# Eigenvector centrality

In graph theory and network analysisindicators of centrality identify the most important vertices within a graph. Applications include identifying the most influential person s in a social networkkey infrastructure nodes in the Internet or urban networksand super-spreaders of disease. Centrality concepts were first developed in social network analysisand many of the terms used to measure centrality reflect their sociological origin.

Centrality indices are answers to the question "What characterizes an important vertex? The word "importance" has a wide number of meanings, leading to many different definitions of centrality.

Two categorization schemes have been proposed. This allows centralities to be classified by the type of flow they consider important.

This allows centralities to be classified based on how they measure cohesiveness. A further conclusion is that a centrality which is appropriate for one category will often "get it wrong" when applied to a different category.

When centralities are categorized by their approach to cohesiveness, it becomes apparent that the majority of centralities inhabit one category. The count of the number of walks starting from a given vertex differs only in how walks are defined and counted. Restricting consideration to this group allows for a soft characterization which places centralities on a spectrum from walks of length one degree centrality to infinite walks eigenvalue centrality.

A network can be considered a description of the paths along which something flows. This allows a characterization based on the type of flow and the type of path encoded by the centrality. A flow can be based on transfers, where each indivisible item goes from one node to another, like a package delivery going from the delivery site to the client's house. A second case is serial duplication, in which an item is replicated so that both the source and the target have it.

An example is the propagation of information through gossip, with the information being propagated in a private way and with both the source and the target nodes being informed at the end of the process.

The last case is parallel duplication, with the item being duplicated to several links at the same time, like a radio broadcast which provides the same information to many listeners at once. An alternative classification can be derived from how the centrality is constructed. This again splits into two classes. Centralities are either radial or medial. The degree and eigenvalue centralities are examples of radial centralities, counting the number of walks of length one or length infinity.

Medial centralities count walks which pass through the given vertex. The canonical example is Freeman's betweenness centrality, the number of shortest paths which pass through the given vertex. Likewise, the counting can capture either the volume or the length of walks. Volume is the total number of walks of the given type.

The three examples from the previous paragraph fall into this category. Length captures the distance from the given vertex to the remaining vertices in the graph. Freeman's closeness centrality, the total geodesic distance from a given vertex to all other vertices, is the best known example. Borgatti and Everett propose that this typology provides insight into how best to compare centrality measures.

Measures from different boxes, however, are categorically distinct. Any evaluation of relative fitness can only occur within the context of predetermining which category is more applicable, rendering the comparison moot. The characterization by walk structure shows that almost all centralities in wide use are radial-volume measures. These encode the belief that a vertex's centrality is a function of the centrality of the vertices it is associated with.

Centralities distinguish themselves on how association is defined. Bonacich showed that if association is defined in terms of walksthen a family of centralities can be defined based on the length of walk considered. Alternative definitions of association are also reasonable.

Alpha centrality allows vertices to have an external source of influence.Centrality and power This page is part of an on-line text by Robert A. Feel free to use and distribute this textbook, with citation. Your comments and suggestions are very welcome. Send me e-mail. There is much less agreement about what power is, and how we can describe and analyze its causes and consequences. In this chapter we will look at some of the main approaches that social network analysis has developed to study power, and the closely related concept of centrality.

Network thinking has contributed a number of important insights about social power. Perhaps most importantly, the network approach emphasizes that power is inherently relational. An individual does not have power in the abstract, they have power because they can dominate others -- ego's power is alter's dependence. Because power is a consequence of patterns of relations, the amount of power in social structures can vary. If a system is very loosely coupled low density not much power can be exerted; in high density systems there is the potential for greater power.

Power is both a systemic macro and relational micro property. The amount of power in a system and its distribution across actors are related, but are not the same thing. Two systems can have the same amount of power, but it can be equally distributed in one and unequally distributed in another.

## Network Centrality Using Eigenvectors

Power in social networks may be viewed either as a micro property i. Network analysts often describe the way that an actor is embedded in a relational network as imposing constraints on the actor, and offering the actor opportunities. Actors that face fewer constraints, and have more opportunities than others are in favorable structural positions. Having a favored position means that an actor may extract better bargains in exchanges, have greater influence, and that the actor will be a focus for deference and attention from those in less favored positions.

But, what do we mean by "having a favored position" and having "more opportunities" and "fewer constraints? But, network analysis has made important contributions in providing precise definitions and concrete measures of several different approaches to the notion of the power that attaches to positions in structures of social relations.

To understand the approaches that network analysis uses to study power, it is useful to first think about some very simple systems. Consider the three simple graphs of networks in figures But, exactly why is it that actor A has a "better" position than all of the others in the star network? What about the position of A in the line network?In graph theoryeigenvector centrality also called eigencentrality or prestige score [1] is a measure of the influence of a node in a network.

Relative scores are assigned to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.

A high eigenvector score means that a node is connected to many nodes who themselves have high scores. Google 's PageRank and the Katz centrality are variants of the eigenvector centrality. With a small rearrangement this can be rewritten in vector notation as the eigenvector equation.

However, the additional requirement that all the entries in the eigenvector be non-negative implies by the Perronâ€”Frobenius theorem that only the greatest eigenvalue results in the desired centrality measure. The eigenvector is only defined up to a common factor, so only the ratios of the centralities of the vertices are well defined. To define an absolute score one must normalise the eigen vector e. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector.

Google 's PageRank is based on the normalized eigenvector centrality, or normalized prestige, combined with a random jump assumption. Eigenvector centrality is a measure of the influence a node has on a network. If a node is pointed to by many nodes which also have high eigenvector centrality then that node will have high eigenvector centrality.

The earliest use of eigenvector centrality is by Edmund Landau in an paper on scoring chess tournaments. More recently, researchers across many fields have analyzed applications, manifestations, and extensions of eigenvector centrality in a variety of domains:.

From Wikipedia, the free encyclopedia. A measure of the influence of a node in a network.

### Eigenvector Centrality (Centrality Measure)

Cambridge University Press. Retrieved Negre, Uriel N. Morzan, Heidi P. Hendrickson, Rhitankar Pal, George P. Lisi, J. Proceedings of the National Academy of Sciences.

International Journal of Neural Systems. Deutsches Wochenschach 11 : â€” Retrieved 17 April Ranking systems. The Econometric Society. AIP Publishing. American Economic Review. University of Chicago Press. Journal of Political Economy. Categories : Graph theory Network analysis.

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Help Learn to edit Community portal Recent changes Upload file.Eigenvector Centrality is an algorithm that measures the transitive influence or connectivity of nodes. Relationships to high-scoring nodes contribute more to the score of a node than connections to low-scoring nodes.

A high score means that a node is connected to other nodes that have high scores. This algorithm is in the alpha tier. It was the first of the centrality measures that considered the transitive importance of a node in a graph, rather than only considering its direct importance.

Eigenvector Centrality can be used in many of the same use cases as the Page Rank algorithm. The following will run the algorithm and write back results:. The number of concurrent threads used for running the algorithm. Also provides the default value for 'readConcurrency' and 'writeConcurrency'. The type of normalization to apply to the results.

Valid values are maxl1norml2norm. The following will run the algorithm and stream results:. Also provides the default value for 'readConcurrency'. As we might expect, the Home page has the highest Eigenvector Centrality because it has incoming links from all other pages. By default, the scores returned by the Eigenvector Centrality are not normalized. We can specify a normalization using the normalization parameter.

The algorithm supports the following options:. The following will run the algorithm and stream results using max normalization:. If node label and relationship type are not selective enough to create the graph projection to run the algorithm on, you can use Cypher queries to project your graph.

This can also be used to run algorithms on a virtual graph. Use nodeQuery and relationshipQuery in the config:. Eigenvector Centrality. History and explanation Use-cases - when to use the Eigenvector Centrality algorithm Syntax Eigenvector Centrality algorithm sample Cypher projection Graph type support. History and explanation. Use-cases - when to use the Eigenvector Centrality algorithm. Configuration Name Type Default Optional Description concurrency int 4 yes The number of concurrent threads used for running the algorithm.

Results Name Type Description nodes int The number of nodes considered.EigenvectorCentrality [ g ]. EigenvectorCentrality [ g"In" ]. EigenvectorCentrality [ g"Out" ]. Rank the vertices. Highest-ranked vertices are connected to many well-connected vertices:. EigenvectorCentrality works with undirected graphs:.

EigenvectorCentrality works with large graphs:. By default, EigenvectorCentrality finds centralities using machine-precision computations:. Highlight the eigenvector centrality for CycleGraph :.

GridGraph :. CompleteKaryTree :. PathGraph :. Find the most influential members of a student government network with connections from student A to student B, if student A consults student B for opinions:. Find the top 10 important articles:.

## Centrality

Hubs are well-connected vertices and have the highest eigenvector centralities:. Find a protein whose deletion will result in lethality in a protein interaction network of yeast:. A Saccharomyces cerevisiae protein interaction network. The frequency of the eigenvector centrality follows a power-law distribution:. Obtain the maximum likelihood parameter estimates, assuming a Pareto distribution:.

A human-computer system of an organization that deals with internet orders and sends goods by mail. Find departments that should be given the most resources:. Most services rely on a system administration department as well as a register orders department:.

**What is an Eigenvector?**

For undirected graphs, the centrality vector satisfies the equation :. For directed graphs, in-centrality vector satisfies the equation :.

Out-centrality vector satisfies the equation :. For disconnected graphs, centralities are normalized with respect to the connected components:. EigenvectorCentrality is a special case of KatzCentrality :.A natural extension of degree centrality is eigenvector centrality. In-degree centrality awards one centrality point for every link a node receives. But not all vertices are equivalent: some are more relevant than others, and, reasonably, endorsements from important nodes count more.

The eigenvector centrality thesis reads:. Eigenvector centrality differs from in-degree centrality: a node receiving many links does not necessarily have a high eigenvector centrality it might be that all linkers have low or null eigenvector centrality. Moreover, a node with high eigenvector centrality is not necessarily highly linked the node might have few but important linkers.

Eigenvector centrality, regarded as a ranking measure, is a remarkably old method. Early pioneers of this technique are Wassily W. Leontief The Structure of American Economy, Harvard University Press, and John R. Seeley The net of reciprocal influence: A problem in treating sociometric data. The Canadian Journal of Psychology, The power method can be used to solve the eigenvector centrality problem. Hence, if the sub-dominant eigenvalue is small compared to the dominant one, then the method quickly converges.

The built-in function evcent RC computes eigenvector centrality. An undirected connected graph with nodes labelled with their eigenvector centrality follows:.In graph theory, eigenvector centrality also called eigencentrality is a measure of the influence of a node in a network. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes.

For a given graph with vertices let be the adjacency matrix, i. The relative centrality score of vertex can be defined as:. With a small rearrangement this can be rewritten in vector notation as the eigenvector equation. In general, there will be many different eigenvalues for which a non-zero eigenvector solution exists.

However, the additional requirement that all the entries in the eigenvector be non-negative implies by the Perronâ€”Frobenius theorem that only the greatest eigenvalue results in the desired centrality measure. The component of the related eigenvector then gives the relative centrality score of the vertex in the network.

The eigenvector is only defined up to a common factor, so only the ratios of the centralities of the vertices are well defined. To define an absolute score one must normalise the eigen vector e. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector.

Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix. Following is the code for the calculation of the Eigen Vector 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 eigen vector centrality of a node. The above result is a dictionary depicting the value of eigen vector centrality of each node.

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