Möbius Markov Chain - The Classics Revival
Non-Euclidean Probabilistic Systems with Dynamic Geometry
Experimental Research Code - Functional but unoptimized, expect rough edges
What Is This?
Möbius Markov Chain operates Markov processes in complex space with dynamically warped geometry via Möbius transformations. The state space itself evolves based on current states, creating probabilistic dynamics that adapt their geometric structure during evolution.
Core Innovation : Markov transition probabilities computed in continuously warped complex space, where the geometry itself learns to optimize transition dynamics.
Architecture Highlights
Complex State Space : States positioned in the complex planeDynamic Möbius Transformations : Learnable conformal mappings f(z) = (az+b)/(cz+d)Geometric Transition Matrices : Distances computed in transformed spaceState-Dependent Warping : Geometry evolves based on current state distributionConformal Invariance : Preserves angles while warping distancesDistance Kernels : Multiple kernel options for probability computation
Quick Start
1 from mobius_markov import MobiusMarkovSystem
2
3 # Create non-Euclidean Markov system
4 system = MobiusMarkovSystem (
5 num_states = 8 ,
6 state_embedding_dim = 32 ,
7 evolution_steps = 5
8 )
9
10 # Initialize state distribution
11 initial_state = torch . zeros ( batch_size , num_states )
12 initial_state [ : , 0 ] = 1.0 # Start in state 0
13
14 # Evolve through warped space
15 output = system ( initial_state , return_full_trajectory = True )
16
17 # Generate sequence predictions
18 sequence = system . predict_sequence ( initial_state , sequence_length = 10 )
Current Status
Working : Complex plane dynamics, Möbius transforms, distance-based transitions, state evolutionRough Edges : Conservative parameter defaults, could use more dramatic geometry warping for demosStill Missing : Advanced kernel functions, multi-scale temporal dynamics, visualization toolsPerformance : Mathematically sound and stable, good for research foundationMemory Usage : Moderate, dominated by complex number operationsSpeed : Reasonable for small state spaces, optimization needed for large systems
Mathematical Foundation
Möbius transformations are defined as:
where a,b,c,d are learnable complex parameters with ad - bc ≠ 0.
State positions in complex space determine transition probabilities:
P_ij = kernel(d_transformed(z_i, z_j))Distance kernels include:
Gaussian : P ∝ exp(-d²/2σ²)Inverse : P ∝ 1/d^αLinear : P ∝ max(0, 1-d)
The geometry evolves according to:
∂θ/∂t = η × state_embedding_evolution(current_distribution)where θ represents the Möbius transformation parameters.
Research Applications
Non-Euclidean machine learning Adaptive probabilistic models Complex systems with geometric constraints Hyperbolic neural networks Conformal prediction systems
Installation
1 pip install torch numpy matplotlib
2 # Download mobius_markov.py from this repo
The Classics Revival Collection
Möbius Markov Chain is part of a larger exploration of foundational algorithms enhanced with modern neural techniques:
Evolutionary Turing Machine
Hebbian Bloom Filter
Hopfield Decision Graph
Liquid Bayes Chain
Liquid State Space Model
Möbius Markov Chain ← You are hereMemory Forest
Citation
1 @misc{mobiusmarkov2025,
2 title={Möbius Markov Chain: Non-Euclidean Probabilistic Systems},
3 author={Jae Parker 𓅸 1990two},
4 year={2025},
5 note={Part of The Classics Revival Collection}
6 }
