TCEC-FCM: Causal Effects in FCMs

• The paper introduces a novel max–min operator combined with binary search to efficiently compute direct and indirect causal effects in Fuzzy Cognitive Maps.
• It details a scalable algorithm that reduces computational complexity from factorial to O(n·e·log e) by avoiding exhaustive path enumeration.
• The work bridges fuzzy causal algebra with explainable AI, enhancing both the interpretability and scalability of causal analysis in complex networks.

Total Causal Effect Calculation for FCMs (TCEC-FCM) addresses the efficient quantification of the causal impact that one concept exerts on another within the structural framework of Fuzzy Cognitive Maps (FCMs). TCEC-FCM encompasses not only the theoretical closed-form Neumann series approach for summing all direct and indirect path effects, but also scalable algorithms—specifically, fast binary-search graph traversal—that make such calculations feasible on large-scale networks. These methods, drawing from the literature in fuzzy causal algebra and recent innovations in explainable artificial intelligence (XAI), permit comprehensive and interpretable causal analysis of FCMs with hundreds or thousands of nodes, overcoming the prohibitive computational costs of brute-force enumeration of all simple paths (Tyrovolas et al., 2024, Osoba et al., 2019).

1. Formal Foundations of Causal Effect in FCMs

A Fuzzy Cognitive Map (FCM) is a weighted, directed graph $G = (C,E)$ with concept nodes $C = \{C_1, \dots, C_n\}$ and edge weights $W = (w_{ij})$ where $w_{ij} \in [-1,1]$ quantifies the signed influence from $C_i$ to $C_j$. Causality is "fuzzy": edge weights reflect the degree and direction of influence, not strict necessity or probabilistic dependency as in Bayesian networks.

The total causal effect $T(C_i, C_j)$ is defined as the maximum, over all simple (cycle-free) paths $\ell$ from $C_i$ to $C_j$, of the minimum weight along each path: $$T(C_i, C_j) = \max_{\ell} \min_{(p,q)\in \ell} w_{pq}$$ This "max–min" operator reflects a “weakest link” interpretation—any indirect effect along a path cannot exceed its minimal edge. This calculus ensures that feedback and cycles can be systematically included or excluded, with the standard implementation considering only acyclic simple paths for the max–min definition (Tyrovolas et al., 2024, Osoba et al., 2019).

In contrast, for certain models (especially under linearization where the transition matrix $W$ satisfies $\rho(|W|)<1$), the sum-over-paths approach using the Neumann series yields: $T = \sum_{k=1}^\infty W^k = (I-W)^{-1} - I$ where the $(i,j)$ entry $t_{ij}$ aggregates the products of all path weights from $C_i$ to $C_j$ over all path lengths (Osoba et al., 2019).

2. Algorithmic Workflow for TCEC-FCM

Computing $T(C_i, C_j)$ via brute-force enumeration of all simple paths is factorial in the worst case. TCEC-FCM leverages graph traversal and binary search to avoid exhaustive path enumeration. The key steps are as follows (Tyrovolas et al., 2024):

1. Sort Nonzero Weights: Assemble an array $W_{\text{sorted}}$ of all nonzero edge weights, sorted in descending order. Let $e = |W_{\text{sorted}}|$.
2. Binary Search: For each source node $C_i$, perform a binary search over $W_{\text{sorted}}$:
   • At each iteration (midpoint index $k$), construct a subgraph $G_{\text{mid}}$ retaining only edges with weight at least $W_{\text{sorted}}[k]$.
   • Use BFS or DFS to test if $C_j$ is reachable from $C_i$ in $G_{\text{mid}}$.
   • If reachable, update $T(C_i, C_j)$ and continue search in the upper range; else, lower the threshold.
3. Termination: On termination, $T(C_i, C_j)$ is the largest threshold for which $C_j$ is still reachable.

This approach avoids enumeration of all paths, reducing the time complexity from $O(n!)$ (exhaustive DFS) to $O(n e \log e)$ per target node, making the method scalable to large graphs.

Algorithm	Complexity	Path Type
TCEC-FCM (binsearch)	$O(n e \log e)$	Simple paths
Exhaustive DFS	up to $O(n!)$	Simple paths
Matrix Inverse	$O(n^3)$	All walks

3. Matrix Formulations and Theoretical Properties

Under the alternative "all-walks" (not just simple paths) total effect formalism—appropriate for linear FCMs with spectral radius $\rho(|W|)<1$—the closed-form for total causal effect between every pair $(i, j)$ is given by the $(i,j)$ entry of $(I-W)^{-1} - I$ (Osoba et al., 2019).

$$t_{ij} = \bigl[(I-W)^{-1}\bigr]_{ij} - \delta_{ij} = \sum_{k=1}^\infty (W^k)_{ij}$$

This formula includes feedback loops and recurrent influences, converging absolutely when the spectral radius condition holds. For non-linear FCMs or those focused on simple path effects, the max–min operator with binary search is required.

4. Experimental Results and Scalability

Empirical evaluation demonstrates orders-of-magnitude speedup for TCEC-FCM compared to exhaustive approaches:

# Concepts	TCEC-FCM-LS (s)	TCEC-FCM (s)	DFSB-FCM (s)
10	0.025	0.027	5.820
100	0.697	0.030	588.410
1000	558.386	634.807	6822.280

At 1000 nodes, exhaustive DFS is intractable, while TCEC-FCM completes in under 11 minutes in MATLAB for dense graphs. With moderate density ($10\%-100\%$), TCEC-FCM remains practical up to several thousand nodes (Tyrovolas et al., 2024).

5. Practical Assumptions, Limitations, and Extensions

• Static network assumption: The FCM and its weights are assumed fixed.
• Simple-path focus: The "max–min" method considers simple (acyclic) paths only; the Neumann approach sums over all walks, including cycles, under convergence conditions.
• Scalability: TCEC-FCM is polynomial but may incur long runtimes for extremely large graphs ($n \gg 10^4$) unless implemented in compiled languages or parallelized.
• Context-specific effects: The current methods compute global causal effects; contextual or dynamic local effect analysis (per input pattern) would require model extensions.
• Numerical issues: For matrix inversion, ill-conditioning of $I-W$ can affect stability; truncation and reweighting may be necessary if $\rho(|W|)$ is not sufficiently less than 1 (Osoba et al., 2019).

Potential extensions include parallelization (multi-core, GPU), compiled-graph traversal, and adaptation for time-varying or state-dependent weights (Tyrovolas et al., 2024).

6. Relationship to Broader Causal Inference and FCM Theory

TCEC-FCM builds upon foundational results in fuzzy causal algebra [Kosko 1986] and generalizes prior theoretical work on the accumulation of causal influence in signed directed graphs. In contrast to DAG-based models, FCMs naturally incorporate feedback cycles, and their total causal effect calculations embrace both direct and recursive (feedback-mediated) influences. The methodology aligns with transitive causal influence theorems and extends computational methods to the XAI domain, enabling interpretable causal explanations on high-dimensional, expert-elicited cognitive structures (Osoba et al., 2019, Tyrovolas et al., 2024).

7. Summary of Key Results and Significance

TCEC-FCM provides a theoretically rigorous and computationally tractable solution for calculating total causal effects in FCMs:

• Establishes the max–min weakest-link definition as the core for simple-path causal effect.
• Delivers an $O(n e \log e)$ algorithm for feasible computation on large, dense FCMs.
• Includes a closed-form, walk-summing alternative for linear FCMs with spectral constraint.
• Enables quantification of indirect, recursive, and feedback-mediated effects within explainable AI systems and expert-structured causal models.

This framework extends the practical reach of FCM-based causal analysis into modern, large-scale settings emergent in XAI, decision analysis, and systems modeling (Tyrovolas et al., 2024, Osoba et al., 2019).