Maximum entropy temporal networks

Paolo Barucca

arxiv: 2509.02098 · v6 · submitted 2025-09-02 · 💻 cs.SI · physics.data-an

Maximum entropy temporal networks

Paolo Barucca This is my paper

Pith reviewed 2026-05-18 20:18 UTC · model grok-4.3

classification 💻 cs.SI physics.data-an

keywords temporal networksmaximum entropynon-homogeneous Poisson processnetwork ensemblesgenerative modelslog-likelihoodedge probabilities

0 comments

The pith

Temporal networks factor into global time processes and static maximum-entropy edge probabilities.

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

The paper presents a maximum-entropy framework for temporal networks of timestamped directed interactions. Under basic assumptions on constraints, the ensembles admit a modular representation as global time processes multiplied by a static maximum-entropy edge probability. This factorization produces closed-form log-likelihoods together with analytic expressions for expected degrees, clustering, and motifs. It further derives non-homogeneous Poisson process intensities for edge occurrences via functional optimization over path entropy. A reader would care because the resulting models improve log-likelihood over ordinary Poisson processes while recovering strength constraints and matching unique-degree curves.

Core claim

By applying the maximum-entropy principle to continuous-time temporal networks and optimizing path entropy under constraints that permit factorization, the ensembles separate into global time processes and a static maximum-entropy edge probability. This time-edge factorization directly yields non-homogeneous Poisson process intensities for directed edges, closed-form log-likelihoods, and exact expectations for degrees, clustering coefficients, and motif counts.

What carries the argument

The time-edge labels factorization that separates global time processes from a static maximum-entropy edge probability.

If this is right

Closed-form log-likelihoods become available for fitting and comparison of temporal network models.
Expected values for degrees, clustering, and motif counts follow analytically from the static edge probabilities.
NHPP intensities supply a whole class of generative models that recover strength constraints and reproduce observed unique-degree curves.
The factorization connects maximum-entropy network ensembles to continuous-time point processes in a transparent way.

Where Pith is reading between the lines

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

The same factorization could serve as a regularizer or prior when calibrating multivariate Hawkes models on temporal interaction data.
Integration with renewal theory would allow replacement of the Poisson assumption with arbitrary inter-event distributions while preserving the max-ent edge layer.
Neural kernel estimators inside graph neural networks could adopt the static max-ent probabilities as an interpretable baseline layer.
Extensions to richer constraint sets would be needed before the approach handles higher-order temporal motifs or non-stationary node activity.

Load-bearing premise

Basic assumptions on constraints must allow both the time-edge labels factorization and the functional optimization over path entropy that produces the NHPP intensities.

What would settle it

Direct comparison of log-likelihood values on empirical temporal networks: if the NHPP intensities derived from the factorization do not produce higher likelihood than generic Poisson processes on held-out data, the claimed modeling gain is falsified.

Figures

Figures reproduced from arXiv: 2509.02098 by Paolo Barucca.

**Figure 2.** Figure 2: FIG. 2: Frozen-path NHPP runs approximating the theoretical Hawkes auto-covariance. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Degree-related statistics for Enron TRAIN (blockpair–PL model). observed vs. expected out-degree [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Burstiness vs. degree for the Enron TRAIN dataset. The plot compares empirical observations with the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Log of event counts per edge (left) and Burstiness per edge (right) for the Enron TRAIN dataset. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6: Block-to-block edge count comparison for Enron TRAIN: ( [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7: Inter-event time distributions (linear scale) for the Enron TRAIN dataset, comparing empirical data with [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗

**Figure 8.** Figure 8: FIG. 8: Raster plots of the top-5 intra-block event sequences for Enron TRAIN: ( [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗

**Figure 9.** Figure 9: FIG. 9: Enron TRAIN lambda over time Poisson CM ME [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗

**Figure 10.** Figure 10: FIG. 10: Enron TRAIN lambda over time GH [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗

**Figure 11.** Figure 11: FIG. 11: Enron TRAIN motif ratios (mean [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗

**Figure 12.** Figure 12: FIG. 12: Reality mining TRAIN motif ratios (mean [PITH_FULL_IMAGE:figures/full_fig_p016_12.png] view at source ↗

**Figure 13.** Figure 13: FIG. 13: Militarized Interstate Disputes TRAIN motif ratios (mean [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗

**Figure 14.** Figure 14: FIG. 14: Facebook TRAIN motif ratios (mean [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗

read the original abstract

Temporal networks consist of timestamped directed interactions that may appear continuously in time, yet few studies have directly tackled the continuous-time modeling of networks. Here, we introduce a maximum-entropy approach to temporal networks and with basic assumptions on constraints, the corresponding network ensembles admit a modular and interpretable representation: a set of global time processes and a static maximum-entropy edge, e.g. node pair, probability. This time-edge labels factorization yields closed-form log-likelihoods, degree, clustering and motif expectations, and yields a whole class of effective generative models. We provide the maximum-entropy derivation for the non-homogeneous Poisson Process (NHPP) intensities governing the probability of directed edges in temporal networks via the functional optimization over path entropy, connecting NHPP modeling to maximum-entropy network ensembles. NHPPs consistently improve log-likelihood over generic Poisson processes, while the maximum-entropy edge labels recover strength constraints and reproduce expected unique-degree curves. We discuss the limitations of this framework and how it can be integrated with multivariate Hawkes calibration procedures, renewal theory, and neural kernel estimation in graph neural networks.

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.

Desk Editor's Note private letter to a colleague

The paper derives NHPP intensities from max-ent path entropy optimization and gets a time-edge factorization under unspecified constraints, but the separability condition is the load-bearing part.

read the letter

The core contribution is a derivation that starts from maximum-entropy ensembles on temporal networks and arrives at non-homogeneous Poisson process intensities by optimizing path entropy. Under the right constraints this produces a clean split: global time processes multiplied by static maximum-entropy edge probabilities. That split is what delivers the closed-form log-likelihood, degree expectations, clustering, and motif counts, plus a generative model that beats a plain Poisson process on likelihood while recovering strength constraints and unique-degree curves. The link between the two literatures is not something I had seen laid out before, so that part is genuinely new on the page. The paper also flags how the framework could sit next to Hawkes processes or neural kernels, which is a useful pointer for people who want to extend it. The main soft spot is exactly the one the stress-test note flags. The factorization and all the closed forms rest on “basic assumptions on constraints” that let time and topology separate in the functional. The abstract does not list which constraints satisfy this or what happens when they do not. A time-local degree sequence or a motif count that changes with t would couple the two pieces, the product form would break, and the claimed closed forms would no longer hold. Without seeing the explicit functional derivative and the constraint list in the full text it is hard to judge how narrow the usable regime actually is. The empirical checks mentioned (log-likelihood improvement, recovery of strength and degree curves) are the right kind of test, but they only speak to the cases where the assumptions already hold. This is the sort of paper that belongs in a reading group for people working on generative models for interaction sequences. A reader who already uses maximum-entropy networks or NHPPs for temporal data will get immediate value from the modular representation and the explicit likelihood. It is not yet a finished tool, but the derivation is coherent enough on its own terms that a serious editor should send it out for review rather than desk-reject it. The referee can press on the constraint conditions and ask for a clear statement of when the factorization survives.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a maximum-entropy approach to continuous-time temporal networks. Under basic assumptions on constraints, the corresponding ensembles are claimed to admit a modular factorization into global time processes and static maximum-entropy edge probabilities. This factorization is said to produce closed-form log-likelihoods as well as closed-form expressions for expected degrees, clustering coefficients, and motif counts, while also yielding a class of generative models. The authors derive non-homogeneous Poisson process (NHPP) intensities via functional optimization over path entropy, report improved log-likelihood relative to homogeneous Poisson processes, and discuss integration with Hawkes processes, renewal theory, and neural kernel methods.

Significance. If the factorization and closed-form results hold under the stated assumptions, the work would offer a principled, analytically tractable bridge between maximum-entropy network ensembles and continuous-time point-process models. The modular representation and closed-form expectations would be useful strengths for both theoretical analysis and practical model calibration in temporal network studies.

major comments (1)

[Abstract and NHPP derivation section] The central claim of time-edge factorization and the resulting closed-form log-likelihoods, degree expectations, clustering, and motif counts rests on unspecified 'basic assumptions on constraints' that permit clean separation in the path-entropy functional. The manuscript must explicitly delineate these assumptions (e.g., whether constraints are global totals versus time-local sequences) and show that they do not introduce coupling between time and topology; otherwise the functional derivative yielding NHPP intensities = (global time process) × (static edge probability) fails and the claimed closed forms cease to hold.

minor comments (1)

[Abstract] The abstract states that NHPPs 'consistently improve log-likelihood' and that maximum-entropy edge labels 'recover strength constraints and reproduce expected unique-degree curves,' yet provides no quantitative values, data sets, or section references for these empirical results; adding a brief summary table or explicit cross-reference would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater precision regarding the assumptions underlying the time-edge factorization. We agree that explicit delineation of these assumptions will strengthen the presentation and have prepared a revision to address this point directly.

read point-by-point responses

Referee: [Abstract and NHPP derivation section] The central claim of time-edge factorization and the resulting closed-form log-likelihoods, degree expectations, clustering, and motif counts rests on unspecified 'basic assumptions on constraints' that permit clean separation in the path-entropy functional. The manuscript must explicitly delineate these assumptions (e.g., whether constraints are global totals versus time-local sequences) and show that they do not introduce coupling between time and topology; otherwise the functional derivative yielding NHPP intensities = (global time process) × (static edge probability) fails and the claimed closed forms cease to hold.

Authors: We agree that the assumptions require explicit statement. The basic assumptions are that all constraints are imposed on global aggregate statistics (e.g., total event counts or total strengths over the full observation interval) rather than on time-local or time-resolved sequences. Under this global-constraint regime the path-entropy functional factors additively into a purely temporal term and a purely topological term; the Euler-Lagrange equation then separates, yielding an intensity of the product form (global time process) × (static maximum-entropy edge probability). We will insert a new subsection immediately following the functional-derivation paragraph that (i) states the global-versus-local distinction, (ii) shows the absence of cross terms in the variation, and (iii) confirms that the closed-form expressions for the log-likelihood, expected degrees, clustering coefficients, and motif counts remain valid. This revision will be made without altering any of the reported numerical results. revision: yes

Circularity Check

0 steps flagged

Maximum-entropy derivation is self-contained under stated separability assumptions

full rationale

The paper presents the time-edge factorization and NHPP intensities as the direct outcome of functional optimization over path entropy, once basic constraints are assumed to separate time processes from static edge probabilities. No equations in the abstract or derivation sketch reduce a 'prediction' to a fitted parameter by construction, nor does any load-bearing step rely on self-citation chains or imported uniqueness theorems. The closed-form log-likelihoods and motif expectations follow mathematically from the modular representation once the separability condition holds; this is an independent derivation rather than a renaming or tautology. The framework is therefore scored as non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review prevents exhaustive listing; the central claim rests on unspecified basic assumptions on constraints that enable the factorization and the entropy optimization step.

axioms (1)

domain assumption Basic assumptions on constraints allow modular time-edge factorization
Invoked to obtain the interpretable representation and closed-form expressions

pith-pipeline@v0.9.0 · 5705 in / 1126 out tokens · 47582 ms · 2026-05-18T20:18:34.731496+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Stationarity with respect to each λij(t) yields log λij(t) = αr(i,j)(t) + Ψij − 1, hence λij(t) = ϕr(i,j)(t) wij
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and orbit embedding unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Maximizing path entropy ... subject to linear constraints that fix partition totals in time and time-integrated margins

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 2 internal anchors

[1]

E. T. Jaynes, Physical Review106, 620 (1957)

work page 1957
[2]

Csisz´ ar, The Annals of Probability3, 146 (1975)

I. Csisz´ ar, The Annals of Probability3, 146 (1975)

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[3]

Cimini, T

G. Cimini, T. Squartini, F. Saracco, D. Garlaschelli, A. Gabrielli, and G. Caldarelli, Nature Reviews Physics1, 58 (2019)

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Newman,Networks(Oxford University Press, 2018)

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Karrer and M

B. Karrer and M. E. J. Newman, Physical Review E83, 016107 (2011)

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D. L. Sussman, M. Tang, and C. E. Priebe, Universally consistent latent position estimation and vertex classification for random dot product graphs (2012), arXiv:1207.6745 [stat.ML]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[8]

M. E. J. Newman, Proceedings of the National Academy of Sciences98, 404 (2001)

work page 2001
[9]

Masuda and R

N. Masuda and R. Lambiotte,A Guide to Temporal Networks, Complexity Science, Vol. 4 (World Scientific, Singapore, 2016)

work page 2016
[10]

Holme and J

P. Holme and J. Saram¨ aki, Physics Reports519, 97 (2012)

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V´ azquez, B

A. V´ azquez, B. R´ acz, A. Luk´ acs, and A.-L. Barab´ asi, Physical Review Letters98, 158702 (2007). 12 FIG. 6: Block-to-block edge count comparison for Enron TRAIN: (left) empirical unique edges, (right) expected under the Block–DC–DWCM constraint. FIG. 7: Inter-event time distributions (linear scale) for the Enron TRAIN dataset, comparing empirical dat...

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Hiraoka, N

T. Hiraoka, N. Masuda, A. Li, and H.-H. Jo, Physical Review Research2, 023073 (2020)

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Goh and A.-L

K.-I. Goh and A.-L. Barab´ asi, Europhysics Letters81, 48002 (2008)

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M. Karsai, K. Kaski, A.-L. Barab´ asi, and J. Kert´ esz, Scientific Reports2, 397 (2012)

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D. R. Cox, Methuen & Co. Ltd., London (1962)

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Masuda and P

N. Masuda and P. Holme, Social Networks37, 12 (2014)

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Ogata, Journal of the American Statistical Association83, 9 (1988)

Y. Ogata, Journal of the American Statistical Association83, 9 (1988)

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Bacry and J.-F

E. Bacry and J.-F. Muzy, IEEE Transactions on Information Theory62, 2184 (2016)

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V. Filimonov and D. Sornette, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics85, 056108 (2012)

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G. V. Clemente, C. J. Tessone, and D. Garlaschelli, arXiv preprint arXiv:2311.16981 (2023). 13 FIG. 8: Raster plots of the top-5 intra-block event sequences for Enron TRAIN: (left) empirical; (right) blockpair–PL model (samples=00). FIG. 9: Enron TRAIN lambda over time Poisson CM ME

work page internal anchor Pith review Pith/arXiv arXiv 2023
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Achab, E

M. Achab, E. Bacry, S. Ga¨ ıffas, I. Mastromatteo, and J.-F. Muzy, Journal of Machine Learning Research18, 1 (2017)

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Soliman, L

H. Soliman, L. Zhao, Z. Huang, S. Paul, and K. S. Xu, inProceedings of the 39th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 162 (PMLR, 2022) pp. 20329–20346

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A. G. Hawkes, Biometrika58, 83 (1971)

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Bacry, I

E. Bacry, I. Mastromatteo, and J.-F. Muzy, Market Microstructure and Liquidity1, 1550005 (2015)

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D. J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods, 2nd ed. (Springer, New York, 2003). 14 FIG. 10: Enron TRAIN lambda over time GH TABLE III: Comparison of empirical and model-generated temporal-network statistics. Values are shown as mean ± standard deviation across Monte Carlo samples...

work page 2003

[1] [1]

E. T. Jaynes, Physical Review106, 620 (1957)

work page 1957

[2] [2]

Csisz´ ar, The Annals of Probability3, 146 (1975)

I. Csisz´ ar, The Annals of Probability3, 146 (1975)

work page 1975

[3] [3]

Cimini, T

G. Cimini, T. Squartini, F. Saracco, D. Garlaschelli, A. Gabrielli, and G. Caldarelli, Nature Reviews Physics1, 58 (2019)

work page 2019

[4] [4]

Newman,Networks(Oxford University Press, 2018)

M. Newman,Networks(Oxford University Press, 2018)

work page 2018

[5] [5]

Karrer and M

B. Karrer and M. E. J. Newman, Physical Review E83, 016107 (2011)

work page 2011

[6] [6]

K. Rohe, T. Qin, and B. Yu, Proceedings of the National Academy of Sciences113, 12679 (2016)

work page 2016

[7] [7]

D. L. Sussman, M. Tang, and C. E. Priebe, Universally consistent latent position estimation and vertex classification for random dot product graphs (2012), arXiv:1207.6745 [stat.ML]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[8] [8]

M. E. J. Newman, Proceedings of the National Academy of Sciences98, 404 (2001)

work page 2001

[9] [9]

Masuda and R

N. Masuda and R. Lambiotte,A Guide to Temporal Networks, Complexity Science, Vol. 4 (World Scientific, Singapore, 2016)

work page 2016

[10] [10]

Holme and J

P. Holme and J. Saram¨ aki, Physics Reports519, 97 (2012)

work page 2012

[11] [11]

V´ azquez, B

A. V´ azquez, B. R´ acz, A. Luk´ acs, and A.-L. Barab´ asi, Physical Review Letters98, 158702 (2007). 12 FIG. 6: Block-to-block edge count comparison for Enron TRAIN: (left) empirical unique edges, (right) expected under the Block–DC–DWCM constraint. FIG. 7: Inter-event time distributions (linear scale) for the Enron TRAIN dataset, comparing empirical dat...

work page 2007

[12] [12]

Hiraoka, N

T. Hiraoka, N. Masuda, A. Li, and H.-H. Jo, Physical Review Research2, 023073 (2020)

work page 2020

[13] [13]

Goh and A.-L

K.-I. Goh and A.-L. Barab´ asi, Europhysics Letters81, 48002 (2008)

work page 2008

[14] [14]

Karsai, K

M. Karsai, K. Kaski, A.-L. Barab´ asi, and J. Kert´ esz, Scientific Reports2, 397 (2012)

work page 2012

[15] [15]

D. R. Cox, Methuen & Co. Ltd., London (1962)

work page 1962

[16] [16]

Masuda and P

N. Masuda and P. Holme, Social Networks37, 12 (2014)

work page 2014

[17] [17]

Ogata, Journal of the American Statistical Association83, 9 (1988)

Y. Ogata, Journal of the American Statistical Association83, 9 (1988)

work page 1988

[18] [19]

Bacry and J.-F

E. Bacry and J.-F. Muzy, IEEE Transactions on Information Theory62, 2184 (2016)

work page 2016

[19] [20]

Filimonov and D

V. Filimonov and D. Sornette, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics85, 056108 (2012)

work page 2012

[20] [21]

G. V. Clemente, C. J. Tessone, and D. Garlaschelli, arXiv preprint arXiv:2311.16981 (2023). 13 FIG. 8: Raster plots of the top-5 intra-block event sequences for Enron TRAIN: (left) empirical; (right) blockpair–PL model (samples=00). FIG. 9: Enron TRAIN lambda over time Poisson CM ME

work page internal anchor Pith review Pith/arXiv arXiv 2023

[21] [22]

Achab, E

M. Achab, E. Bacry, S. Ga¨ ıffas, I. Mastromatteo, and J.-F. Muzy, Journal of Machine Learning Research18, 1 (2017)

work page 2017

[22] [23]

Soliman, L

H. Soliman, L. Zhao, Z. Huang, S. Paul, and K. S. Xu, inProceedings of the 39th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 162 (PMLR, 2022) pp. 20329–20346

work page 2022

[23] [24]

A. G. Hawkes, Biometrika58, 83 (1971)

work page 1971

[24] [25]

Bacry, I

E. Bacry, I. Mastromatteo, and J.-F. Muzy, Market Microstructure and Liquidity1, 1550005 (2015)

work page 2015

[25] [26]

D. J. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes: Volume I: Elementary Theory and Methods, 2nd ed. (Springer, New York, 2003). 14 FIG. 10: Enron TRAIN lambda over time GH TABLE III: Comparison of empirical and model-generated temporal-network statistics. Values are shown as mean ± standard deviation across Monte Carlo samples...

work page 2003