Network Data

Networks

Network data consists of two types of objects:

  • nodes, the primary object of study
  • edges, connections between nodes

These two things can in theory be any set of objects with links between them. Some common examples include:

  • social networks: nodes are people and edges describe relationships
  • family trees: nodes are people and edges describe marriage or child-parent relationships
  • citation networks: nodes are papers/articles/books and edges are citations from one work to another
  • similarity network: nodes are almost any entity and edges describe objects that are particularly similar

Today we will see ways of working with network data. I’ll start here with the general concepts; you’ll see in the tutorials how to work with networks in Python.

Note that you may have heard of what I call networks under the term of a graph. This is the common name in the mathematical study. In math nodes are often called verticies. You’ll also hear edges called both links and connections. These all represent the same structures, though the kinds of questions mathematicians are concerned with are generally distinct from those of interest in statistics.

Kite example

Our first network dataset is an artificial network created to illustrate some of the primary properties. The graph is called a “kite” because of its shape. We will move to more interesting data soon.

Once installed, load the smodels package and pass the edges and nodes to the function graph_data:

We can see the graph below. Notice that the nodes are spread out over the space in a reasonably nice pattern.

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Notice that the layout made linked nodes close together and minimizes crossing of edges. The algorithm actually uses a physics simulation where edges are treated as springs and nodes as positively charged particles. Hence, the nodes want to spread out but the edge push together neighboring nodes.

Centrality

There are a few different ways of measuring how central or important a node appears within a network. Eigenvalue centrality constructs a score for each node such that each nodes centrality score is proportional to the sum of its neighbors scores. In math:

So the more neighbors a node has the higher it’s centrality, though neighbors that are central themselves contribute more than nodes that are not otherwise very central. We won’t go into the math today, but this can be solved by computing the eigenvalue problem for the adjacency matrix of the graph.

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In this measurement, we see that Dieter is the most central node. He is right in the middle of the cluster of most densely connection points.

Another centrality measurement is called betweenness. Consider all shortest paths between all pairs of nodes in the network. Betweenness measures, for each node, the proportion of these paths that run through a given node. We can plot this as well:

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Notice now that Hellen is much more central because she connects everyone in the graph to Ida.

Another measurement of centrality is called closeness. It is defined as the average distance (number of hops) from each node to a given node of interest.

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Here Gaby, Flavia, and Dieter are equally central. Flavia and Gaby have fewer direct links, but are closer to Ida than Dieter.

Finally, we can also use degree centrality. This simply measures how many neighbors a graph has.

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Notice that these measures agree to some extent but do measure slightly different aspects of centrality. Nodes that have a high betweenness but relatively low eigen centrality are known as gatekeepers (such as Hellen above). These are the influencial people who link clusters within a graph.

Clusters

Speaking of clusters, we can also define clustering algorithms over a network. Here is an algorithm applied to our network:

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There is a large primary group (the blue one) and the small subgroup of Hellen and Ida (the red one).