Separable models for dynamic signed networks

Sepa rable mo dels fo r dynamic signed net w o rks Alb erto Caimo ∗ and Isab ella Gollini Scho ol of Mathematics and Statistics, Universit y College Dublin, Belﬁeld, D04 V1W8, Ireland ∗ Corresponding author. alberto.caimo1@ucd.ie Abstract Signed netw orks capture the p ola rit y of relationships b etw een no des, providing valuable insights into complex systems where both supp ortive and antagonistic interactions play a critical role in shaping the netw ork dynamics. W e propose a separable temp oral generative framewo rk based on multi-lay er exp onential random graph mo dels, cha racterised b y the assumption of conditional indep endence b etw een the sign and interaction eﬀects. This structure preserves the ﬂexibly and explanato ry p ow er inherent in the binary netw ork sp eciﬁcation while adhering to consistent balance theory assumptions. Using a fully probabilistic Bay esian paradigm, we infer the doubly intractable posterior distribution of model parameters via an adaptive Metropolis-Hastings appro ximate exchange algo rithm. We illustrate the interpretabilit y of our mo del by analysing signed relations among U.S. Senators during Ronald Reagan’s second term (1985–1989). Sp eciﬁcally , w e aim to understand whether these relations a re consistent and balanced or reﬂect patterns of supportive or antagonistic alliances. Key w o rds: Dynamic signed netwo rks, structural balance, exp onential random graph mo dels, Bay esian inference 1. Intro duction Netw ork data c haracterise the connectivity structures of complex systems, whic h often inv olv e a delicate balance b etw een tw o t yp es of interactions. In biological systems, for example, these in teractions may b e reﬂected in co-expression similarities and dissimilarities betw een genes (Mason et al., 2009) that are explained b y the sign of the correlation of their expression proﬁles, while in social or p olitical netw orks, they manifest as alliances and riv alries betw een individuals (F on tan and Altaﬁni, 2021). Signed net works, which consist of p ositive and negative edges, pro vide a p ow erful to ol for capturing the n uanced relationships within these systems. In fact, by using signed net works, w e can identify patterns and trends that reveal deep er insigh ts into the dynamics of these complex systems, which ma y not be immediately visible through other analytical methods. A central question in the analysis of signed netw orks is how w ell p ositive and negativ e edges align with Structural Balance Theory (SBT) (Cart wrigh t and Harary, 1956). SBT asserts that p ositive relations should exhibit transitivity (t wo p ositive paths should close with a p ositive edge) while negative relations follow an ti- transitivity (tw o negativ e paths should close with a positive edge). Con versely , a p ositive tw o-path should not b e closed b y a negativ e relation, and a negativ e t wo-path should not be closed b y another negativ e relation, as these patterns pro duce unbalanced triads. T o explore these dynamics, our approac h utilises the exponential random graph mo delling (ER GM) framew ork (Lusher et al., 2013), which is particularly suited to mo delling the complex interpla y b etw een p ositiv e and negative edge conﬁgurations. ER GMs allow us to incorp orate terms that capture relational tendencies, such as the clustering of p ositive edges or the formation of unexpected negative edges b et w een individuals with similar no dal cov ariate information. By accounting for these relational patterns, w e can accurately assess the statistical signiﬁcance of balanced conﬁgurations and their impact on the ov erall structure of the netw ork (F ritz et al., 2025). In this paper, w e examine undirected signed netw ork panel data, which consists of cross-sectional snapshots of netw ork relationships observed at regular in terv als. In particular, w e focus on the analysis of political relationships among U.S. Senators, measured across successive U.S. Congresses (Neal, 2014, 2020). Eac h 1 2 Caimo and Gollini snapshot captures the state of signed in teractions within the Senate during eac h legislativ e y ear, oﬀering insight into the evolving dynamics of p olitical alliances and riv alries ov er time. W e extend the temporal exp onen tial random graph mo del (TERGM) framework (Robins and Pattison, 2001; Hanneke et al., 2010) to better accommodate the dynamics of signed in teractions in longitudinal netw ork data. Speciﬁc ally , w e incorp orate a separable sp eciﬁcation (Krivitsky and Handco ck, 2014) that distinguishes betw een the formation and p ersistence of edges, enabling a more detailed understanding of temp oral dependence. Building on the decomp osition introduced by Lerner (2016), we further enhance the TERGM by mo delling the signed netw ork as the outcome of t wo interrelated pro cesses: an interaction pro cess and a conditional sign pro cess. This extension allo ws for a more eﬃcient and interpretable parameterisation that captures not only how relationships form and persist ov er time, but also ho w structural balance, through both balanced and imbalan ced signed conﬁgurations, shap es the ev olution of netw ork in teractions. This p ersp ective aligns with recent work sho wing that micro-level mechanisms can accumulate to pro duce pronounced patterns at the macro level (Duxbury, 2024). Ev en mo dest biases in the formation or sign of individual edges can hav e substantial eﬀects on ov erall netw ork structure, inﬂuencing cohesion, clustering, and balance (Amati et al., 2018). Incorp orating endogenous micro-level eﬀects in to our separable framew ork ensures that b oth the local dynamics and their consequences for global net work patterns are explicitly represen ted. The pap er is structured as follo ws. In Sections 2 and 3, we provide an o verview of ERGMs and temporal ER GMs resp ectively . Section 4 oﬀers a comprehensive deﬁnition of signed netw orks and presents an o v erview of SBT. In Sections 5 a nd 6, we introduce a multi-la yer in terpretation of the signed proc ess, distinguishing b et w een the interaction pro cess and the conditional sign pro cess. W e describ e how these tw o comp onents are incorp orated into the ERGM generative framework to capture the dynamics of signed relationships. Section 7 com bines temp oral separabilit y with la yer separability to deﬁne 2-la yer separable TER GMs. Here, w e deﬁne a Mark ov pro cess based on binary ERGM terms, which allows for a principled handling of Structural Balance Theory conﬁgurations. In Section 8, w e outline a Bay esian inferential approach, which quan tiﬁes the uncertaint y in the p osterior mo del parameter estimates by using an adaptiv e approximate exchange algorithm (Haario et al., 2001; Caimo and F riel, 2011) to facilitate sampling from the doubly-intractable p osterior distribution. In Section 9, we present a comparative simulation study to demonstrate parameter reco verabilit y and the robustness of the multi-la yer conditional approach, and to highligh t the diﬀerences b et w een this method, the TER GM approac h of F ritz et al. (2025), and its separable temporal v ariant. Finally , Section 10 demonstrates the utility of our mo delling approac h b y applying it to the analysis of p olitical relationships among U.S. Senators. W e assess mo del ﬁt using p osterior predictiv e chec ks to ev aluate the mo del performance. Section 11 concludes with a discussion on p otential extensions and future directions. 2. Exp onential random graph mo dels The relational structure of a netw ork graph is describ ed by a random adjacency matrix y of a graph on n nodes (actors) and a set of edge v ariables such that y ij = 1 , if no de i is connected to no de j ( i ∼ j ); y ij = 0 , if no de i is not connected to node j ( i ∼ j ). Exponential random graph models (ERGMs) (Holland and Leinhardt, 1981; Strauss and Ikeda, 1990) are a particular class of discrete linear exponential families with probability mass function: p ( y | φ ) = exp { φ ⊤ s ( y ) − κ ( φ ) } , (1) where s ( y ) ∈ R Q > 0 is a known vector of Q suﬃcient statistics (Besag, 1974), φ ∈ R Q is the asso ciated parameter v ector, and κ ( φ ) a normalising constant whic h is computationally intractable to ev aluate for all but trivially small graphs. The dependence h yp othesis at the basis of these mo dels is that edges form small structures called conﬁgurations. There is a wide range of possible netw ork conﬁgurations (Robins et al., 2007) which give ﬂexibility to adapt to diﬀeren t con texts. A p ositive (or negativ e) v alue of the parameter φ indicates a tendency for the conﬁguration represented b y s ( y ) to app ear more (or less) frequen tly in the data than would b e expected under an Erd˝ os–R ´ enyi random graph model with an edge probabilit y of 0.5. The ERGM likelihoo d deﬁned in Equation (1) can b e generalised by allowing the parameter to v ary non- linearly within the exp onential family , resulting in a curv ed exp onential family (Hun ter, 2007). Mon te Carlo Separable mo dels for dynamic signed netw orks 3 metho ds (Hunter and Handco ck, 2006) can b e used to estimate the decay parameter of the geometrically w eighted netw ork statistics in tro duced b y Snijders et al. (2006) and currently used to alleviate ERGM degeneracy issues (Handcock, 2003). 3. T emp o ral exp onential random graph mo dels T emp oral exponential random graph mo dels (TER GMs) (Robins and P attison, 2001; Hannek e et al., 2010) describ e the joint distribution of a dynamic net work sequence b y assuming a Marko v pro cess on the net w ork from one time step to the next: p ( y 1 , . . . , y T | y 0 , φ ) = T Y t =1 p ( y t | y t − 1 , φ ) = T Y t =1 exp { φ ⊤ s ( y t ; y t − 1 ) − κ ( φ ; y t − 1 ) } , where φ parametrises the inﬂuence of suﬃcient netw ork statistics s ( y t ; y t − 1 ) on the likelihoo d and κ ( φ , y t − 1 ) is a normalising constan t. The pro cess can be extended to incorp orate higher-order Mark ov dependence by assuming that the net work y t depe nds on K ∈ { 0 , 1 , · · · , T − 1 } previously observed netw orks. This is ac hieved b y including lagged netw orks in the net w ork statistics s ( · ) with K selected appropriately (Leifeld et al., 2018). The separable temp oral parametrisation (STERGM) prop osed b y Krivitsky and Handco ck (2014) pro vides a conv enient control ov er incidence and duration of edges and separate interpretation betw een consecutive netw ork observ ations. W e deﬁne the formation netw ork y F = y t − 1 ∪ y t ; and the p ersistence net w ork y P = y t − 1 ∩ y t ; and assume y F and y P are conditionally independent given y t − 1 so that: p  y t | y t − 1 , φ F , φ P  = p  y F | y t − 1 , φ F  × p  y P | y t − 1 , φ P  , (2) where φ F and φ P describ e resp ectively the edge formation and persistence pro cess from t − 1 to t. Then, giv en y F , y P and y t − 1 , we hav e y t = y P ∪  y F \ y t − 1  . The STERGM framew ork addresses a key limitation of standard TERGMs b y separating the mo delling of edge formation and persistence. This separation allo ws for more in terpretable mo delling of dynamic net works, esp ecially when we assume that edge formation and dissolution are gov erned by diﬀerent mec hanisms. F or example, in a friendship netw ork, a new friendship (formation) may b e inﬂuenced by proximit y while the maintenan ce of an existing friendship (persistence) ma y supported b y the presence of m utual friends. 4. Signed netw ork structures A signed net work consists of a set of n no des and signed edges b etw een pairs of them. Its relational structure is describ ed by a random adjacency matrix with elements of the set of edge v ariables y ij ∈ { +1 , − 1 , 0 } resp ectiv ely indicating a p ositive, negative or no relation b etw een no de i and no de j. As for binary net w orks, signed netw orks may b e directed or undirected dep ending on the nature of the relationships b etw een the node s. In this pap er, we fo cus on signed undirected netw orks, although the metho dologies introduced can be easily extended to the directed case. Structural balance the ory (SBT) examines ho w positive and negativ e relationships b etw een actors evolv e and ultimately stabilise within a netw ork. It fo cuses on the dynamics and even tual equilibrium of these relationships in a signed netw ork (Cartwrigh t and Harary, 1956). SBT is commonly illustrated using the familiar concepts of friendship and enmity . These terms provide a clear and intuitiv e understanding of the fundamen tal principles of the theory . Balanced triadic conﬁgurations include: (1) a friend of a friend is a friend, and (2) an enemy of a friend is an enemy . In contrast, unbalanced conﬁgurations are: (3) a friend of an enemy is an enem y , and (4) an enem y of an enemy is a friend. Conﬁguration (1) corresp onds to p ositiv e triangles (Figure 1 (a)), while conﬁguration (2) corresp onds to triangles with tw o negative edges (Figure 1 (b)). These tw o patterns are part of the so-called strong balance structure (Harary, 1953). Da vis (1967) later in tro duced the w eak balance structure, under which conﬁgurations comprising triangles with all negativ e edges (Figure 1 (c)) are also regarded as balanced. Conv ersely , triadic conﬁgurations with exactly tw o p ositive edges and one negative edge (Figure 1 (d)) are considered un balanced. 4 Caimo and Gollini AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E34OMY9OIxonlAsoTZyWwyZHZ2mekVQsgnePGgiFe/yJt/4yTZgyYWNBRV3XR3BYkUBl3328mtrW9sbuW3Czu7e/sHxcOjpolTzXiDxTLW7YAaLoXiDRQoeTvRnEaB5K1gdDvzW09cGxGrRxwn3I/oQIlQMIpWeigH571iya24c5BV4mWkBBnqveJXtx+zNOIKmaTGdDw3QX9CNQom+bTQTQ1PKBvRAe9YqmjEjT+ZnzolZ1bpkzDWthSSufp7YkIjY8ZRYDsjikOz7M3E/7xOiuG1PxEqSZErtlgUppJgTGZ/k77QnKEcW0KZFvZWwoZUU4Y2nYINwVt+eZU0qxXvsnJxXy3VbrI48nACp1AGD66gBndQhwYwGMAzvMKbI50X5935WLTmnGzmGP7A+fwBjNCNUw== (b) AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E34OMY9OIxonlAsoTeyWwyZHZ2mZkVQsgnePGgiFe/yJt/4yTZgyYWNBRV3XR3BYng2rjut5NbW9/Y3MpvF3Z29/YPiodHTR2nirIGjUWs2gFqJrhkDcONYO1EMYwCwVrB6Hbmt56Y0jyWj2acMD/CgeQhp2is9FDG816x5FbcOcgq8TJSggz1XvGr249pGjFpqECtO56bGH+CynAq2LTQTTVLkI5wwDqWSoyY9ifzU6fkzCp9EsbKljRkrv6emGCk9TgKbGeEZqiXvZn4n9dJTXjtT7hMUsMkXSwKU0FMTGZ/kz5XjBoxtgSp4vZWQoeokBqbTsGG4C2/vEqa1Yp3Wbm4r5ZqN1kceTiBUyiDB1dQgzuoQwMoDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4Ai0uNUg== (a) AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E34OMY9OIxonlAsoTZSW8yZHZ2mZkVQsgnePGgiFe/yJt/4yTZgyYWNBRV3XR3BYng2rjut5NbW9/Y3MpvF3Z29/YPiodHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWMbmd+6wmV5rF8NOME/YgOJA85o8ZKD2V23iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSSSPU/mR+6pScWaVPwljZkobM1d8TExppPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGllCmuL2VsCFVlBmbTsGG4C2/vEqa1Yp3Wbm4r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjlWNVA== (c) AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E34OMY8CKeIpoHJEuYnZ1NhszOLDOzQljyCV48KOLVL/Lm3zhJ9qCJBQ1FVTfdXUHCmTau++0U1tY3NreK26Wd3b39g/LhUVvLVBHaIpJL1Q2wppwJ2jLMcNpNFMVxwGknGN/M/M4TVZpJ8WgmCfVjPBQsYgQbKz1Uw/NBueLW3DnQKvFyUoEczUH5qx9KksZUGMKx1j3PTYyfYWUY4XRa6qeaJpiM8ZD2LBU4ptrP5qdO0ZlVQhRJZUsYNFd/T2Q41noSB7Yzxmakl72Z+J/XS0107WdMJKmhgiwWRSlHRqLZ3yhkihLDJ5Zgopi9FZERVpgYm07JhuAtv7xK2vWad1m7uK9XGnd5HEU4gVOoggdX0IBbaEILCAzhGV7hzeHOi/PufCxaC04+cwx/4Hz+AJJCjV0= (d) Fig. 1: T riadic conﬁgurations of a signed netw ork. Solid lines represent p ositive interactions, dashed lines represen t negative interactions. The underlying motiv ation for structural balance is dynamic and based on how unbalanced triangles should resolve to balanced ones. This situation has led naturally to a search for a full dynamic theory of structural balance. Y et ﬁnding systems that reliably guide netw orks to balance has prov ed to be a challenge in itself (Marv el et al., 2011). 5. Multi-lay er interpretation Multi-lay er ERGMs (Krivitsky et al., 2020) extend the traditional ER GM framework to model netw orks with multiple types of interactions b etw een no des, with each lay er representing a distinct kind of relationship. Eac h lay er is modelled using its own set of net work statistics, capturing the structural tendencies speciﬁc to that type of interaction, while allowing for dependencies across la y ers. F or example, supp ose Y 1 represen ts a binary friendship netw ork and Y 2 represen ts a binary collab oration netw ork among the same set of individuals. In this setting, any dyad can simultaneously exhibit b oth t yp es of relations (e.g., t w o individuals may b e friends, collab orators, or b oth). A multi-la yer ERGM can include statistics such as transitivity or homophily within each lay er (e.g., friends-of-friends in Y 1 , shared collab orators in Y 2 ), while also incorp orating cross-lay er eﬀects, such as whether an edge in Y 1 increases the likelihoo d of an edge in Y 2 . In the context of signed net works, where a pair of no des cannot sim ultaneously share b oth a p ositive and a negative edge, the signed edge pro cess can b e decomp osed as Y = XZ , where X , with dyadic elements X ij = 1 if an edge exists b etw een nodes i and j and X ij = 0 otherwise, is a binary adjacency matrix enco ding edge presence, and Z , with dyadic el emen ts Z ij = +1 for p ositive edges and Z ij = − 1 for negative edges, enco des edge signs. This form ulation ensures that eac h dy ad has at most one edge with a single sign. Clearly , this can b e view ed as a special case of a multi-la y er net work, where the sign information is meaningful only in the presence of an actual interaction. In other w ords, the sign lay er is eﬀectively constrained b y the intera ction lay er. This dependency will be further discussed in the following section. How ever, one could also argue that signed relations may exist indep enden tly of observed in teractions, i.e., the sign of a relationship could b e latent, with the interaction merely serving to reveal or activ ate a pre-existing ev aluative stance. In this view, the act of in teraction exp oses underlying sen timen ts suc h as trust, hostility , or alliance that already shap e the relational landscape. This interpretation aligns, for example, with so ciological theories of latent aﬀective edges (Heider, 1946) and is particularly relev ant in contexts where relationships are long-standing or em b edded in broader social structures. In such settings, the absence of interaction do es not necessarily imply neutrality or indiﬀerence, but rather a dorman t state of the relationship. W e assume that the observed signed netw ork follows a 2-lay er ER GM with the probability mass function: p ( y | ϑ ) = exp  ϑ ⊤ s ( x , z ) − κ ( ϑ , x , z )  , (3) where s ( x , z ) incorporates netw ork statistics that model the tw o lay ers jointly (see Figure 2 (a)). The mode lling approach for signed net w orks prop osed by F ritz et al. (2025) is equiv alent to a 2-lay er ER GM which represen ts an ERGM with a uniform categorical reference measure where eac h dyad can tak e one of three p ossible states: p ositive, negative, or absent. The conditional probability that a dyad ( i, j ) is p ositive, Separable mo dels for dynamic signed netw orks 5 giv en the rest of the net w ork y − ( ij ) and the model parameter ϑ , is Pr  Y ij = +1 | y − ( ij ) , ϑ  = exp n ϑ ⊤ s  y +1 ij o P y ∈{ +1 , − 1 , 0 } exp n ϑ ⊤ s  y y ij o , where s ( y y ij ) is the vector of netw ork statistics ev aluated when dyad ( i, j ) is in state y . 6. Separable mo delling Lerner (2016) highlights the distinction betw een marginal and conditional tests of structural balance, sho wing that other net work eﬀects inﬂuencing the likelihoo d of in teraction can mask the presence of balance. In suc h cases, structural balance ma y appear absent even when no des actually prefer balanced triangles o ver unbalanced ones. T o address this, a prop er test of balance theory requires mo delling the conditional probabilit y of an edge b eing p ositiv e or negative, given that the edge exists. More recently , Lerner and Lomi (2020) hav e shown that, when conditioning on activ e users within a large relational even t framework, balance theory provides a compelling explanation for the netw ork structure of teams engaged in con tentious tasks. In line with this insight, our mo del explicitly separates the modelling of the interaction pro cess ( X ) from the sign pro cess ( Z ). Unlike the joint mo del described in the previous section, this separable approach allo ws the edge structure and edge signs to b e modelled conditionally indep endently , while still accounting for their dep endence through the conditional formulation. As shown by Lerner (2016) through analyses of multiple b enc hmark datasets commonly used in the literature, this separation oﬀers b oth practical and in terpretativ e adv antages. It simpliﬁes model estimation, and it reﬂects situations where the drivers of edge formation, such as aﬃliations, or spatial proximit y , are distinct from those inﬂuencing edge polarity , such as trust, co operation, or past conﬂict. F or example, in a so cial netw ork, whether an edge exists may be determined by organisational or logistical constrain ts, whereas the sign of that edge dep ends on the nature of the relationship. F ormally , the marginal probability of the netw ork Y can b e decomp osed as the pro duct of the probabilit y of intera ction X and the conditional probabilit y of the sign given interaction Z | X : Pr( Y = y ) = Pr( Z = z | X = x ) Pr( X = x ) . The fact that Z is conditional on X does not imply a temp oral ordering. The conditionality reﬂects the structural dependence of edge signs on the presence of edges and does not indicate that Z ev olves after X . This explicit separation clariﬁes how structural eﬀects can b e estimated for eac h pro cess without conﬂating the mechanisms driving edge formation and edge sign. This leads to the following formulation of a separable ER GM: p ( y | ζ , ξ ) = p ( x | ξ ) × p ( z | x , ζ ) = exp  ξ ⊤ s ( x ) − κ ( ξ )  × exp  ζ ⊤ s ( z ; x ) − κ ( ζ , x )  , (4) where ξ gov erns the marginal interaction pro cess, while ζ gov erns the conditional sign pro cess within the intera ction structure. Figure 2 (b) provides a graphical representation of the mo del, illustrating that the tw o parameters, ξ and ζ , are conditionally independent given the interaction pro cess x . Consequently , we deﬁne Mo del (4) as a separable 2-la yer ERGM. 7. Lay ered separable temp oral mo dels The separable temp oral assumption deﬁned in Equation (2), when combined with the 2-lay er separability assumption speciﬁed in Equation (4), results in a 2-la yer STERGM, with the transition probabilit y deﬁned 6 Caimo and Gollini AAAB+XicbVC7TsMwFL3hWcorwMhiUSExVQniNVawMBaJPqQmqhzHaa06cWQ7FVXUP2FhACFW/oSNv8FpM0DLkSwfnXOvfHyClDOlHefbWlldW9/YrGxVt3d29/btg8O2EpkktEUEF7IbYEU5S2hLM81pN5UUxwGnnWB0V/idMZWKieRRT1Lqx3iQsIgRrI3Ut20vEDxUk9hcuffEpn275tSdGdAycUtSgxLNvv3lhYJkMU004Vipnuuk2s+x1IxwOq16maIpJiM8oD1DExxT5eez5FN0apQQRUKak2g0U39v5DhWRTgzGWM9VIteIf7n9TId3fg5S9JM04TMH4oyjrRARQ0oZJISzSeGYCKZyYrIEEtMtCmrakpwF7+8TNrndfeqfvlwUWvclnVU4BhO4AxcuIYG3EMTWkBgDM/wCm9Wbr1Y79bHfHTFKneO4A+szx9PIZQd ω AAAB+3icbVA7T8MwGHTKq5RXKCOLRYXEVCWI11jBwlgk+pCaqHIcp7Xq2JHtIEqUv8LCAEKs/BE2/g1OmwFaTrJ8uvs++XxBwqjSjvNtVVZW19Y3qpu1re2d3T17v95VIpWYdLBgQvYDpAijnHQ01Yz0E0lQHDDSCyY3hd97IFJRwe/1NCF+jEacRhQjbaShXfcCwUI1jc2VeU9Eo3xoN5ymMwNcJm5JGqBEe2h/eaHAaUy4xgwpNXCdRPsZkppiRvKalyqSIDxBIzIwlKOYKD+bZc/hsVFCGAlpDtdwpv7eyFCsinhmMkZ6rBa9QvzPG6Q6uvIzypNUE47nD0Upg1rAoggYUkmwZlNDEJbUZIV4jCTC2tRVMyW4i19eJt3TpnvRPL87a7Suyzqq4BAcgRPggkvQAregDToAg0fwDF7Bm5VbL9a79TEfrVjlzgH4A+vzB+cblQQ= ω AAAB/3icbVC7TsMwFHV4lvIKILGwRFRITFWCeI0VLIxFog+piSrHdVqrjh3ZN5Wq0IFfYWEAIVZ+g42/wWkzQMuRLB+dc+/19QkTzjS47re1tLyyurZe2ihvbm3v7Np7+00tU0Vog0guVTvEmnImaAMYcNpOFMVxyGkrHN7mfmtElWZSPMA4oUGM+4JFjGAwUtc+9EPJe3ocmyvzR1jBgAKedO2KW3WncBaJV5AKKlDv2l9+T5I0pgIIx1p3PDeBIDPzGOF0UvZTTRNMhrhPO4YKHFMdZNP9J86JUXpOJJU5Apyp+rsjw7HOVzSVMYaBnvdy8T+vk0J0HWRMJClQQWYPRSl3QDp5GE6PKUqAjw3BRDGzq0MGWGECJrKyCcGb//IiaZ5Vvcvqxf15pXZTxFFCR+gYnSIPXaEaukN11EAEPaJn9IrerCfrxXq3PmalS1bRc4D+wPr8ASPKltc= ω AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E34OMY9OIxonlAsoTeyWwyZHZ2mZkVQsgnePGgiFe/yJt/4yTZgyYWNBRV3XR3BYng2rjut5NbW9/Y3MpvF3Z29/YPiodHTR2nirIGjUWs2gFqJrhkDcONYO1EMYwCwVrB6Hbmt56Y0jyWj2acMD/CgeQhp2is9FDG816x5FbcOcgq8TJSggz1XvGr249pGjFpqECtO56bGH+CynAq2LTQTTVLkI5wwDqWSoyY9ifzU6fkzCp9EsbKljRkrv6emGCk9TgKbGeEZqiXvZn4n9dJTXjtT7hMUsMkXSwKU0FMTGZ/kz5XjBoxtgSp4vZWQoeokBqbTsGG4C2/vEqa1Yp3Wbm4r5ZqN1kceTiBUyiDB1dQgzuoQwMoDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4Ai0uNUg== (a) AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E34OMY9OIxonlAsoTZyWwyZHZ2mekVQsgnePGgiFe/yJt/4yTZgyYWNBRV3XR3BYkUBl3328mtrW9sbuW3Czu7e/sHxcOjpolTzXiDxTLW7YAaLoXiDRQoeTvRnEaB5K1gdDvzW09cGxGrRxwn3I/oQIlQMIpWeigH571iya24c5BV4mWkBBnqveJXtx+zNOIKmaTGdDw3QX9CNQom+bTQTQ1PKBvRAe9YqmjEjT+ZnzolZ1bpkzDWthSSufp7YkIjY8ZRYDsjikOz7M3E/7xOiuG1PxEqSZErtlgUppJgTGZ/k77QnKEcW0KZFvZWwoZUU4Y2nYINwVt+eZU0qxXvsnJxXy3VbrI48nACp1AGD66gBndQhwYwGMAzvMKbI50X5935WLTmnGzmGP7A+fwBjNCNUw== (b) AAAB8XicbVDLSgMxFL3js9ZX1aWbYBFclRnxtSy6cVnBPrAtJZPeaUMzmSHJiGXoX7hxoYhb/8adf2OmnYW2HggczrmXnHv8WHBtXPfbWVpeWV1bL2wUN7e2d3ZLe/sNHSWKYZ1FIlItn2oUXGLdcCOwFSukoS+w6Y9uMr/5iErzSN6bcYzdkA4kDzijxkoPnZCaoR+kT5NeqexW3CnIIvFyUoYctV7pq9OPWBKiNExQrdueG5tuSpXhTOCk2Ek0xpSN6ADblkoaou6m08QTcmyVPgkiZZ80ZKr+3khpqPU49O1kllDPe5n4n9dOTHDVTbmME4OSzT4KEkFMRLLzSZ8rZEaMLaFMcZuVsCFVlBlbUtGW4M2fvEgapxXvonJ+d1auXud1FOAQjuAEPLiEKtxCDerAQMIzvMKbo50X5935mI0uOfnOAfyB8/kDAIiRJg== x AAAB8XicbVDLSgMxFL3js9ZX1aWbYBFclRnxtSy6cVnBPrAtJZPeaUMzmSHJCHXoX7hxoYhb/8adf2OmnYW2HggczrmXnHv8WHBtXPfbWVpeWV1bL2wUN7e2d3ZLe/sNHSWKYZ1FIlItn2oUXGLdcCOwFSukoS+w6Y9uMr/5iErzSN6bcYzdkA4kDzijxkoPnZCaoR+kT5NeqexW3CnIIvFyUoYctV7pq9OPWBKiNExQrdueG5tuSpXhTOCk2Ek0xpSN6ADblkoaou6m08QTcmyVPgkiZZ80ZKr+3khpqPU49O1kllDPe5n4n9dOTHDVTbmME4OSzT4KEkFMRLLzSZ8rZEaMLaFMcZuVsCFVlBlbUtGW4M2fvEgapxXvonJ+d1auXud1FOAQjuAEPLiEKtxCDerAQMIzvMKbo50X5935mI0uOfnOAfyB8/kDA5KRKA== z AAAB8XicbVDLSgMxFL3js9ZX1aWbYBFclRnxtSy6cVnBPrAtJZPeaUMzmSHJiGXoX7hxoYhb/8adf2OmnYW2HggczrmXnHv8WHBtXPfbWVpeWV1bL2wUN7e2d3ZLe/sNHSWKYZ1FIlItn2oUXGLdcCOwFSukoS+w6Y9uMr/5iErzSN6bcYzdkA4kDzijxkoPnZCaoR+kT5NeqexW3CnIIvFyUoYctV7pq9OPWBKiNExQrdueG5tuSpXhTOCk2Ek0xpSN6ADblkoaou6m08QTcmyVPgkiZZ80ZKr+3khpqPU49O1kllDPe5n4n9dOTHDVTbmME4OSzT4KEkFMRLLzSZ8rZEaMLaFMcZuVsCFVlBlbUtGW4M2fvEgapxXvonJ+d1auXud1FOAQjuAEPLiEKtxCDerAQMIzvMKbo50X5935mI0uOfnOAfyB8/kDAIiRJg== x AAAB8XicbVDLSgMxFL3js9ZX1aWbYBFclRnxtSy6cVnBPrAtJZPeaUMzmSHJCHXoX7hxoYhb/8adf2OmnYW2HggczrmXnHv8WHBtXPfbWVpeWV1bL2wUN7e2d3ZLe/sNHSWKYZ1FIlItn2oUXGLdcCOwFSukoS+w6Y9uMr/5iErzSN6bcYzdkA4kDzijxkoPnZCaoR+kT5NeqexW3CnIIvFyUoYctV7pq9OPWBKiNExQrdueG5tuSpXhTOCk2Ek0xpSN6ADblkoaou6m08QTcmyVPgkiZZ80ZKr+3khpqPU49O1kllDPe5n4n9dOTHDVTbmME4OSzT4KEkFMRLLzSZ8rZEaMLaFMcZuVsCFVlBlbUtGW4M2fvEgapxXvonJ+d1auXud1FOAQjuAEPLiEKtxCDerAQMIzvMKbo50X5935mI0uOfnOAfyB8/kDA5KRKA== z Fig. 2: Graphical representation of (a) 2-lay er ER GMs deﬁned in Equation (3) and (b) separable 2-lay er ER GMs deﬁned in Equation (4). Squares represent observed v ariables (netw ork lay ers); circles represent parameters and arro ws represen t directed dep endencies. In panel (a), b oth x and z are inﬂuenced by a common parameter ϑ . In panel (b), x and z each hav e their own parameters, denoted by ξ and ζ , resp ectively . These parameters are distinct, meaning the t wo pro cesses are conditionally indep endent given x . as: p  y t | y t − 1 , ξ F , ζ F , ξ P , ζ P  = p  y F t | y t − 1 , ξ F , ζ F  × p  y P t | y t − 1 , ξ P , ζ P  = p  z F t | x F t , y t − 1 , ζ F  × p  x F t | y t − 1 , ξ F  × p  z P t | x P t , y t − 1 , ζ P  × p  x P t | y t − 1 , ξ P  = exp n ζ F ⊤ s  z F ; x F , y t − 1  − κ ( ζ F ; x F , y t − 1 ) o × exp n ξ F ⊤ s  x F ; y t − 1  − κ ( ξ F ; y t − 1 ) o × exp n ζ P ⊤ s  z P ; x P , y t − 1  − κ ( ζ P ; x P , y t − 1 ) o × exp n ξ P ⊤ s  x P ; y t − 1  − κ ( ξ P ; y t − 1 ) o , (5) where ξ F and ζ F describ e the con tribution of the in teraction net w ork eﬀects and the edge sign net w ork eﬀects to the formation pro cess, resp ectively; and where ξ P and ζ P describ e the contribution of the interaction netw ork eﬀects and the edge sign netw ork eﬀects to the p ersistence process, respectively . Mo del (5) inherits the separability property within each time interv al from Mo del (2) in a wa y that, in each interv al [ t − 1 , t ] the parameters ξ and ζ are conditionally indep endent given the current in teraction structure x t and the past information y t − 1 . This means that while the formation and sign lay ers are modelled as conditionally indep endent within each time in terv al, Mo del (5) do es not necessarily imply full temporal indep endence across time. In fact, the structure of x t ma y b e inﬂuenced not only by x t − 1 , but also by z t − 1 , and similarly , z t ma y dep end not only on z t − 1 but also on x t − 1 . In other words, the presence or absence of an interaction at time t − 1 may explain not only the formation or dissolution of an edge at time t , but also changes in its sign; likewise, the sign structure at t − 1 may inﬂuence not only whether sign p ersist or switc h, but also whether new edges app ear or existing ones disappear. This form of cross-la yer temp oral dep endence is a k ey feature of the mo del, and reﬂects the fact that in teraction and sign dynamics are often intert wined in practice. T o achiev e full separabilit y ov er time betw een the tw o pro cesses, we need to assume that at any time p oint t, x F and x P (and therefore x ) do not dep end on z t − 1 , i.e., at any time p oin t t, s  x F ; y t − 1  = s  x F ; x t − 1  and s  x P ; y t − 1  = s  x P ; x t − 1  . Figure 3 provides a graphical represen tation of the 2-lay er STER GM and illustrates that, given the intera ction pro cesses x 1 and x 2 , the corresp onding formation and p ersistence interaction pro cesses, x F 1 , x F 2 , x P 1 , and x P 2 , are ﬁxed. This implies that ξ F and ζ F are conditionally indep enden t, as are ξ P and ζ P . How ever, the temp oral dependence betw een ξ F and ξ P , as w ell as b etw een ζ F and ζ P , still holds. T able 1 presents the structure of all p ossible dyadic transitions in the signed net wor k pro cess ov er a given time interv al, capturing ho w the relationships betw een no de pairs ev olv e in terms of sign and connectivity . 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Fig. 3: Graphical represen tation of 2-lay er STER GMs for signed netw ork data conditional on an initial netw ork state (not included). W e can therefore interpret the conditional probability of observing a p ositive relation b etw een no des i and j at time t in either the formation or p ersistence pro cess by examining the corresp onding log-o dds expression: log Pr( Z U ij,t = +1 | y U − ( ij ) ,t , X U ij,t = 1 , y t − 1 ) Pr( Z U ij,t = − 1 | y U − ( ij ) ,t , X U ij,t = 1 , y t − 1 ) ! = ζ U ⊤ ∆ ij s ( z U t ; x U t , y t − 1 ) , for U = {F , P } , (6) where Z U ij,t is the potential sign b etw een i and j at time t in either formation or dissolution process, given that the interaction at time t exists in the corresp onding pro cess ( X U ij,t = 1); and the rest of the signed netw ork in the respective pro cess ( y U − ( ij ) ,t ). The right-hand side of the Equation (6) is the net w ork c hange statistics that would result from toggling the sign of Z ij,t , weigh ted by the parameter ζ F or ζ P , resp ectively . A higher log-o dds means the formation or persistence of the sign of the in teraction is more lik ely under the mo del, based on how w ell it matc hes the netw ork conﬁgurations encouraged by the corresponding eﬀects. 7.1. Model terms As with an y ERGM-based mo delling framew ork, the selection of netw ork statistics is highly ﬂexible and con text-dep enden t. Commonly used statistics, suc h as edges, homophily , and transitivit y , are t ypically included in the mo del to capture key structural features. STERGMs are sp eciﬁed by tw o p otentially diﬀeren t sets of net work statistics for formation and persistence. The netw ork statistics s  z F ; x F , y t − 1  and s  z P ; x P , y t − 1  include standard binary eﬀects. These eﬀects are based on the assumption that the la yers z F and z P are endogenous, i.e., structures that are determined within the generative pro cess itself, rather than being sp eciﬁed by external cov ariate information, while x F , x P , and y t − 1 are treated as exogenous. The net w ork statistics s  x F ; y t − 1  and s  x P ; y t − 1  for the 2-lay er STER GM (5) assume that the la y ers x F and x P are endogenous, conditional on y t − 1 . As clearly demonstrated by F ritz et al. (2025), using lagged statistics that combine past and presen t edges can misrepresen t the eﬀects of SBT. F or example, treating the presence of m utual friends or enemies at time t − 1 as an exogenous cov ariate inﬂuencing the sign structure at time t ov erlo oks the fact that such cov ariates ma y not persist at time t , and may evolv e in conjunction with other edge signs to form new balanced 8 Caimo and Gollini T able 1. (a) Possible transitions of a single signed edge variable Y ij from t − 1 to t ; (b) corresponding transitions according to the separable mo del deﬁned in Equation (5) . When no interaction is present ( X ij = 0) the asso ciated sign Z ij is not identiﬁable, which in turn do es not allow us to mo del p ossible changes in its value. (a) (b) Y ij,t − 1 → Y ij,t Y ij,t − 1 Y F ij Y P ij Y ij,t X ij,t − 1 Z ij,t − 1 → X F ij Z F ij X P ij Z P ij → X ij,t Z ij,t 0 0 0 0 0 0 0 +1 0 1 +1 0 1 +1 0 − 1 0 1 − 1 0 1 − 1 +1 0 1 +1 1 +1 0 0 +1 +1 1 +1 1 +1 1 +1 1 +1 +1 − 1 1 +1 1 +1 1 − 1 1 − 1 − 1 0 1 − 1 1 − 1 0 0 − 1 +1 1 − 1 1 − 1 1 +1 1 +1 − 1 − 1 1 − 1 1 − 1 1 − 1 1 − 1 AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E34OMY9OIxonlAsoTeyWwyZHZ2mZkVQsgnePGgiFe/yJt/4yTZgyYWNBRV3XR3BYng2rjut5NbW9/Y3MpvF3Z29/YPiodHTR2nirIGjUWs2gFqJrhkDcONYO1EMYwCwVrB6Hbmt56Y0jyWj2acMD/CgeQhp2is9FDG816x5FbcOcgq8TJSggz1XvGr249pGjFpqECtO56bGH+CynAq2LTQTTVLkI5wwDqWSoyY9ifzU6fkzCp9EsbKljRkrv6emGCk9TgKbGeEZqiXvZn4n9dJTXjtT7hMUsMkXSwKU0FMTGZ/kz5XjBoxtgSp4vZWQoeokBqbTsGG4C2/vEqa1Yp3Wbm4r5ZqN1kceTiBUyiDB1dQgzuoQwMoDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4Ai0uNUg== (a) AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E34OMY9OIxonlAsoTZyWwyZHZ2mekVQsgnePGgiFe/yJt/4yTZgyYWNBRV3XR3BYkUBl3328mtrW9sbuW3Czu7e/sHxcOjpolTzXiDxTLW7YAaLoXiDRQoeTvRnEaB5K1gdDvzW09cGxGrRxwn3I/oQIlQMIpWeigH571iya24c5BV4mWkBBnqveJXtx+zNOIKmaTGdDw3QX9CNQom+bTQTQ1PKBvRAe9YqmjEjT+ZnzolZ1bpkzDWthSSufp7YkIjY8ZRYDsjikOz7M3E/7xOiuG1PxEqSZErtlgUppJgTGZ/k77QnKEcW0KZFvZWwoZUU4Y2nYINwVt+eZU0qxXvsnJxXy3VbrI48nACp1AGD66gBndQhwYwGMAzvMKbI50X5935WLTmnGzmGP7A+fwBjNCNUw== (b) AAAB6nicbVDLSgNBEOyNrxhfUY9eBoMQL2E34OMY9OIxonlAsoTZSW8yZHZ2mZkVQsgnePGgiFe/yJt/4yTZgyYWNBRV3XR3BYng2rjut5NbW9/Y3MpvF3Z29/YPiodHTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWMbmd+6wmV5rF8NOME/YgOJA85o8ZKD2V23iuW3Io7B1klXkZKkKHeK351+zFLI5SGCap1x3MT40+oMpwJnBa6qcaEshEdYMdSSSPU/mR+6pScWaVPwljZkobM1d8TExppPY4C2xlRM9TL3kz8z+ukJrz2J1wmqUHJFovCVBATk9nfpM8VMiPGllCmuL2VsCFVlBmbTsGG4C2/vEqa1Yp3Wbm4r5ZqN1kceTiBUyiDB1dQgzuoQwMYDOAZXuHNEc6L8+58LFpzTjZzDH/gfP4AjlWNVA== (c) AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0jEr2PRi8cK1hbaUDbbSbt0dxN2N0Ip/QtePCji1T/kzX9j0uagrQ8GHu/NMDMvTAQ31vO+ndLK6tr6RnmzsrW9s7tX3T94NHGqGTZZLGLdDqlBwRU2LbcC24lGKkOBrXB0m/utJ9SGx+rBjhMMJB0oHnFGbS65rlvpVWue681AlolfkBoUaPSqX91+zFKJyjJBjen4XmKDCdWWM4HTSjc1mFA2ogPsZFRRiSaYzG6dkpNM6ZMo1lkpS2bq74kJlcaMZZh1SmqHZtHLxf+8Tmqj62DCVZJaVGy+KEoFsTHJHyd9rpFZMc4IZZpntxI2pJoym8WTh+AvvrxMHs9c/9K9uD+v1W+KOMpwBMdwCj5cQR3uoAFNYDCEZ3iFN0c6L8678zFvLTnFzCH8gfP5A4O+jUA= ... 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Fig. 4: Endogenous edgewise shared partners conﬁgurations: (a) esf + : positive edgewise shared friends; (b) ese + : p ositive edgewise shared enemies; (c) ese − : negative edgewise shared enemies; (d) esf − : negative edgewise shared friends. conﬁgurations. F or this reason, in order to eﬀectively model SBT eﬀects, it is necessary to use endogenous netw ork statistics that are implicitly dynamic, that is, they capture temporal dep endencies without explicitly referencing past time p oints. It is imp ortant to emphasise that this reasoning also applies in our context, where SBT eﬀects are estimated conditionally on the interaction structure. Indeed, in our case, assuming that suc h cov ariates remain constan t would imply that b oth the in teractions and their asso ciated signs p ersist from time t − 1 to time t , and are link ed b y an underlying structural dep endency . T o test SBT eﬀects, we adopt the geometrically-weigh ted statistics prop osed by F ritz et al. (2025). These statistics can be included as binary terms s  z F ; x F  and s  z P ; x P  in 2-la yer STER GMs (5) as the edgewise shared partner conﬁgurations sho wn in Figure 4 are used conditionally on the presence of interactions, resulting in binary conﬁgurations within a constrained supp ort. F or example, the geometrically weigh ted distribution of negativ e edgewise shared friends, gwesf − ( y t , α ), where α is the deca y parameter, is equiv alent to the binary geometrically weigh ted non-edgewise shared partners term, gwnsp ( z t ; x t , α ), measured on z t giv en x t . In fact, conditional on x t , eac h dyad in z t is binary , and therefore a negative edge that anchors the p ositiv e tw o-paths in the gwesf − conﬁguration plays the same structural role as a non-edge anchoring the connected tw o-paths coun ted b y the gwnsp conﬁguration in standard binary netw orks. Separable mo dels for dynamic signed netw orks 9 8. Bay esian inference W e adopt a fully probabilistic Bay esian framework to estimate the 2-lay er STERGM by deﬁning a prior distribution ov er the mo del parameters π ( ξ F , ζ F , ξ P , ζ P ), such that the p osterior distribution is given b y: π  ξ F , ζ F , ζ P , ξ P | y 0: T  ∝ π ( ξ F , ζ F , ξ P , ζ P ) T Y t =1 p  y t | y t − 1 , ξ F , ζ F , ξ P , ζ P  = π ( ξ F , ζ F , ξ P , ζ P ) T Y t =1 p  z t | x t , y t − 1 , ζ F , ζ P  p  x t | y t − 1 , ξ F , ξ P  . By assuming indep endent priors π ( ξ F , ζ F , ξ P , ζ P ) = π ( ζ F , ζ P ) × π ( ξ F , ξ P ) we preserv e separability a p osteriori: π  ξ F , ζ F , ξ P , ζ P | y 0: T  = π ( ζ F , ζ P | z 0: T , x 0: T ) × π ( ξ F , ξ P | x 0: T ) . The t w o p osterior terms are doubly intractable since neither the corresp onding lik eliho o ds nor the marginal likelihoo ds are av ailable. How ever, we can estimate b oth by adapting the approximate exc hange algorithm (AEA) as proposed by Caimo and F riel (2011). Algorithm 1 outlines the AEA algorithm for p osterior sampling of parameters ζ F and ζ P given observ ations z 0: T and x 0: T . Sampling from π ( ξ F , ξ P | x 0: T ) follo ws a similar procedure, but omits the dep endence on signed edge states z t during prop osal generation for in teraction structures x F ′ and x P ′ . Algorithm 1 Approximate exc hange algorithm for π ( ζ F , ζ P | z 0: T , x 0: T ) Require: Initial v alues ζ F (0) , ζ P (0) Ensure: Samples ζ F ( i ) , ζ P ( i ) for i = 1 , . . . , I iterations 1: Compute x F t and x P t for all t = 1 , . . . , T 2: for i = 1 to I do 3: Propose ζ F ′ ∼ h F ( · ) , and ζ P ′ ∼ h P ( · ) 4: for t = 1 to T do 5: Sim ulate z F ′ t ∼ p  · | x F t , y t − 1 , ζ F ′  6: Sim ulate z P ′ t ∼ p  · | x P t , y t − 1 , ζ P ′  7: Set z ′ t = z P ′ t ∪  z F ′ t \ z t  8: end for 9: Compute acceptance probabilit y: α = min 1 , T Y t =1 ˜ f  z t | x t , y t − 1 , ζ F ′ , ζ P ′  ˜ f  z t | x t , y t − 1 , ζ F ( i − 1) , ζ P ( i − 1)  × π  ζ F ′ , ζ P ′  π  ζ F ( i − 1) , ζ P ( i − 1)  ! (7) 10: With probability α , set ( ζ F ( i ) , ζ P ( i ) ) ← ( ζ F ′ , ζ P ′ ) 11: Otherwise, set ( ζ F ( i ) , ζ P ( i ) ) ← ( ζ F ( i − 1) , ζ P ( i − 1) ) 12: end for T o address the sampling challenges arising from highly correlated parameters, including correlations b et w een the formation and p ersistence parameter groups, we emplo yed an adaptive Metrop olis–Hastings (AMH) prop osal scheme (Haario et al., 2001). Although the adaptive direction sampling scheme (used by default in Caimo and F riel 2011) necessitates multiple chains and is less eﬃcient o ver the long run, it remains v aluable for sampling initial parameter v alues during the burn-in p erio d. 10 Caimo and Gollini The approximate likelihoo d ratio in (7) is computed as follo ws: T Y t =1 ˜ f  z t | x t , y t − 1 , ζ F ′ , ζ P ′  ˜ f  z t | x t , y t − 1 , ζ F ( i − 1) , ζ P ( i − 1)  = T Y t =1 p  z t | x t , y t − 1 , ζ F ′ , ζ P ′  p  z t | x t , y t − 1 , ζ F ( i − 1) , ζ P ( i − 1)  × p  z ′ t | x t , y t − 1 , ζ F ( i − 1) , ζ P ( i − 1)  p  z ′ t | x t , y t − 1 , ζ F ′ , ζ P ′  = exp (  ζ F ( i − 1) − ζ F ′  ⊤ T X t =1 h s  z F ′ t ; x F t , y t − 1  − s  z F t ; x F t , y t − 1  i +  ζ P ( i − 1) − ζ P ′  ⊤ T X t =1 h s  z P ′ t ; x P t , y t − 1  − s  z P t ; x P t , y t − 1  i ) . A key adv antage of this approach is that netw ork sampling from the likelihoo d can b e conducted with signiﬁcan tly fewer auxiliary iterations and low er computational cost, thanks to the sampling space constraint and reduced complexit y of the statistics which are binary . 8.1. Soft wa re The B2Lstergm pack age for R (R Developmen t Core T eam, 2025), dev eloped alongside this paper, represents a signiﬁcant extension of existing tools for Ba yesian inference on temp oral net work mo dels. It builds on and extends the mo delling and computational capabilities of the Bergm pack age (Caimo et al., 2022, 2025), whic h implements Bay esian inference for static ERGMs. Additionally , the multi.ergm pack age (Krivitsky, 2025), which supp orts m ulti-lay er ER GMs (Krivitsky et al., 2020), is emplo yed here to sim ulate netw ork statistics corresponding to the interaction and conditional sign processes of the tw o separable likelihoo d components describ ed in Equation 5. This simulation relies on the eﬃcient MCMC algorithms implemented in the ergm pac k age (Hunter et al., 2008; Krivitsky et al., 2023; Handcock et al., 2025). Notably , although the tergm() function within the tergm pac k age (Krivitsky and Handcock, 2025) typically relies on sto chastic maximum likelihoo d estimation (MLE), it can alternativ ely use contrastiv e div ergence when computational eﬃciency is a priority or conv ergence issues arise. The current implementation of the B2Lstergm pack age can handle netw orks with a few hundred interacting edges. F or larger netw orks, how ever, more eﬃcient approximate metho ds, such as the pseudo-posterior adjustmen t proposed in Bouranis et al. (2017), w ould be necessary . 9. Comparative simulation study Among existing approaches, only the TERGM by F ritz et al. (2025) and our approach directly extend the ER GM framework to signed netw orks. In this simulation study , we compare three mo delling approaches with the aim of examining ho w their interpretativ e prop erties diﬀer. W e consider the TERGM proposed by F ritz et al. (2025), describ ed in Section 5, as well as our 2-lay er STER GM. In addition, we introduce an intermediate mo delling approach: a separable temp oral extension of the TERGM in F ritz et al. (2025). In this STERGM form ulation, the transition probability is deﬁned as p  y t | y t − 1 , ϑ F , ϑ P  = p  y F t | y t − 1 , ϑ F  × p  y P t | y t − 1 , ϑ P  . It is imp ortant to note that when x is complete, i.e., all nodes interact with all others, the STER GM and the 2-lay er STER GM coincide, as y is binary , enco ding the sign of eac h edge, and therefore equiv ale n t to z . T o better illustrate the in terpretative diﬀerences b etw een the three approaches, w e fo cus on a simple sp eciﬁcation including density and positive transitivit y terms. Sp eciﬁcally , we generate a random graph on 40 no des for the initial interaction netw ork x 1 , where each interaction o ccurs with probabilit y 0 . 2, and each realised edge is indep endently assigned a p ositive or negative sign with probabilit y 0 . 5. The interaction netw ork x 2 is then generated using a TER GM, with b oth the baseline edge density and the change-of-state density parameters set to − 2. Conditional on the transition from x 1 to x 2 w e sim ulate a 2-la y er STER GM Separable mo dels for dynamic signed netw orks 11 to generate z 2 sp eciﬁed as follo ws: ζ F edges + = 0 , ζ F gwesf + ( α ) = 0 . 2 , ζ P edges + = 0 , ζ P gwesf + ( α ) = 0 . Here, edges + denotes the num b er of positive edges in the formation and dissolution pro cesses, and gwesf + is the geometrically w eighted p ositive edgewise shared friends (see Figure 4) representing the statistics related to positive triadic closure. W e set the deca y parameter α for the gwesf + terms to 0 . 6 In practice, the only non-zero eﬀect is therefore the formation parameter ζ F gwesf + , whic h encourages the creation of p ositive triangles from t 1 to t 2 . The TERGM approac h includes terms for the stability of p ositive edges, changes in p ositive edges, and cha nges in transitive triads. The STER GM version of F ritz et al. (2025) uses the same t ype of statistics employ ed in the simulation, as do es the 2-lay er STER GM. In Figure 5, we can observe the MLE estim ates and their corresp onding 95% conﬁdence interv als for b oth the TERGM and STER GM approaches, computed using the ergm.sign pack age for R (Schalberger et al., 2025). W e also show the p osterior means and 95% credible interv als for the tw o-lay er STERGM, obtained with the B2Lstergm pack age using ﬂat priors on all parameters, in order to make the comparison easier. -3 . 0 -2 . 5 -2 . 0 -1 .5 -1 .0 -0 .5 0 . 0 gw es f ( + ) edges ( - ) edges ( + ) -5 -4 -3 -2 -1 0 1 gw es f ( + ) _P edges ( + ) _P gw es f ( + ) _F edges ( + ) _F -1 . 0 -0 . 5 0 . 0 0 . 5 1 . 0 Po i n t Es t i m a t e s w i t h 9 5 % C I s th e t a _ P _ 2 th e t a _ P _ 1 th e t a _ F _ 2 th e t a _ F _ 1 AAACEXicbVDLSgNBEJyNrxhfqx69LAYhIIRd8XUMCuJRwWggm4TZSScZMvtgpleMw/6CF3/FiwdFvHrz5t84iTloYkFDUdVNd1eQCK7Qdb+s3Mzs3PxCfrGwtLyyumavb1yrOJUMqiwWsawFVIHgEVSRo4BaIoGGgYCboH869G9uQSoeR1c4SKAR0m7EO5xRNFLLLvn3gLSp/ZBij1Ghz7KspX2EO0TU0O6Cypp614h20S27IzjTxBuTIhnjomV/+u2YpSFEyARVqu65CTY0lciZgKzgpwoSyvq0C3VDIxqCaujRR5mzY5S204mlqQidkfp7QtNQqUEYmM7h4WrSG4r/efUUO8cNzaMkRYjYz6JOKhyMnWE8TptLYCgGhlAmubnVYT0qKUMTYsGE4E2+PE2u98reYfngcr9YORnHkSdbZJuUiEeOSIWckwtSJYw8kCfyQl6tR+vZerPef1pz1nhmk/yB9fEN9F2fAA== ω F edges + AAACEXicbVDJSgNBEO1xjXGLevQyGISAEGbE7Rj04jGCWSCThJ5OJWnSs9BdI8ZmfsGLv+LFgyJevXnzb+wsB018UPB4r4qqen4suELH+bYWFpeWV1Yza9n1jc2t7dzOblVFiWRQYZGIZN2nCgQPoYIcBdRjCTTwBdT8wdXIr92BVDwKb3EYQzOgvZB3OaNopHau4D0A0pb2Aop9RoUup2lbewj3iKih0wOVtvSREXN5p+iMYc8Td0ryZIpyO/fldSKWBBAiE1SphuvE2NRUImcC0qyXKIgpG9AeNAwNaQCqqccfpfahUTp2N5KmQrTH6u8JTQOlhoFvOkeHq1lvJP7nNRLsXjQ1D+MEIWSTRd1E2BjZo3jsDpfAUAwNoUxyc6vN+lRShibErAnBnX15nlSPi+5Z8fTmJF+6nMaRIfvkgBSIS85JiVyTMqkQRh7JM3klb9aT9WK9Wx+T1gVrOrNH/sD6/AEEep8K ω P edges + 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 ω P gwesf + ( ω ) 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 ω F gwesf + ( ω ) AAACB3icbVDJSgNBEO2JW4zbqEdBBoMgCGFG3I5BLx4jmAUyMfR0apImPQvdNcEwzM2Lv+LFgyJe/QVv/o2d5aCJDwoe71VRVc+LBVdo299GbmFxaXklv1pYW9/Y3DK3d2oqSiSDKotEJBseVSB4CFXkKKARS6CBJ6Du9a9Hfn0AUvEovMNhDK2AdkPuc0ZRS21z3x1QiT1A2k5dhAdETKHTBZXdp8dZ1jaLdskew5onzpQUyRSVtvnldiKWBBAiE1SppmPH2Er1Ds4EZAU3URBT1qddaGoa0gBUKx3/kVmHWulYfiR1hWiN1d8TKQ2UGgae7gwo9tSsNxL/85oJ+petlIdxghCyySI/ERZG1igUq8MlMBRDTSiTXN9qsR6VlKGOrqBDcGZfnie1k5JzXjq7PS2Wr6Zx5MkeOSBHxCEXpExuSIVUCSOP5Jm8kjfjyXgx3o2PSWvOmM7skj8wPn8AsoSafQ== ω edges + AAACB3icbVDJSgNBEO2JW4zbqEdBBoPgxTAjbsegF48RzAKZGHo6NUmTnoXummAY5ubFX/HiQRGv/oI3/8bOctDEBwWP96qoqufFgiu07W8jt7C4tLySXy2srW9sbpnbOzUVJZJBlUUikg2PKhA8hCpyFNCIJdDAE1D3+tcjvz4AqXgU3uEwhlZAuyH3OaOopba57w6oxB4gbacuwgMiptDpgsru0+Msa5tFu2SPYc0TZ0qKZIpK2/xyOxFLAgiRCapU07FjbKV6B2cCsoKbKIgp69MuNDUNaQCqlY7/yKxDrXQsP5K6QrTG6u+JlAZKDQNPdwYUe2rWG4n/ec0E/ctWysM4QQjZZJGfCAsjaxSK1eESGIqhJpRJrm+1WI9KylBHV9AhOLMvz5PaSck5L53dnhbLV9M48mSPHJAj4pALUiY3pEKqhJFH8kxeyZvxZLwY78bHpDVnTGd2yR8Ynz+1kJp/ ω edges → AAACB3icbVDJSgNBEO1xjXEb9SjIYBAEIcyI2zHoxWMEs0ASQ0+nJmnSs9BdEw3D3Lz4K148KOLVX/Dm39hJ5qCJDwoe71VRVc+NBFdo29/G3PzC4tJybiW/ura+sWlubVdVGEsGFRaKUNZdqkDwACrIUUA9kkB9V0DN7V+N/NoApOJhcIvDCFo+7Qbc44yiltrmXnNAJfYAaTtpIjwgYtK9B+Wld8lRmrbNgl20x7BmiZORAslQbptfzU7IYh8CZIIq1XDsCFuJ3sGZgDTfjBVElPVpFxqaBtQH1UrGf6TWgVY6lhdKXQFaY/X3REJ9pYa+qzt9ij017Y3E/7xGjN5FK+FBFCMEbLLIi4WFoTUKxepwCQzFUBPKJNe3WqxHJWWoo8vrEJzpl2dJ9bjonBVPb04KpcssjhzZJfvkkDjknJTINSmTCmHkkTyTV/JmPBkvxrvxMWmdM7KZHfIHxucP0ZeakQ== ω gwesf + AAACHXicbVDLSgNBEJz1bXxFPXpZDIIihF3xdRQF8ahgVMjG0DvpTQZnH8z0qmHYH/Hir3jxoIgHL+LfOIk5+CoYKKq66akKMyk0ed6HMzQ8Mjo2PjFZmpqemZ0rzy+c6TRXHGs8lam6CEGjFAnWSJDEi0whxKHE8/DqoOefX6PSIk1OqZthI4Z2IiLBgazULG8G16CogwSXJoiBOhykOSyKpgkIb4nItG9QR8WlWS9WA5BZB9aKZrniVb0+3L/EH5AKG+C4WX4LWinPY0yIS9C67nsZNYy9LLjEohTkGjPgV9DGuqUJxKgbpp+ucFes0nKjVNmXkNtXv28YiLXuxqGd7AXQv72e+J9XzynabRiRZDlhwr8ORbl0KXV7VbktoZCT7FoCXAn7V5d3QAEnW2jJluD/jvyXnG1U/e3q1slmZW9/UMcEW2LLbJX5bIftsSN2zGqMszv2wJ7Ys3PvPDovzuvX6JAz2FlkP+C8fwJT5KPq ω F gwesf + ( ω ) AAACFXicbVDLSgNBEJz1GeNr1aOXxSAIStgVX0dREI8RjArZJMzOdpIhsw9mesUw7E948Ve8eFDEq+DNv3ES96DGgoaiqpvuriAVXKHrfloTk1PTM7OlufL8wuLSsr2yeqWSTDKos0Qk8iagCgSPoY4cBdykEmgUCLgO+qdD//oWpOJJfImDFJoR7ca8wxlFI7XtHf+WSuwB0pb2I4o9RoU+y/O29hHuEFFD2AWVt/S2Ee2KW3VHcMaJV5AKKVBr2x9+mLAsghiZoEo1PDfFpjYbOROQl/1MQUpZn3ahYWhMI1BNPfoqdzaNEjqdRJqK0RmpPyc0jZQaRIHpHB6u/npD8T+vkWHnqKl5nGYIMfte1MmEg4kzjMgJuQSGYmAIZZKbWx3Wo5IyNEGWTQje35fHydVu1Tuo7l/sVY5PijhKZJ1skC3ikUNyTM5JjdQJI/fkkTyTF+vBerJerbfv1gmrmFkjv2C9fwFWOqDT ω F edges + AAACFXicbVDJSgNBEO1xjXGLevQyGARBCTPidgx68RjBLJBJQk+nkjTpWeiuCYZmfsKLv+LFgyJeBW/+jZ3loIkPCh7vVVFVz48FV+g439bC4tLyympmLbu+sbm1ndvZragokQzKLBKRrPlUgeAhlJGjgFosgQa+gKrfvxn51QFIxaPwHocxNALaDXmHM4pGauVOvAGV2AOkTe0FFHuMCl1K05b2EB4QUUO7Cypt6mMj5vJOwRnDnifulOTJFKVW7strRywJIEQmqFJ114mxoc1GzgSkWS9REFPWp12oGxrSAFRDj79K7UOjtO1OJE2FaI/V3xOaBkoNA990jg5Xs95I/M+rJ9i5amgexglCyCaLOomwMbJHEdltLoGhGBpCmeTmVpv1qKQMTZBZE4I7+/I8qZwW3IvC+d1Zvng9jSND9skBOSIuuSRFcktKpEwYeSTP5JW8WU/Wi/VufUxaF6zpzB75A+vzB2ZIoN0= ω P edges + AAACHXicbVDLSgNBEJz1bXxFPXpZDIIihF3xdRS9eIxgVMjG0DvpTQZnH8z0qmHYH/Hir3jxoIgHL+LfOIk5+CoYKKq66akKMyk0ed6HMzI6Nj4xOTVdmpmdm18oLy6d6TRXHOs8lam6CEGjFAnWSZDEi0whxKHE8/DqqO+fX6PSIk1OqZdhM4ZOIiLBgazUKm8H16CoiwSXJoiBuhykqRVFywSEt0RkOjeoo+LSbBbrAcisCxtFq1zxqt4A7l/iD0mFDVFrld+CdsrzGBPiErRu+F5GTWMvCy6xKAW5xgz4FXSwYWkCMeqmGaQr3DWrtN0oVfYl5A7U7xsGYq17cWgn+wH0b68v/uc1cor2m0YkWU6Y8K9DUS5dSt1+VW5bKOQke5YAV8L+1eVdUMDJFlqyJfi/I/8lZ1tVf7e6c7JdOTgc1jHFVtgqW2c+22MH7JjVWJ1xdsce2BN7du6dR+fFef0aHXGGO8vsB5z3T2RCo/Q= ω P gwesf + ( ω ) AAAB7HicbVBNS8NAEJ3Ur1q/oh69LBbBU0kKfhyLInoRqjRtoQ1ls920SzebsLsRSuhv8OJBEa/+IG/+G7dtDtr6YODx3gwz84KEM6Ud59sqrKyurW8UN0tb2zu7e/b+QVPFqSTUIzGPZTvAinImqKeZ5rSdSIqjgNNWMLqe+q0nKhWLRUOPE+pHeCBYyAjWRvIaN4+39z277FScGdAycXNShhz1nv3V7cckjajQhGOlOq6TaD/DUjPC6aTUTRVNMBnhAe0YKnBElZ/Njp2gE6P0URhLU0Kjmfp7IsORUuMoMJ0R1kO16E3F/7xOqsNLP2MiSTUVZL4oTDnSMZp+jvpMUqL52BBMJDO3IjLEEhNt8imZENzFl5dJs1pxzytnD9Vy7SqPowhHcAyn4MIF1OAO6uABAQbP8ApvlrBerHfrY95asPKZQ/gD6/MHDkiOMw== TER GM AAAB7XicbVDLSgNBEOyNrxhfUY9eBoPgKewGfByDInoRouYFyRJmJ5NkzOzMMjMrhCX/4MWDIl79H2/+jZNkD5pY0FBUddPdFUScaeO6305maXlldS27ntvY3Nreye/u1bWMFaE1IrlUzQBrypmgNcMMp81IURwGnDaC4eXEbzxRpZkUVTOKqB/ivmA9RrCxUv2henV/fdvJF9yiOwVaJF5KCpCi0sl/tbuSxCEVhnCsdctzI+MnWBlGOB3n2rGmESZD3KctSwUOqfaT6bVjdGSVLupJZUsYNFV/TyQ41HoUBrYzxGag572J+J/Xik3v3E+YiGJDBZkt6sUcGYkmr6MuU5QYPrIEE8XsrYgMsMLE2IByNgRv/uVFUi8VvdPiyV2pUL5I48jCARzCMXhwBmW4gQrUgMAjPMMrvDnSeXHenY9Za8ZJZ/bhD5zPH7KfjpA= STER GM AAAB8HicbVDLSgNBEOyNrxhfUY9eFoPgxbAb8HEMiuhBIWpekixhdjKbDJmZXWZmhbDkK7x4UMSrn+PNv3GS7EETCxqKqm66u/yIUaUd59vKLCwuLa9kV3Nr6xubW/ntnboKY4lJDYcslE0fKcKoIDVNNSPNSBLEfUYa/uBi7DeeiFQ0FFU9jIjHUU/QgGKkjfRYujl6qF7eX9128gWn6ExgzxM3JQVIUenkv9rdEMecCI0ZUqrlOpH2EiQ1xYyMcu1YkQjhAeqRlqECcaK8ZHLwyD4wStcOQmlKaHui/p5IEFdqyH3TyZHuq1lvLP7ntWIdnHkJFVGsicDTRUHMbB3a4+/tLpUEazY0BGFJza027iOJsDYZ5UwI7uzL86ReKronxeO7UqF8nsaRhT3Yh0Nw4RTKcA0VqAEGDs/wCm+WtF6sd+tj2pqx0pld+APr8wcpa49Z 2L-STER GM Fig. 5: MLE estimates (black dots) with 95% conﬁdence interv als are shown for the TERGM and STERGM based on F ritz et al. (2025), while the 2-lay er STER GM displa ys p osterior means (black dots) with 95% credible in terv als. In the 2-la y er STER GM plot, the white dots with blac k outline indicate the true parameter v alues used to generate the data. W e can see that the p ositiv e and negativ e density parameters in the TER GM correctly capture the drop in o verall density caused b y the reduced level of in teraction from t 1 to t 2 . The parameter ϑ gwesf + , asso ciated with p ositiv e triangles, is not signiﬁcan tly diﬀerent from zero. In this mo del, without additional controls for other forms of transitiv e structure, it is not p ossible to determine whether this v alue reﬂects a real coun teracting eﬀect, or whether the c hange instead arises from substantially greater formation or prop ortionally less dissolution of transitive structures, relative to the others. In practice, this model determines dyad change rates according to the observ ed probability of each dyad switching state. Therefore, when the initial netw ork has a densit y below 0 . 5, there are more opp ortunities for formation, whereas a density abov e 0 . 5 results in more opp ortunities for dissolution. In the STERGM, the parameter ϑ F edges + is negativ e, as it reﬂects the fact that the prop ortion of p ositive edges formed b etw een t 1 and t 2 is negativ e due to the non signiﬁcan t increase in the prop ortion of p ositive 12 Caimo and Gollini edges in the formation process (relativ e to the prop ortion of negativ e edges and empt y dy ads). The parameter ϑ F gwesf + is not signiﬁcant. If we add a control for transitive interaction in the formation pro cess, ϑ F gwesf + b ecomes signiﬁcan tly p ositive (5 . 46 with a standard error of ab out 0 . 17), balancing a signiﬁcantly negative parameter ( − 5 . 65 with a standard error of about 0 . 16). W e can therefore conclude that this categorical model captures the increase in p ositiv e transitive structures, but we cannot determine precisely whether this is due to the formation of genuinely new p ositive interactions or to the transformation of existing non-positive conﬁgurations in to p ositiv e ones. In other w ords, w e cannot clearly separate the eﬀect of sign changes from the formation in teraction process itself. The 2L-STERGM mo del, through the parameter ζ F gwesf + , captures the increase in the proportion of p ositive transitiv e relations at the expense of other t yp es of triadic signed conﬁgurations, allowing us to deduce that there has b een a rise in p ositive transitive structures, net of ov erall structural changes in the in teraction netw ork. The greater uncertaint y in the persistence pro cess, arises from the reduced sample space determined b y the lo w n umber of p ersistent in teractions from t 1 to t 2 . 10. Application to US Congressional data W e employ our model to examine the evolution of p olitical coop eration and opposition in the United States Congress using the signed netw ork data inferred by Neal (2014, 2020) from bill co-sp onsorship data via stochastic degree sequence mo del (SDSM). Our focus is on the 99th-101st Congresses (1985-1991), c haracterised b y a hung go vernmen t with a split ma jority . This p erio d sa w a republican president, Ronal Reagan, gov erning alongside a Demo cratic ma jority in the Senate. An examination of the Senate b ehaviour during this time can pro vide v aluable insights into the dynamics of the Senate and its role in shaping p olicy during a p erio d of signiﬁcan t uphea v al (Rohde, 1991). This perio d encompasses sev eral pivotal even ts, including the collapse of the So viet Union (1989-1991), the Gulf W ar (1990-1991), and the Savings and Loan Crisis (1989-1991). When Congressional co-sp onsorship data are mo delled using the SDSM framework, eac h legislator has exp ected p ositive and negativ e degree tendencies. F or each dyad, the probability of a p ositive, negativ e, or no edge is computed based on the product of their resp ectiv e tendencies. Edges are then assigned probabilistically: dyads with higher p ositive tendencies are more likely to become positive edges, those with higher negative tendencies are more lik ely to b ecome negative edges, and others remain unconnected. This produces a signed net work that reﬂects legislators’ expected co-sponsorship and av oidance patterns. W e concen trate our analysis on the 70 Senators who remained in oﬃce throughout the 99th-101st Congresses, in order to examine the b ehaviour of the netw ork eﬀects related to their part y aﬃliation and structural balance. AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRoCWJjSVGQRK4kLllDzbs7V1290zIhZ9gY6Extv4iO/+NC1yh4EsmeXlvJjPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTjo5TRVmbxiJW3QA1E1yytuFGsG6iGEaBYI/B5GbuPz4xpXksH8w0YX6EI8lDTtFY6b6Kl4Nyxa25C5B14uWkAjlag/JXfxjTNGLSUIFa9zw3MX6GynAq2KzUTzVLkE5wxHqWSoyY9rPFqTNyYZUhCWNlSxqyUH9PZBhpPY0C2xmhGetVby7+5/VSE177GZdJapiky0VhKoiJyfxvMuSKUSOmliBV3N5K6BgVUmPTKdkQvNWX10mnXvMatcZdvdKs53EU4QzOoQoeXEETbqEFbaAwgmd4hTdHOC/Ou/OxbC04+cwp/IHz+QOGzY1D (a) AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRoCWJjSVGQRK4kL1lDjbs7V1290zIhZ9gY6Extv4iO/+NC1yh4EsmeXlvJjPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTjo5TxbDNYhGrbkA1Ci6xbbgR2E0U0igQ+BhMbub+4xMqzWP5YKYJ+hEdSR5yRo2V7qvB5aBccWvuAmSdeDmpQI7WoPzVH8YsjVAaJqjWPc9NjJ9RZTgTOCv1U40JZRM6wp6lkkao/Wxx6oxcWGVIwljZkoYs1N8TGY20nkaB7YyoGetVby7+5/VSE177GZdJalCy5aIwFcTEZP43GXKFzIipJZQpbm8lbEwVZcamU7IheKsvr5NOveY1ao27eqVZz+MowhmcQxU8uIIm3EIL2sBgBM/wCm+OcF6cd+dj2Vpw8plT+APn8weIUo1E (b) AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzHBhtxRoCWJjSVGQRK4kL1lDjbs7V1290zIhZ9gY6Extv4iO/+NC1yh4EsmeXlvJjPzgkRwbVz32ylsbG5t7xR3S3v7B4dH5eOTjo5TxbDNYhGrbkA1Ci6xbbgR2E0U0igQ+BhMbub+4xMqzWP5YKYJ+hEdSR5yRo2V7qvsclCuuDV3AbJOvJxUIEdrUP7qD2OWRigNE1Trnucmxs+oMpwJnJX6qcaEsgkdYc9SSSPUfrY4dUYurDIkYaxsSUMW6u+JjEZaT6PAdkbUjPWqNxf/83qpCa/9jMskNSjZclGYCmJiMv+bDLlCZsTUEsoUt7cSNqaKMmPTKdkQvNWX10mnXvMatcZdvdKs53EU4QzOoQoeXEETbqEFbWAwgmd4hTdHOC/Ou/OxbC04+cwp/IHz+QOJ141F (c) Fig. 6: US Senators signed net w ork. Blue and red no des represent Demo cratic and Republican Senators, resp ectiv ely . Blac k and orange edges represent p ositiv e and negative relationships, respectively . Graphs: (a) signed relationships during the 99th Congress (initial state); (b) signed relationships during the 100th Congress; (c) signed relationships during the 101st Congress. Separable mo dels for dynamic signed netw orks 13 The observ ed signed netw orks displa yed in Figure 6 exhibit considerable interaction density . Sp eciﬁcally , the formation interaction density rises from just below 0.60 to nearly 0.61, while the p ersistence in teraction density increases from 0.26 to almost 0.31. Most in teractions are negative, ev en within parties, across b oth the formation and p ersistence pro cesses. 10.1. Model speciﬁcation W e adopt a time-homogeneous 2-lay er STERGM where the formation and persistence conﬁgurations are the same and the list of statistics used is sp eciﬁed b elow. Obviously , in the case of excessive non-stationarity across multiple netw ork observ ations, the use of a homogeneous model could pro v e problematic. As mentioned in the previous section, our focus is to assess the extent of the imp ortance of w eak balance structures, and thus we include (non-degenerate) conﬁgurations corresp onding to the weakly balanced triads (Figure 1) in b oth the formation and persistence pro cesses. The in teraction netw ork exhibits suc h high densit y that it lac ks discernible structural heterogeneit y . Consequen tly , w e restrict our analysis to the conditional signed pro cess and the p osterior distribution of its asso ciated parameters π ( ζ F , ζ P | z 0: T , x 0: T ). W e deﬁne a 2-lay er STER GM using the follo wing net work statistics: 1. n umber of positive edges, edges + , measuring the density of p ositive interactions; 2. n umber of p ositiv e edges within the Republican part y , homophily + ( rep ) , measuring the densit y of positive in teractions within the Republican part y; 3. geometrically-w eighted p ositive degrees, gwdegree + ( α ) , measuring the tendency of actors to form multiple positi v e edges; 4. geometrically-w eighted positive edgewise shared friends, gwesf + ( α ) , capturing the p ositive triadic closure conﬁgurations; 5. geometrically-w eighted positive edgewise shared enemies, gwese + ( α ) , capturing the tendency for negative triadic closure among p ositive edges; 6. geometrically-w eighted negative edgewise shared enemies, gwese − ( α ) , capturing negativ e triadic closure conﬁgurations. W e set the decay parameter α to 0 . 2 for the geometrically-weigh ted degree statistic, and to 0 . 6 for all geometrically-w eighted edgewise shared partner-based statistics (Snijders et al., 2006). W e assign indep endent, weakly informative Normal priors to all mo del parameters, incorp orating mild guidance from the structural characteristics of the initial netw ork. Speciﬁcally , the prior means for b oth ζ F edges + and ζ P edges + are set to –1, reﬂecting a prior b elief that the baseline probability of a p ositive edge (not in volv ed in triadic or degree-based conﬁgurations) is low, but without imposing a strong constraint on its magnitude. Similarly , the prior mean for ζ F gwesf + ( α ) is set to reﬂect moderate triadic closure, consistent with exp ectations of structural balance in co operative interactions. All priors are assigned a large v ariance of 25, ensuring they remain w eakly informative, v ague enough to let the data dominate inference, while still prev enting numerical instabilities and implausible para meter v alues during MCMC estimation. 10.2. P osterior inference P osterior inference is conducted using the adaptiv e AEA, as outlined in Algorithm (1). Each iteration emplo ys 5000 auxiliary dra ws for the net work sim ulation, and the main Mark ov c hain comprises 35000 iterations, with the initial 10000 discarded as burn-in. MCMC trace plots (Figure 7) of the simulated posterior parameters indicate that the MCMC chain conv erges to the stationary p osterior distribution. T able 2 presen ts a summary of the posterior densit y parameter estimates. The qualit y of the posterior estimates w as ev aluated using autocorrelation plots (Figure 8) and eﬀective sample sizes. All parameters exhibited eﬀectiv e sample sizes greater than 200, suggesting adequate c hain con vergence and mixing. P osterior predictiv e chec ks are p erformed to assess ho w well the mo del captures the observ ed data. Simulated netw ork statistics were generated from the p osterior predictiv e distribution and compared to the observ ed statistics using b oxplots (Figure 9). In these plots, the medians of the p osterior predictive statistics 14 Caimo and Gollini AAACEXicbVDLSgNBEJyNrxhfqx69LAYhIIRd8XUMCuJRwWggm4TZSScZMvtgpleMw/6CF3/FiwdFvHrz5t84iTloYkFDUdVNd1eQCK7Qdb+s3Mzs3PxCfrGwtLyyumavb1yrOJUMqiwWsawFVIHgEVSRo4BaIoGGgYCboH869G9uQSoeR1c4SKAR0m7EO5xRNFLLLvn3gLSp/ZBij1Ghz7KspX2EO0TU0O6Cypp614h20S27IzjTxBuTIhnjomV/+u2YpSFEyARVqu65CTY0lciZgKzgpwoSyvq0C3VDIxqCaujRR5mzY5S204mlqQidkfp7QtNQqUEYmM7h4WrSG4r/efUUO8cNzaMkRYjYz6JOKhyMnWE8TptLYCgGhlAmubnVYT0qKUMTYsGE4E2+PE2u98reYfngcr9YORnHkSdbZJuUiEeOSIWckwtSJYw8kCfyQl6tR+vZerPef1pz1nhmk/yB9fEN9F2fAA== ω F edges + AAACEXicbVDJSgNBEO1xjXGLevQyGISAEGbE7Rj04jGCWSCThJ5OJWnSs9BdI8ZmfsGLv+LFgyJevXnzb+wsB018UPB4r4qqen4suELH+bYWFpeWV1Yza9n1jc2t7dzOblVFiWRQYZGIZN2nCgQPoYIcBdRjCTTwBdT8wdXIr92BVDwKb3EYQzOgvZB3OaNopHau4D0A0pb2Aop9RoUup2lbewj3iKih0wOVtvSREXN5p+iMYc8Td0ryZIpyO/fldSKWBBAiE1SphuvE2NRUImcC0qyXKIgpG9AeNAwNaQCqqccfpfahUTp2N5KmQrTH6u8JTQOlhoFvOkeHq1lvJP7nNRLsXjQ1D+MEIWSTRd1E2BjZo3jsDpfAUAwNoUxyc6vN+lRShibErAnBnX15nlSPi+5Z8fTmJF+6nMaRIfvkgBSIS85JiVyTMqkQRh7JM3klb9aT9WK9Wx+T1gVrOrNH/sD6/AEEep8K ω P edges + AAACHHicbVDLSgNBEJz1bXxFPXpZDIIihF3fR1EQjwpGhWwMvZNOMjj7YKZXjcN+iBd/xYsHRbx4EPwbJzEHXwUDRVU3PVVhKoUmz/twBgaHhkdGx8YLE5NT0zPF2blTnWSKY4UnMlHnIWiUIsYKCZJ4niqEKJR4Fl7ud/2zK1RaJPEJdVKsRdCKRVNwICvVi+vBLRJcmCACanOQ5iDP6yYgvCEi07puYEsh5hdmNV8OQKZtWMnrxZJX9npw/xK/T0qsj6N68S1oJDyLMCYuQeuq76VUM6BIcIl5Icg0psAvoYVVS2OIUNdML1zuLlml4TYTZV9Mbk/9vmEg0roThXaym0H/9rrif141o+ZOzYg4zQhj/nWomUmXErfblNsQCjnJjiXAlbB/dXkbFHCyfRZsCf7vyH/J6VrZ3ypvHm+Udvf6dYyxBbbIlpnPttkuO2RHrMI4u2MP7Ik9O/fOo/PivH6NDjj9nXn2A877J1sMo2M= ω F gwdegree + ( ω ) AAACHHicbVDLSgNBEJz1bXxFPXpZDIIihF3fR9GLxwhGhWwMvZNOMjj7YKZXjcN+iBd/xYsHRbx4EPwbJzEHXwUDRVU3PVVhKoUmz/twhoZHRsfGJyYLU9Mzs3PF+YVTnWSKY5UnMlHnIWiUIsYqCZJ4niqEKJR4Fl4e9vyzK1RaJPEJdVOsR9CORUtwICs1ipvBLRJcmCAC6nCQppLnDRMQ3hCRaV83sa0Q8wuznq8GINMOrOWNYskre324f4k/ICU2QKVRfAuaCc8ijIlL0LrmeynVDSgSXGJeCDKNKfBLaGPN0hgi1HXTD5e7K1Zpuq1E2ReT21e/bxiItO5GoZ3sZdC/vZ74n1fLqLVXNyJOM8KYfx1qZdKlxO015TaFQk6yawlwJexfXd4BBZxsnwVbgv878l9yulH2d8rbx1ul/YNBHRNsiS2zVeazXbbPjliFVRlnd+yBPbFn5955dF6c16/RIWews8h+wHn/BGuIo20= ω P gwdegree + ( ω ) 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 ω P gwesf + ( ω ) 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 ω F gwesf + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIihF3xdRQF8ahgVMjG0DvpJENmH8z0qnHY3/Dir3jxoIhHPfk3TmIOvgoGiqpueqrCVApNnvfhDA2PjI6NT0wWpqZnZueK8wtnOskUxwpPZKIuQtAoRYwVEiTxIlUIUSjxPOwc9PzzK1RaJPEpdVOsRdCKRVNwICvVi15wiwSXJoiA2hykOczzugkIb4jItK5RY35p1vPVAGTahrW8Xix5Za8P9y/xB6TEBjiuF9+CRsKzCGPiErSu+l5KNQOKBJeYF4JMYwq8Ay2sWhpDhLpm+slyd8UqDbeZKPticvvq9w0DkdbdKLSTvQD6t9cT//OqGTV3a0bEaUYY869DzUy6lLi9mtyGUMhJdi0BroT9q8vboICTLbNgS/B/R/5LzjbK/nZ562SztLc/qGOCLbFltsp8tsP22BE7ZhXG2R17YE/s2bl3Hp0X5/VrdMgZ7CyyH3DePwHi3aIW ω F gwese + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIihF3xdQx68RjBqJCNoXfSSQZnH8z0qnHY3/Dir3jxoIhHPfk3TmIOvgoGiqpueqrCVApNnvfhjIyOjU9MTk0XZmbn5heKi0unOskUxxpPZKLOQ9AoRYw1EiTxPFUIUSjxLLw87PtnV6i0SOIT6qXYiKATi7bgQFZqFr3gFgkuTBABdTlIU83zpgkIb4jIdK5RY35hNvP1AGTahY28WSx5ZW8A9y/xh6TEhqg2i29BK+FZhDFxCVrXfS+lhgFFgkvMC0GmMQV+CR2sWxpDhLphBslyd80qLbedKPticgfq9w0Dkda9KLST/QD6t9cX//PqGbX3G0bEaUYY869D7Uy6lLj9mtyWUMhJ9iwBroT9q8u7oICTLbNgS/B/R/5LTrfK/m5553i7VDkY1jHFVtgqW2c+22MVdsSqrMY4u2MP7Ik9O/fOo/PivH6NjjjDnWX2A877J/M7oiA= ω P gwese + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIeDLvi6xj04jGCUSEbQ++kkwzOPpjpVeOwv+HFX/HiQRGPevJvnMQcfBUMFFXd9FSFqRSaPO/DGRkdG5+YnJouzMzOzS8UF5dOdZIpjjWeyESdh6BRihhrJEjieaoQolDiWXh52PfPrlBpkcQn1EuxEUEnFm3BgazULHrBLRJcmCAC6nKQpprnTRMQ3hCR6VyjxvzCbObrAci0Cxt5s1jyyt4A7l/iD0mJDVFtFt+CVsKzCGPiErSu+15KDQOKBJeYF4JMYwr8EjpYtzSGCHXDDJLl7ppVWm47UfbF5A7U7xsGIq17UWgn+wH0b68v/ufVM2rvN4yI04ww5l+H2pl0KXH7NbktoZCT7FkCXAn7V5d3QQEnW2bBluD/jvyXnG6V/d3yzvF2qXIwrGOKrbBVts58tscq7IhVWY1xdsce2BN7du6dR+fFef0aHXGGO8vsB5z3T/ZXoiI= ω P gwese → ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIeDLvi6ygK4lHBqJCNoXfSSYbMPpjpVeOwv+HFX/HiQRGPevJvnMQcfBUMFFXd9FSFqRSaPO/DGRoeGR0bn5gsTE3PzM4V5xfOdJIpjhWeyERdhKBRihgrJEjiRaoQolDiedg56PnnV6i0SOJT6qZYi6AVi6bgQFaqF73gFgkuTRABtTlIc5jndRMQ3hCRaV2jxvzSrOerAci0DWt5vVjyyl4f7l/iD0iJDXBcL74FjYRnEcbEJWhd9b2UagYUCS4xLwSZxhR4B1pYtTSGCHXN9JPl7opVGm4zUfbF5PbV7xsGIq27UWgnewH0b68n/udVM2ru1oyI04ww5l+Hmpl0KXF7NbkNoZCT7FoCXAn7V5e3QQEnW2bBluD/jvyXnG2U/e3y1slmaW9/UMcEW2LLbJX5bIftsSN2zCqMszv2wJ7Ys3PvPDovzuvX6JAz2FlkP+C8fwLl+aIY ω F gwese → ( ω ) AAACJHicbVDLSgMxFM34tr6qLt0Ei6AIZUZ8gRtREJcKVoVOLZn01oZmJkNyR6xhPsaNv+LGhQ9cuPFbTGsFXwcCh3Pu5eacKJXCoO+/eQODQ8Mjo2PjhYnJqemZ4uzcqVGZ5lDhSip9HjEDUiRQQYESzlMNLI4knEXt/a5/dgXaCJWcYCeFWswuE9EUnKGT6sUdGt4AsgsbxgxbnEl7kOd1GyJcI6JtqVilLSE7+YVdzZe/ZA1pvpLXiyW/7PdA/5KgT0qkj6N68TlsKJ7FkCCXzJhq4KdYs0yj4BLyQpgZSBlvs0uoOpqwGEzN9kLmdMkpDdpU2r0EaU/9vmFZbEwnjtxkN4r57XXF/7xqhs3tmhVJmiEk/PNQM5MUFe02RhtCA0fZcYRxLdxfKW8xzTi6XguuhOB35L/kdK0cbJY3jtdLu3v9OsbIAlkkyyQgW2SXHJIjUiGc3JJ78kievDvvwXvxXj9HB7z+zjz5Ae/9A51Bp0g= ω F homophily + ( rep ) AAACJHicbVDLSgMxFM34tr6qLt0Ei6AIZUZ8gRvRjcsKVoVOLZn01oZmJkNyR6xhPsaNv+LGhQ9cuPFbTGsFXwcCh3Pu5eacKJXCoO+/eUPDI6Nj4xOThanpmdm54vzCqVGZ5lDlSip9HjEDUiRQRYESzlMNLI4knEWdw55/dgXaCJWcYDeFeswuE9ESnKGTGsU9Gt4AsgsbxgzbnElbyfOGDRGuEdG2VazStpDd/MKu56tfsoY0X8sbxZJf9vugf0kwICUyQKVRfA6bimcxJMglM6YW+CnWLdMouIS8EGYGUsY77BJqjiYsBlO3/ZA5XXFKk7aUdi9B2le/b1gWG9ONIzfZi2J+ez3xP6+WYWu3bkWSZggJ/zzUyiRFRXuN0abQwFF2HWFcC/dXyttMM46u14IrIfgd+S853SgH2+Wt483S/sGgjgmyRJbJKgnIDtknR6RCqoSTW3JPHsmTd+c9eC/e6+fokDfYWSQ/4L1/AK4Dp1I= ω P homophily + ( rep ) Fig. 7: MCMC histograms and trace plots for eac h marginal p osterior parameter distribution. are c lose to 0 , indicating that the observ ed data are consisten t with what the mo del predicts. This suggests that the model pro vides a go od ﬁt to the data in terms of model statistics. Although the analysis is restricted to the subset of U.S. Senators who serv ed con tinuously from the 99th to the 101st Congress, the p osterior parameter estimates, presented in T able 2, yield several key insigh ts into the dynamics of p olitical interactions: • The strongly negative 95% p osterior credible interv als for ζ edges + in b oth the formation and p ersistence processes indicate that the baseline probability of forming or maintaining a p ositive edge is very low. F or example, the p osterior mean of ζ edges + = − 3 . 95 in the formation pro cess corresponds to a conditional probabilit y of forming a p ositive edge giv en an interaction of logit − 1 ( − 3 . 95) ≈ 0 . 019, and for p ersistence, ζ edges + = − 4 . 77 gives logit − 1 ( − 4 . 77) ≈ 0 . 008 . • The 95% p osterior credible interv al for ζ homophily + ( rep ) is sligh tly p ositiv e for formation but not signiﬁcan tly diﬀeren t from 0 for p ersistence. Marginally , this translates to a mo dest increase in the probability of Separable mo dels for dynamic signed netw orks 15 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on 05 0 1 0 0 1 5 0 2 0 0 2 5 0 -1 .0 0 .0 1. 0 Lag Autocor r el ati on AAACEXicbVDLSgNBEJyNrxhfqx69LAYhIIRd8XUMCuJRwWggm4TZSScZMvtgpleMw/6CF3/FiwdFvHrz5t84iTloYkFDUdVNd1eQCK7Qdb+s3Mzs3PxCfrGwtLyyumavb1yrOJUMqiwWsawFVIHgEVSRo4BaIoGGgYCboH869G9uQSoeR1c4SKAR0m7EO5xRNFLLLvn3gLSp/ZBij1Ghz7KspX2EO0TU0O6Cypp614h20S27IzjTxBuTIhnjomV/+u2YpSFEyARVqu65CTY0lciZgKzgpwoSyvq0C3VDIxqCaujRR5mzY5S204mlqQidkfp7QtNQqUEYmM7h4WrSG4r/efUUO8cNzaMkRYjYz6JOKhyMnWE8TptLYCgGhlAmubnVYT0qKUMTYsGE4E2+PE2u98reYfngcr9YORnHkSdbZJuUiEeOSIWckwtSJYw8kCfyQl6tR+vZerPef1pz1nhmk/yB9fEN9F2fAA== ω F edges + AAACHHicbVDLSgNBEJz1bXxFPXpZDIIihF3fR1EQjwpGhWwMvZNOMjj7YKZXjcN+iBd/xYsHRbx4EPwbJzEHXwUDRVU3PVVhKoUmz/twBgaHhkdGx8YLE5NT0zPF2blTnWSKY4UnMlHnIWiUIsYKCZJ4niqEKJR4Fl7ud/2zK1RaJPEJdVKsRdCKRVNwICvVi+vBLRJcmCACanOQ5iDP6yYgvCEi07puYEsh5hdmNV8OQKZtWMnrxZJX9npw/xK/T0qsj6N68S1oJDyLMCYuQeuq76VUM6BIcIl5Icg0psAvoYVVS2OIUNdML1zuLlml4TYTZV9Mbk/9vmEg0roThXaym0H/9rrif141o+ZOzYg4zQhj/nWomUmXErfblNsQCjnJjiXAlbB/dXkbFHCyfRZsCf7vyH/J6VrZ3ypvHm+Udvf6dYyxBbbIlpnPttkuO2RHrMI4u2MP7Ik9O/fOo/PivH6NDjj9nXn2A877J1sMo2M= ω F gwdegree + ( ω ) 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 ω F gwesf + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIihF3xdRQF8ahgVMjG0DvpJENmH8z0qnHY3/Dir3jxoIhHPfk3TmIOvgoGiqpueqrCVApNnvfhDA2PjI6NT0wWpqZnZueK8wtnOskUxwpPZKIuQtAoRYwVEiTxIlUIUSjxPOwc9PzzK1RaJPEpdVOsRdCKRVNwICvVi15wiwSXJoiA2hykOczzugkIb4jItK5RY35p1vPVAGTahrW8Xix5Za8P9y/xB6TEBjiuF9+CRsKzCGPiErSu+l5KNQOKBJeYF4JMYwq8Ay2sWhpDhLpm+slyd8UqDbeZKPticvvq9w0DkdbdKLSTvQD6t9cT//OqGTV3a0bEaUYY869DzUy6lLi9mtyGUMhJdi0BroT9q8vboICTLbNgS/B/R/5LzjbK/nZ562SztLc/qGOCLbFltsp8tsP22BE7ZhXG2R17YE/s2bl3Hp0X5/VrdMgZ7CyyH3DePwHi3aIW ω F gwese + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIeDLvi6ygK4lHBqJCNoXfSSYbMPpjpVeOwv+HFX/HiQRGPevJvnMQcfBUMFFXd9FSFqRSaPO/DGRoeGR0bn5gsTE3PzM4V5xfOdJIpjhWeyERdhKBRihgrJEjiRaoQolDiedg56PnnV6i0SOJT6qZYi6AVi6bgQFaqF73gFgkuTRABtTlIc5jndRMQ3hCRaV2jxvzSrOerAci0DWt5vVjyyl4f7l/iD0iJDXBcL74FjYRnEcbEJWhd9b2UagYUCS4xLwSZxhR4B1pYtTSGCHXN9JPl7opVGm4zUfbF5PbV7xsGIq27UWgnewH0b68n/udVM2ru1oyI04ww5l+Hmpl0KXF7NbkNoZCT7FoCXAn7V5e3QQEnW2bBluD/jvyXnG2U/e3y1slmaW9/UMcEW2LLbJX5bIftsSN2zCqMszv2wJ7Ys3PvPDovzuvX6JAz2FlkP+C8fwLl+aIY ω F gwese → ( ω ) AAACJHicbVDLSgMxFM34tr6qLt0Ei6AIZUZ8gRtREJcKVoVOLZn01oZmJkNyR6xhPsaNv+LGhQ9cuPFbTGsFXwcCh3Pu5eacKJXCoO+/eQODQ8Mjo2PjhYnJqemZ4uzcqVGZ5lDhSip9HjEDUiRQQYESzlMNLI4knEXt/a5/dgXaCJWcYCeFWswuE9EUnKGT6sUdGt4AsgsbxgxbnEl7kOd1GyJcI6JtqVilLSE7+YVdzZe/ZA1pvpLXiyW/7PdA/5KgT0qkj6N68TlsKJ7FkCCXzJhq4KdYs0yj4BLyQpgZSBlvs0uoOpqwGEzN9kLmdMkpDdpU2r0EaU/9vmFZbEwnjtxkN4r57XXF/7xqhs3tmhVJmiEk/PNQM5MUFe02RhtCA0fZcYRxLdxfKW8xzTi6XguuhOB35L/kdK0cbJY3jtdLu3v9OsbIAlkkyyQgW2SXHJIjUiGc3JJ78kievDvvwXvxXj9HB7z+zjz5Ae/9A51Bp0g= ω F homophily + ( rep ) AAACEXicbVDJSgNBEO1xjXGLevQyGISAEGbE7Rj04jGCWSCThJ5OJWnSs9BdI8ZmfsGLv+LFgyJevXnzb+wsB018UPB4r4qqen4suELH+bYWFpeWV1Yza9n1jc2t7dzOblVFiWRQYZGIZN2nCgQPoYIcBdRjCTTwBdT8wdXIr92BVDwKb3EYQzOgvZB3OaNopHau4D0A0pb2Aop9RoUup2lbewj3iKih0wOVtvSREXN5p+iMYc8Td0ryZIpyO/fldSKWBBAiE1SphuvE2NRUImcC0qyXKIgpG9AeNAwNaQCqqccfpfahUTp2N5KmQrTH6u8JTQOlhoFvOkeHq1lvJP7nNRLsXjQ1D+MEIWSTRd1E2BjZo3jsDpfAUAwNoUxyc6vN+lRShibErAnBnX15nlSPi+5Z8fTmJF+6nMaRIfvkgBSIS85JiVyTMqkQRh7JM3klb9aT9WK9Wx+T1gVrOrNH/sD6/AEEep8K ω P edges + AAACJHicbVDLSgMxFM34tr6qLt0Ei6ALy4z4AjeiG5cVrAqdWjLprQ3NTIbkjljDfIwbf8WNCx+4cOO3mNYKvg4EDufcy805USqFQd9/84aGR0bHxicmC1PTM7NzxfmFU6MyzaHKlVT6PGIGpEigigIlnKcaWBxJOIs6hz3/7Aq0ESo5wW4K9ZhdJqIlOEMnNYp7NLwBZBc2jBm2OZO2kucNGyJcI6Jtq1ilbSG7+YVdz1e/ZA1pvpY3iiW/7PdB/5JgQEpkgEqj+Bw2Fc9iSJBLZkwt8FOsW6ZRcAl5IcwMpIx32CXUHE1YDKZu+yFzuuKUJm0p7V6CtK9+37AsNqYbR26yF8X89nrif14tw9Zu3YokzRAS/nmolUmKivYao02hgaPsOsK4Fu6vlLeZZhxdrwVXQvA78l9yulEOtstbx5ul/YNBHRNkiSyTVRKQHbJPjkiFVAknt+SePJIn78578F6818/RIW+ws0h+wHv/ALErp1Q= ω P homophily → ( rep ) AAACHHicbVDLSgNBEJz1bXxFPXpZDIIihF3fR9GLxwhGhWwMvZNOMjj7YKZXjcN+iBd/xYsHRbx4EPwbJzEHXwUDRVU3PVVhKoUmz/twhoZHRsfGJyYLU9Mzs3PF+YVTnWSKY5UnMlHnIWiUIsYqCZJ4niqEKJR4Fl4e9vyzK1RaJPEJdVOsR9CORUtwICs1ipvBLRJcmCAC6nCQppLnDRMQ3hCRaV83sa0Q8wuznq8GINMOrOWNYskre324f4k/ICU2QKVRfAuaCc8ijIlL0LrmeynVDSgSXGJeCDKNKfBLaGPN0hgi1HXTD5e7K1Zpuq1E2ReT21e/bxiItO5GoZ3sZdC/vZ74n1fLqLVXNyJOM8KYfx1qZdKlxO015TaFQk6yawlwJexfXd4BBZxsnwVbgv878l9yulH2d8rbx1ul/YNBHRNsiS2zVeazXbbPjliFVRlnd+yBPbFn5955dF6c16/RIWews8h+wHn/BGuIo20= ω P gwdegree + ( ω ) AAACGXicbVBNT9tAFFwDLTT9IMCRi0VUKVWlyEZAOSK4cAxSA0ixYz1vnpNV1h/afaaElf8GF/4KFw6tEMf2xL9hE3IA0pFWGs28p7czcSGFJs97dBYWl969X175UPv46fOX1fra+qnOS8Wxw3OZq/MYNEqRYYcESTwvFEIaSzyLR0cT/+wClRZ59pPGBYYpDDKRCA5kpajuBVdI0DNBCjTkIE27qiITEF4SkRn8Qp1UPfO9agYgiyF8q6J6w2t5U7jzxJ+RBpuhHdX/Bv2clylmxCVo3fW9gkIDigSXWNWCUmMBfAQD7FqaQYo6NNNklfvVKn03yZV9GblT9eWGgVTrcRrbyUkA/dabiP/zuiUl+6ERWVESZvz5UFJKl3J3UpPbFwo5ybElwJWwf3X5EBRwsmXWbAn+28jz5HS75e+1dk92GgeHszpW2CbbYk3msx/sgB2zNuswzq7ZLfvN/jg3zp1z7zw8jy44s50N9grOvyf0zaIh ω P gwesf + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIihF3xdQx68RjBqJCNoXfSSQZnH8z0qnHY3/Dir3jxoIhHPfk3TmIOvgoGiqpueqrCVApNnvfhjIyOjU9MTk0XZmbn5heKi0unOskUxxpPZKLOQ9AoRYw1EiTxPFUIUSjxLLw87PtnV6i0SOIT6qXYiKATi7bgQFZqFr3gFgkuTBABdTlIU83zpgkIb4jIdK5RY35hNvP1AGTahY28WSx5ZW8A9y/xh6TEhqg2i29BK+FZhDFxCVrXfS+lhgFFgkvMC0GmMQV+CR2sWxpDhLphBslyd80qLbedKPticgfq9w0Dkda9KLST/QD6t9cX//PqGbX3G0bEaUYY869D7Uy6lLj9mtyWUMhJ9iwBroT9q8u7oICTLbNgS/B/R/5LTrfK/m5553i7VDkY1jHFVtgqW2c+22MVdsSqrMY4u2MP7Ik9O/fOo/PivH6NjjjDnWX2A877J/M7oiA= ω P gwese + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIeDLvi6xj04jGCUSEbQ++kkwzOPpjpVeOwv+HFX/HiQRGPevJvnMQcfBUMFFXd9FSFqRSaPO/DGRkdG5+YnJouzMzOzS8UF5dOdZIpjjWeyESdh6BRihhrJEjieaoQolDiWXh52PfPrlBpkcQn1EuxEUEnFm3BgazULHrBLRJcmCAC6nKQpprnTRMQ3hCR6VyjxvzCbObrAci0Cxt5s1jyyt4A7l/iD0mJDVFtFt+CVsKzCGPiErSu+15KDQOKBJeYF4JMYwr8EjpYtzSGCHXDDJLl7ppVWm47UfbF5A7U7xsGIq17UWgn+wH0b68v/ufVM2rvN4yI04ww5l+H2pl0KXH7NbktoZCT7FkCXAn7V5d3QQEnW2bBluD/jvyXnG6V/d3yzvF2qXIwrGOKrbBVts58tscq7IhVWY1xdsce2BN7du6dR+fFef0aHXGGO8vsB5z3T/ZXoiI= ω P gwese → ( ω ) Fig. 8: MCMC auto correlation plots for each parameter. T able 2. Summary of the posterior parameter density estimates of the conditional sign process. F ormation ( F ) P ersistence ( P ) P arameters Mean 2.5% Median 97.5% Mean 2.5% Median 97.5% ζ edges + -3.95 -5.81 -3.98 -2.00 -4.77 -5.90 -4.76 -3.68 ζ homophily + ( rep ) 0.53 0.02 0.53 1.06 -0.28 -0.83 -0.27 0.25 ζ gwdegree + ( α =0 . 2) -0.29 -3.11 -0.31 2.49 0.81 -0.28 0.79 1.87 ζ gwesf + ( α =0 . 6) 2.42 1.46 2.44 3.38 1.56 1.05 1.56 2.11 ζ gwese + ( α =0 . 6) -1.41 -2.11 -1.40 -0.70 -0.58 -0.96 -0.58 -0.20 ζ gwese − ( α =0 . 6) -1.05 -1.41 -1.03 -0.75 -0.62 -0.81 -0.61 -0.48 forming a positive edge among republican senators, e.g., logit − 1 ( − 3 . 95 + 0 . 53) ≈ 0 . 027, while the eﬀect on p ersistence remains negligible. • The estimates for ζ gwdegree + ( α =0 . 2) are not signiﬁcan tly diﬀeren t from zero in either process, implying that high-degree no des do not systematically form or main tain more positive edges. This suggests a relatively ev en distribution of interactions across the net work. • The consistently p ositive 95% p osterior credible interv als for ζ gwesf + ( α =0 . 6) support the role of triadic closure in b oth forming and sustaining p ositive interactions. F or instance, adding one unit of balanced triadic closure increases the conditional probability of positive formation from ≈ 0 . 019 to logit − 1 ( − 3 . 95 + 2 . 42) ≈ 0 . 18 and the probability of p ersistence from ≈ 0 . 008 to logit − 1 ( − 4 . 77 + 1 . 56) ≈ 0 . 039. • The negative p osterior estimates for ζ gwese + ( α =0 . 6) and ζ gwese − ( α =0 . 6) , where the former corresp onds to a balanced triadic conﬁguration and the latter to an unbalanced one, indicate that these structures reduce the likelihoo d of forming or main taining positive edges. F or example, a dyad in volv ed in an un balanced negative triad would ha ve its conditional formation probability reduced from ≈ 0 . 019 to logit − 1 ( − 3 . 95 − 1 . 05) ≈ 0 . 0067 . 16 Caimo and Gollini AAACEXicbVDLSgNBEJyNrxhfqx69LAYhIIRd8XUMCuJRwWggm4TZSScZMvtgpleMw/6CF3/FiwdFvHrz5t84iTloYkFDUdVNd1eQCK7Qdb+s3Mzs3PxCfrGwtLyyumavb1yrOJUMqiwWsawFVIHgEVSRo4BaIoGGgYCboH869G9uQSoeR1c4SKAR0m7EO5xRNFLLLvn3gLSp/ZBij1Ghz7KspX2EO0TU0O6Cypp614h20S27IzjTxBuTIhnjomV/+u2YpSFEyARVqu65CTY0lciZgKzgpwoSyvq0C3VDIxqCaujRR5mzY5S204mlqQidkfp7QtNQqUEYmM7h4WrSG4r/efUUO8cNzaMkRYjYz6JOKhyMnWE8TptLYCgGhlAmubnVYT0qKUMTYsGE4E2+PE2u98reYfngcr9YORnHkSdbZJuUiEeOSIWckwtSJYw8kCfyQl6tR+vZerPef1pz1nhmk/yB9fEN9F2fAA== ω F edges + AAACEXicbVDJSgNBEO1xjXGLevQyGISAEGbE7Rj04jGCWSCThJ5OJWnSs9BdI8ZmfsGLv+LFgyJevXnzb+wsB018UPB4r4qqen4suELH+bYWFpeWV1Yza9n1jc2t7dzOblVFiWRQYZGIZN2nCgQPoYIcBdRjCTTwBdT8wdXIr92BVDwKb3EYQzOgvZB3OaNopHau4D0A0pb2Aop9RoUup2lbewj3iKih0wOVtvSREXN5p+iMYc8Td0ryZIpyO/fldSKWBBAiE1SphuvE2NRUImcC0qyXKIgpG9AeNAwNaQCqqccfpfahUTp2N5KmQrTH6u8JTQOlhoFvOkeHq1lvJP7nNRLsXjQ1D+MEIWSTRd1E2BjZo3jsDpfAUAwNoUxyc6vN+lRShibErAnBnX15nlSPi+5Z8fTmJF+6nMaRIfvkgBSIS85JiVyTMqkQRh7JM3klb9aT9WK9Wx+T1gVrOrNH/sD6/AEEep8K ω P edges + AAACHHicbVDLSgNBEJz1bXxFPXpZDIIihF3fR1EQjwpGhWwMvZNOMjj7YKZXjcN+iBd/xYsHRbx4EPwbJzEHXwUDRVU3PVVhKoUmz/twBgaHhkdGx8YLE5NT0zPF2blTnWSKY4UnMlHnIWiUIsYKCZJ4niqEKJR4Fl7ud/2zK1RaJPEJdVKsRdCKRVNwICvVi+vBLRJcmCACanOQ5iDP6yYgvCEi07puYEsh5hdmNV8OQKZtWMnrxZJX9npw/xK/T0qsj6N68S1oJDyLMCYuQeuq76VUM6BIcIl5Icg0psAvoYVVS2OIUNdML1zuLlml4TYTZV9Mbk/9vmEg0roThXaym0H/9rrif141o+ZOzYg4zQhj/nWomUmXErfblNsQCjnJjiXAlbB/dXkbFHCyfRZsCf7vyH/J6VrZ3ypvHm+Udvf6dYyxBbbIlpnPttkuO2RHrMI4u2MP7Ik9O/fOo/PivH6NDjj9nXn2A877J1sMo2M= ω F gwdegree + ( ω ) AAACHHicbVDLSgNBEJz1bXxFPXpZDIIihF3fR9GLxwhGhWwMvZNOMjj7YKZXjcN+iBd/xYsHRbx4EPwbJzEHXwUDRVU3PVVhKoUmz/twhoZHRsfGJyYLU9Mzs3PF+YVTnWSKY5UnMlHnIWiUIsYqCZJ4niqEKJR4Fl4e9vyzK1RaJPEJdVOsR9CORUtwICs1ipvBLRJcmCAC6nCQppLnDRMQ3hCRaV83sa0Q8wuznq8GINMOrOWNYskre324f4k/ICU2QKVRfAuaCc8ijIlL0LrmeynVDSgSXGJeCDKNKfBLaGPN0hgi1HXTD5e7K1Zpuq1E2ReT21e/bxiItO5GoZ3sZdC/vZ74n1fLqLVXNyJOM8KYfx1qZdKlxO015TaFQk6yawlwJexfXd4BBZxsnwVbgv878l9yulH2d8rbx1ul/YNBHRNsiS2zVeazXbbPjliFVRlnd+yBPbFn5955dF6c16/RIWews8h+wHn/BGuIo20= ω P gwdegree + ( ω ) 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 ω P gwesf + ( ω ) 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 ω F gwesf + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIihF3xdRQF8ahgVMjG0DvpJENmH8z0qnHY3/Dir3jxoIhHPfk3TmIOvgoGiqpueqrCVApNnvfhDA2PjI6NT0wWpqZnZueK8wtnOskUxwpPZKIuQtAoRYwVEiTxIlUIUSjxPOwc9PzzK1RaJPEpdVOsRdCKRVNwICvVi15wiwSXJoiA2hykOczzugkIb4jItK5RY35p1vPVAGTahrW8Xix5Za8P9y/xB6TEBjiuF9+CRsKzCGPiErSu+l5KNQOKBJeYF4JMYwq8Ay2sWhpDhLpm+slyd8UqDbeZKPticvvq9w0DkdbdKLSTvQD6t9cT//OqGTV3a0bEaUYY869DzUy6lLi9mtyGUMhJdi0BroT9q8vboICTLbNgS/B/R/5LzjbK/nZ562SztLc/qGOCLbFltsp8tsP22BE7ZhXG2R17YE/s2bl3Hp0X5/VrdMgZ7CyyH3DePwHi3aIW ω F gwese + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIihF3xdQx68RjBqJCNoXfSSQZnH8z0qnHY3/Dir3jxoIhHPfk3TmIOvgoGiqpueqrCVApNnvfhjIyOjU9MTk0XZmbn5heKi0unOskUxxpPZKLOQ9AoRYw1EiTxPFUIUSjxLLw87PtnV6i0SOIT6qXYiKATi7bgQFZqFr3gFgkuTBABdTlIU83zpgkIb4jIdK5RY35hNvP1AGTahY28WSx5ZW8A9y/xh6TEhqg2i29BK+FZhDFxCVrXfS+lhgFFgkvMC0GmMQV+CR2sWxpDhLphBslyd80qLbedKPticgfq9w0Dkda9KLST/QD6t9cX//PqGbX3G0bEaUYY869D7Uy6lLj9mtyWUMhJ9iwBroT9q8u7oICTLbNgS/B/R/5LTrfK/m5553i7VDkY1jHFVtgqW2c+22MVdsSqrMY4u2MP7Ik9O/fOo/PivH6NjjjDnWX2A877J/M7oiA= ω P gwese + ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIeDLvi6xj04jGCUSEbQ++kkwzOPpjpVeOwv+HFX/HiQRGPevJvnMQcfBUMFFXd9FSFqRSaPO/DGRkdG5+YnJouzMzOzS8UF5dOdZIpjjWeyESdh6BRihhrJEjieaoQolDiWXh52PfPrlBpkcQn1EuxEUEnFm3BgazULHrBLRJcmCAC6nKQpprnTRMQ3hCR6VyjxvzCbObrAci0Cxt5s1jyyt4A7l/iD0mJDVFtFt+CVsKzCGPiErSu+15KDQOKBJeYF4JMYwr8EjpYtzSGCHXDDJLl7ppVWm47UfbF5A7U7xsGIq17UWgn+wH0b68v/ufVM2rvN4yI04ww5l+H2pl0KXH7NbktoZCT7FkCXAn7V5d3QQEnW2bBluD/jvyXnG6V/d3yzvF2qXIwrGOKrbBVts58tscq7IhVWY1xdsce2BN7du6dR+fFef0aHXGGO8vsB5z3T/ZXoiI= ω P gwese → ( ω ) AAACGXicbVDLSgNBEJz1bXxFPXpZDIIeDLvi6ygK4lHBqJCNoXfSSYbMPpjpVeOwv+HFX/HiQRGPevJvnMQcfBUMFFXd9FSFqRSaPO/DGRoeGR0bn5gsTE3PzM4V5xfOdJIpjhWeyERdhKBRihgrJEjiRaoQolDiedg56PnnV6i0SOJT6qZYi6AVi6bgQFaqF73gFgkuTRABtTlIc5jndRMQ3hCRaV2jxvzSrOerAci0DWt5vVjyyl4f7l/iD0iJDXBcL74FjYRnEcbEJWhd9b2UagYUCS4xLwSZxhR4B1pYtTSGCHXN9JPl7opVGm4zUfbF5PbV7xsGIq27UWgnewH0b68n/udVM2ru1oyI04ww5l+Hmpl0KXF7NbkNoZCT7FoCXAn7V5e3QQEnW2bBluD/jvyXnG2U/e3y1slmaW9/UMcEW2LLbJX5bIftsSN2zCqMszv2wJ7Ys3PvPDovzuvX6JAz2FlkP+C8fwLl+aIY ω F gwese → ( ω ) AAACJHicbVDLSgMxFM34tr6qLt0Ei6AIZUZ8gRtREJcKVoVOLZn01oZmJkNyR6xhPsaNv+LGhQ9cuPFbTGsFXwcCh3Pu5eacKJXCoO+/eQODQ8Mjo2PjhYnJqemZ4uzcqVGZ5lDhSip9HjEDUiRQQYESzlMNLI4knEXt/a5/dgXaCJWcYCeFWswuE9EUnKGT6sUdGt4AsgsbxgxbnEl7kOd1GyJcI6JtqVilLSE7+YVdzZe/ZA1pvpLXiyW/7PdA/5KgT0qkj6N68TlsKJ7FkCCXzJhq4KdYs0yj4BLyQpgZSBlvs0uoOpqwGEzN9kLmdMkpDdpU2r0EaU/9vmFZbEwnjtxkN4r57XXF/7xqhs3tmhVJmiEk/PNQM5MUFe02RhtCA0fZcYRxLdxfKW8xzTi6XguuhOB35L/kdK0cbJY3jtdLu3v9OsbIAlkkyyQgW2SXHJIjUiGc3JJ78kievDvvwXvxXj9HB7z+zjz5Ae/9A51Bp0g= ω F homophily + ( rep ) AAACJHicbVDLSgMxFM34tr6qLt0Ei6AIZUZ8gRvRjcsKVoVOLZn01oZmJkNyR6xhPsaNv+LGhQ9cuPFbTGsFXwcCh3Pu5eacKJXCoO+/eUPDI6Nj4xOThanpmdm54vzCqVGZ5lDlSip9HjEDUiRQRYESzlMNLI4knEWdw55/dgXaCJWcYDeFeswuE9ESnKGTGsU9Gt4AsgsbxgzbnElbyfOGDRGuEdG2VazStpDd/MKu56tfsoY0X8sbxZJf9vugf0kwICUyQKVRfA6bimcxJMglM6YW+CnWLdMouIS8EGYGUsY77BJqjiYsBlO3/ZA5XXFKk7aUdi9B2le/b1gWG9ONIzfZi2J+ez3xP6+WYWu3bkWSZggJ/zzUyiRFRXuN0abQwFF2HWFcC/dXyttMM46u14IrIfgd+S853SgH2+Wt483S/sGgjgmyRJbJKgnIDtknR6RCqoSTW3JPHsmTd+c9eC/e6+fokDfYWSQ/4L1/AK4Dp1I= ω P homophily + ( rep ) Fig. 9: Posterior predictive b oxplots of centred cumulativ e diﬀerences ov er time b et ween sim ulated and observ ed netw ork statistics: P T t =1 s q  z U ′ t ; x U t , y t − 1 , ζ U q  − s  z U t ; x U t , y t − 1 , ζ U k  , U = {F , P } , q = 1 , · · · , Q. 10.3. Ma rginal estimation The netw ork in teraction structure x 0: T wa s analysed b y estimating the follo wing STER GM posterior π  ξ F , ξ P | x 0: T  ∝ T Y t =1 exp n ξ F ⊤ s  x F t ; y t − 1  o exp n ξ P ⊤ s  x P t ; y t − 1  o π  ξ F  π  ξ P  using a similar version of Algorithm 1. F or both the formation and persistence, we used the follo wing three net work statistics that are matc hing the conﬁgurations used in the conditional signed pro cess: (1) edges , the baseline prop ensit y for an y interaction to form or p ersist; (2) homophily(rep) , the prop ensity for in teractions be t w een republican Senators to form or persist; (3) gwesp ( α = 0 . 2) , the tendency for interactions to form or p ersist in closed triads; and (4) gwdegree ( α = 0 . 6) , the inﬂuence of no de degree on the formation and pe rsistence of edges. P osterior sampling w as performed using a prior speciﬁcation and initialisation pro cedure similar to that describ ed in Section 10.2. The results summarised in T able 3 indicate that new interaction formation was relatively infrequen t, with a strongly negative baseline tendency ( ξ F edges ), suggesting that in the absence of other eﬀects, new Separable mo dels for dynamic signed netw orks 17 T able 3. Summary of the posterior parameter density estimates of the interaction process. F ormation ( F ) P ersistence ( P ) Pa rameters Mean 2.5% Median 97.5% Mean 2.5% Median 97.5% ξ edges -6.40 -7.45 -6.38 -5.46 -4.78 -5.46 -4.80 -4.12 ξ homophily ( rep ) 0.08 -0.15 0.08 0.31 0.06 -0.30 0.06 0.43 ξ gwdegree ( α =0 . 2) -1.18 -3.65 -1.11 0.89 -0.10 -0.94 -0.08 0.66 ξ gwesp ( α =0 . 6) 2.52 2.04 2.51 3.07 1.55 1.24 1.56 1.88 edges were unlikely to form. There was only weak evidence of Republican homophily in edge formation ( ξ F homophily ( rep ) ), indicating minimal preference for within-group connections. Actors with higher existing degrees w ere somewhat less lik ely to create additional edges ( ξ F gwdegree ( α ) ) consisten t with degree saturation eﬀects limiting new edge formation. By contrast, there was strong and credible evidence of transitive closure in the formation pro cess ( ξ F gwesp ( α ) ), indicating that new edges tend to form within cohesive triads. In contrast, existing in teractions display a strong tendency to p ersist, although ov erall edge stability remained selective ( ξ P edges ). Homophily again play ed little role in p ersistence ( ξ P homophily ( rep ) ). Degree eﬀects w ere near zero ( ξ P gwdegree ( α ) ), implying that high-degree individuals were only marginally less likely to sustain their existing edges. T ransitiv e closure eﬀects were positive ( ξ P gwesp ( α ) ), showing that edges embedded in triadic structures w ere substan tially more stable o ver time. Ov erall, these results suggest a relatively stable netw ork c haracterised b y limited new edge creation, strong clustering tendencies within triads, and modest degree-related constrain ts on b oth the formation and maintenance of in teractions. New connections tend to emerge within existing cohesive subgroups, while established relationships persist preferen tially within these clustered structures. T o obtain marginal probabilities for positive edges, we m ultiply the in teraction probabilities by the conditional probabilities according to Equation (4). The estimated probabilities suggest that the formation of new interactions in the netw ork is rare. The strongly negative formation parameter ξ edges implies that, in the absence of additional structural eﬀects, the probabilit y of an y new edge emerging is extremely lo w, logit − 1 ( − 6 . 40) ≈ 0 . 0017. When this is combined with the small conditional probabilit y that a newly formed interaction is positive ( ≈ 0 . 019), the resulting marginal probabilit y of observing a new p ositive edge b ecomes eﬀectively negligible. How ever, this pattern changes mark edly once triadic closure is considered. Incorp orating the transitivity eﬀect ξ gwesp ( α =0 . 6) increases the probabilit y of forming an interaction to ab out logit − 1 ( − 6 . 40 + 2 . 52) ≈ 0 . 02, and the probability that such an intera ction is positive to roughly logit − 1 ( − 3 . 95 + 2 . 42) ≈ 0 . 18. The resulting marginal probability of forming a p ositive edge within a triad ( ζ gwesf + ) rises to around 0 . 18 × logit − 1 ( − 6 . 40 + 2 . 52) ≈ 0 . 0036. This clearly indicates that the formation of p ositive edges is concentrated within cohesive p ositive triadic structures. P ersistence follo ws a similar but sligh tly stronger pattern. Existing interactions are more lik ely to con tinue than new ones are to form, with a baseline p ersistence probability of around 0 . 0083. Y et, the probability that a persisting edge remains p ositive is v ery close to 0. When triadic eﬀects are present, both p ersistence and conditional p ositivit y rise to approximately logit − 1 ( − 4 . 78 + 1 . 55) ≈ 0 . 038, resulting in a marginal p ositive p ersistence of ab out 0 . 038 × logit − 1 ( − 4 . 77 + 1 . 56) ≈ 0 . 0015. These ﬁndings imply that while the ov erall netw ork formation structure remains sparse, p ositive relationships, though rare, are substan tially more likely to b oth form and endure within tigh tly clustered triads. T aken together, the results portray a netw ork cha racterised by isolated sets of stable, m utually reinforcing p ositive in teractions. While these estimates provide a useful illustration of the patterns in the netw ork, the true uncertaint y around these marginal probabilities ma y b e larger, and credible in terv als deriv ed from p osterior dra ws pro vide a more complete picture. 11. Discussion This pap er introduces a nov el separable temp oral generativ e framework for mo delling the dynamics of signed netw orks. By distinguishing betw een in teraction eﬀects and conditional sign processes, our approach preserves 18 Caimo and Gollini the ﬂexibility and interpretabilit y of traditional binary exp onential random graph mo dels while remaining consistent with the assumptions of structural balance theory (SBT). When mo delling signed net works, the separable approac h used in this paper is particularly adv antageous when formation and dissolution pro cesses are plausibly driven b y diﬀerent mec hanisms, and when the goal is to rigorously test structural balance conﬁgurations while accounting for the p otentially strong inﬂuence of ov erall interaction density on the observ ed signed edges. In such settings, some signed edges may b e unobserv ed or laten t, and failing to accoun t for this can bias the results. T o empirically test SBT, we speciﬁcally incorp orate endogenous eﬀects that capture patterns predicted by the theory . Nonetheless, the framew ork remains fully compatible with the broader class of binary statistics commonly used in separable temp oral ERGM pro cesses, including exogenous and lagged co v ariates. This separation allows for a deeper understanding of how net w ork relations evolv e ov er time, making it suitable for complex systems in whic h the in terplay of both p ositive and negativ e relationships is critical to the system o verall dynamics. When the assumptions underlying separability are not met, alternativ e approaches that relax one of these assumptions can be emplo yed. W e employ ed a fully probabilistic Ba yesian approach to infer mo del parameters, utilising an adaptiv e Metropoli s-Hastings appro ximate exchange algorithm to eﬃciently estimate the doubly in tractable p osterior distribution. This metho d enables us to accoun t for the complexity of signed net work structures and provides robust uncertaint y estimates despite the computational challenges. The Bay esian framew ork also enables assessmen t of p otential violations by comparing p osterior predictive p erformance across alternativ e mo dels, for instance, b y examining whether a model that allo ws conditional dep endence b etw een the pro cesses yields a substantially b etter ﬁt. W e analysed the p olitical relationships among U.S. Senators during Ronald Reagan’s second term (1985–1989), uncov ering the evolving dynamics of p olitical alliances and riv alries within the Senate. By applying our framework with a fo cus on endogenous structural eﬀects grounded in SBT, we iden tiﬁed k ey patterns of b oth supportive and an tagonistic alliances, shedding light on the structural mechanisms underpinning shifts in p olitical coalitions. One limitation of this empirical analysis is that the signed relationships in the net work are derived from a pro jection of the underlying bipartite net work, based on the similarity of in teraction patterns within that same structure (Neal et al., 2024). As such, the sign attributed to edges reﬂects inferred structural similarity rather than directly observ ed relational con tent. A more direct approac h would inv olve the use of longitudinal relational data in which tie signs carry an explicit substantiv e meaning, for example, recorded friendships and antagonistic relationships betw een individuals. The ﬂexibilit y and in terpretability of our model op en up sev eral a v en ues for future researc h. One promising direction inv olves extending our approach to a contin uous-time framework, such as those based on longitudinal ER GMs (Koskinen and Lomi, 2013; Koskinen et al., 2015), which could signiﬁcantly enhance the mo del applicabilit y to real-time, dynamic net work data. This extension ma y inv olv e the dev elopmen t of data augmen tation strategies in which MCMC algorithms alternate betw een sampling from the conditional p osterior distribution of the mo del parameters and generating plausible interaction paths that connect consecutive net work snapshots. Another direction concerns the adaptation of the metho dology to dynamic w eighted signed net works, where the transition probabilities of the signed pro cess follow a partially separable temp oral model, as recently prop osed by Kei et al. (2023). Crucially , the feasibilit y of these extensions will dep end on the dev elopmen t of more eﬃcient and scalable estimation algorithms. As netw ork data grow in complexity and temporal resolution, computational b ottlenecks become increasingly signiﬁcan t. Adv ancing the underlying inferential algorithm, through improv ed MCMC schemes, v ariational approximations (T an and F riel, 2020), will b e essential to ensure the practical usabilit y of these models in applied settings. 12. Acknowledgments The authors thank Marc Schalberger and Cornelius F ritz for sharing an early version of the ergm.sign pac k age, which enabled us to complete the comparative sim ulation study presen ted in this pap er. W e thank the anon ymous review ers for their insigh tful commen ts and co nstructive suggestions, which hav e substan tially impro ved the clarit y and quality of the man uscript. Separable mo dels for dynamic signed netw orks 19 13. Data availability The dataset and co de used in this paper are a v ailable for do wnload at github.com/acaimo/B2Lstergm. References V. Amati, A. Lomi, and A. Mira. So cial net work mo deling. Annual R eview of Statistics and Its Application , 5 (1):343–369, 2018. J. E. Besag. 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Separable models for dynamic signed networks

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