Average Network Degree

up:: Node Degree

Given a network $G$ with $N$ nodes and $m$ edges, its average degree is simply

$$⟨k⟩=\frac{\sum _{i=1}^N{k}_i}{N}=\frac{\sum _i\sum _j{A}_{ij}}{N}$$

using the Adjacency Matrix.

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For undirected Simple Graphs, we always have that

$${⟨k⟩}_{undirected}=\frac{2m}{N}$$

since all edges have two ends. In other words, we are counting all edges $\{i-j\}$ twice: $\{i\to j\}$ and $\{j\to i\}$.

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For directed Simple Graphs, we always have that

$${⟨k⟩}_{directed}=\frac{m}{N}$$

in which we don’t have the “ambiguity” that undirected networks have.

Note, also, that this value is unambiguous even for directed graphs, since

$$\begin{array}{rl}{⟨k⟩}_{directed}& =\frac{\sum _i\sum _j{A}_{ij}}{N}\\ & =\frac{{⟨k⟩}_{in}}{N}\\ & =\frac{{⟨k⟩}_{out}}{N}\end{array}$$

Thus, the average degree of a directed network is unambiguous, since it is the average In Degree and also the average Out Degree, by changing the order of the sums¹.

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References

• NEWMAN, Mark. Networks. Oxford University Press, 2018.

Footnotes

1. With no problem, since we are summing over finite graphs. ↩