Degree Centrality


Number of direct neighbors of node v
Degree Centrality
where N(v) is the set of direct neighbors of node v.
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 maximum degree of a graph G, denoted by Δ(G), and the minimum degree of a graph, denoted by δ(G), are the maximum and minimum degree of its vertices. In a regular graph, all degrees are the same, and so we can speak of the degree of the graph.

Degree centrality is a local and static metric, since it considers only the directly connected neighbors of a vertex in a static state. Nonetheless, it serves as a useful indicator of the extent of attachment of a vertex to the graph. [ZHANG, A. 2009]

In degree/Out degree
In directed networks two variants of the degree centrality may be appropriate: the in-degree centrality ciD(v) = d−(v) and the out-degree centrality coD(v) = d+(v). [BRANDES, U. 2005]

Degree Weighted
A more significant measure of the network properties in terms of the actual weights is obtained by extending the definition of vertex degree ki = Σjaij in terms of the vertex strength si, defined as Formula
Degree Centrality
This quantity measures the strength of vertices in terms of the total weight of their connections. [BARRAT, A. 2004]

In biological terms
The degree allows an immediate evaluation of the regulatory relevance of the node. For instance, in signaling networks, proteins with very high degree are interacting with several other signaling proteins, thus suggesting a central regulatory role, that is they are likely to be regulatory hubs. For instance, signaling proteins encoded by oncogenes, such as HRAS, SRC or TP53, are hubs. De- pending on the nature of the protein, the degree could indicate a central role in amplification (kinases), diversification and turnover (small GTPases), signaling module assembly (docking proteins), gene expression (transcription factors), etc. Signaling networks have typically a scale-free architecture. [SCARDONI, G.]

Degree centrality can be interpreted as a measure of immediate influence, as opposed to long-term effect in the network. For instance, if a certain proportion of nodes in the network are infected, those nodes having a direct connection with them will also be infected. However, although a node in a network may be linked to only one node, the risk of infection to the first node remains high if the latter is connected to many others. [RODRÍGUEZ, J. A. 2007]


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