# assortativity

##### Assortativity coefficient

The assortativity coefficient is positive is similar vertices (based on some external property) tend to connect to each, and negative otherwise.

- Keywords
- graphs

##### Usage

```
assortativity (graph, types1, types2 = NULL, directed = TRUE)
assortativity.nominal (graph, types, directed = TRUE)
assortativity.degree (graph, directed = TRUE)
```

##### Arguments

- graph
- The input graph, it can be directed or undirected.
- types
- Vector giving the vertex types. They as assumed to be
integer numbers, starting with one. Non-integer values are
converted to integers with
`as.integer`

. - types1
- The vertex values, these can be arbitrary numeric values.
- types2
- A second value vector to be using for the incoming
edges when calculating assortativity for a directed graph.
Supply
`NULL`

here if you want to use the same values for outgoing and incoming edges. This argument is ignored (with - directed
- Logical scalar, whether to consider edge directions
for directed graphs. This argument is ignored for undirected
graphs. Supply
`TRUE`

here to do the natural thing, i.e. use directed version of the measure for directed graphs and th

##### Details

The assortativity coefficient measures the level of homophyly of the
graph, based on some vertex labeling or values assigned to
vertices. If the coefficient is high, that means that connected
vertices tend to have the same labels or similar assigned values.
M.E.J. Newman defined two kinds of assortativity coefficients, the
first one is for categorical labels of
vertices. `assortativity.nominal`

calculates this measure. It is
defines as

$$r=\frac{\sum_i e_{ii}-\sum_i a_i b_i}{1-\sum_i a_i b_i}$$

where $e_{ij}$ is the fraction of edges connecting vertices
of type $i$ and $j$,
$a_i=\sum_j e_{ij}$ and
$b_j=\sum_i e_{ij}$.
The second assortativity variant is based on values assigned to the
vertices. `assortativity`

calculates this measure. It is defined
as

$$r=\frac1{\sigma_q^2}\sum_{jk} jk(e_{jk}-q_j q_k)$$

for undirected graphs ($q_i=\sum_j e_{ij}$) and as

$$r=\frac1{\sigma_o\sigma_i}\sum_{jk}jk(e_{jk}-q_j^o q_k^i)$$

for directed ones. Here $q_i^o=\sum_j e_{ij}$, $q_i^i=\sum_j e_{ji}$, moreover, $\sigma_q$, $sigma_o$ and $sigma_i$ are the standard deviations of $q$, $q^o$ and $q^i$, respectively.

The reason of the difference is that in directed networks the relationship is not symmetric, so it is possible to assign different values to the outgoing and the incoming end of the edges.

`assortativity.degree`

uses vertex degree (minus one) as vertex
values and calls `assortativity`

.

##### Value

- A single real number.

##### concept

Assortativity coefficient

##### References

M. E. J. Newman: Mixing patterns in networks, *Phys. Rev. E* 67,
026126 (2003)

M. E. J. Newman: Assortative mixing in networks,
*Phys. Rev. Lett.* 89, 208701 (2002)

##### Examples

```
# random network, close to zero
assortativity.degree(erdos.renyi.game(10000,3/10000))
# BA model, tends to be dissortative
assortativity.degree(ba.game(10000, m=4))
```

*Documentation reproduced from package igraph, version 0.6.5-2, License: GPL (>= 2)*