In graph theory, the degree (or valency) of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice. The degree of a vertex is denoted or . The maximum degree of a graph G, denoted by (G), and the minimum degree of a graph, denoted by (G), are the maximum and minimum degree of its vertices. In the graph on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, all degrees are the same, and so we can speak of the degree of the graph.
Historically first and conceptually simplest is degree centrality, which is defined as the number of links incident upon a node (i.e., the number of ties that a node has). The degree can be interpreted in terms of the immediate risk of a node for catching whatever is flowing through the network (such as a virus, or some information). In the case of a directed network (where ties have direction), we usually define two separate measures of degree centrality, namely indegree and outdegree. Accordingly, indegree is a count of the number of ties directed to the node and outdegree is the number of ties that the node directs to others. When ties are associated to some positive aspects such as friendship or collaboration, indegree is often interpreted as a form of popularity, and outdegree as gregariousness.
The degree centrality of a vertex , for a given graph with vertices and edges, is defined as
Calculating degree centrality for all the nodes in a graph takes in a dense adjacency matrix representation of the graph, and for edges takes in a sparse matrix representation.
The definition of centrality on the node level can be extended to the whole graph, in which case we are speaking of graph centralization. Let be the node with highest degree centrality in . Let be the node connected graph that maximizes the following quantity (with being the node with highest degree centrality in ):
Correspondingly, the degree centralization of the graph is as follows:
The value of is maximized when the graph contains one central node to which all other nodes are connected (a star graph), and in this case
Following is the code for the calculation of the degree centrality of the graph and its various nodes.
The above function is invoked using the networkx library and once the library is installed, you can eventually use it and the following code has to be written in python for the implementation of the Degree centrality of a node.
The result is as follows:
The above result is a dictionary depicting the value of degree centrality of each node. The above is an extension of my article series on the centrality measures. Keep networking!!!
You can read more about the same at
This article is contributed by Jayant Bisht. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
- Betweenness Centrality (Centrality Measure)
- Katz Centrality (Centrality Measure)
- Statistical Functions in Python | Set 2 ( Measure of Spread)
- Statistical Functions in Python | Set 1 (Averages and Measure of Central Location)
- Degree of a Cycle Graph
- Find the Degree of a Particular vertex in a Graph
- Print the degree of every node from the given Prufer sequence
- Spanning Tree With Maximum Degree (Using Kruskal's Algorithm)
- Print the node with the maximum degree in the prufer sequence
- Python | Measure similarity between two sentences using cosine similarity
- Python | Blackman in numpy
- Smart calculator in Python
- Intesting Fact about Python Multi-line Comments
- ML | Auto-Encoders
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.