Introduction

The study of network dynamics has gained significant attention across various scientific disciplines, encompassing biological, physical, informational, and social systems. Among the diverse range of complex networks, social networks involve relationships between entities. These relationships can have different characteristics, including weighted and directional edges, reciprocity, shared traits between entities, and dual natures such as friendship and enmity1. Moreover, such relationships can form motifs or structural features like transitive triplets2. Over time, relationships may develop, dissolve, and change, leading to evolving network structures. These complex systems often exhibit recognizable patterns of network ties, and researchers employ various network models to understand their evolutionary dynamics.

Recently, signed networks, where edges are labeled with different signs, have garnered significant attention3,4. By assigning a positive or negative sign to edges, the structure of these networks can be modeled as signed graphs. Signed networks differ from unsigned networks not only due to the additional complexity introduced by the signs associated with edges but also because certain principles, such as balance theory, play a fundamental role in driving their dynamics. Balance theory describes relationships between individuals in terms of triads, where each triad comprises three individuals and three signed ties therebetween5. Traditional social balance studies primarily focus on static states and examine the harmony or tension within specific local regions of the system. Analyzing the properties of signed social networks allows for a better understanding of the key roles that the network structure plays when some relations between social entities are positive6 while others are negative7.

However, real social systems are dynamic and continuously changing over time. The dynamics of temporal signed networks stem from the dynamical relationships specified by the positive and negative links. Studies have attempted to capture how the social balance of a signed network evolves through changes in the signs of links over time8,9. By focusing on the formation and dissolution of balanced local structures, such analyses provide valuable insights into the evolution of social systems and whether they will reach a stable state. These structures are the result of rational or behavioral choices made by individuals, and individual preferences are expected to influence changes in social networks over time10,11,12. For instance, research has shown how indirect contacts can lead to direct connections, enhancing triadic closures in social networks13. Modeling complex systems as temporal signed networks enables us to gain insights into the evolutionary patterns of real-world social systems, identify key factors affecting social balance, and potentially predict the future of these networks14. Furthermore, interdisciplinary approaches combining insights from sociology, computer science, and physics are increasingly being utilized to tackle the complexities of network evolution. These perspectives not only enhance our understanding of how different types of interactions influence each other but also improve the accuracy of models predicting network changes. The ongoing convergence of these diverse fields underscores the necessity of holistic approaches in studying the dynamic processes of signed networks15.

Physics-based models and techniques, such as cellular automata and mean-field solutions, have been applied to investigate the dynamics of structural balance in social networks with signed ties. These works highlight phase transitions, thermal properties, and the role of homophily in shaping network evolution, demonstrating how principles from statistical physics can be used to model the progression toward balanced or imbalanced states16. Recent studies have significantly expanded our understanding of structural balance and the dynamics of social networks with signed ties. Antal et al. examined the evolution of social networks with friendly and unfriendly links, revealing a dynamic phase transition towards balance17. Further exploration by Malarz et al.18 and Malarz & Wołoszyn19 introduced novel models like cellular automata and heat-bath algorithms to study the progression of Heider balance and the impact of structural changes. Wołoszyn & Malarz explored the thermal properties of structurally balanced systems, emphasizing the role of network density in reaching balanced states20. Meanwhile, Rabbani et al. developed models addressing the phase transitions in networks under the influence of external tensions, using statistical physics frameworks21. Marvel et al. provided insights into the division of social networks into opposing factions, highlighting the inevitable emergence of either global harmony or conflict22. Kułakowski et al. applied continuous models and statistical physics to explain the persistence of triadic relationships23. Lastly, Górski et al. showed how homophily and structural balance interplay, sometimes hindering the attainment of global cooperation24. These studies collectively underscore the importance of balance theory in understanding complex systems and social network dynamics.

In recent years, several statistical approaches have emerged to analyze static and dynamic networks, addressing challenges like interdependence and covariate effects at nodal and dyadic levels. Snijders et al. introduced stochastic actor-based models to capture dynamic behaviors within networks, considering both endogenous and exogenous factors influencing tie formation25. Hunter et al. developed the exponential-family random graph model (ERGM), which models networks through local structures such as reciprocated ties and triangles26. Robins et al. highlighted ERGMs’ flexibility in modeling social networks by incorporating actor attributes and network configurations27. Lusher et al. further emphasized the theory-driven nature of ERGMs, noting their ability to test competing social processes like reciprocity, transitivity, and homophily28. Leifeld et al. expanded this with the temporal ERGM (TERGM), which accounts for longitudinal dependencies and network evolution over time29. Cranmer et al. compared ERGMs with other network models, demonstrating their advantages in handling interdependent relationships in complex networks30. These models collectively provide robust frameworks for understanding both static and dynamic network behaviors.

Recent advancements in machine learning and data analytics have further enriched the study of network dynamics. Techniques such as dynamic network embedding and deep learning have been employed to model and predict the evolution of complex networks more accurately31. These methods leverage the power of neural networks to capture intricate patterns and dependencies in network data, providing new insights into the structural and temporal characteristics of networks. Additionally, traditional methods like supervised temporal link prediction, as shown by de Bruin et al., demonstrate that using temporal information can significantly improve prediction performance in large-scale real-world networks by leveraging past events systematically32. In temporal networks, sampling-based algorithms such as the one proposed by Ahmed et al. use random walks and weighted graphs to achieve faster and more accurate link prediction, integrating both temporal and topological information33. In signed networks, approaches such as the Signed Latent Factor model34 and graph kernel-based methods35 have proven effective in addressing the challenges of predicting both links and their signs. Meanwhile, comprehensive reviews, like the one by Daud et al., underscore the growing complexity and diversity in link prediction applications, highlighting advancements in areas like anomaly detection and community detection36. Furthermore, studies by Chiang et al. have shown that exploiting longer cycles in signed networks can enhance link prediction accuracy, particularly when dealing with networks rich in positive and negative interactions4.

Regarding the investigation of temporal signed networks, balance theory has been supported and challenged by empirical research. For instance, Davis demonstrated that balance theory could effectively predict the evolution of small social groups, suggesting that groups tend to polarize into two cohesive subgroups37. However, Hummon and Doreian found that the application of balance theory to larger, more complex networks led to inconsistent results, as the theory often oversimplified the social processes at play38. To address these limitations, ERGMs have been introduced to incorporate additional covariates and interaction terms, offering a more nuanced understanding of network dynamics. Goodreau et al. applied ERGMs to adolescent social networks and demonstrated the utility of triadic closure and demographic factors in predicting social ties39. Yap and Harrigan also showed that incorporating balance theory into ERGM frameworks can improve predictive accuracy, though they emphasize that other factors such as status and homophily are equally important in signed network formation40. Similarly, Rambaran et al. found that while balance theory provides some explanatory power, it alone is insufficient to fully predict network evolution in heterogeneous settings, particularly during adolescence41. More recently, Dinh et al. extended balance theory to signed and directed graphs, demonstrating moderate levels of balance in real-world networks and underscoring the importance of considering both transitivity and sign consistency in directed social ties42. These studies collectively highlight the need for models that incorporate balance theory alongside other social factors to accurately predict network dynamics.

In this study, our objective is to incorporate TERGM specifications into signed networks and investigate whether utilizing covariates and structures from different networks can improve the prediction of tie formation in a signed social network. We will separately consider the positive and negative parts of the network and further divide them into subsets that are consistent with or deviate from balance theory. Additionally, we will analyze the effect of directionality on prediction accuracy by comparing the directed and undirected versions of different networks. By increasing the levels of covariates such as mutual reflexivity, transitivity, and information about triangles, we aim to enhance the prediction of network dynamics in signed networks across these subsets. We will examine the influence of dynamic triadic structures on different layers of signed networks and assess our prediction accuracy on subsets that vary in size, density, and directionality.

Methods

In this section, we outline the methods used to analyze the dynamic interactions within signed networks, leveraging principles from balance theory and advanced network modeling techniques. By integrating balance theory with advanced network modeling techniques, our approach aims to capture the complex interplay of positive and negative ties, enhancing the prediction of future network formations. Fritz Heider’s Balance Theory explains how individuals strive for consistency in their attitudes and relationships. It uses a triadic structure, often referred to as “P-O-X”, where P is the person, O is another individual, and X is an object or idea. A balanced state occurs when the relationships between all three entities are logically consistent, meaning positive or negative attitudes align. For example, if P likes O and O likes X, P should also like X for balance. An imbalanced state, such as when P likes O but dislikes X while O likes X, creates psychological discomfort. To restore balance, individuals may change their attitudes, the relationships between others, or even end certain relationships. The theory is widely applied in understanding social networks such as interpersonal relationships, marketing, and the resolution of cognitive dissonance7,8,16. Balance theory suggests that the relationships between any three nodes will eventually reach a balanced status, forming a triangle with an odd number of positive edges. For example, if nodes \(i\) and \(k\) are positively connected with a node \(j\) through edges \((i,j)\) and \((j,k)\), a positive tie will likely form between nodes \(i\) and \(k\). This principle can be summarized as “the friend of my friend is my friend.” To capture such relationships, we first define the notation used for the signed networks discussed herein. In this paper, at time t, \(P_{ij}^t\) represents the relationship between nodes \(i\) and \(j\) on the positive network, while \(N_{ij}^t\) represents their relationship in the negative network. We simultaneously consider two adjacency matrices, that is, P for the positive relationships and N for the negative relationships. Equation 1 provides a clear illustration of the notation. For details on the formalism and data processing, please refer to Supplementary Materials.

$$\begin{aligned}{\left\{ \begin{array}{ll} {\mathbf{P}}_{ij}^t= 1, & i\, \text{has a positive tie with}\, j\, \text{at time}\, t.\\ {\mathbf{P}}_{ij}^t= 0, & i\, \text{has no positive tie with}\, j\, \text{at time}\, t.\\ {\mathbf{N}}_{ij}^t=1, & i\, \text{has a negative tie with}\, j\,\text{at time}\, t.\\ {\mathbf{N}}_{ij}^t=0, & i\, \text{has no negative tie with}\, j\,\text{at time}\, t. \end{array}\right. }\end{aligned}$$
(1)

Dynamic triadic structures

This article focuses on the one-step change of the relations within a triple of vertices. Specifically, we distinguish between two types of dynamic triadic structures: triangle formation and triangle break. For triangle formation, we included all types of triadic structures that could form a triangle at time \(t\) but did not exist as a triangle at time \(t-1\) (i.e., there were no ties, one tie, or two ties at time \(t-1\)). Conversely, for any three vertices forming a triangle at time \(t-1\) but breaking out in \(t\), the structure is defined as a triangle break. In Fig. 1, we use PForm to illustrate how we define the one-step change to a triangle, considering transitions from 0 ties, 1 tie, and 2 ties.

Fig. 1
figure 1

Examples of the definition of PForm.

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As there are positive and negative networks, we distinguish the triadic behaviors on the single-layer network from the signed networks. On PForm and PBrk, we calculate the triangle formation and triangle break structures solely on the positive network and do not consider the node’s behaviors on the negative network, NForm and NBrk vise versa. More specifically, for PForm and PBrk, we take a set of three nodes, say \((i,j,k),\) arbitrarily and observe their behavior in the positive network from time \(t-1\) to \(t\). If the set of nodes forms a triadic closure, regardless of the direction, at time \(t\) from any non-triangle structure, then such three nodes are classified as “forming a triangle on the positive network.” Conversely, when the node set \((i,j,k)\) constitutes a triadic closure in time \(t-1\) but breaks out at time \(t\), then it is defined as “breaking a triangle in the positive network.” When such triangle formation and break happens in the negative network, they are NForm or NBrk. The network structures above are classified as “single-layer triangle structures.” As for SFrom and SBrk, “signed triangle structures,” which collect the vertex’s behavior across both positive and negative networks. Figure 2 shows some examples of temporal signed triangle structures. When counting the triangle form or break, we consider ties that are either in the positive or the negative network. To illustrate, if \(P_{ij}^{t-1},N_{jk}^{t-1},P_{jk}^{t-1}\)=(1,0,1) and \(P_{ij}^{t},N_{jk}^{t},P_{jk}^{t}\)=(1,1,1), then \((i,j,k)\) are considered as composing an SForm structure.

Fig. 2
figure 2

Examples of signed triangle structures.

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Temporal ERGM in a multilayer network

In this study, we utilized statistical modeling techniques and analyses to investigate the research question. We employed baseline models to capture the initial dynamics of the networks, focusing on fundamental characteristics such as the number of edges and node degrees. We then expanded these models to incorporate indicators related to balance theory, examining their influence on network evolution. Furthermore, we extended our analysis to incorporate signed networks, treating them as multilayer networks comprising a positive and a negative layer. By transforming the multilayer networks into large monotonic networks, we employed the bootstrap TERGM (BTERGM)29 to analyze the dynamic patterns of the longitudinal networks.

To represent the transformation, we constructed a large adjacency matrix of size \(2n\times 2n\), with the positive network positioned in the upper-right and the negative network in the lower-left. This reconstruction facilitated the application of multilayer modeling methods proposed43. Equation 2 illustrates the process of reconstructing the networks.

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbf{X}}_{ij}^t \subset {\mathbf{P}}_{ij}^t, & \text{if and only if}\,i,j \in [1,n]\,\text{at time}\,t.\\ {\mathbf{X}}_{ij}^t \subset {\mathbf{N}}_{ij}^t, & \text{if and only if}\,i,j \in [n+1,2n]\,\text{at time}\,t.\\ \end{array}\right. } \end{aligned}$$
(2)

In our analysis, we included dynamic triangle structures as edge covariates in the model and examined their effects on the network processes. This approach allowed us to investigate the interplay between triangle structures and other network characteristics within the context of the dynamic network modeling framework.

Key characteristics and variability across datasets

We apply our methods to five distinct signed networks, each offering unique structural and temporal features, to test the robustness and adaptability of our approach. Starting with directed networks, the Bitcoin trust network44 is a directed and sparse network characterized by its decentralized structure and asymmetric flow of trust, reflecting the dynamics of trust and distrust among users. We then extend our analysis to undirected networks, ensuring a comprehensive evaluation of our method across varying network types. Similarly, the Fraternity network9, also directed but denser, captures the hierarchical and complex social interactions within a tightly-knit community. In contrast, the CoW (Correlates of War) network45 is undirected and balanced in density, representing the intricate and shifting alliances and conflicts in international relations. The High School Contact and Friendship Networks dataset46 blends multiple dimensions of social interaction, including physical proximity, self-reported friendships, and online connections, modeling the interplay of positive and negative ties in a dynamic social environment. Finally, the MIDIP 5.0 (Militarized Interstate Disputes) dataset47 captures highly dynamic geopolitical relationships, showcasing how nations oscillate between alliances and enmities over time. Together, these datasets vary in size, density, directedness, and positive-negative tie ratios, demonstrating the versatility of our models in capturing diverse network dynamics. Further details on these networks are provided in Supplementary Table 1.

Model types and variable classification

Our analysis began with a comprehensive exploration, estimating a total of 24 distinct models for each signed network. However, it is essential to clarify that these 24 models do not represent unique predictive models. Instead, they encompass various configurations, allowing for a comprehensive exploration of network dynamics. In reality, we have four fundamental models at the core of our analysis: the baseline model, the GWESP model (considering both positive and negative networks), the positive triangle model, and the signed triangle model. In addition to these models, we delved into the effects of balance theory by introducing variables that classified the networks into two categories: those that are consistent with the balance theory and those that are not. Specifically, we further categorized the already classified SForm and SBrk networks into SF-Bal, SB-Bal, SF-NBal, and SB-NBal. For instance, networks consistent with balance theory in SForm were classified as SF-Bal, while those inconsistent with balance theory were classified as SF-NBal. A similar classification was applied to SBrk networks.

To illustrate this further, let us consider Fig. 3 as an example. In the first scenario, a positive connection is formed, but according to balance theory, a triangle with two positive edges and one negative edge is considered unstable and prone to break. Therefore, the first scenario is inconsistent with balance theory. Similarly, in the second scenario, a triangle with three negative edges is also considered unstable according to balance theory, leading to a classification as not consistent with balance theory. However, in the third and fourth scenarios, a triangle with one positive edge and two negative edges and a triangle with three positive edges can both exist stably, which is consistent with balance theory. Based on this logic, we further classify the originally categorized SForm and SBrk networks based on whether they are consistent with balance theory.

Fig. 3
figure 3

Illustration of the transition of four signed triangles as classified by balance theory.

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Table 1 provides an overview of the core model types considered in our analysis, including the baseline model, GWESP model, single-layer triangle model, and signed triangle model, along with their corresponding variables. In our study of network dynamics, a crucial aspect involves the classification of single-layer triangles, an operation that unveils intriguing insights into the underlying relationships within the network. This classification process serves as a cornerstone in comprehending the dynamics and behaviors exhibited by various entities within the network.

Table 1 Model types.
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To evaluate model performance, we use the Precision-Recall Area Under Curve (PR-AUC), a metric ideal for imbalanced data48. PR-AUC values range from 0 to 1, with higher values indicating better performance. PR-AUC combines precision (correct positive predictions) and recall (true positives identified) and is preferred over ROC-AUC for sparse networks, as it avoids inflated scores from many zero entries49. We estimate the BTERGM model using data from \([t-4,t-1]\) and then test the prediction accuracy on the network of time \(t\).

Results

Our analysis explores five datasets-Bitcoin, Fraternity, CoW, MIDIP, and High School networks-to evaluate the performance of the PosTriangle/NegTriangle and SignedTriangle models across diverse social and geopolitical contexts. This section presents the key findings, focusing on model performance trends and the influence of network characteristics such as directionality, density, and dynamic changes.

Bitcoin network

The Bitcoin network, a directed and sparse system, demonstrates the pivotal role of directionality in predicting tie formations. As shown in Fig. 4, PR-AUC values across positive and negative networks highlight the superior performance of the SignedTriangle model, particularly in positive networks. Balanced and unbalanced triadic structures are also examined, with consistent trends underscoring the SignedTriangle model’s robustness in capturing the network’s complex dynamics. However, when directional information is removed, the accuracy of the SignedTriangle model declines significantly, as illustrated in Fig. 5. This underscores the essential role of directionality in capturing the intricate flow of trust and influence in the Bitcoin network. These findings suggest that the absence of directional cues disrupts the model’s ability to identify key relational patterns, highlighting the importance of preserving directional information for robust predictive accuracy in sparse, decentralized systems.

Fig. 4
figure 4

PR-value in the Bitcoin network. (a) Positive, (b) Positive and balanced, (c) Positive and unbalanced, (d) Negative, (e) Negative and balanced, (f) Negative and unbalanced.

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Fig. 5
figure 5

Comparison of signed triangle models on directed and undirected Bitcoin networks. (a) Positive: SignedTriangle versus undirected SignedTriangle, (b) Negative: SignedTriangle versus undirected SignedTriangle.

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Fraternity network

The Fraternity network, a small yet densely directed network, highlights the influence of social hierarchies and power dynamics on network modeling. Figure 6 demonstrates that the SignedTriangle model achieves the highest prediction accuracy across both positive and negative networks. The inclusion of directional information is shown to be critical; its removal results in a marked decline in model performance, particularly for positive ties, as depicted in Fig. 7. These findings reaffirm the importance of directionality in understanding the complex dynamics of close-knit social groups and emphasize the SignedTriangle model’s ability to capture these nuanced interactions. Additionally, the performance variation underscores the unique challenges posed by dense, directed networks where power asymmetries and layered social structures heavily influence relational dynamics.

Fig. 6
figure 6

PR-value in the fraternity network. (a) Positive, (b) Positive and balanced, (c) Positive and unbalanced, (d) Negative, (e) Negative and balanced, (f) Negative and unbalanced.

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Fig. 7
figure 7

Comparison of signed triangle models on directed and undirected fraternity networks. (a) Positive: SignedTriangle versus undirected SignedTriangle, (b) Negative: SignedTriangle versus undirected SignedTriangle.

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Correlates of war (CoW) network

The CoW network, an undirected network reflecting alliances and conflicts, reveals consistent model performance irrespective of directional information. Figure 8 illustrates the superior performance of the SignedTriangle model in capturing both positive (alliances) and negative (conflicts) relationships. Notably, the PosTriangle/NegTriangle model demonstrates competitive performance in contexts where clustering patterns are more distinct. Balanced and unbalanced triadic configurations are also visualized, showing consistent trends that validate the adaptability of the SignedTriangle model. This dataset underscores the SignedTriangle model’s ability to generalize effectively to undirected networks, where balanced interactions often reflect stability and conflict resolution mechanisms. The performance across these scenarios highlights the model’s versatility in capturing the dynamic interplay of alliances and hostilities in global political systems.

Fig. 8
figure 8

PR-value in the CoW network. (a) Positive, (b) Positive and balanced, (c) Positive and unbalanced, (d) Negative, (e) Negative and balanced, (f) Negative and unbalanced.

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MIDIP 5.0 dataset

In the MIDIP 5.0 dataset, which tracks militarized disputes between countries, both models exhibit complementary strengths. The SignedTriangle model demonstrates superior performance during transitional periods when states oscillate between alliances and enmities (Supplementary Fig. 4). Conversely, the PosTriangle/NegTriangle model excels during periods of heightened geopolitical tension, such as periods 9 and 10, when alliances and conflicts are clearly defined. These results highlight the models’ ability to adapt to temporal shifts in geopolitical relationships.

High school contact and friendship networks

The High School Contact and Friendship Networks dataset captures rich social dynamics, including friendships, rivalries, and online connections. Here, the SignedTriangle model consistently outperforms others, particularly during periods of social flux, as shown in Supplementary Fig. 5. In contrast, the PosTriangle/NegTriangle model performs better during periods where positive and negative ties are distinctly defined, underscoring its limitations in handling fluctuating social dynamics.

Common features and patterns

Across all datasets, a consistent pattern emerges: the SignedTriangle model excels in capturing complex and fluid network dynamics, particularly during periods of significant transitions. These transitions include geopolitical shifts in the MIDIP dataset, the evolution of alliances and conflicts in the CoW network, and dynamic changes in social structures observed in the High School and Fraternity networks. In contrast, the PosTriangle/NegTriangle model performs better in scenarios characterized by stable and well-defined relationships, such as heightened geopolitical tensions in the CoW and MIDIP datasets or strongly defined social hierarchies in the Fraternity network. These results demonstrate how each model leverages its strengths to address different types of network dynamics.

The role of directionality emerges as a dataset-specific factor that significantly impacts model performance. In directed networks like Bitcoin and Fraternity, maintaining directional cues greatly enhances predictive accuracy. This is especially evident in positive networks, where the flow of trust, influence, or social hierarchy depends heavily on the directionality of ties. When directional information is removed, the SignedTriangle model exhibits notable declines in accuracy, underscoring the importance of capturing this critical feature. In contrast, undirected networks, such as CoW and MIDIP, show consistent performance regardless of directionality, reflecting the inherently balanced nature of alliances and conflicts in these datasets.

These findings also highlight the complementary strengths of the two models. The SignedTriangle model excels in capturing the nuanced interplay of positive and negative ties during periods of flux, where relationships are rapidly shifting or undergoing reorganization. By integrating both types of interactions within triadic structures, the model provides a comprehensive framework for understanding dynamic and fluid network behaviors. Conversely, the PosTriangle/NegTriangle model is more suited for contexts with clear and stable distinctions between positive and negative relationships, where its separation of these two layers enables robust predictions of network behavior.

Together, the results underline the importance of tailoring predictive approaches to the structural and temporal characteristics of the network. The adaptability of the SignedTriangle model to diverse and dynamic systems highlights its utility in capturing intricate and evolving relationships. At the same time, the PosTriangle/NegTriangle model offers reliable performance in scenarios with well-defined relationship patterns. These complementary strengths validate the versatility of our methods, demonstrating their applicability to a wide range of social and geopolitical systems.

Discussion

Our study demonstrates that signed triangle structure models consistently outperform other models in predicting network behaviors across all five examined networks. This finding aligns with previous research suggesting the importance of triadic structures in understanding network dynamics3,4. By incorporating both positive and negative interactions, signed triangle models provide a more holistic and accurate representation of the complexities within social networks. This comprehensive approach enables better prediction and deeper insights into the underlying mechanisms driving network evolution.

Our research distinguishes itself from previous work by focusing on the dynamic interplay of positive and negative interactions within triadic structures in signed networks, with particular emphasis on temporal evolution. While previous approaches, such as the Signed Latent Factor model34 and graph kernel-based methods35, have achieved significant advancements in predicting signed links, they often lack a focus on the temporal dynamics of triadic transformations. Additionally, methods like those proposed by de Bruin et al.32 and Ahmed et al.33 improve temporal link prediction but do not adequately address the complexities of signed interactions. By incorporating both temporal and directional information, our signed triangle structure model outperforms traditional methods, especially in networks where the directionality and duality of relationships are critical. Moreover, while models exploiting longer cycles4 offer enhanced prediction in static signed networks, our approach goes further by dynamically capturing how triadic structures evolve and contribute to network stability or instability over time. This gives our model a broader scope and stronger predictive accuracy in multilayer networks, making it a significant step forward in both understanding and predicting the dynamics of signed social networks.

In interpreting the results, it is important to connect the network features with the real-world systems represented by each dataset. In the Bitcoin network, the importance of “multiple layers of network features and interactions” refers to the complex and decentralized nature of trust relationships, where both positive (trust) and negative (distrust) interactions occur simultaneously. These dynamics are critical in shaping the network’s evolution, as users base their decisions on prior interactions, reputation, and trustworthiness. The directional nature of these ties, where one user may trust another without reciprocity, adds complexity, which our signed triangle model captures more effectively than other methods. Similarly, in the Fraternity network, the strong cohesion within a small, tightly connected group, combined with occasional conflicts, reflects the dual nature of relationships that balance theory seeks to explain. The performance of the signed triangle model demonstrates the importance of incorporating both positive and negative interactions to predict social dynamics in real communities. The CoW network, representing international relations, showcases the coexistence of alliances and conflicts, emphasizing the need to account for both trust and enmity in network modeling. In the MIDIP dataset, the dynamic oscillation between cooperation (alliances) and hostility (conflicts) highlights how international disputes unfold over time. Here, the model effectively captures the shifts in balance and imbalance, providing insights into geopolitical stability and disruption. Finally, the High School Contact and Friendship Networks dataset represents a distinctly personal and dynamic social environment, where physical proximity, friendships, and rivalries coexist. The model’s ability to adapt to both rapidly fluctuating and more stable social ties underscores its versatility. Across all five systems, the dynamic interplay between positive and negative ties is a crucial feature that our method effectively captures, offering deeper insights into the real-world mechanisms driving network evolution across diverse domains.

The observed variation in the performance of single-layer models, particularly the poor performance of GWESP models in negative networks, underscores the limitations of traditional network models that do not account for the dual nature of relationships. Our results suggest that models focusing solely on positive interactions may miss critical dynamics present in networks with significant negative interactions38. These findings emphasize the necessity of integrating multiple dimensions of network data to capture the full spectrum of social interactions, a need echoed by recent advancements in network science. Importantly, the results of our study provide key theoretical insights related to the role of directional information in signed networks. Directional information is crucial in networks where the flow of influence, trust, or relationships is inherently asymmetric, such as in financial systems (e.g., the Bitcoin network) or hierarchical social structures. The significant impact of directional information on the prediction accuracy of signed triangle models highlights the importance of considering directionality in network analysis. The loss of directional cues in the Bitcoin and Fraternity networks led to a noticeable decrease in model performance, particularly in positive networks. This aligns with studies indicating that directional ties provide essential context for understanding the flow and influence patterns within a network50. However, this does not apply uniformly to all directed networks. In networks where relationships are largely reciprocal or symmetric, the absence of directional information may not significantly impact predictive accuracy.

While this study provides valuable insights into network dynamics, there are several caveats to consider. The importance of signed network structures, including both positive and negative ties, may not apply universally to all networks. In systems where conflict or competition is minimal, focusing solely on positive interactions might suffice. Similarly, the application of balance theory to explain triadic structures is more relevant in stable, cohesive networks, and may offer limited insights in more fluid or less structured environments. Another limitation is the reliance on temporal modeling using TERGM, which is critical for capturing rapidly evolving networks, but may not be necessary for networks that change at slower or more predictable intervals. In such cases, simpler static models could perform equally well. These considerations highlight the need to tailor our approach depending on the specific characteristics and dynamics of the network being studied.

In conclusion, our study validates the effectiveness of signed triangle structure models in capturing the nuanced dynamics of social networks. These models offer significant improvements over traditional approaches by integrating both positive and negative interactions and accounting for directional information. The findings suggest several avenues for future research, including the refinement of models to incorporate additional factors such as temporal changes and contextual influences. Furthermore, future work could benefit from incorporating both micro- and macro-level dynamics to provide a more comprehensive understanding of network evolution, as suggested by45. Exploring the application of signed triangle models in other types of networks and domains could further validate their utility and uncover new insights vis-à-vis network evolution and behavior. These advancements hold promise for developing more accurate predictive models, thus enhancing our understanding and anticipation of changes in complex social systems.