Appendix F: Notation and Symbols Guide

This appendix provides a comprehensive reference for all mathematical, statistical, set-theoretic, graphical, information-theoretic, and machine learning notation used throughout the textbook. Symbols are organized by domain and accompanied by plain-language definitions, the mathematical form where relevant, and chapter references. This appendix is especially useful when reading Chapters 22–24 (computational methods) and Chapter 28 (Bayesian reasoning).

F.1 Probability Notation

Probability statements are fundamental to Chapters 3, 13, 16, 28, and Appendix A.

Bayes' Theorem in Full Notation

P(A|B) = P(B|A) × P(A) / P(B)

Expanded form using the law of total probability:

P(A|B) = P(B|A) × P(A) / [P(B|A)×P(A) + P(B|Aᶜ)×P(Aᶜ)]

Additional Probability Notation

F.2 Statistical Notation

Used throughout Chapters 27–29 and Appendix A.

Descriptive Statistics

Inference and Hypothesis Testing

Correlation and Regression

Effect Sizes

F.3 Set Notation

Used in formal definitions throughout the textbook, particularly in Chapters 22–24.

F.4 Graph and Network Notation

Used in Chapter 23 (Network Analysis) and Chapter 16 (Information Diffusion).

F.5 Information Theory Notation

Used in Chapters 23 and Appendix A.

F.6 Machine Learning Notation

Used in Chapters 22–24.

Data Representation

Model Parameters and Training

Evaluation Metrics

NLP Notation

F.7 Notation from Chapters 22–24 and 28

This section provides quick look-up for notation introduced in specific chapters.

Chapter 22: Computational Misinformation Detection

TF-IDF(t, d) = tf(t, d) × log(N / (1 + df(t)))

Logistic regression: P(y=1|x) = σ(wᵀx + b) = 1 / (1 + exp(−(wᵀx + b)))

Cross-entropy loss: L = −[y log(ŷ) + (1−y) log(1−ŷ)]

BERT tokenization: input = [CLS] token₁ token₂ ... tokenₙ [SEP]

Classification head: ŷ = softmax(W · h_[CLS] + b)

Chapter 23: Network Analysis

Degree of node v: k(v) = |{u : (u,v) ∈ E}|

Betweenness centrality: C_B(v) = Σ_{s≠v≠t} [σ_st(v) / σ_st]
  where σ_st = number of shortest paths from s to t
        σ_st(v) = those paths that pass through v

PageRank: PR(v) = (1−d)/N + d × Σ_{u→v} [PR(u) / k_out(u)]
  where d ≈ 0.85 is the damping factor

Modularity: Q = (1/2m) Σ_{ij} [A_ij − k_i k_j / 2m] × δ(c_i, c_j)
  where δ(c_i, c_j) = 1 if i and j are in the same community

SIR model differential equations:
  dS/dt = −β S I / N
  dI/dt = β S I / N − γ I
  dR/dt = γ I
  R₀ = β / γ

Chapter 24: Coordinated Behavior Detection

Cosine similarity: sim(A, B) = (A · B) / (|A| × |B|)

Jaccard similarity: J(A, B) = |A ∩ B| / |A ∪ B|

Co-retweet matrix C: C_ij = number of tweets retweeted by both account i and account j

Synchrony score for account pair (i,j):
  S_ij = number of posts within window Δt / total posts by i and j

Chapter 28: Bayesian Reasoning Applied

Posterior ∝ Likelihood × Prior
P(θ|data) ∝ P(data|θ) × P(θ)

Bayes factor: BF = P(data|H₁) / P(data|H₀)
  BF > 3: moderate evidence for H₁
  BF > 10: strong evidence for H₁
  BF > 100: decisive evidence for H₁

Credibility updating:
  Odds(H|E) = Odds(H) × [P(E|H) / P(E|¬H)]
  Odds(H) = P(H) / P(¬H)

F.8 Greek Alphabet Reference

F.9 Notation Conventions

Scalars: Lowercase italic letters — a, b, x, y.

Vectors: Lowercase bold letters — x, w, θ — or lowercase letters with an arrow: x⃗.

Matrices: Uppercase bold letters — X, A, W.

Sets: Uppercase calligraphic or italic letters — V, E, S, Ω.

Random variables: Uppercase italic letters — X, Y, Z.

Realizations of random variables: Corresponding lowercase letters — x, y, z.

Estimated quantities: Hat notation — θ̂ is an estimate of θ; ŷ is a prediction.

Transpose: Superscript T — xᵀ is the transpose of x.

Norms: ‖x‖₂ is the L2 (Euclidean) norm; ‖x‖₁ is the L1 norm.

Infinity: ∞ denotes an unbounded limit.

Proportionality: ∝ means "is proportional to" (differs by a positive constant).

Approximately: ≈ means approximately equal; ~ (in probability) means "distributed as."