Centrality Metrics for Graphs

A graph is akin to a visual illustration of the relationships between multiple nodes, also known as entities, which signify the interconnected individuals or components. By exploring the connections between nodes linked by edges, a process known as graph analytics, valuable insights can be derived from the graph. This article concentrates on one such notion, centrality, and its significance in obtaining a holistic comprehension of the graph.

The Definition of the Word “Central”

Crucial to graph analytics is the identification of significant nodes within a network, a task that necessitates an understanding of centrality. Centrality is a concept employed to pinpoint the most prominent or central nodes of a network, enabling their distinction.

Evaluating the significance of a particular node entails the scrutiny of several metrics to ensure an accurate assessment. Centrality offers crucial analytical information regarding both the node and the graph, enabling the derivation of conclusions based on the observations made.

Centrality metrics provide a range of measurements that allow for a multi-faceted interpretation of a given node. These metrics equip us with the necessary information to extract data from a network, consequently enabling a comprehensive comprehension of the network. It is also imperative to comprehend the various available measures, as they all attribute a unique definition to the importance of a node, for the purpose of graph visualisation.

Variations in Significance

Before delving into degree centrality, let’s quickly review the concept of node degree in a graph.

When constructing a graph, it is feasible to create both directed and undirected versions. In an undirected graph, the degree of a node is equivalent to the number of edges interconnecting with other nodes. On the other hand, in a directed graph, the degree is divided into in-degree and out-degree. The number of linkages leading into a node is known as its in-degree, while the quantity of linkages arising from the node and connecting to other nodes is its out-degree.

Degree centrality is an effective means of precisely quantifying the importance of a node within a network. This analytical process is premised on the notion that the significance of a node increases proportionally with its degree, which is the number of connections it possesses. By utilising this metric, it is possible to identify influential individuals in a network, including those with the most extensive contact networks, those transitioning rapidly between peers, and those with access to the most essential information.

Degree centrality is one of the most straightforward measurements of node connectivity, making it a valuable tool for analysing financial information, account activity, and other related data sets.

Distance to the Centre

To determine a node’s significance, closeness centrality examines the extent to which it is interconnected with all other nodes in a network. This is done by determining the node’s geodesic distance (GD), which refers to the number of edges required to connect one node to another.

To obtain a node’s closeness or GD, you must add up the GD between that node and every other node in the network.

Closeness Centrality is particularly useful in identifying individuals who have the potential to exert significant influence on the entire network rapidly. It can also uncover isolated yet influential individuals or organisations. Those who hold utmost importance within the organisation have access to and can influence vital data. Additionally, a graph-based approach to keyphrase extraction can predict the significance of specific words in a given text.

Harmonic Equivalence Centre

Harmonic centrality is a measure of closeness that quantifies the ability to achieve global reachability (GD) across links. When some nodes are beyond the limits of reachability, harmonic centrality measurements can provide a more precise estimate of proximity.

Position in the Centre

The betweenness centrality of an edge defines its role as a mediator between other edges in the network. This metric is used to establish a node’s proximity to the shortest path through a network and the percentage of that path it occupies.

A node with a high betweenness centrality plays an outsized role in influencing other nodes in the network. This metric is valuable in identifying which links in the network are critical to its effective operation, making it an essential tool for maintaining system stability.

The use of betweenness centrality enables anti-terrorism authorities to examine terrorist networks on a global scale. The information gathered by these measures is then used to identify and neutralise potential threats, as well as to track data transmission speeds in telecommunications networks and package deliveries online. Additionally, microbloggers can leverage this hub status to expand their Twitter following with the assistance of a recommendation engine. This helps them determine which contacts to engage with in order to expand their sphere of influence.

Eigenvector Centrality

A node’s significance can be measured using eigenvector centrality, which takes into account the influence of its neighbouring nodes. For instance, if there is a node in a network, its linked nodes should be evaluated to determine their eigenvector centrality. A high score suggests that the node is surrounded by and connected to influential nodes, making it a crucial component of the network.

When calculating a node’s eigenvector centrality score, connections to nodes with higher scores are deemed to be more significant than those with lower ones. As a result, the node’s score is determined by the level of influence it exerts on other nodes, which is determined by the number and quality of connections it shares with other nodes.

The assessment of a node’s centrality can help identify those nodes which exert broad influence on the entire network, rather than just on their immediate neighbours. Nodes with high centrality scores tend to be in close proximity to other nodes with similarly high scores. This measure of centrality has numerous applications, such as Google’s PageRank algorithm, as well as in the analysis of virus and social networks.


PageRank is a type of eigenvector centrality that is particularly useful for directed graphs. It is determined by the impact of nodes on the directed graph, making it possible to evaluate the influence of nodes in the specified direction.

Eigenvector centrality is most effective when used on undirected graphs, making PageRank the first viable option for directed graphs. A popular social media platform, Twitter, uses this centrality metric to suggest additional accounts to users that may be of interest.

The healthcare and insurance industries use PageRank algorithms to pinpoint flaws in the fraud detection system. By conducting iterations that navigate a network of intersections and other junction points, the system is even able to forecast traffic flow in public locations and on roads.


ArticleRank is a type of PageRank that gauges a node’s influence in a network by measuring its capacity to impact other nodes through transitive citation links. While this may seem to contradict a key principle of network theory, which argues that relationships with nodes with a low number of outgoing connections are more important than those with a high number, both PageRank and ArticleRank provide effective methods for generating meaningful results when analysing a network with numerous nodes. The ability to calculate different types of centrality enables nodes in a network to be evaluated from multiple perspectives, allowing for more accurate and reliable conclusions.

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