aiq-quivers 1.2.1

Impact Automata on Quivers (AIQ) — An algebraic framework for modeling dynamics on complex systems

AIQ — Impact Automata on Quivers

An algebraic framework for modeling dynamics on complex systems using quiver representations, path algebras, and Brauer configuration algebras.

Based on: Zainea Maya & Moreno Cañadas, "Autómatas de Impacto sobre Quivers: Un Marco Algebraico para Modelar Dinámicas de Sistemas Complejos".

Features

Module	Description
quiver	Quiver Q=(Q₀,Q₁,s,t) with adjacency matrices, BFS distances, O(1) degree queries
path_algebra	Path algebras kQ, ideals I, quotient algebras kQ/I
impact	Impact degree, vector, Fundamental Neighborhood System (SFV), impact rates
automaton	Impact automata (AIQ) with SI/SIS/SIR evolution rules, Monte Carlo
brauer	Brauer configuration algebras: δ_B, entropy H(B), dimension formulas
morphisms	Morphisms of quivers and AIQs (base, dynamic, algebraic)
datasets	Paper examples + Cora + cit-HepPh citation networks
validation	Temporal validation with real citation data
visualization	Quiver drawing, heatmaps, evolution grids, validation dashboard

Installation

pip install aiq-quivers

For development:

git clone https://github.com/izaineam/aiq-quivers.git
cd aiq-quivers
pip install -e ".[dev]"

Quick Start

Impact Automaton (AIQ)

from aiq import Quiver, AIQ

# Define a citation quiver
Q = Quiver(
    vertices=["P1", "P2", "P3", "P4", "P5"],
    arrows=[
        ("a1", "P1", "P2"), ("a2", "P1", "P3"),
        ("a3", "P2", "P4"), ("a4", "P3", "P4"),
        ("a5", "P3", "P5"), ("a6", "P4", "P5"),
    ],
)

# Create AIQ with SIR rule
aiq = AIQ(Q, states=["S", "R", "O"], evolution_rule="SIR",
          beta=1.0, recovery_prob=0.3)

# Set initial config: P1 is Relevant, rest Susceptible
aiq.set_initial_config({"P1": "R", "P2": "S", "P3": "S", "P4": "S", "P5": "S"})

# Run 10 steps
aiq.run(10, seed=42)
print(aiq.state_counts())

Brauer Configuration Analysis

from aiq.brauer import example_partitions_of_10

bc = example_partitions_of_10()
print(bc.summary())
# BrauerConfiguration(|Γ₀|=4, |Γ₁|=3, δ_B=13, H(B)=1.8339)
#   dim_k Λ_M = 45
#   dim_k Z(Λ_M) = 14
#   δ_B = 13
#   H(B) = 1.833927

# From citation network JSON
from aiq.brauer import brauer_from_citation_json
bc = brauer_from_citation_json("data/canadas_citation_network.json")
analysis = bc.brauer_analysis()

Path Algebra

from aiq import Quiver, PathAlgebra

Q = Quiver([1, 2, 3, 4],
           [("α₁", 1, 2), ("α₂", 1, 2), ("γ", 1, 3), ("β", 2, 4), ("δ", 3, 4)])

pa = PathAlgebra(Q)
assert pa.verify_matrix_equivalence()  # dim(e_i·kQ_k·e_j) == (A^k)_{ij}
print(f"dim(kQ) = {pa.total_dimension()}")  # 12

Key Concepts

• Impact degree g(cᵢ, cⱼ): shortest directed path length (Def. 2.2)
• Impact vector vec_g(cᵢ, cⱼ): walk counts by length (Def. 2.5)
• SFV {A_k(c)}: fundamental neighborhood system (Def. 2.3)
• Impact rate i^t(c): weighted influence from neighbors (Def. 2.9)
• Brauer factor δ_B = Σ μ(m)·val(m): algebraic impact invariant
• Brauer entropy H(B): Shannon entropy of the configuration

Running Tests

pytest tests/ -v

Citation

@article{zainea2025aiq,
  title={Aut{\'o}matas de Impacto sobre Quivers:
         Un Marco Algebraico para Modelar Din{\'a}micas de Sistemas Complejos},
  author={Zainea Maya, Isaac and Moreno Ca{\~n}adas, Agust{\'i}n},
  year={2025},
  institution={Universidad Nacional de Colombia}
}

License

MIT License. See LICENSE.