Controlled Causal Hallucinations Can Estimate Phantom Nodes in Multiexpert Mixtures of Fuzzy Cognitive Maps

TL;DR

The paper tackles identifying missing causal variables in large-scale multivariable causal models with feedback by proposing a multiexpert mixture of fuzzy cognitive maps (FCMs) whose phantom nodes augment each expert's model. Each expert proposes phantom nodes and the models are combined with convex weights to form a joint dynamics that approximates the target limit cycles of the underlying dynamical system. A gradient-based phantom-edge learning scheme minimizes $L=\sum_{t=1}^k ||C(t) - \tilde{C}(t)||^2$ to train the phantom connections, enabling the mixture to align with observed limit cycles. Experiments on a dolphin-like system demonstrate that phantom-augmented FCM mixtures can closely reproduce the target limit cycles, offering a scalable approach to capturing hidden causal factors and forecasting future trajectories in dynamical systems.

Abstract

An adaptive multiexpert mixture of feedback causal models can approximate missing or phantom nodes in large-scale causal models. The result gives a scalable form of \emph{big knowledge}. The mixed model approximates a sampled dynamical system by approximating its main limit-cycle equilibria. Each expert first draws a fuzzy cognitive map (FCM) with at least one missing causal node or variable. FCMs are directed signed partial-causality cyclic graphs. They mix naturally through convex combination to produce a new causal feedback FCM. Supervised learning helps each expert FCM estimate its phantom node by comparing the FCM's partial equilibrium with the complete multi-node equilibrium. Such phantom-node estimation allows partial control over these causal hallucinations and helps approximate the future trajectory of the dynamical system. But the approximation can be computationally heavy. Mixing the tuned expert FCMs gives a practical way to find several phantom nodes and thereby better approximate the feedback system's true equilibrium behavior.

Figures (9)

• Figure 1: Causal phantom nodes augment a FCM and help approximate the FCM dynamics and its target equilibrium limit cycles. The figures on the left show the FCMs and the figures on the right show their corresponding limit cycle. (a) The original FCM with 4 nodes. (b) The augmented FCM with 4 observable nodes and one phantom node. The original FCM does not approximate the limit cycles of the system it models. The FCM with the phantom node can approximate the target limit cycles with its own limit cycles.
• Figure 2: Phantom nodes augment experts' FCMs that then combine to approximate the limit cycles of a dynamical system through controlled hallucinations. (a) Causal Dynamical system that the expert FCMs model and its target limit cycle that the FCMs try to approximate. (b) Three 6-node FCMs from 3 different experts and their corresponding limit cycles. The limit cycles do not match those of the dynamical system. This indicates the presence of phantom nodes. (c) Phantom nodes $A$, $B$, and $C$ respectively augment FCMs from experts 2, 3, and 1. Phantom nodes allow controlled hallucinations and approximate the dynamical system's limit cycle. (d) The 3 7-node FCMs mix into a 10-node FCM through convex combination. The combined FCM hallucinates a limit cycle even closer to that of the dynamical system.
• Figure 3: Dolphin FCM of dolphins near a threat such as a shark.
• Figure 4: Limit cycles from the Dolphin FCM. The time step is along the $x$-axis. The nodes are along the $y$-axis. The images have 5 rows of pixels because the FCM has 5 nodes and each row represents the time evolution of one node. The color of the node represents its value. A bright color corresponds to a high value. White nodes have value 1. Black nodes have value 0. The top figure starts with the initial state $00010$ and the bottom figure starts with the initial state $01010$. They both fall into the same limit cycle: $00010\rightarrow10001\rightarrow01000\rightarrow00100\rightarrow00010$. The middle figure starts with initial state $00011$ and falls into a different limit cycle: $10010\rightarrow11001\rightarrow01100\rightarrow00110\rightarrow10010$.
• Figure 5: The Dolphin FCM with a phantom node $C_4$ that corresponds to survival threats like sharks. The domain expert or causal learning system does not observe $C_4$ or know the edges that correspond to $C_4$.