contextual stochastic block model

[[concept]]

contextual stochastic block model (C-SBM)

The (binary) contextual stochastic block model (C-SBM) is like an SBM, but includes node features that are drawn from a Gaussian distribution. Here,

The adjacency matrix is given as $A : A_{i j} = A_{j i} \sim Bernoulli (P_{i j})$ where

P \in R^{(n / 2 + n / 2) \times (n / 2 + n / 2)} = [\begin{array}{cccccc} p & \dots & p & q & \dots & q \\ ⋮ & ⋮ & ⋮ \\ p & \dots & p & q & \dots & q \\ q & \dots & q & p & \dots & p \\ ⋮ & ⋮ & ⋮ & ⋮ \\ q & \dots & q & p & \dots & p \end{array}]

$y \in {- 1, 1}^{n}$ (or ${0, 1}$ ) and $B \in R^{2 \times 2}$
And node features $X$ are drawn
$x_{i} \sim \sqrt{\frac{μ}{n}} y_{i} u + \frac{z_{i}}{\sqrt{d}}$ , $u \sim Normal (0, \frac{I_{d}}{d})$ , $z_{i} \sim Normal (0, I_{d})$

X_{i} | Y_{i}, u \sim Normal (\pm \sqrt{\frac{μ}{n}} u, \frac{I_{d}}{d})

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feature-aware spectral embeddings
2025-02-12 graphs lecture 7