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**NETWORK > CENTRALITY > DEGREE**

**PURPOSE **Calculates the degree and normalized degree centrality of each vertex and gives the overall network degree centralization.

**DESCRIPTION** The number of vertices adjacent to a given vertex in a symmetric graph is the degree of that vertex. For non-symmetric data the in-degree of a vertex u is the number of ties received by u and the out-degree is the number of ties initiated by u. In addition if the data is valued then the degrees (in and out) will consist of the sums of the values of the ties. The normalized degree centrality is the degree divided by the maximum possible degree expressed as a percentage. The normalized values should only be used for binary data.

For a given binary network with vertices v1....vn and maximum degree centrality cmax, the network degree centralization measure is S(cmax - c(vi)) divided by the maximum value possible, where c(vi) is the degree centrality of vertex vi.

The routine calculates these measures and some descriptive statistics based on these measures. Directed graphs may be symmetrized and the analysis is performed as above, or an analysis of the in and out degrees can be performed.

**PARAMETERS**

**Input dataset:**

** **Name of file containing network to be analyzed. Data type: Valued Graph

**Treat data as symmetric:** (Default = Yes).

If **Yes** directed data is automatically converted to undirected by taking the underlying graph.

**No** gives a separate analysis for in and out-degrees.

**Count reflexive ties (diagonal values)?** (Default = No).

No means that self loops are ignored.

**Output dataset:** (Default = 'FreemanDegree').

Name of file which will contain degree and normalized degree centrality of each vertex.

**LOG FILE **A table which contains a list of the degree and normalized degree (n Degree) centralities expressed as a percentage for each vertex, together with the share. The share is the centrality measure of the actor divided by the sum of all the actor centralities in the network. These have been ordered so that the actor with the highest centrality appears first. Note the stored UCINET output file retains the original order.

Descriptive statistics which give the mean, standard deviation, variance, minimum value and maximum value for each list generated. This is followed by the degree network centralization index expressed as a percentage. For the treating data as symmetric option a heterogeneity measure is produced. This is the sum of the squares of the proportion of the total centrality held by each actor.

For directed data the tables are the same as for undirected except that separate values are calculated for in and out degrees.

**TIMING** O(N).

**COMMENTS** Degree centrality measures network activity. For valued data the non-normalized values should be used and the degree centralization should be ignored.

**REFERENCES** Freeman L C (1979). 'Centrality in Social Networks: Conceptual clarification', Social Networks 1, 215-239.