Node centrality is a family of metrics that quantify how important, influential, or strategically positioned a particular node is within a network. Whether the network is a web of friendships, a power grid, or a map of protein interactions inside a cell, centrality measures answer the same underlying question: which nodes matter most, and why? The concept sounds simple, but there is no single “centrality score.” Researchers have developed dozens of distinct measures, each capturing a different aspect of what it means to be central, and the results they produce can disagree sharply depending on the structure of the network being studied.
What Centrality Actually Measures
At its core, every centrality measure evaluates how a node participates in the paths and connections that run through a network. A foundational description from graph theory frames it this way: what all centrality measures do is assess a node’s involvement in the walk structure of a network, where “walks” are the sequences of steps you can take along edges from one node to another.1Social Networks. A Graph-theoretic perspective on centrality That definition is deliberately broad. Some measures care only about direct connections. Others care about the shortest route between two distant nodes. Still others consider every possible path, weighted by length. The differences are not academic hairsplitting; they reflect genuinely different ideas about what makes a node “important.”
Think of a city’s road network. A highway interchange might be important because many roads converge there (that is one kind of centrality). A different intersection might not have many roads, but it sits on the only bridge across a river, making it a critical bottleneck (a different kind). A third location might matter because it connects two wealthy commercial districts whose traffic is especially valuable (yet another kind). All three are “central,” but for completely different reasons. That is why network analysts rarely rely on a single centrality measure.
Degree Centrality
Degree centrality is the simplest and most intuitive measure. It counts how many direct connections a node has. In a social network, a person who knows 200 people has a higher degree centrality than someone who knows 20. No calculation of paths or distances is needed; you just count edges. Despite that simplicity, degree turns out to be surprisingly powerful. In studies of network attacks, removing nodes in order of highest degree is one of the most effective ways to break a network apart, sometimes outperforming more sophisticated measures.2PLoS ONE. Attack Robustness and Centrality of Complex Networks
When analysts talk about “hubs” in a network, they usually mean high-degree nodes. Normalized degree centralization, which zooms out from individual nodes to characterize the whole network, captures how concentrated hub-like structure is across the graph.3PubMed Central. Evaluating global measures of network centralization: Axiomatic and numerical assessments A network where one node connects to almost everyone while the rest have few connections has high centralization. A network where everyone has roughly the same number of connections has low centralization.
The limitation of degree centrality is that it is entirely local. A node could have 500 connections, but if all 500 of those neighbors are isolated from the rest of the network, that node is not especially useful for reaching distant parts of the graph. For that, you need measures that look beyond the immediate neighborhood.
Closeness Centrality
Closeness centrality asks: how quickly can a node reach every other node in the network? It is calculated from the average shortest-path distance between a node and all other nodes. A node with high closeness centrality can, on average, get information to anyone in the network in fewer hops. If you imagine spreading a rumor, the person with the highest closeness centrality could reach the entire population fastest.
An interesting finding from research linking closeness and degree is that the two measures are not independent. Across both synthetic network models and 130 real-world data sets, researchers found a consistent nonlinear relationship: the inverse of closeness is roughly proportional to the logarithm of degree.4Nature (Communications Physics). Linking the network centrality measures closeness and degree In plain terms, adding more direct connections improves your closeness centrality, but with diminishing returns. Going from 5 connections to 50 helps a lot. Going from 500 to 5,000 helps much less. This makes intuitive sense: once you are well-connected enough, adding yet another friend does not shave much off your average distance to the rest of the world.
Closeness centrality has a practical limitation in disconnected networks, where some nodes cannot reach others at all. In those cases, the measure either breaks or requires workarounds like harmonic closeness, which handles infinite distances more gracefully.
Betweenness Centrality
Betweenness centrality measures something fundamentally different from degree or closeness. It asks: how often does a node sit on the shortest path between two other nodes? A high-betweenness node acts as a bridge or bottleneck. Remove it, and communication between other parts of the network gets disrupted or cut off entirely.
This bridge role maps neatly onto the idea of a gatekeeper. In social networks, people with high betweenness centrality connect groups that would otherwise be isolated from each other. Research on information sharing suggests that individuals who occupy these bridging positions show greater neural sensitivity to their gatekeeper role, with increased brain activation in regions associated with thinking about whether and how to share information with others.5Oxford Academic. Being the Gatekeeper: How Thinking about Sharing Affects Neural Encoding of Information They are, in a real psychological sense, aware that they control a flow of information.
Betweenness centrality behaves differently from the other common measures in an important way. Empirical analysis across real networks found that while closeness and degree tend to correlate with their respective cousins (closeness with eccentricity, degree with eigenvector centrality), betweenness varied according to network topology and did not demonstrate a consistent pattern.6PLOS ONE. Towards a Methodology for Validation of Centrality Measures in Complex Networks A node can have moderate degree and moderate closeness but extremely high betweenness if it happens to sit at a structural chokepoint. This unpredictability is exactly why betweenness captures something the other measures miss.
Eigenvector Centrality and PageRank
Degree centrality treats all connections as equally valuable. Eigenvector centrality does not. It gives a node a high score if that node is connected to other nodes that are themselves highly central. Being friends with five well-connected people counts for more than being friends with five loners. The logic is recursive: your importance depends on the importance of your neighbors, and their importance depends on the importance of their neighbors, and so on. The math resolves this circularity into a stable set of scores. In brain imaging research, eigenvector centrality has been applied at the level of individual brain voxels, assigning high values to regions that are strongly correlated with many other central regions within the brain network.7PubMed Central. Eigenvector centrality mapping for analyzing connectivity patterns in fMRI data of the human brain
PageRank, the algorithm that originally powered Google’s search rankings, is a variant of eigenvector centrality designed for directed networks, where links have a direction (page A links to page B, but not necessarily the other way around). PageRank models a random walker who follows links from page to page, occasionally jumping to a random page entirely. A page’s centrality score is the fraction of time the random walker spends there in the long run.8arXiv. Random Walks on Directed Networks: the Case of PageRank Pages that receive links from many important pages tend to be visited often and score highly. The random-jump component prevents the walker from getting stuck in dead ends, which is a real problem in directed networks where some pages have no outgoing links.
Beyond Shortest Paths
Classical betweenness centrality assumes that information travels only along the shortest path between two nodes. In many real networks, that assumption is shaky. Messages in social networks bounce around through friends of friends, not always by the most efficient route. Electrical current in a circuit flows along all available paths simultaneously. Diseases can spread through any contact, not just the most direct chain of transmission.
Random-walk betweenness relaxes this assumption. Instead of counting how often a node lies on the single shortest path between two others, it counts how often a random walker passes through the node while wandering between those two others. This approach includes contributions from essentially all paths, not just the shortest, while still giving more weight to short ones.9Social Networks. A measure of betweenness centrality based on random walks The result is a measure that better reflects how information or influence actually flows in messy, real-world systems where nobody navigates optimally.
When Networks Change Over Time
Most classic centrality measures treat a network as a frozen snapshot. But real networks evolve: friendships form and dissolve, airline routes get added and cut, neural connections strengthen and weaken. A node that is central today may be peripheral tomorrow. Dynamic centrality approaches account for this by tracking how a node’s importance changes as the network’s links shift over time. If you want to quantify and compare the overall time-dependent centrality of network nodes, you need to account for the global flow of information through time, not just at a single moment.10PubMed Central. Sparse matrix computations for dynamic network centrality
One framework for this generalizes the Katz centrality measure, which already accounts for information spreading along paths of all lengths with a discount for longer ones. The dynamic version adds a second kind of discount: attenuation over time, reflecting the fact that old information becomes stale. This allows researchers to identify and track the most influential nodes in terms of broadcasting and receiving information through time-dependent links.11PubMed Central. A dynamical systems view of network centrality If you are monitoring a rapidly changing contact network during a disease outbreak, for instance, a static centrality snapshot might tell you who was important last week, while a dynamic measure can tell you who is important right now.
How Different Measures Relate to Each Other
With dozens of centrality measures available, a natural question is whether they tend to agree or produce wildly different rankings. A large-scale study examining 17 centrality measures across 212 real-world networks found that centrality measures are generally positively correlated, meaning a node that ranks high on one measure tends to rank reasonably high on others. But the strength of those correlations varies across networks, and network modularity, the degree to which a network breaks into distinct clusters, plays a key role in driving those cross-network variations.12PubMed Central. Consistency and differences between centrality measures across distinct classes of networks
Separate work looking at topology metrics more broadly found that density-related metrics and distance-based metrics tend to cluster into two groups that are roughly independent of each other.13PubMed. Overview of metrics and their correlation patterns for multiple-metric topology analysis on heterogeneous graph ensembles In practical terms, this means that degree-based measures and closeness-based measures are telling you partially different things, and you may need both to get a full picture. The fact that betweenness does not cluster cleanly with either group reinforces the point made earlier: it captures a structural feature, the bridging role, that neither raw connectivity nor average distance fully reflects.
Centrality in Biology
One of the most striking real-world applications of node centrality is in molecular biology. Proteins in a cell interact with each other in complex networks, and researchers have found that a protein’s position in its interaction network predicts its biological importance. A pattern known as the centrality-lethality rule holds that proteins with many interaction partners, that is, high-degree hubs, tend to be essential for an organism’s survival. Research into yeast protein interaction networks showed that the majority of these essential hubs are important because they participate in densely connected groups of proteins that share a biological function.14PLoS Computational Biology. Why Do Hubs in the Yeast Protein Interaction Network Tend To Be Essential: Reexamining the Connection between the Network Topology and Essentiality
This is not limited to yeast. A comparative study across three different organisms found that proteins occupying a more central position in all three interaction networks, regardless of how many direct partners they had, evolve more slowly and are more likely to be essential for survival.15Molecular Biology and Evolution. Comparative Genomics of Centrality and Essentiality in Three Eukaryotic Protein-Interaction Networks Evolution, in effect, is more careful about changing proteins that sit at the network’s core. This insight has practical consequences for drug discovery. In type 1 diabetes research, centrality analysis of protein interaction networks has been used to prioritize drug targets, identifying central proteins and then screening for drugs that interact with them.16PubMed Central. Centrality Analysis of Protein-Protein Interaction Networks and Molecular Docking Prioritize Potential Drug-Targets in Type 1 Diabetes
Centrality in the Brain
The human brain can be mapped as a network of interconnected regions, and centrality analysis has become a standard tool for understanding how that network is organized. A study of resting-state brain imaging data from over a thousand healthy adults showed that different centrality measures capture different aspects of brain connectivity, highlighting both global and local connectivity properties. Beyond identifying hub regions, centrality measures in the brain also showed reliable associations with age and sex, suggesting they reflect real biological variation.17PubMed. Network centrality in the human functional connectome
Brain hubs also shift during development. In early life, the most functionally central brain regions are in primary sensory areas, the parts that process basic sights, sounds, and touch. By late childhood, hub status has migrated to association areas, the regions that integrate information across senses and support complex thought. Structural hubs in the brain’s physical wiring, however, appear to be established much earlier, suggesting that the anatomical infrastructure comes first and the functional refinement follows during years of postnatal maturation.18PubMed Central. The development of brain network hubs Researchers have also proposed specialized centrality measures tailored to brain networks, such as leverage centrality, which identifies high-leverage nodes that tend to be provincial hubs holding together the modular structure of brain networks.19PLoS ONE. A New Measure of Centrality for Brain Networks
Protecting Infrastructure by Finding Critical Nodes
Power grids, transportation networks, and communication systems are all networks, and centrality analysis helps engineers figure out which components are most critical to protect. A study of the Nordic transmission grid used betweenness, closeness, degree, and a combined centrality measure to identify the most structurally vulnerable nodes. The combined measure performed best, and the nodes it flagged as critical coincided geographically with regions already slated for major grid expansion by the Swedish transmission system operator.20Physica A: Statistical Mechanics and its Applications. Power grid vulnerability analysis using complex network theory: A topological study of the Nordic transmission grid That independent confirmation, where a purely mathematical analysis identified the same weak points that experienced grid operators had already recognized, is a strong validation of the method.
Similar approaches have been applied to power grid resilience planning more broadly, using centrality metrics from both static and dynamic perspectives to identify key nodes and then formulating emergency plans around protecting them.21International Journal of Critical Infrastructure Protection. Assessing the comprehensive importance of power grid nodes based on DEA The logic works in reverse for adversarial thinking, too. If you are trying to assess how vulnerable a network is to deliberate attack, centrality tells you where the damage will be greatest if a targeted node goes down.
Targeted Attacks and Network Resilience
Complex networks tend to be resilient against random failures. If a random node goes offline, the rest of the network usually holds together. Targeted attacks, where an adversary deliberately removes the most central nodes, are a different story. Research on scale-free networks found that certain combinations of centrality measures can produce more severe network disruption than any single measure alone.22Journal of Complex Networks. Exploring the effectiveness of node attacks based on combined centrality measures in scale-free networks
The choice between degree and betweenness for targeting also depends on the attack strategy. When you remove many nodes all at once based on their pre-attack scores, degree is often the most effective targeting criterion. But when you remove nodes one at a time and recalculate centrality after each removal, betweenness becomes the most effective measure for every network type studied.2PLoS ONE. Attack Robustness and Centrality of Complex Networks This makes sense: betweenness identifies bottlenecks, and once you remove one bottleneck, the traffic reroutes and creates new bottlenecks that betweenness can find. Degree, being local, does not adapt to structural changes as well in sequential scenarios. Understanding this distinction matters for both defense planners trying to protect infrastructure and security researchers probing vulnerabilities.
Centrality in Social Networks and Its Limits
Centrality measures are perhaps most popularly associated with social network analysis. Identifying influential users, detecting opinion leaders, and mapping information diffusion all rely on some form of centrality. But there is an important caveat: high centrality in a graph does not automatically translate into social influence. A review of opinion-leader detection methods found that node centrality in graph theory does not necessarily align with the opinion leader concepts in social psychology, and that existing research lacks cross-disciplinary consensus on what makes someone an opinion leader versus simply well-connected.23Applied Intelligence. How to find opinion leader on the online social network?
This gap matters in practice. A marketing team that targets the highest-degree users for a product launch may be reaching people who have many connections but little persuasive power. A public health campaign that identifies betweenness-central individuals for a messaging campaign may find that those structural bridges do not actually influence the groups they connect. Centrality describes network position; influence is a psychological and social phenomenon that position alone does not fully explain.
Computational Challenges at Scale
Computing centrality is straightforward for small networks, but modern networks can contain millions or billions of nodes. Calculating betweenness centrality for every node in a network with a billion edges is extremely time-consuming. Even approximation algorithms designed to speed up closeness and betweenness computation still require substantial processing time when applied to large-scale networks.24PubMed Central. Inductive Graph Neural Networks for Node Centrality Approximation in Complex Networks This has driven interest in machine-learning approaches, including graph neural networks, that learn to predict centrality scores without performing the full calculation.
Degree centrality, by contrast, is trivially fast to compute at any scale, which partly explains its enduring popularity despite its theoretical simplicity. For many practical applications, the question is not “which measure is most theoretically appropriate?” but “which measure can I actually compute on this data set in a reasonable time frame?”
Multilayer Networks and New Frontiers
Real-world systems often involve multiple types of relationships layered on top of each other. You might interact with the same people through email, social media, and face-to-face meetings, and each layer of interaction could produce a different network structure. In these multiplex networks, centrality becomes a two-level question: which nodes are most important, and which layers are most influential? One framework addresses this by coupling the two questions together, assigning more centrality to nodes that receive links from highly influential layers and from already central nodes.25Journal of Complex Networks. Centralities of nodes and influences of layers in large multiplex networks
Semi-local centrality measures represent another active area of development. These try to split the difference between purely local measures like degree, which are cheap to compute but miss broader structure, and purely global measures like betweenness, which capture more but are expensive to compute. Recent work on a semi-local measure based on the average shortest path within a node’s extended neighborhood showed a modest improvement over the best existing comparable method in identifying influential spreaders in simulated epidemics.26SpringerLink / CrossRef. A new semi-local centrality for identifying influential nodes based on local average shortest path with extended neighborhood The tradeoff between accuracy and computational cost ensures this middle ground will remain an active research front for the foreseeable future.