥ n.S is a lower bound used to detect non-valid solutions; it misses only 4 of 44 valid solutions after 5% is observed and identifies all valid solutions above 10%.Source: §5.2 · p. 6, Difficulty of estimating N · p. 7
  • The estimation study fixes µ = 0, κ = 5, θ = 0.2, and N = 100, and compares SIR prefixes containing infection and recovery events with HawkesN prefixes containing infections only.It uses 100 simulated cascades and increasingly long prefixes spanning 5% to 100% of each cascade.Source: Difficulty of estimating N · p. 7
  • At 5% observed, 56% of HawkesN cascades lack a valid N; SIR has valid solutions for all cascades once more than 10% is observed.HawkesN solutions exist once more than 50% of a cascade is observed; SIR estimates N correctly earlier because recovery-event timing carries information about population size.Source: Difficulty of estimating N · p. 7

6 EXPERIMENTS AND RESULTS

Across three Twitter retweet datasets, the experiments test HawkesN on held-out events and examine distributions over cascade sizes. HawkesN has lower median holdout negative log-likelihood at higher observed proportions, while its cascade-size distributions illustrate early-extinction and large-cascade possibilities.

6.1 Datasets

  • The evaluation uses ActiveRT, Seismic, and News retweet cascades, with event times and retweeter follower counts available for each event.The experiments compare performance on held-out events after fitting models to an initial proportion of each cascade.Source: §6.1 · pp. 7–8, §6.2 · p. 8

6.2 Generalization to unobserved data

  • At 40% and 80% observation, HawkesN has lower median holdout negative log-likelihood; means are comparable on News and Seismic but higher for HawkesN on ActiveRT.For higher observed proportions, the mean negative log-likelihood improves for HawkesN and degrades for Hawkes; below 40% observed, HawkesN performance has high variance.Source: §6.2 · p. 8, Figure 6 · p. 8

6.3 Explaining popularity unpredictability

  • A News cascade’s a priori size distribution has peaks near very small sizes and around 65, corresponding to early extinction or a large cascade relative to N.After observed events, the a posteriori distribution shifts toward larger sizes and narrows, indicating less uncertainty as more events are observed.Source: §6.3 · pp. 8–9

7 RELATED WORK

Prior work applies point-process and epidemic-model frameworks to online diffusion, but the cited approaches do not combine point processes with epidemic models. The paper distinguishes its work by introducing finite-population Hawkes modeling and cascade-size distributions.

Point process approaches

  • Point-process models have been used for popularity modeling and prediction [25] [33] [34] [45], including models that estimate cascade size [34].Crane and Sornette link popularity bursts and decays to a Hawkes self-exciting process; other models address social contagion and rich-get-richer effects [45].Source: §7 · p. 9

Stochastic differential equation approaches

  • Stochastic differential equation approaches connect event-level data with macroscopic inference, but the cited work does not link point processes to epidemic models [40] [47] [48].The paper identifies the use of epidemic-model tools for information diffusion as an advantage of its connection.Source: §7 · p. 9

Epidemic model approaches

  • Epidemic models have been applied to knowledge diffusion, computer viruses, rumors, topic diffusion, and retweet cascades [24] [27] [29].The cited work does not leverage epidemic-model tools such as cascade-size probability distributions or link epidemic models to point-process models.Source: §7 · p. 9

8 CONCLUSION

The paper links Hawkes and SIR models, introduces HawkesN for finite populations, and uses it to estimate cascade-size distributions. The covered appendices also derive NHPP results, describe HawkesN fitting, and state scope limits.

8 CONCLUSION

  • The conclusion connects the expected event rate in extended Hawkes to the SIR infection rate after marginalizing recovery events, and introduces HawkesN and a cascade-size distribution method.The method applies SIR Markov-chain tools to partially observed cascades fitted with HawkesN.Source: §8 · p. 9
  • The work assumes a fixed population, with users neither entering nor leaving; it notes that user-specific behavior or non-exponential kernels require more advanced SIR formulations.An agent-based formulation is given as an example of such an advanced formulation.Source: §8 · p. 9

A.1 Inter-arrival times probabilities

A.2 Two follow-up conclusions

  • A non-homogeneous Poisson process is generally not Markovian: its next event can depend on the full event history, though specific rate functions may be exceptions.For a homogeneous Poisson process, inter-arrival times are independent and exponentially distributed.Source: §A.2 · p. 12

B.2 Used solvers and optimization setup

  • The fitting setup runs ten optimization rounds from varied initializations and selects the solution with the highest training log-likelihood.It combines LGO's approximate global search with IPOPT refinement, alongside multiple IPOPT initializations.Source: §B.2 · p. 13

B.3 Interfacing AMPL with R

  • The R–AMPL interface generates model and data files, invokes AMPL, retrieves results, and handles solver errors.Temporary files use process IDs and ram-drives to support parallel runs and reduce disk-access costs.Source: §B.3 · p. 13

C.1 Kernel function and branching factor

  • HawkesN uses a modified exponential kernel incorporating local user influence, and its branching factor indicates cascade speed and final-size distribution.The local-influence distribution is described as a power law with exponent α = 2.016.Source: §C.1 · pp. 13–14

C.2 Log-likelihood function.

  • HawkesN parameters are fitted by maximizing a nonlinear log-likelihood with constraints on θ, κ, and η, using AMPL and IPOPT.The model assumes zero background intensity after the first event when modeling diffusion cascades.Source: §C.2 · p. 14

C.3 Equivalence to stochastic SIR

The paper establishes an expected-rate equivalence between HawkesN and stochastic SIR, then examines that connection through simulation and fitting. It also describes how stochastic and deterministic SIR are fitted from observed data.

C.3 Equivalence to stochastic SIR

  • The expected infection rate in stochastic SIR equals the expected HawkesN event rate under parameter equivalence and zero background rate; their basic reproduction number and branching factor also match.The equivalence averages over SIR recovery times and HawkesN user influences.Source: §C.3 · p. 14

D RELATION BETWEEN DETERMINISTIC SIR AND STOCHASTIC SIR

E SIR-HAWKESN EQUIVALENCE THROUGH SIMULATION AND FITTING

E.1 Maximum likelihood estimates for SIR

  • For stochastic SIR, each event likelihood combines its inter-arrival time with the probability of its event class; deterministic SIR is fitted from population time series.The deterministic approach predicts S, I, and R by simulating the differential equations, then uses squared error or a likelihood metric.Source: §E.1 · p. 15

E.2 Equivalence on Synthetic Data

Synthetic-data experiments test HawkesN–SIR equivalence in both directions. Fitted parameters closely match simulation parameters, and expected SIR population sizes and event rates closely trace observed values.

E.2 Equivalence on Synthetic Data

  • The experiments simulate 20 realizations of stochastic SIR and fit HawkesN to the new-infection process before mapping its parameters to SIR.The reverse experiment fits deterministic SIR to population sizes at fixed intervals.Source: §E.2 · p. 15
  • Fitted parameters were very close to simulation parameters in both directions of the synthetic-data experiment.HawkesN was fitted to stochastic-SIR infections and mapped to SIR parameters; HawkesN simulations were also fitted with deterministic SIR because recovery times were unobserved.Source: §E.2 · p. 15
  • For a stochastic SIR realization, expected population sizes and infection and recovery rates closely trace their observed values.The expectations are computed when only new infection events are observed.Source: §E.2 · p. 16, Figure 9 · p. 16

F ROBUSTNESS OF FIT – ADDITIONAL GRAPHICS

HawkesN estimates become reliable only after observing a larger cascade prefix than basic Hawkes estimates. The section also examines Hawkes generalization and the difficulty of estimating finite population size.

F ROBUSTNESS OF FIT – ADDITIONAL GRAPHICS

  • Hawkes requires less than 30% of a cascade for reliable estimates, while HawkesN requires more than 40% before median branching-factor and N estimates approach their true values.HawkesN is more sensitive because N is estimated from the observed data; Figure 10 shows estimates over increasing prefixes.Source: F ROBUSTNESS OF FIT – ADDITIONAL… · p. 16, Figure 10 · p. 16

G GENERALIZATION PERFORMANCE – HAWKES

  • The News experiment evaluates Hawkes holdout negative log-likelihood over 1,000 cascades as the observed training share ranges from 10% to 95%.Lower holdout negative log-likelihood is better; the supplied passage describes the evaluation but gives no outcome.Source: G GENERALIZATION PERFORMANCE – HAWKES · p. 16, Figure 12 · p. 17

H ESTIMATING POPULATION SIZE (N) IN HAWKESN

  • Estimating HawkesN's fixed population size is challenging, motivating analytic and empirical investigations of N estimation.The investigation covers a simplified intensity function and empirical retrieval of N under both that function and HawkesN.Source: H ESTIMATING POPULATION SIZE (N) IN… · p. 16

H.1 Step 1: Analytic Results for A Simplified Intensity Function

  • Under the simplified intensity function, a positive derivative test means the likelihood increases monotonically, so no valid maximum exists within the parameter range.When the test is negative, a valid maximum may exist.Source: §H.1.2 · p. 18

H.2 Step 2: Experiments on Simplified Intensity Function

The experiments examine valid likelihood roots and population-size estimation in simplified-intensity and HawkesN settings, then relate HawkesN’s branching factor to the SIR reproduction number.

H.2 Step 2: Experiments on Simplified Intensity Function

  • A valid root is an optimum N*>n with zero likelihood derivative; more observed events made valid roots more likely, and simulations yielded one or none.Source: §H.2.1 · p. 18, §H.2.2 · p. 18

H.2.4 Correctness of Estimated

H.2.5 Difficulty of Estimating

  • Eq. (40) motivates a fast test for identifying likelihood functions without maximum values; Table 4 evaluates its informativeness across observed-cascade percentages.Source: §H.2.5 · pp. 18–19, Table 4 · p. 19

H.3 Step 3: Experiments on HawkesN

  • With κ and θ fixed, N was the only HawkesN parameter estimated, and likelihood trends supported the valid roots as maxima.Source: §H.3 · p. 19, Figure 17 · p. 20

H.3.2 Difficulty of estimating

H.4 Estimating N in the Deterministic SIR

  • For deterministic SIR, the final-size prediction is R(∞)=N−S(∞), with the root of Eq. (41) found numerically in [0,N].Source: §H.4 · p. 20

I NARRATIVE ON BRANCHING FACTORS

I.1 Observations on branching factor

Fitting HawkesN to 80% of cascades reveals different branching-factor patterns across datasets and video categories, alongside category differences in fitted population size.

I.1 Observations on branching factor

  • Seismic and News branching-factor densities peak around 0.2 with long tails, while ActiveRT has a secondary peak around 1.5, possibly related to Youtube-video cascades.HawkesN was fitted using 80% of each cascade.Source: §I.1 · p. 21, Figure 19 · p. 22
  • Sports, People & Blogs, and Film & Animation cascades tend to have above-median n*, while Gaming, Howto & Style, and Nonprofit & Activism tend to be below; Gaming has relatively high N.Source: §I.1 · p. 21, Figure 19 · p. 22
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