How Do Emotions Spread And Affect Decision-Making In Social Networks? An Opinion Dynamics Perspective

The gist The proposed model formalizes the idea that emotional alignment between agents enhances receptivity or willingness to agree to others’ opinions, whereas emotional misalignment suppresses such responsiveness in an ongoing discussion<ref:2610.22334#pg6>

Model Representation of Emotions

Emotions are represented on a circle by utilizing the circumplex model This representation of emotions can also be done alternatively using a two-dimensional (2D) real-valued representation The origin in this circle represents the neutral state of emotions, x-axis represents the valence or pleasantness, ranging from sadness (negative valence) to happiness (positive valence), while the y-axis represents the degree of arousal, ranging from high arousal (alertness) to low arousal (calmness) Following the circumplex model, we denote e ∈ C as the emotion of an individual as; e = v+ιa This representation of emotions can also be done alternatively using a two-dimensional (2D) real-valued representation We prefer the complex-valued representation because the 2D real-valued representations treat these dimensions independently, whereas a complex-valued representation combines them into a single variable while preserving their orthogonality through the real and imaginary components

Emotional Contagion Dynamics

Emotional contagion is modeled as a diffusion process on a digraph The graph Laplacian provides a mathematically principled representation of pairwise influence and has been widely adopted in diffusion models due to its ability to capture the network topology and convergence properties We propose the following Laplacian-based framework to capture the dynamics of emotions via emotional contagion in a network: e˙ = −∆(D+ f −AEC)e = −Le˜, where L˜ is the modified Laplacian matrix with a modified degree matrix The parameter, emotional susceptibility quantifies how strongly an agent is influenced by the emotional states of its neighbors A maximum value of δi = 1 indicates maximum susceptibility, meaning the agent is fully influenced by its neighbors The digraph, GEC is considered to be equitable partitioned; this implies that every agent within a cluster receives the same aggregate emotional influence from every other cluster Consequently, every cluster is treated as a single emotional unit in further analysis The feedback vector in our model ensures that the emotional states converge to steady state outcomes characterized by convergence such as consensus, or clustering of emotions

Convergence and Consensus

Theorem 1 states that the emotions and opinions converge to consensus if and only if all the agents in GEC have fi = 0,∀ i and the matrix,−L is ˜marginally stable of corank 1 The final state vector is given by ef = exp(−∆LECt)e0 = vECwEC Hx0 This implies that consensus of emotions at steady state leads to consensus of opinions in the resultant complex-weighted networks We obtain complex-valued opinions at the steady state, which represent the influence of emotions, allowing agents to express opinions with varying levels of support or opposition rather than being limited to completely affirmative or negative opinions

Multi-partite Consensus

Theorem 2 presents conditions to achieve multi-partite consensus in complex-weighted networks This is achieved if the values of fp in f = Lk p=1 fpp = diag(fp) are chosen such that the matrices L and L ˜ π are marginally stable of corank 1 The final opinions converge to multi-partite consensus The resultant right eigenvector corresponding to the zero eigenvalue is ve = E1n, where E = M k p=1 epp, epp = diag(ep), where ep ∈ C denotes the steady state emotion of the pth agent

Implications for Social Relationships

The modified edge-weight, (ae)i j implying complex-valued opinions later, can be expressed as (ae)i j = R((ae)i j)+I((ae)i j), where R((ae)i j) signifies social alignment, whereas I((ae)i j) signifies emotional control between a pair of agents According to Table3 I, the emotional state of happiness increases co-operation and decreases antagonism among the agents This is in line with the theory of positive emotions (see Theory 1) Instead, an agent in a sad emotional state converts co-operation to antagonism and vice-versa However, it is worth noting that when both the agents are sad, they tend to co-operate with each other, as shared sadness promotes mutual empathy Finally, the modified edge-weight, (ae)i j implying complex-valued opinions later, can be expressed as (ae)i j = R((ae)i j)+I((ae)i j) The paper proposes a continuous-time linear timeinvariant model that demonstrates the spread of emotions and incorporates the influence of final emotions in the decisionmaking process The proposed framework is novel in the sense that it further enables us to investigate more general forms of agreement beyond consensus; in particular, we establish conditions for achieving multi-partite consensus in complex-weighted networks

Validation and Future Scope

The paper validates its theoretical results using a synthetic social network and a real-world dataset The Zachary Club’s Network consists of 34 agents (nodes) and 78 friendship ties (edges), representing social interactions observed over approximately two years We aim to extend our framework to time-varying network topologies The proposed framework is fully distributed, where each agent updates its emotional state based on its local interactions The paper proposes a novel interpretation to these networks by modeling emotions using complexweights in a social network We plan to analyze the effect of multiple emotions and explore time-varying complex models in our future work The proposed framework is novel in the sense that it demonstrates the spread of emotions and incorporates the influence of final emotions in the decisionmaking process

The gist The proposed model formalizes the idea that emotional alignment between agents enhances receptivity or willingness to agree to others’ opinions, whereas emotional misalignment suppresses such responsiveness in an ongoing discussion The paper presents a novel framework for modeling emotions and studying their impact in social networks using a Laplacian-based flow The key finding shows that agents with similar emotions reach agreement eventually, exhibiting emotional contagion This approach relies on the equitable partitions and spectral properties of Laplacians in complex-weighted networks It is the first study to incorporate emotions in the area of opinion dynamics and rigorously investigate the conditions to achieve multi-partite consensus in complex-weighted networks The framework is a continuous-time linear timeinvariant model that demonstrates the spread of emotions and incorporates the influence of final emotions in the decisionmaking process It is important because it provides a mathematical framework to model signed networks where complex weights represent both intensity and valence of emotional influence This research matters because it shows how collective emotional states can lead to consensus or polarization in social networks, offering insights into collective decision-making processes The paper's findings have practical implications for improving mental health and emotional wellbeing in social groups by suggesting that emotions can be varied through systematic intervention It is a novel contribution because it rigorously investigates the conditions to achieve multi-partite consensus in complex-weighted networks using equitable partitions.

How it works

The model represents emotions on a circle by utilizing the circumplex model This representation of emotions can also be done alternatively using a two-dimensional (2D) real-valued representation Emotions spread among agents as a result of emotional contagion during social interactions and finally affect the decisions in a group of interacting agents Emotional contagion is modeled as a diffusion process on a digraph The graph Laplacian provides a mathematically principled representation of pairwise influence and has been widely adopted in diffusion models due to its ability to capture the network topology and convergence properties The proposed framework is defined by the equation e˙ = −∆(D+ f −AEC)e = −Le˜, where L˜ is the modified Laplacian matrix with a modified degree matrix This model captures emotional contagion by representing AEC as the adjacency matrix that encodes the network structure governing dynamics of emotional contagion The term ∆ ∈ R n×n: a diagonal matrix, consisting of i th diagonal entries with δi ∈ [0,1] as the emotional susceptibility of the i th agent

Key Mechanisms and Assumptions

1. Equitable Partitions (AEP): A partition π of V is said to be an AEP if for all s,t ∈Ci,i, j = 1,…,r, the following equality holds: ∑ k∈Cj lsk = ∑ k∈Cj ltk The digraph GEC is considered to be equitable partitioned; this implies that every agent within a cluster receives the same aggregate emotional influence from every other cluster

1. Emotional State Modeling for Decision Support: Implement an agent model where emotions are represented on a circle by utilizing the circumplex model to capture valence and arousal, allowing AI systems to predict how emotional alignment or misalignment between agents will affect their final opinions and decisions.

2. Complex-Weighted Network Analysis for Consensus Prediction: Utilize the proposed Laplacian-based framework, specifically the equation e˙ = −∆(D+ f −AEC)e = −Le˜, to model emotional contagion and predict convergence outcomes, as the paper shows that agents with similar emotions reach agreement eventually, exhibiting emotional contagion.

3. Multi-Partite Consensus Strategy: Develop algorithms to achieve multi-partite consensus in complex-weighted networks by tuning the feedback vector in the modified Laplacian matrix, ensuring that the matrices L and L ˜ π are marginally stable of corank 1 as specified in Theorem 2.

4. Emotionally Informed Relationship Modeling: Enhance decision-making modules by incorporating the derived relationship dynamics, where the modified edge-weight, (ae)i j implying complex-valued opinions later, can be expressed as R((ae)i j)+I((ae)i j), where R((ae)i j) signifies social alignment.

5. Mental Health and Wellbeing Intervention Systems: Design systems that use the model's findings to suggest interventions, such as an agent in a distressed state surrounded by agents having positively valenced emotional states may, over time, experience emotional uplift through sustained exposure to positive valenced emotions like happiness.

Emotions affect how we think and perceive information. Motivated by this, we present a novel framework for modeling emotions, and study its impact in social networks. We represent emotions on a circle by utilizing the circumplex model. These emotions spread in a group through the process of emotional contagion, and influence the interpersonal relationships therein. We propose a model to explain the process of emotional contagion, and its effect on the decision-making process, using a Laplacian-based flow. Our key finding in this direction shows that agents with similar emotions reach agreement eventually, exhibiting emotional contagion. Through this, the widely observed behaviors of consensus, polarization, and clustering of opinions follow in complex-weighted networks. Our approach relies on the equitable partitions and spectral properties of Laplacians in complex-weighted networks. Lastly, we validate our findings using synthetic and real-world datasets.