Assortativity coefficient

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

Usage

assortativity(
  graph,
  values,
  ...,
  values.in = NULL,
  directed = TRUE,
  normalized = TRUE,
  types1 = NULL,
  types2 = NULL
)

assortativity_nominal(graph, types, directed = TRUE, normalized = TRUE)

assortativity_degree(graph, directed = TRUE)

Arguments

graph: The input graph, it can be directed or undirected.
values: The vertex values, these can be arbitrary numeric values.
...: These dots are for future extensions and must be empty.
values.in: A second value vector to use 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 a warning) if it is not NULL and undirected assortativity coefficient is being calculated.
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 the undirected version for undirected graphs.
normalized: Boolean, whether to compute the normalized assortativity. The non-normalized nominal assortativity is identical to modularity. The non-normalized value-based assortativity is simply the covariance of the values at the two ends of edges.
types1, types2: Deprecated aliases for values and values.in, respectively.
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().

Value

A single real number.

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 defined 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 as vertex values and calls assortativity().

Undirected graphs are effectively treated as directed ones with all-reciprocal edges. Thus, self-loops are taken into account twice in undirected graphs.

References

M. E. J. Newman: Mixing patterns in networks, Phys. Rev. E 67, 026126 (2003) https://arxiv.org/abs/cond-mat/0209450

M. E. J. Newman: Assortative mixing in networks, Phys. Rev. Lett. 89, 208701 (2002) https://arxiv.org/abs/cond-mat/0205405

Author

Gabor Csardi csardi.gabor@gmail.com

igraph_assortativity(), igraph_assortativity_nominal(), igraph_assortativity_degree().

Examples


# random network, close to zero
assortativity_degree(sample_gnp(10000, 3 / 10000))
#> [1] -0.003901217

# BA model, tends to be dissortative
assortativity_degree(sample_pa(10000, m = 4))
#> [1] -0.02572967