template graph

[[concept]]

template graph

A template graph $G_n$ is a way to sample a graph from a graphon. The nodes are defined by partitioning $[0,1]$ in a grid (regular partition) in the same way we partitioned for induced graphons:

[ \frac{k-1}{n}, \frac{k}{n} [\;\;\;1 \leq k \leq n-1 \\ \left[ \frac{n-1}{n}, 1 \right] \;\;\; k=n \end{cases}$$ So that $I_{1} \cup I_{2} \cup\dots \cup I_{n} = [0,1]$ and the node labels are $u_{j}=\frac{j-1}{n}$ for each $j$. The [[Concept Wiki/adjacency matrix]] is then given as $$[A_{n}]_{ij} = W(u_{i},u_{j})$$ Where $W$ is the [[Concept Wiki/graphon]] that we sample from. This is a complete, [[Concept Wiki/unweighted graph\|weighted graph]] with edge weights coming from the [[Concept Wiki/graphon]] evaluated at each node pair $(u_{i},u_{j}) \in [0,1]^2$. This is the simplest way to sample a graph. We can think of it as the graph sampling counterpart to inducing a graphon.

Review

dsg

Q: How do we define the nodes in a template graph? -?- A: We partition $[0,1]$ in a grid (regular partition) in the same way we partitioned for induced graphons $I_k=\{\begin{array}{l}[\frac{k-1}{n},\frac{k}{n}[1\le k\le n-1\\ \left[\frac{n-1}{n},1\right]k=n\end{array}$

We define a template graph’s adjacency matrix as {-as||$[A_n{]}_{ij}=W(u_i,u_j)$-} which results in a {ha||complete, weighted graph.||characteristics}

We can think of a template graph as the {a||graph sampling} counterpart to {a||inducing a graphon}.

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