A Bayesian Dynamic Latent Space Model for Weighted Networks

A Ba y esian Dynamic Laten t Space Mo del for W eigh ted Net w orks ∗ Rob erto Casarin † Matteo Iacopini ‡ An tonio P eruzzi § 26th March 2026 Abstract A new dynamic laten t space eigenmo del (LSM) is prop osed for w eigh ted tem- p oral net w orks. The model accommo dates integer-v alued w eigh ts, excess of zeros, time-v arying node positions (features), and time-v arying netw ork sparsit y . The laten t p ositions evolv e according to a vector autoregressive pro cess that accounts for lagged and con temp oraneous dep endence across no des and features, a c harac- teristic neglected in the LSM literature. A Ba y esian approac h is used to address t w o of the primary sources of inference in tractabilit y in dynamic LSMs: laten t feature estimation and the c hoice of laten t space dimension. W e employ an ef- ficien t auxiliary-mixture sampler that p erforms data augmentation and supp orts conditionally conjugate prior distributions. A p oint-process representation of the net w ork w eights and the finite-dimensional distribution of the latent pro cesses are used to deriv e a multi-mo ve sampler in whic h each feature tra jectory is dra wn in a single blo c k, without recursions. This sampling strategy is new to the net w ork literature and can significantly reduce computational time while impro ving chain mixing. T o a v oid trans-dimensional samplers, a Laplace approximation of the partial marginal lik eliho o d is used to design a partially collapsed Gibbs sampler. Ov erall, our procedure is general, as it can b e easily adapted to static and dynamic settings, as w ell as to other discrete or contin uous weigh t distributions. Keyw ords: Auxiliary mixture sampler; Bay esian inference; Latent space mo dels; T emp oral net w orks. ∗ Supplemen tary Materials are a v ailable up on request to the authors. † Ca’ F oscari Universit y of V enice, Italy . r.casarin@unive.it ‡ Luiss Univ ersit y , Italy . miacopini@luiss.it § Ca’ F oscari Universit y of V enice, Italy . antonio.peruzzi@unive.it 1 1 In tro duction 1.1 Scop e and Con tribution The last tw o decades ha v e seen an increasing interest in the statistical mo delling of net w ork data, leading to the dev elopment of sev eral classes of netw ork mo dels ( Sosa and Buitrago , 2021 ). Eac h of these classes captures different asp ects of net w orks, suc h as hidden no de clustering or the factors driving connectivit y patterns. An imp ortan t class consists of latent space mo dels (LSM, Hoff et al. , 2002 ; Hoff , 2005 , 2008 ), whic h ha v e b ecome a widely adopted framew ork in a broad range of disciplines, including media studies ( Casarin et al. , 2025 ), neuroscience ( Duran te et al. , 2017 ; W ang et al. , 2025 ), p olitical science ( P ark and Sohn , 2020 ; Y u and Ro driguez , 2021 ), and so cial science ( W ang et al. , 2023 ). In an LSM, eac h no de i ∈ { 1 , . . . , N } is represen ted b y a v ector of laten t features, x i ∈ R d , d ≪ N , in a low-dimensional Euclidean space. The probability or the strength of a connection b etw een no des i and j dep ends on a function f ( x i , x j ) of the no des’ laten t features. Hoff et al. ( 2002 ) prop osed distance-based mo dels, where the probabilit y of a connection is in versely related to the distance b et w een no des’ laten t features, and inner-pro duct mo dels, where the connection probability is link ed to the angle of the no des’ latent features. Later, Hoff ( 2008 ) considered eigenmo dels 1 as a generalization of the latter with in teraction probabilities driv en by f ( x i , x j ) = x ′ i Ξ x j , where Ξ is a diagonal matrix enco ding the level of homophily along each laten t dimension (similarit y in one laten t feature rather than another may imply a higher interaction probability). Originally , LSMs w ere designed to model static binary net works, although more general types of netw ork data ma y b e av ailable. Dynamic w eigh ted netw orks provide a refined picture of the time-v arying strength of the relationships among agents. T o study this class of data, w e prop ose a nov el dynamic latent space eigenmo del for time series of coun t-v alued netw orks, in which each edge enco des the num b er of interactions b etw een an y tw o no des, and w e pro vide a nov el and efficient MCMC algorithm for Bay esian inference. Our prop osal is motiv ated by three relev ant issues. First, most existing approac hes analyse binary data (e.g., see Hoff , 2005 ) whereas w eigh ted netw orks ha ve receiv ed little attention, and efficien t estimation sc hemes are lac king. F urthermore, count net- w orks frequen tly display an excess of zeros. Second, single-mo v e Metrop olis-Hastings algorithms (MH) for p osterior inference on the latent v ariables are usually implemen- 1 Eigenmo dels can accommodate b oth p ositive and negative relationships betw een en tities, unlik e distance models. Moreo ver, although laten t distance mo dels may app ear simple to understand, the term f ( x i , x j ) = x ′ i x j in eigenmo dels is conceptually app ealing being in terpretable as a random effect. 2 ted for distance-based dynamic LSMs (e.g., see Sewell and Chen , 2015 ) whic h ma y b e highly inefficien t. Third, the complexit y and p erformance of an LSM are driven by the dimension of the laten t space, d , which is typically fixed or c hosen using an information criterion without an y uncertain ty quantification ( Loy al and Chen , 2023 ). The main con tributions of this article can b e summarised as follo ws. W e propose a new dynamic zero-inflated laten t space eigenmo del for coun t net works, a class that has b een less explored due to the significant computational challenges p osed b y the lac k of conjugacy in P oisson regression mo dels. W e address the computational issues of distance-based dynamic mo dels b y adopting a more general eigenmo del sp ecification. Then, we leverage the auxiliary mixture sampler of ( F rüh wirth-Sc hnatter and W agner , 2006 ; F rühwirth-Sc hnatter et al. , 2009 ) to obtain a conditionally linear Gaussian lik eli- ho o d. T o the b est of our knowledge, this is the first time such a scheme has b een used in LSM and, more generally , in multiv ariate count time-series mo delling. T raditional LSMs imp ose indep endence b et ween no des and features (called no de-wise parametrisation), thereb y ignoring imp ortant c haracteristics of real-w orld data, in whic h actors may share similar features for geographical, cultural, or other reasons. T o account for them, we prop ose a fe atur e-wise parametrisation that allo ws for con temp oraneous dep endence among no des in the laten t space. F rom a computational p ersp ectiv e, w e impro ve the speed and mixing by sampling directly from the smoothed distribution of no de-sp ecific latent features, without lo ops ( McCausland et al. , 2011 ). W e leverage the marginal data augmen tation of Zens et al. ( 2024 ) to enhance the mixing of the probabilit y of structural zeros. A partially collapsed Gibbs sampler ( V an Dyk and P ark , 2008 ) is designed to make inference on the laten t space dimension d without requiring any trans-dimensional mo v e. The efficiency of the prop osed metho dology is assessed via an extensive sim ulation study . Three real- life applications to United Nations v oting, in ternational trade, and brain connectivit y illustrate the p oten tial of our mo delling approac h. It is w orth emphasising that the prop osed metho dologies and MCMC algorithms can b e easily simplified to the static case, whic h represen ts an additional contribution of this article. 1.2 Related literature The ma jority of LSMs in the statistics and mac hine learning literature are static and analyze a single, binary netw ork. Ov er the last decade, sev eral metho ds ha v e b een prop osed to in v estigate temp oral netw orks. Sark ar and Mo ore ( 2005 ) were among the first to design a dynamic LSM for binary net w orks. Later, Durante and Dunson ( 2014 ) prop osed a Bay esian nonparametric approach to infer laten t co ordinates evolving in 3 con tin uous time via Gaussian pro cesses. In a distance-based mo del, Sewell and Chen ( 2015 , 2016 ) used a single-mo v e MH step to infer the path of the time-v arying features, whereas T urn bull et al. ( 2023 ) designed a sequential Monte Carlo pro cedure. Recently , P a v one et al. ( 2025 ) dev elop ed a ph ylogenetic laten t space mo del to c haracterize the generativ e pro cess of the no des’ feature v ectors. Mo dels for coun t netw orks are more scan t in the literature. In a static con text, Lu et al. ( 2025 ) prop osed t w o zero-inflated P oisson latent space sp ecifications. The former pro vides a laten t-space represen tation of the matrix of structural zeros, while the latter enables no de clustering and offers a strategy for imputing non-observ ed interactions. In the context of temp oral net w orks, Artico and Wit ( 2023 ) introduced a dynamic distance- based LSM for count net works in both contin uous- and discrete-time settings. Their approac h relies on a likelihoo d appro ximation to implement an unscented Kalman filter for the laten t features, and on an EM algorithm to estimate static parameters. More recen tly , Kaur and Rastelli ( 2024 ) use on a P oisson log-linear autoregressive mo del augmen ted b y an LSM, whereas Casarin et al. ( 2025 ) prop osed a dynamic P oisson laten t distance mo del featuring Mark ov-switc hing dynamics. Our con tribution to this literature is a no vel dynamic zero-inflated latent space ei- genmo del for time series of count-v alued netw orks (a static mo del is obtained as a sp ecial case). Differen t from existing approaches, our prop osal mo dels the link probability as a function of similarit y , which pro vides a more general framework than a distance-based approac h (see Hoff , 2008 , for a pro of ). One of the main c hallenges in Bay esian estimation of P oisson regressions is the lac k of (conditionally) conjugate prior distributions, whic h requires tailored MH or Hamiltonian Mon te Carlo algorithms. This c hallenge is exacerbated when a Mark o vian sto c hastic pro cess is assumed to drive the Poisson intensit y , leading to a nonlinear, non-Gaussian state-space mo del. Existing metho ds based on single-mo v e MH steps to sample the laten t in tensit y path are b oth computationally intensiv e and p otentially slow-mixing. Recen tly , King and K o w al ( 2025 ) prop osed a semiparametric approac h to mo del m ul- tiv ariate time series of counts, by discretising a nonlinear transform of a laten t Gaussian dynamic linear mo del. One of the main drawbac ks of this approach is a computational b ottlenec k arising from sampling laten t v ariables from a m ultiv ariate Gaussian distri- bution truncated to an N -dimensional h yp ercub e, which b ecomes increasingly sev ere as the probability mass in the region diminishes. D’Angelo and Canale ( 2023 ) pro- p osed a Pólya-Gamma data augmen tation sc heme for Poisson regression, leveraging the limiting connection b et w een the Negativ e Binomial and the Poisson distribution. The resulting approximation is simple, but its computational cost can b e v ery high. A fur- ther approximation step is in tro duced; as highlighted by the authors, the accuracy and 4 computational sp eed nevertheless dep end hea vily on tuning parameters that are diffi- cult to set. An alternativ e approac h has been dev elop ed b y F rühwirth-Sc hnatter and W agner ( 2006 ) and F rühwirth-Sc hnatter et al. ( 2009 ), who introduced the improv ed auxiliary mixture sampler (IAMS) as a tw o-step data augmentation strategy for Pois- son regression mo dels. Theoretical prop erties of the sampler and p ossible improv emen t ha v e been discussed in Gardini et al. ( 2026 ). The adv antage of this approac h is that the final augmented mo del has a linear-Gaussian (conditional) lik eliho o d, allowing con- jugate Gaussian prior distributions to b e defined ov er the mo del’s p arameters, which is particularly app ealing for high-dimensional m ultiple-equation mo dels. The c hoice of the latent dimension d has been a crucial issue in the literature (e.g., see Handco ck et al. , 2007 ; Lu et al. , 2025 ). Some w orks assume a fixed small d (e.g., d = 2 or d = 3 ) to enable a visual represen tation of the latent features ( Gollini and Murph y , 2016 ; Sew ell and Chen , 2016 ), whereas others use an information criterion to select the optimal v alue (e.g., see Handco c k et al. , 2007 ). Ho w ev er, metho ds to prop erly estimate d hav e b een considered only recen tly . Lo y al and Chen ( 2025 ) prop osed to infer the latent space dimension b y specifying a static laten t space eigenmo del. Then, they sp ecified an upp er bound on the dimension of the laten t space, d , and assumed a spik e-and-slab prior distribution for eac h diagonal elemen t of Ξ , with a spike comp onen t corresp onding to a Dirac mass at zero and a shrinking sequence of slab probabilities to induce a sto chastically decreasing ordering of the prior probabilit y of d . Ho wev er, a closed-form expression for the prior distribution of d is not easily obtained. A p oin t estimate of the laten t space dimension is indirectly obtained as the mo de of the p osterior distribution of the allo cation v ariable in tro duced to resolv e the spik e-and-slab mixture. Instead, Gw ee et al. ( 2025 ) proposed a w ay to shrink the laten t dimensions of a static distance-based LSM b y exploiting a multiplicativ e gamma pro cess prior for the precision of the latent features. Despite its effectiv eness, this metho dology is highly sensitiv e to prior hyperparameters and computationally exp ensive for large netw orks. F urthermore, this strategy requires thresholding the v ariance of the laten t co ordinates to obtain an estimate of d . 1.3 Structure and notation The rest of the article is organised as follows. The new mo del sp ecifications are in- tro duced and described in Section 2 . Then, Section 3 presents Bay esian inference and p osterior appro ximation and briefly describ e the p erformance of the prop osed meth- o ds through sim ulated exp eriments whic h hare extensiv ely cov ered in Section C of the Supplemen t. Real-w orld dataset applications are in Section 4 . Section 5 concludes and 5 discusses p ossible extensions. The pro ofs of the results are rep orted in Section A of the Supplemen t. In the follo wing, lo wercase letters denote scalars, b oldface low ercase letters denote (column) v ectors, and upp ercase letters denote matrices. F or a column vector x ∈ R m , let x ′ b e its transp ose and ∥ x ∥ = ∥ x ∥ 2 = ( P m i =1 x 2 i ) 1 / 2 the L 2 -norm. Moreov er, X = diag( x ) ∈ R m × m denotes a diagonal matrix having the elemen ts of the v ector x on the main diagonal. Let S k ++ b e the set of all k × k p ositive definite matrices and let I ( x ∈ S ) b e the indicator function taking v alue 1 if x ∈ S and 0 otherwise. Let I k denote the k -dimensional iden tity matrix and O n × m the n × m n ull matrix, ι k and 0 k the k -dimensional unit and n ull vectors, respectively . Finally , w e denote with δ c ( x ) a Dirac mass at c . W e denote with P oi ( · | λ ) the Poisson distribution with intensit y λ ∈ R + , with N ( · | µ, σ 2 ) the Gaussian distribution with mean µ ∈ R and v ariance σ 2 ∈ R + , with N k ( · | µ , Σ) the multiv ariate Gaussian distribution with mean v ector µ ∈ R k and cov ariance matrix Σ ∈ S k ++ , with MN k,p ( · | M , Σ , Υ) the matrix-v ariate Gaussian distribution with mean matrix M ∈ R k × p , row co v ariance matrix Σ ∈ S k ++ and column co v ariance matrix Υ ∈ S p ++ . 2 A Dynamic Zero-Inflated Coun ts Eigenmo del 2.1 Mo del Consider an observed time series of coun ts, { Y t } t ≥ 0 , with Y t ∈ N N × N b eing a count- v alued matrix. W e prop ose a dynamic latent space mo del for Y t , which is based on the follo wing sp ecification for eac h ( i, j ) th en try of Y t : y ij,t | λ ij,t , z ij,t ind ∼ p ij,t ( z ij,t ) P oi  y ij,t | λ ij,t  + (1 − p ij,t ( z ij,t )) δ { 0 } ( y ij,t ) (1a) log( λ ij,t ) = α i + α j + x ′ i : ,t Ξ x j : ,t (1b) X t = e Φ X t − 1 Φ ′ + H t H t iid ∼ MN N ,d ( H t | O N ,d , e Υ , Υ) (1c) z ij,t = β ′ i v i : ,t + β ′ j v j : ,t + υ ij,t , υ ij,t iid ∼ N ( υ ij,t | 0 , 1) , (1d) where p ij,t ( z ij,t ) = P ( z ij,t > 0) , and z ij,t corresp onds to the latent utilit y v ariable in the utilit y function formulation of the probit ( Alb ert and Chib , 1993 ), v i : ,t ∈ R L is a vector of node-sp ecific observ ed co v ariates, x i : ,t ∈ R d is a v ector of no de-sp ecific unobserv ed dynamic features, X t = ( x 1: ,t , . . . , x N : ,t ) ′ ∈ R N × d is a matrix of laten t v ariables and Ξ = diag ( ξ 1 , . . . , ξ d ) ∈ R d × d is a diagonal matrix of parameters. The inner pro duct in eq. ( 1b ) captures the cosine similarity and postulates that an edge b et w een t wo no des has a higher (count) w eigh t, the more similar their features are. W e assume Ξ = I d and 6 discuss p ossible extensions to a real-v alued diagonal matrix in Section 5 . Eq. ( 1c ) is a matrix autoregressiv e pro cess ( Chen et al. , 2021 ), where e Φ ∈ R N × N and e Υ ∈ S N ++ are resp ectiv ely the ro w autoregressive and ro w co v ariance matrices, while Φ ∈ R d × d and Υ ∈ S d ++ are resp ectively the column autoregressive and column co v ariance matrices. A matrix represen tation of eqs. ( 1a )-( 1d ) is a v ailable in Section A of the Supplement. The laten t space mo del in equations ( 1a ) to ( 1c ) represen ts a nonlinear and non- Gaussian state space mo del. It b elongs to the class of laten t eigenmo dels, whic h includes laten t class and latent distance mo dels as sp ecial cases ( Hoff , 2008 ). As a result, it can represen t b oth sto c hastic equiv alence (i.e., no des can b e divided into groups suc h that mem b ers of the same group hav e similar patterns of relationships) and homophily (i.e., relationships b etw een no des with similar characteristics are stronger), o wing to the fact that it pro vides a lo w-rank approximation to the sociomatrix, and is therefore able to represen t a wide arra y of patterns in the data. This sp ecification is more tractable and computationally more efficient than a Gaussian Pro cess or other non-parametric sp ecifications. Finally , eq. ( 1d ) is the random utility representation of a probit mo del for the (prior) probabilit y of sampling from a random Poisson term (e.g., see Zens et al. , 2024 ). 2.2 Data Augmen tation W e resolve the mixture of the zero-inflated Poisson in eq. ( 1a ) b y in tro ducing the aux- iliary allocation v ariable w ij,t = I ( z ij,t > 0) , such that p ( y ij,t , w ij,t | λ ij,t , z ij,t ) = p ( y ij,t | w ij,t , λ ij,t ) × p ( w ij,t | z ij,t ) is equal to p ( y ij,t | w ij,t , λ ij,t ) =  P oi  y ij,t | λ ij,t   w ij,t  δ { 0 } ( y ij,t )  1 − w ij,t , (2) p ( w ij,t | z ij,t ) = p ij,t ( z ij,t ) w ij,t (1 − p ij,t ( z ij,t )) 1 − w ij,t . (3) F or w ij,t = 1 , one obtains the Poisson laten t eigenmo del. One of the main issues in Pois- son regression mo dels is the lack of conjugate priors for the latent in tensity parameters, λ ij,t , whic h induces a computational b ottlenec k. This issue is further exacerbated in dynamic settings, where the en tire path { λ ij,t } T t =1 m ust b e estimated. T o address this issue, w e exploit the improv ed auxiliary mixture sampler (IAMS) in tro duced by F rühwirth-Sc hnatter et al. ( 2009 ) and F rüh wirth-Sc hnatter and W agner ( 2006 ) to transform our nonlinear non-Gaussian observ ation equation into a (condition- ally) linear Gaussian one. This approac h is a double data augmentation scheme that exploits the property of the P oisson distribution, with intensit y λ t , which is the distri- bution of the total num b er of jumps of a P oisson pro cess with the same in tensit y o ver 7 the unit in terv al [0 , 1] . L emma 1 . Define τ ij,t = ( τ ij, 1 t , τ ij, 2 t ) and r ij,t = ( r ij, 1 t , r ij, 2 t ) and assume w ij,t = 1 , then the Poisson distribution in eq. ( 1a ) is the marginal distribution of p ( y ij,t , τ ij,t , r ij,t | w ij,t = 1 , x i : ,t , x j : ,t , α i , α j ) , which can b e approximated b y q ( y ij,t , τ ij,t , r ij,t | d, w ij,t = 1 , x i : ,t , x j : ,t , α i , α j ) , defined as: g 1 ( τ ij, 1 t ) R ( y ij,t ) Y k =1 c I ( r ij, 1 t = k ) k  g 2 ( τ ij, 2 t ) R ( y ij,t ) Y k =1 c I ( r ij, 2 t = k ) k  I ( y ij t > 0) , (4) where g 1 ( τ ij, 1 t ) and g 2 ( τ ij, 2 t ) are the densities of N ( − log ( τ ij, 1 t ) | log( λ ij,t )+ µ r ij, 1 t (1) , σ 2 r ij, 1 t (1)) and N ( − log( τ ij, 2 t ) | log( λ ij,t ) + µ r ij, 2 t ( y ij,t ) , σ 2 r ij, 2 t ( y ij,t )) , resp ectively , with µ r ij,st ( · ) , σ 2 r ij,st ( · ) , s = 1 , 2 , and c k ( · ) the auxiliary mixture lo cations, scales, and weigh ts. The optimal v alues of the lo cation, scale, and w eigh ts of the auxiliary mixture are tabulated in F rühwirth-Sc hnatter and W agner ( 2006 ) and F rüh wirth-Sc hnatter et al. ( 2009 ). Let us define the collections of auxiliary v ariables τ = { τ ij,t } ij t and r = { r ij,t } ij t , the collections of parameters and latent v ariables α = ( α 1 , . . . , α N ) ′ and x = ( x ′ :: , 1 , . . . , x ′ :: ,T ) ′ resp ectiv elly with x :: ,t = vec( X t ) . Let Q t = { ( i, j ) : i ∈ { 1 , . . . , N } , j ∈ { i + 1 , . . . , N } , w ij,t = 1 } b e the set of edges allo cated to the Poisson comp onent. F rom eq. ( 2 ), ( 3 ), ( 4 ) and eq. ( 1a ), we obtain the complete data likelihoo d p ( y , τ , r , w | x , α ) = T Y t =1 Y ( i,j ) ∈Q t p ( y ij,t , τ ij,t , r ij,t | w ij,t , x i : ,t , x j : ,t , α i , α j ) × Y ( i,j ) / ∈Q t δ { 0 } ( y ij,t ) × Y i 0) . Prop osition 1 requires using a p oint of high p osterior mass for the latent features, b x , to p erform a Laplace approximation. Since the latent features exhibit temp oral dep endence, we prop ose to use the output of the Kalman smo other as v alues for b x i : ,t , i = 1 , . . . , N and t = 1 , . . . , T . Computationally , this approach w ould require running d Kalman smo others at each iteration of the Gibbs sampler, one for each candidate v alue of the latent space dimension. Once a new v alue of d is drawn from eq. ( 10 ), w e sample x : ℓ, : , ℓ = 1 , . . . , d , from the smo othed distribution (i.e., the full conditional p osterior in the static case). Unfortunately , for large net works, b oth single-mo v e MH and forward-filtering backw ard-sampling approaches ( F rühwirth-Sc hnatter , 1994 ) are 11 computationally to o exp ensive, preven ting the use of dynamic latent space mo dels. W e prop ose instead to dra w tra jectories of eac h node’s laten t features, all without a lo op (A W OL, McCausland et al. , 2011 ). This strategy significan tly reduces the computing time and impro v es the mixing. W e sho w that this sampling step can incorp orate different prior dep endence structures ( no de-wise and fe atur e-wise ), exploiting the assumption of join t Gaussian distribution and conditioning. This result is stated in the follo wing prop osition. Pr op osition 2 . Assume the laten t space dimension, d , is giv en, and the quan tities z i : , : , x i : , : , K i , K i , G i , and e Σ i as in the Supplemen t. Then, one obtains the follo wing: (i) Under the no de-wise prior sp ecification in Eq. ( 6 ) and assuming the initial v alue for the pro cess to b e x i : , 0 ∼ N d ( x i : , 0 | x i : , 0 , Ω 0 ) , the p osterior (smo othed) distribution for x i : , : is x i : , : | d, τ , r , α , x − i : , : , x i : , 0 , Φ , Υ ind ∼ N dT  x i : , : | b x i : , : , b K − 1 i  , where b K i = K i + G ′ i e Σ − 1 i G i and b x i : , : = b K i  K i x i : , : + G ′ i e Σ − 1 i z i : , :  . (ii) Under the fe atur e-wise prior sp ecification in Eq. ( 7 ) and assuming the initial v alue for the pro cess to b e x : ℓ, 0 ∼ N N ( x : ℓ, 0 | x ∗ : ℓ, 0 , Ω ∗ 0 ) , the posterior (smo othed) distri- bution for x i : , : is x i : , : | d, τ , r , α , x − i : , : , x i : , 0 , e Φ , e Υ ind ∼ N dT  x i : , : | b x i : , : , b K − 1 i  , where b K i =  K i + G ′ i e Σ − 1 i G i  − 1 , b x i : , : = b K i  K i x i : , : + G ′ i e Σ − 1 i z i  . Thanks to the sparse structure of b K i and b K ∗ i , we can efficien tly sample exactly x i : , : , i = 1 , . . . , N , from their p osterior (smo othed) distribution distribution p ( x , x :: , 0 | d, τ , r , w , α , Φ , Υ) , av oiding time lo ops. Summarising, b y Propositions 1 and 2 , sampling from p ( d | y , τ , r , w , α , Φ , Υ) and p ( x , x :: , 0 | d, τ , r , w , α , Φ , Υ) used in A) is p erformed b y dra wing d and x i : , : with prob- abilit y q ( y , τ , r | d, w , α , Φ , Υ) p ( d ) P d ℓ =1 q ( y , τ , r | ℓ, w , α , Φ , Υ) p ( ℓ ) , N dT  x i : , : | b x i : , : , b K − 1 i  . (12) and N N d  x :: , 0 | b x :: , 0 , b K − 1 0  . These steps m ust b e p erformed in this order since x is in tegrated out from the (marginal) p osterior of d , and together these tw o steps represen t a single dra w from the joint full conditional p osterior p ( d, x | · ) . The remaining parameters in step B) are sampled exactly from their full conditional distributions. Sp ecifically , the distributions of the no de fixed effects p ( α | τ , r , w , x ) and of the laten t autoregressiv e coefficient matrix under the t w o sp ecifications, p ( ϕ | x , x :: , 0 , Υ) and p ( e ϕ | x , x :: , 0 , e Υ) are: N N ( α | µ α , Σ α ) , N d 2  ϕ | µ Φ , Σ Φ  , and N N 2  e ϕ | µ e Φ , Σ e Φ  . (13) 12 Note that the sampling scheme of the node fixed effects α is p erformed join tly instead of elemen t-wise, leading to an improv ed mixing compared to the standard LSM literature (see Section A.5 of the Supplemen t for further details). Under the fe atur e-wise sp ecification and the graphical horsesho e prior assumption, the precision matrix e Ω = e Υ − 1 has a full conditional distribution prop ortional to   e Ω   N d 2 exp  − tr  1 2 S e Ω   Y i 0) (19) and p ( r ij,ℓt = k | y ij,t , w ij,t , x , α , τ ij,t ) , ℓ = 1 , 2 , prop ortional to c k (1) N  − log( τ ij, 1 t ) − log( λ ij,t ) | µ k (1) , σ 2 k (1)  , (20) c k ( y ij,t ) N  − τ ij, 2 t − log( λ ij,t ) | µ k ( y ij,t ) , σ 2 k ( y ij,t )  , (21) resp ectiv ely , the hyperparameters can b e tabulated outside the main lo op of the MCMC, sp eeding up the computations. R emark. The prop osed model and MCMC algorithm can b e easily mo dified to handle other data types, such as binary and categorical, when the observ ation mo del is a logit, binomial, multinomial logit, and negativ e binomial. The data augmen tation strategy in F rüh wirth-Sc hnatter et al. ( 2009 ) yields a conditionally linear Gaussian state-space mo del. Positiv e real-v alued data can b e handled b y introducing a Gaussian data aug- men tation, as in Sew ell and Chen ( 2016 ). The conditional distributions of τ , r c hange according to the observ ation distribution, whereas the full conditionals of the latent pro cess are Gaussian. W e conduct simulation exp erimen ts to assess the sampler’s p erformance under the prior sp ecification detailed in Section C.2 of the Supplemen t. A set of exp eriments demonstrates the accuracy of our sampler in reco vering the ground-truth parameters under the true data-generating pro cess in the presence of mo del missp ecification. The second set of experiments shows the accuracy of our sampler for differen t sample sizes and prop ortions of zeros. A comparison b etw een the IAMS-based sampler and a well- established alternativ e in the literature shows that the former exhibits b etter mixing p erformance, as measured b y effective sample size. See Section C.3 and Section C.4 in the Supplemen t for further details. 4 Real-data Applications This section presen ts applications of the prop osed method to three temp oral net work datasets: UN co-v oting, in ternational trade, and brain connectivit y . The illustration demonstrates our mo del’s ability to provide new evidence on laten t space dimension no de feature dynamics and structural zeros. 14 2014 2024 ARE ARG AUS AUT BEL BGD BRA CAN CHE CHL CHN COL DEU DNK EGY ESP FIN FRA GBR IDN IND IRL IRN IRQ ISR IT A JPN KAZ KOR MEX MYS NGA NLD NOR NZL P AK PER PHL POL PRT ROU RUS SAU SGP SWE THA TUR USA ARE ARG AUS AUT BEL BGD BRA CAN CHE CHL CHN COL DEU DNK EGY ESP FIN FRA GBR IDN IND IRL IRN IRQ ISR IT A JPN KAZ KOR MEX MYS NGA NLD NOR NZL P AK PER PHL POL PRT ROU RUS SAU SGP SWE THA TUR USA RUS IRN CHN BRA NGA CAN ARG IRQ AUS IT A DEU GBR DNK POL ROU USA IND EGY SAU ISR KAZ P AK COL CHL IDN ARE BGD PER THA MYS MEX PHL SGP IRL BEL FRA JPN NZL KOR NOR SWE ESP TUR NLD CHE AUT PRT FIN RUS IRN CHN BRA NGA CAN ARG IRQ AUS IT A DEU GBR DNK POL ROU USA IND EGY SAU ISR KAZ P AK COL CHL IDN ARE BGD PER THA MYS MEX PHL SGP IRL BEL FRA JPN NZL KOR NOR SWE ESP TUR NLD CHE AUT PRT FIN Figure 1: UN General Assembly co-voting net work. Circular pro jection of the laten t space representation for the years 2014 and 2024 (left and middle), and no de correlation p osterior mean (righ t). 4.1 UN Co-v oting Net w orks W e apply our mo del to a dynamic co-voting net w ork constructed from the United Na- tions General Assembly roll-call voting dataset ( United Nations , 2025 ). W e consider a time series of T = 21 net works from 2004 to 2024, as there is minimal v ariation in coun try names ov er this timeframe. Eac h no de represents one of the top-50 coun tries in terms of GDP , and eac h edge w eight represen ts the o ccurrence of aligned v otes (“yes”– “y es”, “no”–“no”, “abstain”–“abstain”) b etw een coun try i and country j in y ear t . V oting net w orks are of in terest b ecause they pro vide a measure of bilateral relations, enabling b oth historical analysis and forecasting (see Lauderdale , 2010 ; Kim et al. , 2023 ). The p osterior distribution of d is highly concentrated at b d = 2 , and the evidence sup- p orts non-trivial netw ork top ologies with time-v arying communities and islands (panel (a) in Figure D.11 of the Supplement). The latent feature in circular pro jections in 2014 (left) and 2024 (right), (see also Section D.1 in the Supplemen t), sho wn in Fig. 1 , exhib- its t wo distinct clusters: the first comprises NA TO and EU coun tries, and the second comprises BRICS coun tries. The cluster comp osition is c hanging slo wly , with the US progressiv ely realigning with the rest of NA TO and Argentina changing its orien tation in 2023-2024, as its foreign p olicy realigned to w ards the US and NA TO (red circles and green dots). The p osterior estimate of the cross-coun try correlation (righ t) pro vides evidence of further blo cking structure in the no des that is finer than the NA TO - BRICS bi-p olarit y . F or instance, the position alignmen t of the China–Iran–Russia blo c k is tigh ter than the alignmen t of other BRICS members. There are other distinct clusters of dynamics: the EU and the Latin American–Asian group. 15 4.2 T rade Net w orks In this section, we apply our zero-inflated latent space mo del to a dynamic trade net work constructed from the CEPII–BACI dataset of i n ternational trade flows ( De Benedictis et al. , 2014 ). W e consider a time series of T = 21 trade net w orks from 2004 to 2024. In these net works, each no de represents one of the top-50 countries b y GDP , and eac h edge w eigh t represen ts the n umber of distinct HS2 pro duct categories exchanged betw een coun tries i and j in year t . Considering trade v ariet y rather than its economic v alue is of in terest b ecause it offers a different p ersp ectiv e on bilateral relations b etw een countries. This measure can b e considered a proxy for the extensiv e margin of trade ( Hummels and Kleno w , 2005 ), reflecting the diversit y and strength of bilateral exc hange relationships in the global trade system ( Gemmetto et al. , 2016 ). The mo del estimates a latent space dimension b d = 1 . F or this reason, we represen t the coun tries on the ( x :1 ,t , α ) -plane. The left panel of Figure 2 reports the ( x :1 ,t , α ) - plane representation for the year 2024. T wo coun tries hav e features p oin ting in the same horizontal direction if they trade many go o ds with one another relative to their total num b er of go o ds exc hanged, and/or if their trade with other countries is similar. Coun tries with higher α i are globally more connected. This enables the iden tification of “core” countries, with man y edges, and “p eripheral” coun tries, more sparsely connected. The core, comp osed of Europ ean countries, the United States, and a few other adv anced economies, is clearly identified and stable (top side of the plot). Other countries dis- coun t some degree of isolation in world trade and hav e p eripheral p ositions in the plane. Other clusters emerge, such as the one related to Latin American countries (Argentina, Chile, Colombia, Mexico, and Peru). There is also evidence of heterogeneity in the no de tra jectories. F or example, from 2022 on ward, Russia exhibits a displacemen t a wa y from the core group, p ossibly reflecting trade disruptions and geop olitical fragmen ta- tion related to the conflict in Ukraine, while Iran recen tly mo v ed far apart after some alternating p erio ds, also in this case discoun ting geop olitical tensions (red circles in the righ t panel of Figure 2 ). 4.3 Brain Net w orks W e illustrate the effectiveness of our dynamic zero-inflated laten t space mo del in cap- turing the structural connectivit y patterns of the human brain. Some w orks fo cused on fMRI connectivit y data (e.g., see W ang et al. , 2025 ). W e consider net works extrac- ted from Diffusion T ensor Imaging (DTI) data. Latent Space models ha ve previously sho wn their p otential in mo delling DTI-inferred netw ork structure. Duran te et al. ( 2017 ) pro vide a cross-sectional binary LSM to infer a common latent structure from a p opu- 16 ARE ARG AUS AUT BEL BRA CAN CHE CHL CHN COL DEU DNK EGY ESP FIN FRA GBR HKG IDN IND IRL IRN ISR IT A JPN KAZ KOR MEX MYS NLD NOR NZL P AK PER PHL POL PRT ROU RUS SAU SGP SWE THA TUR USA 2.0 2.1 2.2 2.3 −3 −2 −1 0 1 ARE ARG AUS AUT BEL BRA CAN CHE CHL CHN COL DEU DNK EGY ESP FIN FRA GBR HKG IDN IND IRL ISR IT A JPN KAZ KOR MEX MYS NLD NOR NZL P AK PER PHL POL PRT ROU SAU SGP SWE THA TUR USA IRN RUS 2.0 2.1 2.2 2.3 −2 −1 0 Time 2005 2010 2015 2020 Figure 2: T rade netw ork. Laten t space representation in the ( x :1 ,t , α ) -plane (left) in 2024 with 95% credible ellipses which are rep orted in blue, and position temporal ev olution (righ t), with lighter no de colours denoting more recent years. lation of sub jects, while Aliv erti and Duran te ( 2019 ) proposes a cross-sectional brain- region LSM clustering. Instead, we consider a single individual and p erform information p o oling for the laten t co ordinates b y lev eraging the temp oral dimension. Our analysis is based on diffusion MRI data pro vided by the NeuroData rep osit- ory ( https://neurodata.io/mri/ ). Sp ecifically , we fo cus on one anonymized sub ject, 0025982 , from the control group of the study by Lu et al. ( 2011 ). The sub ject underwen t fiv e diffusion MRI scanning sessions, eac h appro ximately 6 mon ths apart, yielding a se- quence of brain net works. The brain is segmen ted usin g the Sc haefer 200 atlas ( Sc haefer et al. , 2018 ), which partitions the cerebral cortex into 200 anatomical regions. Eac h re- gion is a no de in the brain netw ork, while edges represen t the strength of white-matter connectivit y b et w een pairs of regions. Connectivit y w eights corresp ond to the num b er of white-matter fib ers b et w een pairs of regions. This representation yields a weigh ted, undirected adjacency matrix with coun t w eights, in whic h structural zeros indicate the absence of detected fib er tracts. Suc h data naturally motiv ate the use of a zero-inflated mo delling framework that can distinguish b etw een true anatomical absences and sampling zeros arising from limited tractograph y sensitivit y . Figure 3 (top panel) rep orts a graphical represen tation of the Sc haefer 200 brain regions coloured according to the p osterior mean of the corresp onding individual effect. F unctional brain areas are c haracterized by high connectivit y , i.e., large α i v alues, where eac h α i can b e interpreted as the relativ e cen trality of the brain regions within the net w ork. The area with the highest connectivity in the middle panel corresp onds to the 17 α −1 0 1 2 0 25 50 75 0.90 0.95 1.00 1.05 1.10 L R Left hemisphere Right hemisphere Left hemisphere Right hemisphere 0 50 100 150 200 0 50 100 150 200 Σ ij −5 0 5 10 Figure 3: Spatial and distributional c haracteristics of the parameter estimates. T op: Brain re- gions (Sc haefer-200 atlas) coloured b y node-sp ecific p osterior mean ˆ α i . Bottom left: Posterior distribution of the a v erage effects o ver left ( P 100 i =1 α i / 100 , ligh t gra y) and righ t hemisphere no des ( P 200 i =101 α i / 100 , dark gra y). Bottom righ t: ˆ Σ , estimated co v ariance matrix among x : ℓ,t , with dashed lines separating hemispheres. motor cortices. The b ottom-left panel rep orts the p osterior distribution of the av erage of the individual effects for eac h hemisphere, i.e., α L = 1 100 P 100 i =1 α i and α R = 1 100 P 200 i =101 α i . Regions in the left hemisphere are more cen tral α L > α R , whic h is consistent with existing evidence on hemispheric netw ork asymmetries ( Iturria-Medina et al. , 2011 ). Our mo del allows us to study fluctuations in hemispheric asymmetry for each individual, as discussed b elow, thereby extending the evidence from the static framework. The estimated cov ariance matrix ˆ Σ (b ottom-right) indicates that fluctuations across regions are dep endent, with clustering effects (blo ck-wise patterns), stronger correlations within hemispheres (red and blue), and w eak er correlations b et w een hemispheres (white). Panel (a) of Figure 4 rep orts the laten t space represen tation along 4 of the 6 laten t co ordinates estimated ov er 5 scanning sessions. In each session, the separation b etw een the left and righ t hemispheres (orange and blue, respectively) aligns with the data, with more closely connected brain regions sho wing higher white fiber counts. Our mo del also captured significan t temp oral fluctuations while filtering out noise due to measuremen t errors, whic h is app ealing for monitoring the ev olution of lesioned circuits and for studying patien t reco v ery . Differen t information on fluctuations in hemisph eric connectivit y is obtained from 18 (a) Latent Co ordinates (b) Structural Zeros Session 2 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 X 1 X 2 Session 2 −2 −1 0 1 2 −2 −1 0 1 2 X 5 X 6 0 50 100 150 200 0 50 100 150 200 Nodes Nodes Session 4 −2 −1 0 1 2 3 −2 −1 0 1 2 X 1 X 2 Session 4 −2 −1 0 1 2 −2 −1 0 1 2 X 5 X 6 0 50 100 150 200 0 50 100 150 200 Nodes Nodes Figure 4: Results across five MRI sessions (rows). (a) Latent space representations. Estimated co ordinate pairs ( ˆ x i, 1 ,t , ˆ x i, 2 ,t ) and ( ˆ x i, 5 ,t , ˆ x i, 6 ,t ) (left and middle columns) for a subsample t = 2 , 4 . Eac h p oint is a brain region as defined b y Sc haefer’s 200 atlas: left hemisphere (orange) and right hemisphere (blue). (b) P osterior probability of structural zero es, P ( z ij,t ≤ 0 | y ) . A dark er colour denotes a higher probabilit y of a structural zero. the structural zeros (panel b). The v ast ma jority of zeros in the observed connectivit y matrices are structural, reflecting genuine anatomical separations b etw een the t w o hemi- spheres, whereas a relativ ely small fraction is attributed to random fluctuations (Poisson comp onen t), suggesting that these arise from measurement v ariability or noise. 5 Conclusion W e prop osed a new dynamic eigenmo del for coun t-v alued temp oral net works. The mo del accoun ts for excess zeros and dynamic latent features via alternative contem- p oraneous cov ariance structures. Our proposal lev erages a data-augmen tation sc heme for the P oisson lik eliho o d to obtain a conditionally Gaussian state-space representation. W e developed a Ba yesian inference procedure for the laten t space dimension using a par- tially collapsed Gibbs sampler, whic h a v oids trans-dimensional mov es. The approac h relies on a Laplace approximation of the marginal lik eliho o d and p erforms w ell in terms of accuracy and computing time. Real-data applications demonstrate the methods’ ef- fectiv eness in capturing dynamic features of the netw ork connectivit y . Extensions of the prop osed mo del that are w orth inv estigating include taking in to accoun t homophily (a 19 matrix Ξ with real diagonal elements), o v erdisp ersion (negativ e binomial distribution), incomplete netw ork (missing edge forecasting in space/time), and mixed-frequency data. References Alb ert, J. H. and S. Chib (1993). Ba y esian analysis of binary and p olyc hotomous resp onse data. Journal of the Americ an Statistic al Asso ciation 88 (422), 669–679. Aliv erti, E. and D. Durante (2019). Spatial mo deling of brain connectivit y data via laten t distance models with no des clustering. Statistic al Analysis and Data Mining: The ASA Data Scienc e Journal 12 (3), 185–196. Artico, I. and E. C. Wit (2023). Dynamic latent space relational even t mo del. Journal of the R oyal Statistic al So ciety Series A: Statistics in So ciety 186 (3), 508–529. Casarin, R., A. Peruzzi, and M. F. Steel (2025). Media bias and p olarization through the lens of a Mark ov switc hing latent space net w ork model. The A nnals of Applie d Statistics 19 (4), 3416–3437. Chen, R., H. Xiao, and D. Y ang (2021). Autoregressiv e mo dels for matrix-v alued time series. Journal of Ec onometrics 222 (1), 539–560. De Benedictis, L., S. Nenci, G. San toni, L. T a joli, and C. Vicarelli (2014). Net- w ork analysis of world trade using the BACI-CEPII dataset. Glob al Ec onomy Journal 14 (03n04), 287–343. Duran te, D. and D. B. Dunson (2014). Nonparametric Bay es dynamic mo delling of relational data. Biometrika 101 (4), 883–898. Duran te, D., D. B. Dunson, and J. T. V ogelstein (2017). Nonparametric Bay es mo deling of p opulations of netw orks. Journal of the A meric an Statistic al Asso ciation 112 (520), 1516–1530. D’Angelo, L. and A. Canale (2023). Efficient p osterior sampling for Bay esian P oisson regression. Journal of Computational and Gr aphic al Statistics 32 (3), 917–926. F rühwirth-Sc hnatter, S. (1994). Data augmentation and dynamic linear mo dels. Journal of Time Series A nalysis 15 (2), 183–202. F rühwirth-Sc hnatter, S., R. F rüh wirth, L. Held, and H. Rue (2009). Impro ved auxil- iary mixture sampling for hierarc hical mo dels of non-Gaussian data. Statistics and Computing 19 , 479–492. 20 F rühwirth-Sc hnatter, S. and H. W agner (2006). Auxiliary mixture sampling for parameter-driv en mo dels of time series of coun ts with applications to state space mo delling. Biometrika 93 (4), 827–841. Gardini, A., F. Greco, and C. T rivisano (2026). A note on auxiliary mixture sampling for Ba y esian Poisson mo dels. Statistics and Computing 36 (1), 27. Gemmetto, V., T. Squartini, F. Picciolo, F. Ruzzenen ti, and D. Garlaschelli (2016). Multiplexit y and m ultirecipro city in directed m ultiplexes. Physic al R eview E 94 (4), 042316. Gollini, I. and T. B. Murphy (2016). Join t mo deling of multiple net work views. Journal of Computational and Gr aphic al Statistics 25 (1), 246–265. Gw ee, X. Y., I. C. Gormley , and M. F op (2025). A latent shrink age position mo del for binary and coun t net work data. Bayesian A nalysis 20 (2), 405–433. Handco c k, M. S., A. E. Raftery , and J. M. T antrum (2007). Model-based clustering for so cial net works. Journal of the R oyal Statistic al So ciety Series A: Statistics in So ciety 170 (2), 301–354. Hoff, P . D. (2005). Bilinear mixed-effects mo dels for dyadic data. Journal of the A mer- ic an Statistic al Asso ciation 100 (469), 286–295. Hoff, P . D. (2008). Modeling homophily and sto chastic equiv alence in symmetric rela- tional data. A dvanc es in Neur al Information Pr o c essing Systems 1 , 00–00. Hoff, P . D., A. E. Raftery , and M. S. Handco ck (2002). Latent space approaches to so cial netw ork analysis. Journal of the Americ an Statistic al Asso ciation 97 (460), 1090–1098. Hummels, D. and P . J. Klenow (2005). The v ariet y and qualit y of a nation’s exp orts. A meric an Ec onomic R eview 95 (3), 704–723. Iturria-Medina, Y., A. Pérez F ernández, D. M. Morris, E. J. Canales-Rodríguez, H. A. Haro on, L. García P en tón, M. Augath, L. Galán García, N. Logothetis, G. J. Park er, et al. (2011). Brain hemispheric structural efficiency and interconnectivit y righ tw ard asymmetry in h uman and nonhuman primates. Cer ebr al Cortex 21 (1), 56–67. Kaur, H. and R. Rastelli (2024). A laten t space mo del f or multiv ariate count data time series analysis. arXiv pr eprint arXiv:2408.13162 . 21 Kim, B., X. Niu, D. Hun ter, and X. Cao (2023). A dynamic additiv e and multiplicativ e effects net w ork mo del with application to the united nations v oting behaviors. The A nnals of Applie d Statistics 17 (4), 3283. King, B. and D. R. Ko wal (2025). W arp ed dynamic linear mo dels for time series of coun ts. Bayesian A nalysis 20 (1), 1331–1356. Lauderdale, B. E. (2010). Unpredictable voters in ideal p oin t estimation. Politic al A nalysis 18 (2), 151–171. Li, Y., B. A. Craig, and A. Bhadra (2019). The graphical horsesho e estimator for in v erse co v ariance matrices. Journal of Computational and Gr aphic al Statistics 28 (3), 747– 757. Lo y al, J. D. and Y. Chen (2023). An eigenmo del for dynamic multila yer net w orks. Journal of Machine L e arning R ese ar ch 24 (128), 1–69. Lo y al, J. D. and Y. Chen (2025). A spik e-and-slab prior for dimension selection in generalized linear net w ork eigenmo dels. Biometrika 112 (3), asaf014. Lu, C., R. Rastelli, and N. F riel (2025). A zero-inflated P oisson latent p osition cluster mo del. arXiv pr eprint arXiv:2502.13790 . Lu, J., H. Liu, M. Zhang, D. W ang, Y. Cao, Q. Ma, D. Rong, X. W ang, R. L. Buc kner, and K. Li (2011). F o cal p on tine lesions provide evidence that in trinsic functional con- nectivit y reflects p olysynaptic anatomical pathw ays. Journal of Neur oscienc e 31 (42), 15065–15071. McCausland, W. J., S. Miller, and D. P elletier (2011). Sim ulation smo othing for state– space mo dels: A computational efficiency analysis. Computational Statistics & Data A nalysis 55 (1), 199–212. P ark, J. H. and Y. Sohn (2020). Detecting structural c hanges in longitudinal net w ork data. Bayesian A nalysis 15 (1), 133–157. P a v one, F., D. Duran te, and R. J. Ryder (2025). Ph ylogenetic laten t space mo dels for net w ork data. arXiv pr eprint arXiv:2502.11868 . Sark ar, P . and A. W. Mo ore (2005). Dynamic so cial netw ork analysis using latent space mo dels. A cm Sigkdd Explor ations Newsletter 7 (2), 31–40. 22 Sc haefer, A., R. K ong, E. M. Gordon, T. O. Laumann, X.-N. Zuo, A. J. Holmes, S. B. Eic khoff, and B. T. Y eo (2018). Lo cal-global p arcellation of the h uman cerebral cortex from in trinsic functional connectivit y MRI. Cer ebr al Cortex 28 (9), 3095–3114. Sew ell, D. K. and Y. Chen (2015). Laten t space mo dels for dynamic netw orks. Journal of the Americ an Statistic al Asso ciation 110 (512), 1646–1657. Sew ell, D. K. and Y. Chen (2016). Latent space mo dels for dynamic netw orks with w eigh ted edges. So cial Networks 44 , 105–116. Sosa, J. and L. Buitrago (2021). A review of laten t space mo dels for so cial netw orks. R evista Colombiana de Estadístic a 44 (1), 171–200. T urnbull, K., C. Nemeth, M. Nunes, and T. McCormick (2023). Sequential estimation of temp orally evolving latent space netw ork mo dels. Computational Statistics & Data A nalysis 179 , 107627. United Nations (2025). United nations general assem bly voting data: Resolutions 1 (11 decem b er 1946) to 79/328 (5 september 2025). A ccessed: 24 F ebruary 2026. V an Dyk, D. A. and T. Park (2008). Partially collapsed Gibbs samplers: Theory and metho ds. Journal of the Americ an Statistic al Asso ciation 103 (482), 790–796. W ang, S., S. Paul, and P . De Bo eck (2023). Join t latent space mo del for so cial net w orks with m ultiv ariate attributes. Psychometrika 88 (4), 1197–1227. W ang, S., Y. W ang, F. H. Xu, L. Shen, Y. Zhao, A. D. N. Initiativ e, et al. (2025). Estab- lishing group-lev el brain structural connectivity incorp orating anatomical knowledge under laten t space mo deling. Me dic al Image Analysis 99 , 103309. Y u, X. and A. Ro driguez (2021). Spatial v oting mo dels in circular spaces: A case study of the US house of representativ es. The A nnals of Applie d Statistics 15 (4), 1897–1922. Zens, G., S. F rühwirth-Sc hnatter, and H. W agner (2024). Ultimate Póly a Gamma samplers – Efficient MCMC for p ossibly im balanced binary and categorical data. Journal of the A meric an Statistic al Asso ciation 119 (548), 2548–2559. 23