Mostly wrapper functions that can be used in conjunction with indirect_relations to fine tune indirect relations.

## Usage

```
dist_2pow(x)
dist_inv(x)
dist_dpow(x, alpha = 1)
dist_powd(x, alpha = 0.5)
walks_limit_prop(x)
walks_exp(x, alpha = 1)
walks_exp_even(x, alpha = 1)
walks_exp_odd(x, alpha = 1)
walks_attenuated(x, alpha = 1/max(x) * 0.99)
walks_uptok(x, alpha = 1, k = 3)
```

## Arguments

- x
Matrix of relations.

- alpha
Potential weighting factor.

- k
For walk counts up to a certain length.

## Details

The predefined functions follow the naming scheme `relation_transformation`

.
Predefined functions `walks_*`

are thus best used with type="walks" in
indirect_relations. Theoretically, however, any transformation can be used with any relation.
The results might, however, not be interpretable.

The following functions are implemented so far:

`dist_2pow`

returns \(2^{-x}\)

`dist_inv`

returns \(1/x\)

`dist_dpow`

returns \(x^{-\alpha}\) where \(\alpha\) should be chosen greater than 0.

`dist_powd`

returns \(\alpha^x\) where \(\alpha\) should be chosen between 0 and 1.

`walks_limit_prop`

returns the limit proportion of walks between pairs of nodes. Calculating
rowSums of this relation will result in the principle eigenvector of the network.

`walks_exp`

returns \(\sum_{k=0}^\infty \frac{A^k}{k!}\)

`walks_exp_even`

returns \(\sum_{k=0}^\infty \frac{A^{2k}}{(2k)!}\)

`walks_exp_odd`

returns \(\sum_{k=0}^\infty \frac{A^{2k+1}}{(2k+1)!}\)

`walks_attenuated`

returns \(\sum_{k=0}^\infty \alpha^k A^k\)

`walks_uptok`

returns \(\sum_{j=0}^k \alpha^j A^j\)

Walk based transformation are defined on the eigen decomposition of the adjacency matrix using the fact that $$f(A)=Xf(\Lambda)X^T.$$ Care has to be taken when using user defined functions.