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arxiv: 2604.07694 · v1 · submitted 2026-04-09 · ⚛️ physics.soc-ph · physics.comp-ph

Modeling non-Poissonian temporal hypergraphs by Markovian node dynamics

Hang-Hyun Jo , Naoki Masuda This is my paper

Pith reviewed 2026-05-10 18:18 UTC · model grok-4.3

classification ⚛️ physics.soc-ph physics.comp-ph
keywords temporal hypergraphsMarkov processesinter-event time distributionburstinessautocorrelationnode activitygroup interactionsnon-Poissonian processes
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The pith

Markovian node activity switches produce non-Poissonian event timing in temporal hypergraphs

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces simple models of temporal hypergraphs driven by individual nodes that switch between low- and high-activity states using Markov chains. Hyperedges generate events according to rules that depend on how many nodes are currently in the high-activity state. Despite the memoryless nature of the node dynamics, the collective event sequences exhibit longer-tailed inter-event time distributions and slower-decaying autocorrelations than Poisson processes. These non-Poissonian features vary with the size of the hyperedge and match observations from various empirical datasets on group interactions. The models thus offer a minimal mechanism to connect fluctuating individual behavior to the bursty timing seen in real social group events.

Core claim

Despite the Markovian dynamics of node states, the event processes on hyperedges emerge as mixtures of Poissonian components. This mixture arises because the effective rate for each hyperedge fluctuates according to the instantaneous number of high-activity nodes it contains, producing interevent time distributions with heavier tails and autocorrelation functions that decay more slowly than those of homogeneous Poisson processes. The strength of these deviations increases with hyperedge size, in agreement with empirical temporal hypergraph data.

What carries the argument

Two-state Markovian node dynamics combined with hyperedge event probability that depends on the number of high-state nodes in the hyperedge.

If this is right

  • The inter-event time distribution for a hyperedge is a weighted sum of exponential distributions, with weights given by the binomial probabilities of having k high nodes.
  • Autocorrelation functions of event sequences decay as a sum of exponentials whose rates are set by the node transition rates.
  • For larger hyperedges the effective event process approaches a Poisson process with average rate because the fraction of high nodes concentrates around its mean.
  • Node-level inter-event times remain shorter-tailed than the induced hyperedge-level times under the same dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Testing the model on hypergraphs with heterogeneous hyperedge sizes could reveal whether size-dependent burstiness holds across domains.
  • If node states are not fully independent the mixture may produce even longer tails, suggesting a direction for model refinement.
  • The same mechanism might apply to modeling temporal networks with higher-order interactions beyond hypergraphs.
  • One could use the analytical expressions to fit node transition rates directly from observed group event data.

Load-bearing premise

The event generation rate for each hyperedge depends only on the current count of high-activity nodes within it and not on any history or external factors.

What would settle it

Collect inter-event time histograms for hyperedges of varying sizes from empirical temporal hypergraph data and check whether the tail heaviness or coefficient of variation scales with size exactly as predicted by the mixture-of-Poissons formula.

Figures

Figures reproduced from arXiv: 2604.07694 by Hang-Hyun Jo, Naoki Masuda.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the dynamics of six nodes, [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Average event probabilities. (a) Hyperedge, AND [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. IET distributions. (a) Hyperedge, AND rule. (b) Hy [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. CVs of IETs. (a) Hyperedge, AND rule. (b) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. ACFs. (a) Hyperedge, AND rule. (b) Hyperedge, [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Results of empirical data analysis. We show the av [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
read the original abstract

Temporal hypergraphs capture time-resolved group interactions among nodes. Empirical data support that time-stamped group interactions show bursty event sequences and non-trivial temporal correlations. In the present study, we introduce node-driven temporal hypergraph models in which each node stochastically alternates between low- and high-activity states, and a hyperedge produces time-stamped events with a probability that depends on the number of high-state nodes in the hyperedge. For two event-generation rules, we analytically derive interevent time distributions and autocorrelation functions of event sequences, both for hyperedges and nodes. Despite Markovian node-state dynamics, the induced event processes become mixtures of Poissonian, short-tailed components, resulting in longer-tailed interevent time distributions and slowly decaying autocorrelation. The theory further shows the dependence of these features on the size of hyperedge, which largely agrees with various empirical data. We expect our models to provide a simple, interpretable framework for connecting individual-level activity fluctuations to the timing patterns observed in real group interactions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 4 minor

Summary. The paper introduces node-driven models for temporal hypergraphs in which each node follows an independent continuous-time Markov chain alternating between low- and high-activity states. A hyperedge generates events according to one of two deterministic rules that depend only on the instantaneous count of high-activity nodes inside it. For both rules the authors derive closed-form expressions for the inter-event time distribution and the autocorrelation function of the event sequence, both at the hyperedge and at the node level. The resulting aggregate processes are mixtures of Poissonian components whose tails are longer than exponential and whose autocorrelations decay slowly; both features strengthen with hyperedge size, in qualitative agreement with several empirical data sets.

Significance. If the derivations are correct, the work supplies a minimal, fully analytical mechanism that converts Markovian individual dynamics into the bursty, long-range-correlated event sequences observed in real group interactions. The explicit dependence on hyperedge cardinality furnishes testable predictions and connects the model to the broader literature on how microscopic fluctuations generate macroscopic temporal heterogeneity.

major comments (2)
  1. [§3.2] §3.2, Eq. (8)–(11): the master equation for the count process N(t) is written, but the explicit transition rates q_{k→k±1} induced by the independent node chains are not displayed; without them the claim that the event process is exactly a mixture of Poisson processes cannot be verified by the reader.
  2. [§5.3] §5.3, Fig. 4: the comparison of model predictions with empirical inter-event time distributions is visual only; no quantitative goodness-of-fit measure (KS statistic, likelihood ratio, or parameter uncertainty) is reported, weakening the statement that the size dependence “largely agrees” with data.
minor comments (4)
  1. The two event-generation rules are introduced in §2.2 but never given short mnemonic names; consistent labels (e.g., “linear rule” and “threshold rule”) would improve readability throughout the derivations and figures.
  2. [§4] Notation for the stationary probability π_k of k active nodes inside a hyperedge of size m is introduced only in §3.1; repeating the definition once in the results section would help readers who skip the methods.
  3. [§1] Several sentences in the introduction cite “temporal hypergraphs” without distinguishing between the static hypergraph and its temporal realization; a brief clarifying sentence would avoid ambiguity.
  4. The supplementary material is referenced for “full derivations,” yet the main text does not indicate which equations are proved there versus derived in place; an explicit pointer at the end of each derivation would be helpful.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3.2] §3.2, Eq. (8)–(11): the master equation for the count process N(t) is written, but the explicit transition rates q_{k→k±1} induced by the independent node chains are not displayed; without them the claim that the event process is exactly a mixture of Poisson processes cannot be verified by the reader.

    Authors: We agree that the explicit transition rates q_{k→k±1} should be displayed to allow direct verification of the mixture-of-Poisson property. In the revised manuscript we will derive and insert these rates from the independent node-level continuous-time Markov chains, showing how they enter the master equation for the count process N(t) and confirming that the resulting event process is indeed a mixture of Poisson processes whose weights depend on hyperedge size. revision: yes

  2. Referee: [§5.3] §5.3, Fig. 4: the comparison of model predictions with empirical inter-event time distributions is visual only; no quantitative goodness-of-fit measure (KS statistic, likelihood ratio, or parameter uncertainty) is reported, weakening the statement that the size dependence “largely agrees” with data.

    Authors: The referee correctly notes that a purely visual comparison limits the strength of the agreement claim. In the revision we will augment Fig. 4 and the accompanying text with quantitative measures: Kolmogorov-Smirnov statistics comparing model and empirical inter-event time distributions for each hyperedge size, together with the fitted parameter values and their uncertainties obtained from the maximum-likelihood procedure. These additions will provide an objective basis for the reported size dependence. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained from model definitions

full rationale

The paper starts from explicit definitions of independent continuous-time Markov chains for each node's activity state (low/high) and two deterministic event-rate rules that depend only on the instantaneous count of high-activity nodes within a hyperedge. Inter-event time distributions and autocorrelation functions are then obtained by solving the resulting higher-level CTMC for the count process and applying standard renewal theory or embedded Markov equations. These steps use only the model primitives and binomial stationary occupancy; no fitted parameters are renamed as predictions, no self-citations supply load-bearing uniqueness theorems, and no ansatz is smuggled in. The size dependence of tail weight and autocorrelation decay follows directly from the hyperedge-size parameter in the binomial count distribution. The derivation therefore contains no circular reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The model relies on several free parameters for transition rates and probability functions, and domain assumptions about independence and functional dependence, which are not derived from first principles but introduced to capture the phenomena.

free parameters (2)
  • node state transition rates
    Rates governing the Markovian switching between low- and high-activity states for each node.
  • event generation probability parameters
    Parameters defining how the probability of event production depends on the number of high-state nodes.
axioms (2)
  • domain assumption Node dynamics are independent Markov chains
    Assumed for the stochastic alternation between activity states.
  • domain assumption Event probability depends only on count of high-state nodes
    Core modeling choice for the two event-generation rules.

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    (7), (8), (10), (11), and (12), we can compactly writeP e(τ) in Eq

    IET DISTRIBUTION FOR HYPEREDGES AS MA TRIX PRODUCTS By combining Eqs. (7), (8), (10), (11), and (12), we can compactly writeP e(τ) in Eq. (6) as Pe(τ) = ⃗Λ⊤ e,f(WeLe)τ−1 We⃗Λe,i,(S12) where ⃗Λe,f ≡   λe(m,0) λe(m,1) ... λe(m, m)   ,(S13) Le ≡   ¯λe(m,0) 0· · ·0 0 ¯λe(m,1)· · ·0 ... ... ... 0 0· · · ¯λe(m, m)   ,(S14) 3 ⃗Λe,i ≡ 1 Ωe  ...

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  79. [81]

    DERIV A TION OF THE IET DISTRIBUTION FOR A NODE IN THE GENERAL CASE The IET distribution for the node,P v(τ), can be written as Pv(τ) = Pr[v(τ) = 1, v(τ−1) = 0, . . . , v(1) = 0|v(0) = 1] = X ⃗ n0,...,⃗ nτ Pr[v(τ) = 1|⃗ µh(τ) =⃗ nτ] "τ−1Y t=1 Pr[⃗ µh(t+ 1) =⃗ nt+1|⃗ µh(t) =⃗ nt] Pr[v(t) = 0|⃗ µh(t) =⃗ nt] # ×Pr[⃗ µh(1) =⃗ n1|⃗ µh(0) =⃗ n0] Pr[⃗ µh(0) =⃗ n...

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  80. [82]

    AND rule Under the AND rule, we consider a simple nontrivial case in which the focal node belongs to two hyperedges of the same sizem, i.e.,k= 2 andm 1 =m 2 =m

    DERIV A TION OF THE IET DISTRIBUTION FOR A NODE IN A MINIMAL NONTRIVIAL CASE 4.1. AND rule Under the AND rule, we consider a simple nontrivial case in which the focal node belongs to two hyperedges of the same sizem, i.e.,k= 2 andm 1 =m 2 =m. It implies that⃗ µ= (1, m−1, m−1). Usingλ e given by Eq. (1), we write λv,AND in Eq. (22) as λv,AND(⃗ µ, ⃗ n) = 1−...

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