Generalized Poisson Dynamic Network Models

Antonio Peruzzi; Giulia Carallo; Roberto Casarin

arxiv: 2604.05838 · v2 · submitted 2026-04-07 · 📊 stat.ME · econ.EM

Generalized Poisson Dynamic Network Models

Giulia Carallo , Roberto Casarin , Antonio Peruzzi This is my paper

Pith reviewed 2026-05-10 18:37 UTC · model grok-4.3

classification 📊 stat.ME econ.EM

keywords dynamic network modelsgeneralized Poisson distributionoverdispersionunderdispersioncount-weighted networkstemporal networksBayesian inferencelatent factor dynamics

0 comments

The pith

Generalized Poisson dynamic network models capture under- and overdispersion in temporal edge counts beyond latent factors.

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

Count-weighted temporal networks often show unequal dispersion in edge weights that cannot be fully explained by latent factors alone in the conditional mean. The paper develops new model classes that embed the Generalized Poisson distribution into three dynamic specifications to handle both underdispersion and overdispersion explicitly. These specifications are latent factor dynamics, autoregressive dynamics, and latent position dynamics. Theoretical results show how the dispersion parameter shapes network connectivity, and Bayesian inference is developed to estimate the models. Applications to bike-sharing and media networks demonstrate improved in-sample fit and out-of-sample performance when dispersion is modeled directly.

Core claim

We propose new dynamic network model classes exploiting the Generalized Poisson distribution to capture both under- and overdispersion in count-weighted temporal networks, considering latent factor, autoregressive, and latent position dynamics, establishing the impact of the dispersion parameter on network connectivity, providing a Bayesian inference procedure, and showing through applications that explicit dispersion modeling reduces misspecification bias and improves fit and forecasting.

What carries the argument

Generalized Poisson distribution integrated with latent factor, autoregressive, or latent position dynamics, where the dispersion parameter adjusts variance independently of the mean to model edge-weight heterogeneity.

If this is right

The dispersion parameter influences the connectivity properties of the random networks generated by the models.
Neglecting unequal dispersion produces quantifiable misspecification bias in parameter estimates and predictions.
Explicit modeling of dispersion yields better in-sample fit for temporal count networks.
Out-of-sample performance improves for forecasting in datasets such as bike-sharing and media interactions.
The Bayesian posterior sampling algorithm enables practical estimation and inference under the new models.

Where Pith is reading between the lines

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

The connectivity results could guide simulation experiments that generate synthetic networks with controlled levels of dispersion for testing other network algorithms.
Similar dispersion adjustments might extend to non-network count time series where latent factors are already used but variance remains heterogeneous.
The models could inform policy analysis in systems like bike-sharing by revealing how overdispersion signals uneven usage patterns across stations over time.

Load-bearing premise

The Generalized Poisson distribution combined with the chosen dynamic specifications sufficiently captures dispersion in edge weights beyond what latent factors alone can explain, without introducing misspecification in the dynamics.

What would settle it

If posterior predictive checks or out-of-sample forecasts on the bike-sharing or media network data show no improvement in accuracy when using the Generalized Poisson models compared to standard Poisson versions with the same dynamics.

Figures

Figures reproduced from arXiv: 2604.05838 by Antonio Peruzzi, Giulia Carallo, Roberto Casarin.

**Figure 2.** Figure 2: Sensitivty Analysis: Value of θ ∈ (−1, 1) (vertical axis) as a function of σ 2 α ∈ (0, 0.2) (horizontal axis). Left: for a given dispersion index value of 0.5 ( ) and 1.5 ( ) and network size N = 10 ( , ), N = 20 ( , ) and N = 22 ( , ). Right: for a given average expected strength of 10 ( ) and 20 ( ) and factor ft = log(10) ( , ), ft = log(20) ( , ) and ft = log(32) ( , ). are compatible with overdispersi… view at source ↗

**Figure 3.** Figure 3: Estimation Bias: Posterior Density ( ) and true value ( ) of f1 for M1, δ1 + δ2 for M2, and x1,1,1 for M3, for the Poisson specification (top) and the Generalized Poisson specification (bottom) when data have been generated assuming a GP with overdispersion. 5.1 Citibike dataset In the first application, we consider the Citibike dataset Citibike (2019), which contains information on rides between any two s… view at source ↗

**Figure 4.** Figure 4: Citi Bike Network, Model M3 Results: (a) posterior mean of the αi parameter associated to each of the 61 neighborhoods. The larger the NTA dot size, the more prominent the neighborhood in the Citi Bike network ( - ). (b) time series of the observed average strength ( ) against the posterior mean ( ) with its 95% credible interval ( ). (c)-(d) latent space representation of the neighborhoods with d = 2 in A… view at source ↗

**Figure 5.** Figure 5: Media Network, Model M3 Results: (a) Geographical distribution of the news outlets for France with node color and size proportional to the posterior mean of αi ( - ). (b) Posterior mean of the latent coordinates with node color proportional to the posterior mean of αi (January 2016). color proportional to αi . As expected, we find that national news outlets, often located in the most prominent cities of th… view at source ↗

read the original abstract

Count-weighted temporal networks often exhibit unequal dispersion in the edge weights, which cannot be fully explained by modelling observational heterogeneity through latent factors in the conditional mean. Therefore, we propose new dynamic network model classes exploiting the Generalized Poisson distribution to capture both under- and overdispersion. We consider three different dynamic specifications: latent factor dynamics, autoregressive dynamics, and latent position dynamics, and study some theoretical properties of the random networks, showing the impact of the dispersion parameter on the random network's connectivity. After discussing the parameter identification strategy, we present a Bayesian inference procedure along with a posterior sampling algorithm. A numerical illustration demonstrates the effectiveness of the designed algorithm and provides estimates of the misspecification bias when unequal dispersion is neglected. Our new models are then applied to two relevant dynamic datasets considered in previous studies: a set of bike-sharing dynamic networks and a set of dynamic media networks. Our results highlight the importance of explicitly modeling overdispersion for both an accurate in-sample fit and out-of-sample performance.

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 adds Generalized Poisson edge weights to three dynamic network setups to handle dispersion in temporal count data, with connectivity results and two applications.

read the letter

This paper puts the Generalized Poisson distribution into dynamic network models for count edges so that under- and overdispersion can be captured separately from mean heterogeneity. It works with three specifications: latent factor dynamics, autoregressive dynamics, and latent position dynamics. The authors derive how the dispersion parameter affects network connectivity, lay out an identification strategy, and give a Bayesian sampler with a posterior algorithm. A small numerical study shows the bias that appears when dispersion is ignored, and the models are fit to bike-sharing networks and media networks, where they improve both in-sample fit and out-of-sample performance over standard Poisson versions. The core construction is self-contained and the applications serve as direct checks on the modeling choice. The soft spots are limited. The connectivity derivations are sketched rather than fully expanded, and the empirical gains rest on two specific datasets without broad Monte Carlo experiments across many network sizes or dependence structures. Identification for the latent-position version could use more practical checks. This is useful for people who already work with temporal count networks in transportation or communication data and need a way to model extra dispersion without adding more latent factors. It deserves a serious referee because the idea is straightforward, the sampler is supplied, and the real-data results are reported plainly.

Referee Report

2 major / 2 minor

Summary. The manuscript develops three classes of dynamic network models (latent factor, autoregressive, and latent position) that employ the Generalized Poisson distribution for edge weights in count-weighted temporal networks. This allows modeling of both under- and overdispersion beyond what can be captured by latent factors in the conditional mean. The authors derive theoretical properties on how the dispersion parameter affects network connectivity, outline an identification strategy, present a Bayesian inference procedure with a posterior sampling algorithm, provide numerical illustrations of the sampler and misspecification bias from ignoring dispersion, and apply the models to bike-sharing and dynamic media networks to show gains in in-sample fit and out-of-sample performance.

Significance. If the central claims hold, the work supplies a flexible and theoretically grounded extension for dynamic network analysis where edge counts exhibit unequal dispersion. The explicit treatment of connectivity as a function of the dispersion parameter, combined with the two real-data applications and the numerical checks on bias, strengthens the contribution for fields such as transportation and media studies. The separation of dispersion modeling from mean heterogeneity is a clear strength when the identification and inference steps are reliable.

major comments (2)

[Theoretical properties section] Theoretical properties section: the claimed impact of the dispersion parameter on random-network connectivity is presented as general across the three dynamic specifications, but the derivation should explicitly show whether (and how) this impact remains uniform when the dynamics are latent-factor, autoregressive, or latent-position; without that, the connectivity results risk being specification-dependent.
[Numerical illustration] Numerical illustration: the reported estimates of misspecification bias when unequal dispersion is neglected require a precise definition of the bias metric (e.g., parameter recovery error or predictive log-score) and the exact simulation design (network size, time length, true dispersion values); without these, it is difficult to judge the practical magnitude of the bias or to replicate the exercise.

minor comments (2)

[Abstract] The abstract refers to 'a set of bike-sharing dynamic networks' without stating the number of time periods or nodes; adding these details would improve context for readers.
[Bayesian inference procedure] The description of the posterior sampling algorithm would benefit from a short statement on convergence diagnostics (e.g., trace plots or effective sample size) in the numerical section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive recommendation for minor revision. The comments are constructive and will help improve the clarity of our theoretical and numerical results. We address each major comment point by point below.

read point-by-point responses

Referee: [Theoretical properties section] Theoretical properties section: the claimed impact of the dispersion parameter on random-network connectivity is presented as general across the three dynamic specifications, but the derivation should explicitly show whether (and how) this impact remains uniform when the dynamics are latent-factor, autoregressive, or latent-position; without that, the connectivity results risk being specification-dependent.

Authors: We appreciate this comment. The connectivity properties are derived from the edge-weight distribution under the Generalized Poisson model, which is shared across all three dynamic specifications; the dispersion parameter therefore affects connectivity independently of the particular dynamics placed on the conditional mean. To eliminate any potential ambiguity, we will revise the theoretical properties section to explicitly note this generality and, where helpful, add a short verification that the same connectivity expressions apply under each of the latent-factor, autoregressive, and latent-position specifications. revision: yes
Referee: [Numerical illustration] Numerical illustration: the reported estimates of misspecification bias when unequal dispersion is neglected require a precise definition of the bias metric (e.g., parameter recovery error or predictive log-score) and the exact simulation design (network size, time length, true dispersion values); without these, it is difficult to judge the practical magnitude of the bias or to replicate the exercise.

Authors: We agree that additional detail is warranted for transparency and replicability. In the revised manuscript we will state the exact bias metric employed (parameter-recovery error together with predictive log-score) and supply the complete simulation design, including network size, time length, and the specific true dispersion values used in the Monte Carlo exercise. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript defines three dynamic network classes with Generalized Poisson edge weights, derives connectivity properties directly from the model's probability mass function and parameter definitions, states an identification strategy based on standard moment conditions for the dispersion parameter, and supplies a Bayesian sampler. These steps are self-contained mathematical constructions and do not reduce any claimed prediction or theoretical result to a fitted input by construction. Numerical checks and applications serve as external validation rather than tautological restatements. No self-citation chains, ansatz smuggling, or renaming of known results appear as load-bearing elements in the derivation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central modeling choice introduces a dispersion parameter as a free parameter to be estimated from data; the core assumption is that edge weights are conditionally Generalized Poisson given the dynamics.

free parameters (1)

dispersion parameter
Extra parameter in Generalized Poisson that controls under- or overdispersion and is fitted to the network data.

axioms (1)

domain assumption Edge weights follow a Generalized Poisson distribution conditional on the latent dynamics or factors.
Fundamental modeling assumption enabling the new model classes.

pith-pipeline@v0.9.0 · 5464 in / 1145 out tokens · 29045 ms · 2026-05-10T18:37:43.448860+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.

We propose new dynamic network model classes exploiting the Generalized Poisson distribution to capture both under- and overdispersion... three different dynamic specifications: latent factor dynamics, autoregressive dynamics, and latent position dynamics
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Proposition 3 (Node Centrality)... Bernstein concentration inequalities... spectral radius ϱ(Yt)

What do these tags mean?

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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.
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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

14 extracted references · 14 canonical work pages

[1]

Aldous, D. J. (2004), A tractable complex network model based on the stochastic mean-field model of distance,in‘Complex Networks’, Springer, pp. 51–87. Aldous, D. & Pitman, J. (1998), ‘Tree-valued markov chains derived from galton-watson processes’,Annales de l’IHP Probabilité et Statistiques34(5), 637–686. Ambagaspitiya, R. & Balakrishnan, N. (1994), ‘On...

work page 2004
[2]

Betzel, R. F. & Bassett, D. S. (2017), ‘Generative models for network neuroscience: prospects and promise’,Journal of The Royal Society Interface14(136), 20170623. Boatwright, P., Borle, S., Kadane, J. B., Minka, T. P. & Shmueli, G. (2006), ‘Conjugate analysis of the Conway-Maxwell-Poisson distribution’,Bayeian Analysis1(2), 00–00. Bräuning, F. & Koopman,...

work page arXiv 2017
[3]

& Moyeed, R

Shahtahmassebi, G. & Moyeed, R. (2016), ‘An application of the generalized Poisson difference distribution to the Bayesian modelling of football scores’,Statistica Neerlandica 70(3), 260–273. Shang, Y. (2009), Exponential random geometric graph process models for mobile wireless networks,in‘2009 International conference on cyber-enabled distributed comput...

work page 2016
[4]

For simplicity, we drop the time index. Let us assumeYij∼ GP(λij,θ),i,j= 1,...,n i̸=jis a sequence of GP variables, with pmf Pr({Yij =y ij}) =λij(λij +θyij)yij−1exp(−(λij +θyij)) yij! , y ij = 0,1,2,..., Following Ambagaspitiya & Balakrishnan (1994), Eq. 2.5, the probability generating function ofYij is GYij(z) =E[z Yij] = exp (−λij θ(W(−θzexp(−θ)) +θ)), ...

work page 1994
[5]

1 is bounded in a neighborhood of 0 and then by applying Theorem 2.7.1 iv) in Vershynin (2018)

i)The proof follows from showing the mgf in Eq. 1 is bounded in a neighborhood of 0 and then by applying Theorem 2.7.1 iv) in Vershynin (2018). SinceW (z) in the mgf is bounded forz≥−1/e then the mgf is bounded for−θexp(u−θ)≥−1/e that is foru∈(−∞,θ−log(θ)). ii)We expand the logarithm of the moment generating function (mgf) and bound the residual term. Let...

work page 2018
[6]

For an alternative derivation method, see Consul (1989)

(1−θ)7 . For an alternative derivation method, see Consul (1989). The Taylor expansion of the log–mgf, for|u|<u0(θ), isg(u) =κ1u+κ2 2u2+κ3 6u3+R4(u)with remainderR4(u) = g(4)(ξ) 24 u4 for someξ∈(0,u). SinceE[Y] =κ1, then logE[exp(u(Y−κ1))] = κ2 2 u2 + κ3 6 u3 +R 4(u)≤κ2 2 u2 +|κ3| 6 |u|3 + Br(θ,λ) 24 |u|4, where we definedBr(θ,λ) = sup{|g(4)(u)|:|u|≤r}for...

work page 1989
[7]

The result in (Vershynin 2018, Lemma 2.7.6, ii)) and the log-mgf bound in Lemma 1 gives ||Y||ψ1≤C (√v+b ) =C( √ λ (1−θ)3/2 + |2θ+ 1| 3(1−θ)2 + (1−θ)3r 12λ Br(θ,λ))

SincethedefinitionofOrlicznorm ||Z||ψ1 = inf{K >0 :E(exp(|Z|/K))≤2}, requires E(exp(|Z|/K))≤2, the log-mgf bound in Lemma 1 can be used to derive, through Chernoff bound and tail-integral representation, a bound for the Orlicz norm. The result in (Vershynin 2018, Lemma 2.7.6, ii)) and the log-mgf bound in Lemma 1 gives ||Y||ψ1≤C (√v+b ) =C( √ λ (1−θ)3/2 +...

work page 2018
[8]

Thus, the conditions of (Tropp et al

In conclusion, the matrixXt is self-adjoint (symmetric), has zero mean, and sub-exponential tails. Thus, the conditions of (Tropp et al. 2015, Sec

work page 2015
[9]

See also (Vershynin 2018, th

are satisfied, and a Bernstein matrix inequality can be obtained as shown in the following. See also (Vershynin 2018, th. 2.8.1 and th. 5.4.1). 25 We follow the same procedure used in the proof of Th. 6.5.1 of Vershynin (2018) for matrices with non-i.i.d. entries and decomposeXt as: Xt = N∑ i=1 N∑ j̸=i XijtEij, where Eij =e ie′ j withe i i = 1,...,N the s...

work page 2018
[10]

The cumulant function is defined asKt(u) = logMt(u) =−λtφ(g(u))

Define, for parametersλ>0and θ >0,Mt(u) = E( ¯St|α∨F∗ t−1) is Mt(u) = exp ( −λtφ(g(u)) ) , φ(g) =W(g) +θ θ , g(u) =−θexp(−θ+u), where W (x)denotes the Lambert’sW-function (principal branch). The cumulant function is defined asKt(u) = logMt(u) =−λtφ(g(u)). In order to find the find the expected value of the cumulant we proceed as follows: i) apply the Faà ...

work page 2022
[11]

Denote withιa∈Ra the unit vector and withIa the a×aidentity matrix

From the first-order log-Taylor expansion of the GP likelihood given in Proposition 9, the full conditional in 8 can be approximated by q(xit)∝exp(−1 2(x′ itΣ−1 x xit−2x′ itΣ−1 x xi,t−1+x′ itΣ−1 x xit−2x′ itΣ−1 x xi,t+1)) exp((xit−˜x)′Cit(˜x)) (A.27) ∝exp(−1 2(2x′ itΣ−1 x xit−2x′ itΣ−1 x (xi,t−1+xi,t+1 + ΣxCit(˜x))∝Nd((xi,t−1+xi,t+1 + ΣxCit(˜x))/2,Σ x/2) ...

work page 2009
[12]

Graphical inspection of Figures B.1-B.2 reveals convergence to the true parameter values and a good mixing of the MCMC chains

We run the algorithm for 5,000 iterations, using the first 2,000 as burn-in samples and thinning every 5 iterations. Graphical inspection of Figures B.1-B.2 reveals convergence to the true parameter values and a good mixing of the MCMC chains. We also report in Tab. B.1 some standard convergence diagnostics (ESS, Geweke CD), which confirm the findings of ...

work page 2000
[13]

The bigger the dot size of each NTA, the more prominent the neighbor- hood in the Citi Bike network (- )

The size of the nodes is proportional toαi. The bigger the dot size of each NTA, the more prominent the neighbor- hood in the Citi Bike network (- ). 38 40.65°N 40.70°N 40.75°N 40.80°N 74.04°W 74.02°W 74.00°W 73.98°W 73.96°W 73.94°W 73.92°W 73.90°W Longitude Latitude 20000 30000 40000 50000 60000 Jan 2019 Apr 2019 Jul 2019 Oct 2019 Time Avg. Expected Stre...

work page 2019
[14]

and posterior estimates of the time-varying latent factorft for France (top) and Germany (bottom). 44 Alto.AdigeANSA.it Avvenire Beppe.Grillo Corriere.Adriatico.it Corriere.del.Mezzogiorno Corriere.della.Sera Gazzetta.di.Mantova Gazzetta.di.Modena Gazzetta.di.Parma Gazzetta.di.Reggio Giornale.di.Brescia Giornale.di.Sicilia HuffPost.Italia Il.Fatto.Quotidi...

work page 2015

[1] [1]

Aldous, D. J. (2004), A tractable complex network model based on the stochastic mean-field model of distance,in‘Complex Networks’, Springer, pp. 51–87. Aldous, D. & Pitman, J. (1998), ‘Tree-valued markov chains derived from galton-watson processes’,Annales de l’IHP Probabilité et Statistiques34(5), 637–686. Ambagaspitiya, R. & Balakrishnan, N. (1994), ‘On...

work page 2004

[2] [2]

Betzel, R. F. & Bassett, D. S. (2017), ‘Generative models for network neuroscience: prospects and promise’,Journal of The Royal Society Interface14(136), 20170623. Boatwright, P., Borle, S., Kadane, J. B., Minka, T. P. & Shmueli, G. (2006), ‘Conjugate analysis of the Conway-Maxwell-Poisson distribution’,Bayeian Analysis1(2), 00–00. Bräuning, F. & Koopman,...

work page arXiv 2017

[3] [3]

& Moyeed, R

Shahtahmassebi, G. & Moyeed, R. (2016), ‘An application of the generalized Poisson difference distribution to the Bayesian modelling of football scores’,Statistica Neerlandica 70(3), 260–273. Shang, Y. (2009), Exponential random geometric graph process models for mobile wireless networks,in‘2009 International conference on cyber-enabled distributed comput...

work page 2016

[4] [4]

For simplicity, we drop the time index. Let us assumeYij∼ GP(λij,θ),i,j= 1,...,n i̸=jis a sequence of GP variables, with pmf Pr({Yij =y ij}) =λij(λij +θyij)yij−1exp(−(λij +θyij)) yij! , y ij = 0,1,2,..., Following Ambagaspitiya & Balakrishnan (1994), Eq. 2.5, the probability generating function ofYij is GYij(z) =E[z Yij] = exp (−λij θ(W(−θzexp(−θ)) +θ)), ...

work page 1994

[5] [5]

1 is bounded in a neighborhood of 0 and then by applying Theorem 2.7.1 iv) in Vershynin (2018)

i)The proof follows from showing the mgf in Eq. 1 is bounded in a neighborhood of 0 and then by applying Theorem 2.7.1 iv) in Vershynin (2018). SinceW (z) in the mgf is bounded forz≥−1/e then the mgf is bounded for−θexp(u−θ)≥−1/e that is foru∈(−∞,θ−log(θ)). ii)We expand the logarithm of the moment generating function (mgf) and bound the residual term. Let...

work page 2018

[6] [6]

For an alternative derivation method, see Consul (1989)

(1−θ)7 . For an alternative derivation method, see Consul (1989). The Taylor expansion of the log–mgf, for|u|<u0(θ), isg(u) =κ1u+κ2 2u2+κ3 6u3+R4(u)with remainderR4(u) = g(4)(ξ) 24 u4 for someξ∈(0,u). SinceE[Y] =κ1, then logE[exp(u(Y−κ1))] = κ2 2 u2 + κ3 6 u3 +R 4(u)≤κ2 2 u2 +|κ3| 6 |u|3 + Br(θ,λ) 24 |u|4, where we definedBr(θ,λ) = sup{|g(4)(u)|:|u|≤r}for...

work page 1989

[7] [7]

The result in (Vershynin 2018, Lemma 2.7.6, ii)) and the log-mgf bound in Lemma 1 gives ||Y||ψ1≤C (√v+b ) =C( √ λ (1−θ)3/2 + |2θ+ 1| 3(1−θ)2 + (1−θ)3r 12λ Br(θ,λ))

SincethedefinitionofOrlicznorm ||Z||ψ1 = inf{K >0 :E(exp(|Z|/K))≤2}, requires E(exp(|Z|/K))≤2, the log-mgf bound in Lemma 1 can be used to derive, through Chernoff bound and tail-integral representation, a bound for the Orlicz norm. The result in (Vershynin 2018, Lemma 2.7.6, ii)) and the log-mgf bound in Lemma 1 gives ||Y||ψ1≤C (√v+b ) =C( √ λ (1−θ)3/2 +...

work page 2018

[8] [8]

Thus, the conditions of (Tropp et al

In conclusion, the matrixXt is self-adjoint (symmetric), has zero mean, and sub-exponential tails. Thus, the conditions of (Tropp et al. 2015, Sec

work page 2015

[9] [9]

See also (Vershynin 2018, th

are satisfied, and a Bernstein matrix inequality can be obtained as shown in the following. See also (Vershynin 2018, th. 2.8.1 and th. 5.4.1). 25 We follow the same procedure used in the proof of Th. 6.5.1 of Vershynin (2018) for matrices with non-i.i.d. entries and decomposeXt as: Xt = N∑ i=1 N∑ j̸=i XijtEij, where Eij =e ie′ j withe i i = 1,...,N the s...

work page 2018

[10] [10]

The cumulant function is defined asKt(u) = logMt(u) =−λtφ(g(u))

Define, for parametersλ>0and θ >0,Mt(u) = E( ¯St|α∨F∗ t−1) is Mt(u) = exp ( −λtφ(g(u)) ) , φ(g) =W(g) +θ θ , g(u) =−θexp(−θ+u), where W (x)denotes the Lambert’sW-function (principal branch). The cumulant function is defined asKt(u) = logMt(u) =−λtφ(g(u)). In order to find the find the expected value of the cumulant we proceed as follows: i) apply the Faà ...

work page 2022

[11] [11]

Denote withιa∈Ra the unit vector and withIa the a×aidentity matrix

From the first-order log-Taylor expansion of the GP likelihood given in Proposition 9, the full conditional in 8 can be approximated by q(xit)∝exp(−1 2(x′ itΣ−1 x xit−2x′ itΣ−1 x xi,t−1+x′ itΣ−1 x xit−2x′ itΣ−1 x xi,t+1)) exp((xit−˜x)′Cit(˜x)) (A.27) ∝exp(−1 2(2x′ itΣ−1 x xit−2x′ itΣ−1 x (xi,t−1+xi,t+1 + ΣxCit(˜x))∝Nd((xi,t−1+xi,t+1 + ΣxCit(˜x))/2,Σ x/2) ...

work page 2009

[12] [12]

Graphical inspection of Figures B.1-B.2 reveals convergence to the true parameter values and a good mixing of the MCMC chains

We run the algorithm for 5,000 iterations, using the first 2,000 as burn-in samples and thinning every 5 iterations. Graphical inspection of Figures B.1-B.2 reveals convergence to the true parameter values and a good mixing of the MCMC chains. We also report in Tab. B.1 some standard convergence diagnostics (ESS, Geweke CD), which confirm the findings of ...

work page 2000

[13] [13]

The bigger the dot size of each NTA, the more prominent the neighbor- hood in the Citi Bike network (- )

The size of the nodes is proportional toαi. The bigger the dot size of each NTA, the more prominent the neighbor- hood in the Citi Bike network (- ). 38 40.65°N 40.70°N 40.75°N 40.80°N 74.04°W 74.02°W 74.00°W 73.98°W 73.96°W 73.94°W 73.92°W 73.90°W Longitude Latitude 20000 30000 40000 50000 60000 Jan 2019 Apr 2019 Jul 2019 Oct 2019 Time Avg. Expected Stre...

work page 2019

[14] [14]

and posterior estimates of the time-varying latent factorft for France (top) and Germany (bottom). 44 Alto.AdigeANSA.it Avvenire Beppe.Grillo Corriere.Adriatico.it Corriere.del.Mezzogiorno Corriere.della.Sera Gazzetta.di.Mantova Gazzetta.di.Modena Gazzetta.di.Parma Gazzetta.di.Reggio Giornale.di.Brescia Giornale.di.Sicilia HuffPost.Italia Il.Fatto.Quotidi...

work page 2015