# Degree and Weight Distributions

Returns the degree, weighted degree and edge weight distributions.

The degree of a node in an undirected networks counts the number of neighbours that node has. Its weighted degree counts the number of interactions each node has. Finally the edge weight counts the number of interactions which have happened across an edge.

For example, in the graph above with all edges aggregated between time $t_1$ and $t_3$, node $A$ has degree 2 (having neighbours $B$ and $C$) but weighted degree 3 (from the interactions at times $t_1$, $t_2$ and $t_3$). In the same way, edge $AB$ has weight 2.

## Returns

• degDist– the degree distribution, an array of dictionaries of the form {"degree": d, "freq": f} counting the number of nodes f (type Int) of degree d (type Int)
• weightDist – the weighted degree distribution, an array of dictionaries of the form {"weight": w, "freq": f} counting the number of nodes with weighted degree w (type Int)
• edgeDist – the edge weight distribution, an array of dictionaries of the form {"eweight": w, "freq": f} counting the number of edges of weight w (type Int)

## Example

In recent work studying the Gab social network as an interaction graph, the degree, weighted degree and edge weight distribution was obtained from the aggregate graph, displayed below.

The edge weight distribution is heavy-tailed, and shows that the majority of interactions between users occurred once and never again.

## Notes

• The edge weight as described here counts the number of times the edge appears (in social interaction graphs the number of interactions that occur over one edge) but the analyser code could be easily modified to compute instead the sum of user defined weights across the edge.
• This algorithm treats the graph as undirected, focusing on just node degree, rather than in/out degree. This can also be modified to address directed networks by considering incoming and outgoing edges rather than the union of these edges.
• This algorithm outputs a relatively large amount of data for each view of the graph (three arrays of dictionaries which could each have a number of entries similar magnitude to the number of nodes in worst case), therefore caution is advised against jobs with finely grained time ranges.

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