Non-normal interactions create socio-economic bubbles

Many complex systems, including financial markets, exhibit periods of stability punctuated by sudden bursts of unsustainable growth, or "bubbles." Existing models often explain these phenomena using the Ising model and its variants, emphasizing the importance of reaching a critical point where collective behavior emerges. These models typically involve two types of traders: fundamentalists who act rationally, and noise traders whose behavior is influenced by others through an Ising-like imitative dynamic. Bubbles, in these models, arise when the imitation strength reaches or exceeds a critical threshold, leading to synchronized trading behavior. This paper challenges this prevailing view, proposing that the emergence of bubbles is not necessarily tied to a critical point but rather to the intrinsic non-normal structure of socio-economic networks. These networks, characterized by directed and hierarchical influences, exhibit transient bursts of herd behavior even below any critical threshold. This is significant because it suggests that bubbles are a more common and inherent feature of these systems than previously thought, arising from the very architecture of social influence rather than solely from a specific level of collective behavior.

The existing literature extensively uses the Ising model and its variations to model collective behavior in social and financial systems. These models posit that bubbles emerge when a system reaches a critical point, where the strength of imitation among agents surpasses a critical threshold. This triggers a phase transition leading to synchronized behavior and the formation of bubbles. Agent-based models incorporating this framework have been developed and analyzed, yet these models require precise fine-tuning of parameters to achieve criticality and thus bubble formation. The authors review this body of work, emphasizing the limitation that these models require a critical point to observe bubbles and crashes. The authors note the extensive body of knowledge around tipping points and critical transitions, but highlight the lack of consideration for the non-normality of complex systems.

The authors employ an agent-based model (ABM) simulating a financial market with fundamentalist and noise traders. Fundamentalists maximize expected utility, while noise traders' investment decisions are influenced by an Ising-like social interaction model. Crucially, unlike previous models that assume a fully connected, symmetric network, this model explicitly incorporates the network topology, represented by a directed adjacency matrix A. The dynamics of noise traders are modeled using the equation Δs(t) = Ms(t), where M is a transition matrix proportional to A, and s(t) represents the state of each noise trader (+1 for holding the risky asset, -1 for the risk-free asset). The collective opinion is defined as the average state. The paper focuses on subcritical regimes where the system is linearly stable (all eigenvalues of M have negative real parts), but the matrix M is non-normal (M^TM ≠ MM^T). Non-normal matrices are characterized by non-orthogonal eigenvectors, leading to transient behavior that differs significantly from the asymptotic behavior determined by the largest eigenvalue. The key metric for assessing the non-normality is the Kreiss constant K(M), which provides a lower bound for transient growth. The authors describe an algorithm for generating non-normal adjacency matrices, incorporating features such as hierarchical levels and level-dependent reciprocity based on empirical observations from Reddit discussion forums. The algorithm incorporates six parameters (N, N₀, m, θ, a, b) allowing the researchers to manipulate different features of the network. They investigate the influence of the network structure (especially the level of reciprocity, θ and the number of top nodes, N₀) on the occurrence and characteristics of bubbles. Finally, they analyze real-world data from Reddit discussions on four meme stocks (Blackberry, Nokia, GameStop, and AMC), extracting dynamically evolving influence networks to empirically assess the relationship between non-normality, as measured by the Kreiss constant, and price bubbles.

The agent-based simulations demonstrate that non-normal networks generate transient bubbles even in the subcritical regime (where the average magnetization fluctuates around zero). The simulations show that strongly non-normal networks (with low reciprocity) exhibit much larger and more persistent deviations from the zero-magnetization state than near-normal networks. These deviations translate into pronounced price peaks in the simulated market, characteristic of bubbles. The authors find a strong correlation between the size of these simulated bubbles and the Kreiss constant K(M) of the interaction matrix M. Larger Kreiss constants are associated with larger bubbles. They also show a relationship between bubble steepness and the numerical abscissa of M. This indicates that the size and shape of bubbles are primarily determined by the non-normality of the social network. The empirical analysis of Reddit data related to meme stocks reveals a positive correlation between the Kreiss constant of the discussion networks and the occurrence of price bubbles. The most prominent price peaks for these meme stocks coincide with peaks in the Kreiss constant, supporting the hypothesis that non-normal social interactions contribute significantly to price instability. This analysis includes the creation of simulated price trajectories using the empirical network data, further supporting this correlation. The effect of a contrary opinion on the size of the bubble is also investigated, showing that a contrary opinion can mitigate the bubble size but only to a limited extent, even when a large fraction of the nodes are receptive to the contrary opinion.

The findings challenge the traditional view that bubbles are solely associated with criticality in social systems. Instead, the results suggest that the non-normality of interaction networks is a fundamental driver of transient price exuberance. The authors demonstrate that this non-normality leads to transient bursts in collective behavior, generating bubbles even when the system is far from a critical phase transition. This mechanism is more general than those based on criticality, potentially explaining the ubiquity of bubbles in various socio-economic systems. The empirical analysis of meme stock data provides strong evidence supporting the theoretical model, highlighting the role of Reddit discussion forums in shaping market dynamics and price instability. This suggests that the architecture of social interactions, rather than the level of social coupling, has a significant impact on the development of bubbles.

This paper presents a novel mechanism for the formation of socio-economic bubbles, based on the non-normality of social interaction networks. The findings challenge the traditional emphasis on critical points and suggest that bubbles are an intrinsic feature of many systems due to their inherent hierarchical and asymmetric structure. The results are robust across simulations and empirical data analysis, highlighting the importance of network topology in understanding price dynamics. Future research could explore the specific mechanisms by which the hierarchical structure influences opinion formation and the possibility of designing interventions to mitigate bubble formation, targeting specific highly influential nodes.

The agent-based model makes certain simplifying assumptions about trader behavior, such as the binary nature of noise trader decisions and the simplified representation of fundamentalist behavior. The empirical analysis relies on Reddit data, which might not fully capture all relevant factors influencing meme stock prices. The model doesn't take into account the effect of other factors which can affect the prices besides those discussed in the paper. External factors not captured in the Reddit data could contribute to price variations. Despite these limitations, the model and analysis present a compelling explanation for bubble formation that merits further investigation.