PaperlayerView exact revision on arXiv

Source-linked AI summary

Fundamental limits to identifying node and tie memory in temporal networks: marginal artefacts and spreading dynamics

Michele Tizzani

arXiv:2609.10399v1physics.soc-phcs.SI

TL;DR

The paper asks whether temporal contact records identify node self-excitation separately from tie reinforcement, and analyzes how initiator labels and timing structure affect that identification. It finds that missing labels couple the mechanisms, while empirical node self-excitation can instead be pinned to the inter-event marginal and remain uninformative about memory.

Abstract

from arXiv · show

Temporal-network models attribute memory in contact data to either node self-excitation (branching ratio n_node) or tie reinforcement (kappa), carrying major consequences for epidemic spreading. We prove that when event initiators are observed, the two mechanisms are orthogonal: the Fisher information is block-diagonal and neither trades off against the other. In undirected proximity data, where initiators are unobserved, marginalising over them couples the mechanisms into a structural confound that survives posterior smoothing. On empirical proximity, messaging, and email records, however, a cruder failure dominates: fitted node memory is pinned to the inter-event marginal law and remains virtually invariant across latent label posterior samples (coefficient of variation below 1%). An inter-event-order shuffle test and burstiness-memory diagnostics reveal that exponential-Hawkes node memory is recovered from none, while tie reinforcement remains identifiable throughout. This near-unidentifiability is intrinsic, not an artefact of the exponential kernel: refitting flexible scale-free (sum-of-exponentials) kernels on synthetic power-law self-exciting processes fails to distinguish genuine node memory from memoryless renewal controls, with identical collapses recurring on algorithmic networks (edit bots, cloud microservices) and cortical spiking. Downstream epidemic consequences are quantitative: simulations fitted to empirical contact records under-predict outbreak sizes by up to a factor of 2.5 and shift the epidemic threshold. We conclude that observational temporal networks face a two-fold identifiability boundary: contact directionality is essential to decouple tie reinforcement, whereas heavy-tailed node self-excitation is intrinsically unidentifiable from contact timings alone.

I. RESULTS

The results separate identifiable tie reinforcement from node self-excitation that, on empirical records, often reflects timing marginals rather than memory. Missing initiator labels couple the mechanisms, while short records and model misspecification create distinct additional problems.

I. RESULTS

  • On empirical data, κ is treated as a measurement, whereas fitted n_node is only a diagnostic because timing tests reject the model in some settings.The target-selection model is adequate in the hospital and conference settings, but not uniformly across datasets.Source: §I · p. 3, Goodness of fit · p. 7

Recovery with observed initiators

  • With initiators observed, n_node and κ are recovered within one standard error and their profile likelihoods show no trade-off.Both profiles are sharply curved and contain the true values within their 95% intervals.Source: Recovery with observed initiators · p. 3

Cost of the missing initiator label

  • Removing initiator labels couples node and tie memory; the confound survives exact posterior smoothing, with correlation −0.63 on synthetic data.The missing-information off-diagonal is set by initiator-posterior entropy and vanishes as labels are supplied.Source: Cost of the missing initiator label · pp. 3–4

Where the estimator is confidently wrong

  • At ∼2–4 events per node, strict intervals under-cover because profiling baseline activities biases n_node downward; a hierarchical prior roughly halves the bias.By ∼30 events per node, coverage is nominal, even in cells with thousands of events.Source: Where the estimator is confidently wrong · p. 4

Empirical estimates

  • Across datasets, κ varies descriptively, but empirical n_node values are not mechanistic estimates because they read the inter-event marginal.The fitted node parameter remains near unity under sessionisation changes, while κ is quantitatively stable within reported between-day dispersion.Source: Empirical estimates · p. 5, Sensitivity and preprocessing artefacts · p. 5

Sensitivity and preprocessing artefacts

  • Sessionising repeated proximity frames removes the sampling-grid spike; fitting across overnight gaps instead distorts decay rates and inflates n_node.The 60 s sessionisation recovers a tie-memory timescale of about 28,300 s for High School Day 0.Source: Sensitivity and preprocessing artefacts · p. 5

Goodness of fit

  • Timing diagnostics reject the model’s temporal structure across tested settings, while target selection passes for the hospital and conference but fails for the high school.Online digital records also show residual shape defects and over-dispersion beyond simultaneity artefacts.Source: Goodness of fit · p. 7

The node axis: a burstiness–memory plane

  • Burstiness and memory are distinct: an exponential Hawkes fit can assign strong self-excitation to a heavy-tailed renewal marginal even when consecutive intervals have no memory.The inter-event-order shuffle preserves each node’s marginal while destroying event order, so persistence under that shuffle signals marginal dependence rather than memory.Source: The node axis: a burstiness–memory plane · p. 8

When nnode means what it says

The inter-event-order shuffle tests whether fitted node self-excitation reads genuine kernel-matched memory or only the inter-event marginal, and its response shows a smooth crossover rather than a sharp threshold.

When nnode means what it says

  • R declines smoothly from approximately 1 at M ≈ 0 toward small values at large M, indicating a crossover rather than a sharp transition.The shuffle ratio tracks kernel-matched memory, not M in general.Source: When nnode means what it says · pp. 8–9
  • The inter-event-order shuffle preserves the marginal distribution while destroying memory, so R ≈ 1 signals a marginal-driven fit and R → 0 signals recovery of kernel-matched memory.The test generates single-node event trains with controlled burstiness and memory, then fits n_node before and after shuffling.Source: When nnode means what it says · p. 8
  • The decline width grows with event count, with a log-log slope of +0.16 over E ≈ 67–3324 events per node.The width is the M interval over which R falls from 0.9 to 0.7.Source: When nnode means what it says · p. 9

When the fitted nnode is set by the marginal

In the tested records, fitted node memory tracks inter-event marginals rather than temporal order, and can miss genuine memory outside the assumed kernel. The empirical node axis is unresolvable even as tie memory remains identifiable, with consequential spreading differences.

When the fitted nnode is set by the marginal

  • Proximity records combine B≈0.7 with M=0.049 (hospital), 0.046 (conference), 0.005 (school), and +0.02 (email); shuffling intervals leaves fitted nnode nearly unchanged.The unchanged fits indicate that nnode reflects the heavy-tailed inter-event marginal, not node self-excitation.Source: When the fitted nnode is set by the… · p. 9
  • Online streams retain serial memory—Stack Overflow M1=0.22 and Wikibooks M1=0.11—yet R≈1 because their memory is not exponential-Hawkes.Wikibooks is an editor–page affiliation log; the analysis treats the editor stream as the node process.Source: When the fitted nnode is set by the… · p. 9
  • Flexible scale-free kernels also fail to distinguish synthetic power-law self-excitation from renewal controls; algorithmic networks likewise yield R≈1 under both kernels.The reported near-unidentifiability is attributed to heavy-tailed node timing, not only human data or the exponential kernel.Source: When the fitted nnode is set by the… · p. 9

merely present, memory

  • Shuffle-invariant nnode coexists with M=+0.34 synchronised bursts or M<0 refractory timing; the inter-event-order shuffle preserves marginals and destroys only event order.The shuffle ratio tests memory of the assumed kernel form, not all temporal memory.Source: merely present, memory · pp. 10–11

A located negative, and when the model would resolve

  • Across ten real and simulated datasets R≈1; kernel-matched injected node memory is recovered (nnode: 0.20→0.204 and 0.80→0.806) and collapses under shuffling.This positive control shows the estimator can recover node memory when it matches the assumed exponential form.Source: A located negative, and when the model… · p. 11

The full parameter space

  • Empirical records occupy the unresolvable node-memory but resolvable tie-memory region; κ remains identifiable with enough events, degrading as targets concentrate.The node axis is resolvable when M dominates B.Source: The full parameter space · p. 11

Why the distinction matters: spreading

  • Fitted-record models under-predict real outbreaks by up to a factor of 2.5; at scale, SIS thresholds differ (βc≈0.14 node pole; ≈0.30 memoryless and tie-reinforced sequences).The threshold shift is reported without a resolved sharp critical point.Source: Why the distinction matters: spreading · p. 12

II. DISCUSSION

The discussion separates a label-dependent structural confound from a deeper timing-marginal failure that persists even with directionality. It maps what instrumentation and diagnostics can establish, while qualifying the model’s scope and empirical fit.

What the separation result does and does not say

  • Without initiator labels, node and tie memory can trade off, but recording directionality decouples them only within the paper’s model class.The authors caution that this result does not establish that the model is correct or that these are the only mechanisms operating.Source: What the separation result does and… · p. 12
  • On empirical streams, heavy-tailed inter-event times peg node self-excitation to the renewal marginal, regardless of whether directionality is observed.The same failure persists in directed human and algorithmic records; directionality isolates tie reinforcement but does not rescue node self-excitation from the timing marginal.Source: What the separation result does and… · p. 13
  • An inter-event-order shuffle, the (B, M) plane, and a hierarchical prior on baseline activities offer diagnostics and an estimator remedy for specific data regimes.The prior is described as restoring honest coverage when there are few events per node; recovering heavy-tailed node dynamics requires access to internal states or external queues.Source: What the separation result does and… · p. 13

Relation to existing mechanism families

  • The joint parametrisation makes node self-excitation and tie reinforcement distinct estimable mechanisms, rather than treating them as competing modelling choices.The discussion relates these mechanisms to activity-memory and edge-memory families, while distinguishing this paper’s inferential question from descriptive work on memory structure.Source: Relation to existing mechanism families · p. 14

Limitations

  • A sum-of-exponentials kernel fits timing better than a single exponential, yet leaves near-critical branching-ratio estimates nearly unchanged.Timing log-likelihood gains are +175 on the hospital ward and +1200 on the high school; nnode changes from 0.91 to 0.95 for the school and 0.99 to 0.97 for the hospital.Source: Limitations · p. 14
  • Sequential imputation can substantially bias latent-initiator estimates, and smoothing corrects only part of that bias.Reported differences include −30% on synthetic data, −12% on the spiking network, and more than +100% on the email log; instrumental directionality avoids this bias.Source: Limitations · p. 14
  • Sessionising contacts discards duration information, while native hyperedge encoding does not remove the residual timing-fit defects or near-critical nnode.Duration is pair-attributed and enriches the tie channel; both duration modelling and remaining limitations are left to future work.Source: Limitations · p. 14

Outlook

  • Timing-block diagnostics reject the model in every empirical setting, so fitted mechanism parameters describe the fitted object rather than measure mechanisms.The outlook proposes extending the framework to distinguish pair-attributed functional similarity from label-dependent co-firing self-excitation in cortical data; testing that extension remains future work.Source: Limitations · p. 14, Outlook · p. 15

III. METHODS

The model represents temporal contacts as marked events and separates memoryless activity, node self-excitation, and tie reinforcement. Node memory changes when a node acts; tie memory changes which target it selects.

Events and mechanisms

  • Each contact is represented by its time, initiator, target, and optional interaction weight, while the contact itself remains undirected.The initiator records which endpoint generated the event; proximity data may lack this label.Source: Events and mechanisms · p. 15
  • The model distinguishes initiator history from the label-free history of an unordered contacting pair.This asymmetry determines whether a mechanism depends on the initiator label.Source: Events and mechanisms · p. 15
  • Memoryless node activity uses baseline initiation rates, while node self-excitation makes a node more likely to act again after recent events.The self-excitation term is Hawkes-type, and its branching ratio is dimensionless with stationarity requiring nnode < 1.Source: Events and mechanisms · p. 15
  • Tie reinforcement makes a recently contacted target more likely to be selected again, with κ setting reinforcement strength.The target-selection mechanism carries memory on links rather than on node activity.Source: Events and mechanisms · p. 15
  • Optional event weights have a separate mark law, and whether a weight inherits initiator confounding depends on whether it is pair-attributed or initiator-attributed.A contact duration is pair-attributed; a node’s engagement intensity when initiating is initiator-attributed. The two memory mechanisms also leave distinct temporal-burst and tie-recurrence signatures.Source: Events and mechanisms · p. 15, Figure 7 · p. 16

Normalisations, and why they are not cosmetic

Kernel and activity normalisations make the node branching ratio and tie-reinforcement strength interpretable. They also avoid scale ambiguity and keep node burstiness distinct from tie memory.

Normalisations, and why they are not cosmetic

  • A unit-integral node kernel makes nnode a branching ratio and gives the stationarity condition nnode < 1 independently of the kernel decay rate.The tie kernel’s unit peak makes a just-contacted target’s excess weight equal to κ against its baseline attractiveness.Source: Normalisations, and why they are not… · p. 16
  • Without a fixed normalisation of baseline attractiveness, κ is defined only up to an arbitrary scale.The present model profiles activities from the likelihood and pins nnode and κ through kernel normalisations and the stated constraint.Source: Normalisations, and why they are not… · pp. 16–17
  • The earlier convex-mixture intensity is unidentified because rescaling node activities can be absorbed by changing its mixture weight.That formulation also conflates node-versus-link with memoryless-versus-memory; separating nnode from κ allows node burstiness and tie reinforcement to vary independently.Source: Normalisations, and why they are not… · p. 17

Induced dyadic intensities and stationary rate

  • For the stationary Hawkes ground process, node i’s expected event rate is ai/(1 − nnode), and target selection does not change how often it acts.This rate relation holds with a unit-integral node kernel and serves as an implementation check.Source: Induced dyadic intensities and… · p. 17

Likelihood

The likelihood is organised into timing, target-selection, and optional weight components. Its additive separation assigns each component to its own parameters and sufficient statistics.

Likelihood

  • The marked point-process likelihood distinguishes timing, selection, and optional weight parameters for events observed over [0, T].The weight term is included only for marked events; dropping it recovers the unweighted likelihood.Source: Complete-data log-likelihood · p. 17
  • The compensator is available in closed form, and substituting the model intensities yields a likelihood decomposition.The decomposition is the basis for the identifiability results.Source: Complete-data log-likelihood · p. 17
  • The likelihood separates additively across parameter blocks, each depending on its own sufficient statistics.When weights are present, the weight block depends on labels only through the history selected by the weight’s attribution.Source: Complete-data log-likelihood · p. 18

Linear-time evaluation

Exponential-kernel recursions make full-record fitting linear in the number of events, but coincident timestamps can make the likelihood unbounded. Rate bounds therefore reflect the data’s time resolution, not merely numerical convenience.

Linear-time evaluation

  • The exponential-kernel states update recursively, reducing evaluation from O(P_i) direct resummation to O(E) and making full-record fitting feasible without truncation.Earlier truncation to a few thousand events selected a contiguous day-long stretch and introduced daily-cycle bias.Source: Linear-time evaluation · p. 18

Unboundedness under coincident events

  • One node initiating at least two events at the same timestamp makes the likelihood unbounded as the node-memory decay rate increases, when nnode > 0 is fixed.An unconstrained optimizer can drive the rate to its upper bound and report a spurious fit with an apparently excellent likelihood; this failure applies to any self-exciting point process fitted to grid-sampled data with an unbounded decay rate.Source: Unboundedness under coincident events · p. 18
  • The upper decay-rate bound is set by the smallest positive inter-event gap because memory faster than the sampling grid cannot be estimated from the data.The lower bound excludes kernels flat over the observation window, which are indistinguishable from a shift in baseline activity.Source: Unboundedness under coincident events · p. 18

Latent initiator

The paper frames identifiability around whether event-attribution labels are observed: observed labels separate mechanism blocks, while shared dependence on latent labels can couple them. Label-free mechanisms remain orthogonal, including pair-attributed weights, and the structural result does not guarantee that each parameter is estimable in practice.

Latent initiator

  • With an unobserved initiator, the observed-data likelihood sums over label assignments and does not factorize over events, so the paper approximates it with Monte Carlo EM.Node states, compensators, and normalizers depend on earlier labels, producing a sum with 2E terms and no product structure.Source: Latent initiator · p. 18

An attribution–orthogonality principle

  • When attribution is observed, separable, variation-independent parameter blocks have block-diagonal Fisher information; with latent attribution, label-free blocks remain orthogonal while blocks sharing labels generically confound.The coupling vanishes as the label posterior concentrates and, under uniform revelation, scales with residual entropy.Source: An attribution–orthogonality principle · p. 19
  • In the unweighted model, observing initiators makes node timing and tie selection orthogonal, whereas a latent initiator couples both label-dependent blocks.The general principle concerns per-event attribution labels, rather than directedness or weighting alone.Source: An attribution–orthogonality principle · p. 19
  • With latent initiators, pair-attributed weights remain orthogonal, but an initiator-attributed weight joins node timing and tie selection in a full three-block coupling.The weight-attribution choice determines whether the mark inherits the confound.Source: An attribution–orthogonality principle · p. 19
  • A contact-duration mark reinforced by tie memory is pair-attributed and therefore remains identifiable with a latent initiator, distinguishing interaction-frequency reinforcement from interaction-length reinforcement.The paper’s numerical test reports near-zero estimator spread for the pair-attributed mark at every tested label-revelation fraction; the test is a proof of concept on one realization family.Source: An attribution–orthogonality principle · pp. 19–20
  • The orthogonality theorem is kernel-agnostic, but the paper’s scale-free-kernel probe finds that the exponential estimator remains effectively blind to scale-free node memory.The result is structural for observed initiators; the empirical failure described here concerns fitting the exponential instance to data whose node memory may have another form.Source: An attribution–orthogonality principle · p. 20
  • Orthogonality does not ensure precise estimation: nuisance timescales are unidentified when their associated mechanism strength is zero.The paper gives κ = 0 as the case where γe is unidentified and nnode = 0 as the case where γn is unidentified.Source: An attribution–orthogonality principle · p. 20

Loss of orthogonality under a latent initiator

Marginalizing over an unobserved initiator mixes timing and selection contributions to the likelihood. This couples node-memory and tie-memory estimates because changing one mechanism can alter the inferred labels and thus the other mechanism’s sufficient statistics.

Loss of orthogonality under a latent initiator

  • With latent initiators, the observed likelihood sums over label configurations, and the mixture couples timing and selection rather than preserving their complete-data separation.The posterior over labels depends on both the node-timing and tie-selection parameters.Source: Loss of orthogonality under a latent… · p. 20
  • The coupled information matrix has nonzero off-diagonal blocks, so node self-excitation and tie reinforcement become confounded.A parameter change that makes node memory more plausible can reassign labels, changing the tie-memory sufficient statistics.Source: Loss of orthogonality under a latent… · p. 20
  • Across-imputation correlation between fitted nnode and κ measures the confound induced by missing labels and is exactly zero when labels are observed.The Results section reports this diagnostic for synthetic and empirical data.Source: Loss of orthogonality under a latent… · p. 20

What a point estimate cannot tell you

A point estimate alone does not establish whether node memory is determined by the data, so the paper uses profile likelihood, observed information, and interval coverage as complementary diagnostics. It profiles nuisance baseline activities exactly, adapting them to each candidate node-memory value.

What a point estimate cannot tell you

  • The profile-likelihood curve’s curvature indicates identifiability; a flat profile means the mechanism is not recoverable even if its argmax appears sharp.The profile re-optimizes the remaining parameters, including nuisance activities, at each value of nnode.Source: What a point estimate cannot tell you · p. 21
  • Observed-information standard errors are compared with across-replicate estimate dispersion to check whether asymptotic uncertainty is reliable at the available sample size.A large discrepancy indicates a data requirement not met at that sample size.Source: What a point estimate cannot tell you · p. 21
  • Coverage distinguishes honest imprecision from confident error: wide intervals with nominal coverage are imprecise, while narrow intervals with subnominal coverage indicate a degenerate regime.The paper maps coverage across system size, observation time, nnode, and κ to locate the boundary between these regimes.Source: What a point estimate cannot tell you · p. 21

Exact profiling of the baseline activities

  • For fixed nnode and γn, the baseline activities are profiled node by node because the timing likelihood is strictly concave and separable in them.The resulting profiling adapts activities to each nnode value rather than leaving a fixed normalization for the mechanism parameter to absorb.Source: Exact profiling of the baseline… · p. 21

Optimisation over the mechanism parameters

The mechanism parameters are fit in separate timing and selection problems, with latent initiators handled by sequential Monte Carlo imputation. Boundary tests use likelihood ratios, but their calibration requires care because the nulls lie on parameter boundaries and timescales are unidentified under the null.

Optimisation over the mechanism parameters

  • The timing and selection blocks are optimised independently after profiling, using bounded quasi-Newton iterations seeded by a coarse logarithmic grid scan.Initialising the node timescale at an O(1) value can underflow memory terms and trap the optimiser at a no-memory estimate; data-implied seeding avoids this failure.Source: Optimisation over the mechanism… · p. 21

Monte Carlo EM for the latent initiator

  • Sequential imputation draws initiator labels forward through the event list, then averages complete-data estimates across the imputed datasets.Standard errors combine within- and between-imputation variance.Source: Monte Carlo EM for the latent initiator · p. 21
  • Because labels are never revised using later events, sequential imputation approximates filtering rather than smoothing and has bias that the authors do not bound.The authors leave a Metropolis-within-Gibbs sampler over labels for future work.Source: Monte Carlo EM for the latent initiator · pp. 21–22

Tests of the two boundary nulls

  • Likelihood-ratio tests address the no-tie-reinforcement and no-node-self-excitation nulls, but boundary parameters and unidentified null timescales complicate calibration.Using the χ2_1 reference yields p-values roughly conservative by a factor of two; simulations check operating characteristics.Source: Tests of the two boundary nulls · p. 22

Implementation

The implementation cross-checks its Python and Julia likelihood calculations, and the empirical design spans proximity, communication, animal, and neural records. Proximity samples are sessionised because the sampling grid can otherwise be mistaken for contact memory.

Implementation

  • Python and Julia implementations are checked on shared event-sequence fixtures and must agree to 10^-8; simulators are separately checked against the analytic stationary rate.Code and fixtures are openly available.Source: Implementation · p. 22

Datasets

  • The dataset contrast spans school, conference, and hospital proximity records alongside baboon co-presence, directed email and messaging, and Allen Brain Network activations.The settings test whether estimates separate across contexts at comparable instrumentation and beyond human contacts.Source: Datasets · p. 22

Sessionisation

  • A continuous dyadic conversation can generate one proximity sample every 20 seconds, so fitting samples directly would mistake the grid for tie memory.The authors collapse consecutive same-dyad samples separated by at most 60 seconds into one event, while noting that this discards contact duration.Source: Sessionisation · p. 23

Resolution cap and tied timestamps

The analysis caps decay rates when shared timestamps would make the timing likelihood unbounded, segments human contacts by day, and filters low-contact nodes. Empirical estimates are treated as mechanistic only when residual checks support model adequacy.

Resolution cap and tied timestamps

  • Shared timestamps from the common sampling grid and group conversations make the timing likelihood unbounded, so decay rates are capped using the grid spacing as the minimum interval.The number of tied events is reported for each fit.Source: Resolution cap and tied timestamps · p. 23

Daily segmentation

  • Human contacts are fit day by day to avoid overnight gaps pulling memory timescales toward the gap duration, with between-day dispersion reported as a consistency check.Wide scatter can indicate that the model absorbs day-specific structure it does not represent.Source: Daily segmentation · p. 23

Node filtering

  • Nodes with fewer than five contacts are removed because near-inactive nodes add little likelihood information but can inflate the target pool and shift κ downward.The threshold is applied explicitly and its sensitivity is reported.Source: Node filtering · p. 23

A caveat on model adequacy

  • Synthetic parameter recovery only shows that the estimator inverts its own generative model; empirical estimates need time-rescaling residual checks before carrying mechanistic weight.The paper reports bootstrap-calibrated Kolmogorov–Smirnov statistics and identifies which settings the model describes adequately.Source: A caveat on model adequacy · p. 23
Loading 2609.10399v1…