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 $A:A_{ij}=A_{ji}\sim \text{Bernoulli}(P_{ij})$ where

p & \dots & p & q & \dots & q \\ & & \vdots & \vdots & & \vdots \\ p & \dots & p & q & \dots & q \\ \hline q & \dots & q & p & \dots & p \\ \vdots & & \vdots & \vdots & & \vdots \\ q & \dots & q & p & \dots & p\end{array}\right]$$ - $y \in \{ -1,1 \}^n$ (or $\{ 0,1 \}$) and $B \in \mathbb{R}^{2 \times 2}$ And [[Concept Wiki/graph signals\|node features]] $X$ are drawn - $x_{i} \sim \sqrt{ \frac{\mu}{n} } y_{i} u + \frac{z_{i}}{\sqrt{ d }}$, $u \sim \text{Normal}\left( 0, \frac{I_{d}}{d} \right)$, $z_{i} \sim \text{Normal}(0, I_{d})$ So $$X_{i}|Y_{i}, u \sim \text{Normal}\left( \pm \sqrt{ \frac{\mu}{n} }u, \frac{I_{d}}{d} \right)$$

Mentions

TABLE
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