# Progress Update Till 14 Oct

## Checkpoints:

* Mainly reading papers and trying to understand them to finish literature review.
* Finished random walks implementation.

Stas prediction was correct :D

## Line Graph Model:

https://arxiv.org/pdf/1705.08415.pdf

This paper introduces 2 elegant ideas, first is for capturing higher order dependecies and 2nd is for capturing directed relationships.

Higher order interactions approach:

Adjacency matrix of a graph $A^1$ basically is a binary matrix. What happens when we multiply it?

$$A^2_{i,j} = \sum_{k \in V} A^1_{i,k} A^1_{k,j}$$

$A^2_{i,j}$ will correspond to the number of paths between $i,j$ with length exactly 2.

Basically we can consider it as binary matrix and use or operations.

Now let's calculate these matrices for each hop $\{2^0,2^1,2^2,...2^{log\,V}\}$

Finding a path now of whatever length is very trivial.


$F=(A^1,A^2,A^4...)$

$[O_i \, . x]$ is basically message passing.

### Full Chinese Mode (Same paper):

Maintain 2 graphs, the original graph and the edges graph.

Each edge would be converted to a node. $(i->j) \& (i' -> j')$ are basically connected if and only if $i'=j \,\&\, i \neq j$

First of all, edges are duplicated and considered as unidirectional to capture information.


$F=(A^1,A^2,A^4...)$
$F'=(B^1,B^2,B^4...)$
where $B$ is basically the adjacency matrix for the edges graph.
$F''=(P,\hat{P})$
$P_{u,(u,v)} = 1$
$P_{u,(v,u)} = 1$
$\hat{P}_{u,(u,v)} = 1$
$\hat{P}_{u,(v,u)} = -1$

$z$ is activations for normal graph, $w$ is for edges.

First equation:
First term is the same as previous one.
Second term adds for each node the activation of its incident edges (oriented and unoriented case is considered).

While I consider it chinese mode, however the ideas used are each very effective and could be used on its own.

## IGMC

https://arxiv.org/pdf/1905.08108.pdf

* Inductive
* No node features whatsoever

Main approach:

* Consider each edge (u,v)
* Sample a hop subgraph using BFS
* Label users with distance $d$ as $d$ and items with distance $2*d$ as $2*d+1$
* Replace nodes with one hot encodings
* Aggregate using R-GCN


* Decode the result
* Apply regularization on probabilities.