This vignette describes how to build different centrality indices on
the basis of indirect relations as described in this vignette. Note, however, that
the primary purpose of the netrankr package is **not** to
provide a great variety of indices, but to offer alternative methods for
centrality assessment. Nevertheless, the package also provides an
Rstudio addin ‘index_builder()’, which allows to create and customize
more than 20 different indices.

## Theoretical Background

A one-mode network can be described as a *dyadic variable*
\(x\in \mathcal{W}^\mathcal{D}\), where
\(\mathcal{W}\) is the value range of
the network (in the simple case of unweighted networks \(\mathcal{W}=\{0,1\}\)) and \(\mathcal{D}=\mathcal{N}\times\mathcal{N}\)
describes the dyadic domain of actors \(\mathcal{N}\).

Observed presence or absence of ties (the value range is binary) is
usually not the relation of interest for network analytic tasks.
Instead, mostly implicitly, relations are *transformed* into a
new set of *indirect* relations on the basis of the
*observed* relations. As an example, consider (shortest path)
distances in the underlying graph. While they are fairly easy to derive
from an observed network of contacts, it is impossible for actors in a
network to answer the question “How far away are you from others you are
not connected with?”. We denote generic transformed networks from an
observed network \(x\) as \(\tau(x)\).

With this notion of indirect relations, we can express all centrality
indices in a common framework as \[
c_\tau(i)=\sum\limits_{t \in \mathcal{N}} \tau(x)_{it}
\] Degree and closeness centrality, for instance, can be obtained
by setting \(\tau=id\) and \(\tau=dist\), respectively. Others need
several additional specifications which can be found in Brandes (2016) or
Schoch & Brandes
(2016).

With this framework, all centrality indices can be characterized as
degree-like measures in a suitably transformed network \(\tau(x)\). To build specific indices, we
follow the *analytic pipeline* for centrality assessment: \[
\text{Observed network}\;(x) \longrightarrow
\text{transformation}\;(\tau(x)) \longrightarrow
\text{aggregation}\;(e.g. \sum_j \tau(x)_{ij})
\]

## Building indices with the `netrankr`

package

The `netrankr`

does, by design, not explicitly implement
any centrality index. It does, however, provide a large set of
components to create indices. Building an index based on an indirect
relation, computed with `indirect_relations()`

, is done with
the function `aggregate_positions()`

.

The usual workflow is as follows:`g %>% indirect_relations() %>% aggregate_positions()`

which is equivalent to
`aggregate_positions(indirect_relations(g))`

.

The former, however, comes with enhanced readability and is in
accordance with the proposed analytic pipeline (see above).`aggregate_position()`

has a parameter `type`

which is used to choose an appropriate aggregation method. Commonly,
this is simply the sum operation.

```
data("dbces11")
g <- dbces11
V(g)$name <- 1:11
#Degree
g %>%
indirect_relations(type="adjacency") %>%
aggregate_positions(type="sum")
#Closeness
g %>%
indirect_relations(type="dist_sp") %>%
aggregate_positions(type="invsum")
#Betweenness Centrality
g %>%
indirect_relations(type="depend_sp") %>%
aggregate_positions(type="sum")
#Eigenvector Centrality
g %>%
indirect_relations(type="walks",FUN=walks_limit_prop) %>%
aggregate_positions(type="sum")
```

For closeness `type="invsum"`

is used since traditional
closeness is defined as \[
c_c(i)=\frac{1}{\sum_t dist(i,t)}.
\] To obtain a slight variant of closeness, i.e. \[
c_c(i)=\sum_t \frac{1}{dist(i,t)},
\] the following code can be used:

```
#harmonic closeness
g %>%
indirect_relations(type="dist_sp",FUN=dist_inv) %>%
aggregate_positions(type="sum")
```

Indices based on shortest path distances constitute the biggest group
of indices in the `netrankr`

package.

```
#residual closeness (Dangalchev,2006)
g %>%
indirect_relations(type="dist_sp",FUN=dist_2pow) %>%
aggregate_positions(type="sum")
#generalized closeness (Agneessens et al.,2017) (alpha>0)
g %>%
indirect_relations(type="dist_sp",FUN=dist_dpow,alpha=2) %>%
aggregate_positions(type="sum")
#decay centrality (Jackson, 2010) (alpha in [0,1])
g %>%
indirect_relations(type="dist_sp",FUN=dist_powd,alpha=0.7) %>%
aggregate_positions(type="sum")
#integration centrality (Valente & Foreman, 1998)
dist_integration <- function(x){
x <- 1 - (x - 1)/max(x)
}
g %>%
indirect_relations(type="dist_sp",FUN=dist_integration) %>%
aggregate_positions(type="sum")
```

The package implements several additional distance measures for
networks, for which no index exists so far. Consult the help of
`indirect_relations()`

for possibilities.

Another large group of indices is based on walk counts.

```
#subgraph centrality
g %>%
indirect_relations(type="walks",FUN=walks_exp) %>%
aggregate_positions(type="self")
#communicability centrality
g %>%
indirect_relations(type="walks",FUN=walks_exp) %>%
aggregate_positions(type="sum")
#odd subgraph centrality
g %>%
indirect_relations(type="walks",FUN=walks_exp_odd) %>%
aggregate_positions(type="self")
#even subgraph centrality
g %>%
indirect_relations(type="walks",FUN=walks_exp_even) %>%
aggregate_positions(type="self")
#katz status
g %>%
indirect_relations(type="walks",FUN=walks_attenuated) %>%
aggregate_positions(type="sum")
```

**Note**: The analytic pipeline can of course be wrapped
into a function.

```
degree_centrality <- function(g){
DC <- g %>%
indirect_relations(type="adjacency") %>%
aggregate_positions(type="sum")
return(DC)
}
```

Additionally, the Rstudio addin `index_builder()`

provides
a convenient way to produce the code for any desired index.