GraphAssortativity

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`GraphAssortativity`

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GraphAssortativity[g]

gives the assortativity coefficient of a graph g using vertex degrees.

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GraphAssortativity[g,"prop"]

gives the assortativity coefficient of the graph g using vertex property "prop".

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GraphAssortativity[g,{{v_{i 1},v_{i 2},…},…}]

gives the assortativity coefficient of the graph g with respect to the vertex partition {{v_{i 1},v_{i 2},…},…}.

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GraphAssortativity[g,{v₁,v₂,…}{x₁,x₂,…}]

gives the assortativity coefficient of the graph g using data {x₁,x₂,…} for vertices {v₁,v₂,…}.

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GraphAssortativity[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options

For a graph with edges and adjacency matrix entries , the assortativity coefficient is given by , where is the out-degree for the vertex v_i and is 1 if there is an edge from v_i to v_j and 0 otherwise.
For quantitative data where x₁,x₂,… are used, is taken to be x_ix_j.
For categorical data where x₁,x₂,… are used, is taken to be 1 if x_i and x_j are equal and 0 otherwise.
In GraphAssortativity[g], x_i is taken to be the vertex out-degree for the vertex v_i.
In GraphAssortativity[g,"prop"], x_i is taken to be AnnotationValue[{g,v_i},"prop"] for the vertex v_i.
In GraphAssortativity[g,{{v_{i 1},v_{i 2},…},…}], vertices in a subset {v_{i 1},v_{i 2},…} have the same categorical data x_{i 1}=x_{i 2}=….
GraphAssortativity[g,Automatic->{x₁,x₂,…}] takes the vertex list to be VertexList[g].
The option "DataType"->type can be used to specify the type for the data x₁,x₂,…. Possible settings are "Quantitative" and "Categorical".
The option "Normalized"->False can be used to compute the assortativity modularity.
For a graph with edges and adjacency matrix entries , the assortativity modularity is given by , where is the out-degree for the vertex v_i.
GraphAssortativity works with undirected graphs, directed graphs, weighted graphs, multigraphs, and mixed graphs.

Examples

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Basic Examples (2)Summary of the most common use cases

Compute the assortativity coefficient of the Zachary karate club network:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-lpethk

Out[1]=1

Distribution of the assortativity coefficient of uniform random graphs:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-ztkee

Out[1]=1

Scope (12)Survey of the scope of standard use cases

GraphAssortativity works with undirected graphs:

In[2]:=2

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-tzs5an

Out[2]=2

Directed graphs:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-d7rrmc

Out[1]=1

Weighted graphs:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-gcbs2g

Out[1]=1

Multigraphs:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-ffb0ew

Out[1]=1

Mixed graphs:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-ffsn8

Out[1]=1

Use rules to specify the graph:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-bndh30

Out[1]=1

Compute the assortativity coefficient using vertex property data:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-msp35

Out[1]=1

A vertex partition:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-gfd9b1

Out[1]=1

A specified dataset:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-dn3zoj

Out[1]=1

A partition or assignment of a subset of VertexList:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-8bjvs

Out[1]=1

GraphAssortativity works with symbolic expressions:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-nwyp8y

Out[1]=1

GraphAssortativity works with large graphs:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-gw1usj

In[2]:=2

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-cuju06

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-gfsfsb

Out[3]=3

Applications (3)Sample problems that can be solved with this function

Compare the assortativity coefficient of vertex partitions by vertex color:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-lt2pno

In[2]:=2

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-j9ko80

Out[2]=2

Disassortativity by number of friends in a friendship network:

In[1]:=1

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-ntpxnb

In[2]:=2

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-echrlr

Out[2]=2

A partition of the network shows assortative mixing:

In[3]:=3

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-lvmgei

In[4]:=4

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-i1l4m5

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0bzranqs7bdmc3ysq-dwtlyz

Out[5]=5

A friendship network at a high school, with vertices color coded by race. Analyze the preference for students to associate with others who are similar:

In[1]:=1