Mathematical Foundations

Multilayer Network

A multilayer network is a graph where nodes and edges are partitioned into layers, with intra-layer (within a layer) and inter-layer (between layers) connections.

Example in MLKN.lab:

• Layers 1-4 of the Polyhierarchy (Core Domains → Thematic Domains).
• Intra-layer edges: Connections between subdisciplines in the same discipline (e.g., Quantum Mechanics → Quantum Field Theory).
• Inter-layer edges: Hierarchical relationships (e.g., Physics → Quantum Mechanics).

Why it matters: Captures the hierarchical structure of academic knowledge.

Hypergraph

A hypergraph is a generalization of a graph where an edge (hyperedge) can connect any number of nodes (not just pairs).

Example in MLKN.lab:

• Layers 4-5 of the Polyhierarchy (Thematic Domains → Concepts).
• Hyperedge: A concept like \"Machine Learning\" connects Computer Science, Mathematics, and Neuroscience simultaneously.

Why it matters: Captures cross-disciplinary relationships that cannot be represented in traditional graphs.

Graph Theory

Knowledge is modeled as a directed, weighted, multi-layered graph \( G = (V, E, A) \), where:

• Nodes (V): Represent disciplines, subdisciplines, and concepts (e.g., "Physics," "Quantum Mechanics").
• Edges (E): Represent hierarchical (e.g., Physics → Quantum Mechanics), interdisciplinary (e.g., Quantum Mechanics → Chemistry), and thematic (e.g., Machine Learning → Neural Networks) relationships.
• Adjacency Matrix (A): A matrix where \( A_{ij} = \omega(e_{ij}) \) if an edge \( e_{ij} \) exists from node \( i \) to \( j \), else \( 0 \).

Key Matrices for MLKN.model:

• Intra-Layer Adjacency \( A^{(l)} \): Connections within layer \( l \) (e.g., subdisciplines in Physics).
• Inter-Layer Adjacency \( A^{(l, l+1)} \): Hierarchical edges between layers \( l \) and \( l+1 \) (e.g., Discipline → Subdiscipline).
• Laplacian Matrix \( L \): Defined as \( L = D - A \), where \( D \) is the degree matrix. Used for spectral clustering of disciplines.

Example in MLKN.lab:

• The Laplacian of Layer 2 (Disciplines) can reveal clusters (e.g., Natural Sciences vs. Social Sciences).
• The eigenvector centrality of \( A^{(2)} \) identifies influential disciplines (e.g., Mathematics).

Literature: Boccaletti et al., 2014 (Multilayer Networks), Newman, 2006 (Modularity and Community Structure).

Key Tools: NetworkX (Python), Vis.js (JavaScript), D3.js.

Topological Data Analysis (TDA)

Uses persistent homology to study the shape of knowledge by analyzing the filtration of the hypergraph incidence matrix \( \mathbf{H} \):

• Filtration of \( \mathbf{H} \): Construct a sequence of simplicial complexes from \( \mathbf{H} \) by thresholding hyperedge weights \( \omega(e) \):

  \[ \mathbf{H}_\theta = \{ H_{v,e} \mid \omega(e) \geq \theta \}, \quad \theta \in [0, 1] \]

• Holes (Betti-1): Represent gaps or missing connections in knowledge (e.g., under-explored interdisciplinary areas between Physics and Biology).
• Connected Components (Betti-0): Represent clusters of related disciplines (e.g., all subfields of Computer Science).
• Betti Numbers: Quantify the dimensionality of knowledge structures (e.g., \( \beta_0 \) = number of connected components, \( \beta_1 \) = number of holes).

Example in MLKN.lab:

• A hole in the filtration at \( \theta = 0.5 \) might reveal a missing connection between Quantum Computing (Layer 3) and Topological Materials (Layer 3).
• A persistent connected component across all \( \theta \) indicates a stable discipline cluster (e.g., Mathematics).

Literature: Carlsson, 2009 (Topology and Data), Petri et al., 2013 (TDA for Networks), Bianconi, 2018 (TDA for Science of Science).

Key Tools: Gudhi, Ripser, Dionysus.

Information Theory

Applies entropy and complexity metrics to quantify the organization and efficiency of knowledge in MLKN.model:

• Shannon Entropy \( H(X) \): Measures the uncertainty or diversity of a discipline’s subfields:

  \[ H(X) = -\sum_{x \in X} p(x) \log p(x) \]

  where \( X \) is a discipline (e.g., Physics) and \( p(x) \) is the probability of subfield \( x \) (e.g., Quantum Mechanics).

• Mutual Information \( I(X;Y) \): Quantifies the shared information between two disciplines \( X \) and \( Y \):

  \[ I(X;Y) = \sum_{x \in X} \sum_{y \in Y} p(x,y) \log \frac{p(x,y)}{p(x)p(y)} \]

• Redundancy: Measures overlap between fields (e.g., shared concepts between Computer Science and Mathematics).
• Efficiency: Assesses how optimally knowledge is structured (e.g., low entropy = highly organized).

Example in MLKN.lab:

• High \( H(X) \) for Interdisciplinary Fields (e.g., Cognitive Science) indicates diverse subfields.
• High \( I(X;Y) \) between Computer Science and Mathematics reflects their strong interplay.

Literature: DeDeo et al., 2019 (Information Theory in Science), Rosvall & Lizana, 2019 (Compression and Information in Networks).

Key Tools: Scikit-learn, NumPy, SciPy.

Dynamical Systems

Models knowledge as a dynamical system where the state evolves via the hypergraph transition matrix \( \mathbf{T}_{\mathcal{H}} \):

• State Space: All possible configurations of the knowledge network, represented as probability distributions \( \mathbf{P}_t \in \mathbb{R}^{|V|} \).
• Trajectories: The evolution of \( \mathbf{P}_t \) over time, governed by:

  \[ \mathbf{P}_{t+1} = \mathbf{P}_t \mathbf{T}_{\mathcal{H}} \]

• Attractors: Stable configurations (e.g., major disciplines like Physics or Mathematics) where \( \mathbf{P}_t \) converges to a stationary distribution \( \mathbf{P}_\infty \).
• Wasserstein Stability: The system is stable if \( W_1(\mathbf{P}_t, \mathbf{P}_{t+1}) \to 0 \) as \( t \to \infty \).

Example in MLKN.lab:

• Simulate the emergence of Bioinformatics by tracking \( \mathbf{P}_t \) from 1980 to 2020.
• Identify attractors (e.g., Mathematics) where \( \mathbf{P}_\infty \) has high probability mass.

Literature: Barrat et al., 2008 (Dynamical Processes on Networks), Shi et al., 2019 (Knowledge Evolution), Norris, 1998 (Markov Chains).

Key Tools: SciPy, MATLAB, custom ODE solvers.

Optimal Transport Theory

Measures the "distance" between disciplines by modeling knowledge as a probability distribution and using the Wasserstein-1 distance to quantify the epistemic effort required to transform one distribution into another.

• Knowledge State Distributions: Let \( \mathbf{P}_t \in \mathbb{R}^{|V|} \) be a probability distribution over nodes \( V \) at time \( t \), generated via the hypergraph random walk transition matrix \( \mathbf{T}_{\mathcal{H}} \):

  \[ \mathbf{P}_t = \mathbf{P}_0 \mathbf{T}_{\mathcal{H}}^t \]

• Ground Metric: The cost \( c(v_i, v_j) \) of moving knowledge from node \( v_i \) to \( v_j \) is defined as:

  \[ c(v_i, v_j) = -\log P(v_j \mid v_i) \]

  Intuition: Nodes with high transition probability are "closer" in epistemic space.

• Wasserstein-1 Distance: The optimal transport distance between two knowledge states \( \mathbf{\mu} \) and \( \mathbf{\nu} \) is:

  \[ W_1(\mathbf{\mu}, \mathbf{\nu}) = \inf_{\mathbf{\gamma} \in \Pi(\mathbf{\mu}, \mathbf{\nu})} \sum_{i,j} \gamma_{i,j} c(v_i, v_j) \]

  where \( \Pi(\mathbf{\mu}, \mathbf{\nu}) \) is the set of transport plans \( \mathbf{\gamma} \) such that:

  \[ \sum_j \gamma_{i,j} = \mathbf{\mu}_i, \quad \sum_i \gamma_{i,j} = \mathbf{\nu}_j \]

Example in MLKN.lab:

• Compare the knowledge distributions of Computer Science (\( \mathbf{\mu} \)) and Neuroscience (\( \mathbf{\nu} \)) to measure their cognitive distance.
• Track the evolution of Physics from 1990 (\( \mathbf{\mu} \)) to 2020 (\( \mathbf{\nu} \)) to quantify disciplinary drift.

Computational Note: For large-scale networks (e.g., OpenAlex), use the Sinkhorn algorithm with entropy regularization:

\[ W_1^\epsilon(\mathbf{\mu}, \mathbf{\nu}) = \min_{\mathbf{\gamma}} \sum_{i,j} \gamma_{i,j} c(v_i, v_j) + \epsilon \sum_{i,j} \gamma_{i,j} \log \gamma_{i,j} \]

Literature: Villani, 2003 (Optimal Transportation), Cuturi, 2013 (Sinkhorn Distances), Peyré & Cuturi, 2019 (Computational OT), Bassetti et al., 2020 (OT for Knowledge Graphs).

Key Tools: POT (Python Optimal Transport), Sinkhorn algorithm.

Causal Inference

Tests meta-science hypotheses by modeling knowledge as a causal graph and using do-calculus to infer relationships:

• Causal Graph: A directed acyclic graph (DAG) where nodes are disciplines/concepts and edges represent causal influences (e.g., "Mathematics → Physics → Engineering").
• Do-Calculus: A framework for identifying causal effects from observational data. For example, to assess:

  \[ P(Y \mid do(X=x)) \quad \text{(Effect of intervention \( X=x \) on \( Y \))} \]

  where \( do(X=x) \) represents an intervention (e.g., "What if we increase funding for interdisciplinarity?").

• Counterfactuals: Answer "what-if" questions by simulating alternative histories of science:

  \[ P(Y_{X=x} \mid X=x', Y=y') \]

  Example: "What would the citation network look like if Computer Science had not emerged?"

Example in MLKN.lab:

• Use do-calculus to test: "Does interdisciplinarity increase citation impact?"
• Use counterfactuals to explore: "How would Physics have evolved without Quantum Mechanics?"

Literature: Pearl, 2009 (Causality), Hernán & Robins, 2020 (Causal Inference).

Key Tools: DoWhy, CausalML, PyMC, Dowhy.