Modeling Social Connectivity with Graph Distributions

Construct a random network and analyze its connectivity to understand how relationships emerge in small social groups

Researchers are studying how individuals in a small community interact and how different social structures affect the overall connectivity of the network. Three network models are explored: a random friendship model, a clustered small-world model, and a degree-based interaction model.

Define a random friendship model for 20 individuals (a Bernoulli distribution means each edge is a simple yes/no event):

In[1]:=

=BernoulliGraphDistribution[20,0.2];

Generate a random graph from the distribution:

In[2]:=

SeedRandom[1234];graph=RandomGraph

‸



Out[2]=

Highlight the three most popular people by selecting the vertices with the highest degrees:

In[3]:=

HighlightGraph[graph,TakeLargestBy[VertexList[graph],VertexDegree[graph,#]&,3]]

Out[3]=

Compute the probability that the network is connected:

In[4]:=

=GraphPropertyDistribution[Boole[ConnectedGraphQ[g]],g];

In[5]:=

NProbability[x1,x]

Out[5]=

0.745797

Next we will explore a larger social connectivity graph distribution. Assume that people mostly connect within their local circles but also maintain a few random acquaintances.

Define a small-world model for 100 individuals with the given parameters:

In[6]:=

=WattsStrogatzGraphDistribution[100,0.1,20/2];

In[7]:=

g=RandomGraph

‸



Out[7]=

Compute the expected number of relations of the least-connected individual using VertexDegree:

In[8]:=

Mean[GraphPropertyDistribution[Min[VertexDegree[g]],g]]//N

Out[8]=

16.462

Calculate the average path length between any two people:

In[9]:=

N[MeanGraphDistance[g]]

Out[9]=

2.04545

Highlight the immediate social circle of a randomly chosen individual:

In[10]:=

v=RandomInteger[{1,VertexCount[g]}];HighlightGraph[g,NeighborhoodGraph[g,v,1]]

Out[10]=

Finally, model a realistic case where each participant reports the number of contacts they typically have.

Define a DegreeGraphDistribution model for seven individuals:

In[11]:=

𝒢=DegreeGraphDistribution[{2,4,4,4,3,3,2}];

Generate a random graph from the distribution:

In[12]:=

SeedRandom[1234];graphs=RandomGraph𝒢,3

Out[12]=





Define a variable that is one if individuals 1 and 2 are connected and is zero otherwise:

In[13]:=

interaction=GraphPropertyDistributionBoole[EdgeQ[q,12]],q𝒢;

Compute the probability that subjects 1 and 2 have directly interacted:

In[14]:=

NProbability[x1,xinteraction]

Out[14]=

0.458383

Choose one realization of the degree-based network:

In[15]:=

gr=Graph[graphs〚1〛,ImageSize350,VertexLabels"Name"]

Out[15]=

Check how uneven the interaction pattern is by computing the coefficient of variation:

In[16]:=

N[StandardDeviation[VertexDegree[gr]]/Mean[VertexDegree[gr]]]

Out[16]=

0.286279

External Links

https://reference.wolfram.com/language/guide/SocialNetworks.html

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Modeling Social Connectivity with Graph Distributions

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Modeling Social Connectivity with Graph Distributions

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