Eigenvector centrality

In graph theory, eigenvector centrality (also called eigencentrality or prestige score ^[1] ) is a measure of the influence of a node in a connected 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. ^{[2] [3]}

Google's PageRank and the Katz centrality are variants of the eigenvector centrality. ^[4]

Using the adjacency matrix to find eigenvector centrality

For a given graph ${\displaystyle G:=(V,E)}$ with ${\displaystyle |V|}$ vertices let ${\displaystyle A=(a_{v,t})}$ be the adjacency matrix, i.e. ${\displaystyle a_{v,t}=1}$ if vertex ${\displaystyle v}$ is linked to vertex ${\displaystyle t}$, and ${\displaystyle a_{v,t}=0}$ otherwise. The relative centrality score, ${\displaystyle x_{v}}$, of vertex ${\displaystyle v}$ can be defined as:

${\displaystyle x_{v}={\frac {1}{\lambda }}\sum _{t\in M(v)}x_{t}={\frac {1}{\lambda }}\sum _{t\in V}a_{v,t}x_{t}}$

where ${\displaystyle M(v)}$ is the set of neighbors of ${\displaystyle v}$ and ${\displaystyle \lambda }$ is a constant. With a small rearrangement this can be rewritten in vector notation as the eigenvector equation

${\displaystyle \mathbf {Ax} =\lambda \mathbf {x} }$

In general, there will be many different eigenvalues ${\displaystyle \lambda }$ for which a non-zero eigenvector solution exists. However, the connectedness assumption and the additional requirement that all the entries in the eigenvector be non-negative imply (by the Perron–Frobenius theorem) that only the greatest eigenvalue results in the desired centrality measure. ^[5] The ${\displaystyle v^{\text{th}}}$ component of the related eigenvector then gives the relative centrality score of the vertex ${\displaystyle v}$ 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 eigenvector e.g. such that the sum over all vertices is 1 or the total number of vertices n. Power iteration is one of many eigenvalue algorithms that may be used to find this dominant eigenvector. ^[4] Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix.

Normalized eigenvector centrality scoring

Google's PageRank is based on the normalized eigenvector centrality, or normalized prestige, combined with a random jump assumption. ^[1] The PageRank of a node ${\displaystyle v}$ has recursive dependence on the PageRank of other nodes that point to it. The normalized adjacency matrix ${\displaystyle N}$ is defined as:

$${\displaystyle N(u,v)={\begin{cases}{1 \over \operatorname {od} (u)},&{\text{if }}(u,v)\in E\\0,&{\text{if }}(u,v)\not \in E\end{cases}}}$$

where ${\displaystyle od(u)}$ is the out-degree of node ${\displaystyle u}$, or in vector form:

${\displaystyle \mathbf {N} =\mathbf {diag} (\mathbf {Ae} )^{-1}\mathbf {A} }$,

where ${\displaystyle \mathbf {e} }$ is the vector of ones, and ${\displaystyle \mathbf {diag} (\mathbf {x} )}$ is the diagonal matrix of vector ${\displaystyle \mathbf {x} }$. ${\displaystyle \mathbf {N} }$ is a row-stochastic matrix.

The normalized eigenvector prestige score is defined as:

${\displaystyle p(v)=\sum _{u}{N^{T}(v,u)\cdot p(u)},}$

or in vector form,

${\displaystyle \mathbf {p} =\mathbf {N} ^{T}\mathbf {p} .}$

Applications

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. ^[6]

The earliest use of eigenvector centrality is by Edmund Landau in an 1895 paper on scoring chess tournaments. ^{[7] [8]}

More recently, researchers across many fields have analyzed applications, manifestations, and extensions of eigenvector centrality in a variety of domains:

• Eigenvector centrality is the unique measure satisfying certain natural axioms for a ranking system. ^{[9] [10]}
• In neuroscience, the eigenvector centrality of a neuron in a model neural network has been found to correlate with its relative firing rate. ^[6]
• Eigenvector centrality and related concepts have been used to model opinion influence in sociology and economics, as in the DeGroot learning model.
• The definition of eigenvector centrality has been extended to multiplex ^[11] and multilayer networks through the concept of versatility ^[12]
• In a study using data from the Philippines, researchers showed how political candidates' families had disproportionately high eigenvector centrality in local intermarriage networks. ^[13]
• Eigenvector centrality has been extensively applied to study economic outcomes, including cooperation in social networks. ^[14] In economic public goods problems, a person's eigenvector centrality can be interpreted as how much that person's preferences influence an efficient social outcome. ^[15]

See also

• Centrality

References

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3. Christian F. A. Negre, Uriel N. Morzan, Heidi P. Hendrickson, Rhitankar Pal, George P. Lisi, J. Patrick Loria, Ivan Rivalta, Junming Ho, Victor S. Batista. (2018). "Eigenvector centrality for characterization of protein allosteric pathways". Proceedings of the National Academy of Sciences. 115 (52): E12201–E12208. arXiv: 1706.02327 . Bibcode:2018PNAS..11512201N. doi: 10.1073/pnas.1810452115 . PMC   6310864 . PMID   30530700.
4. David Austin. "How Google Finds Your Needle in the Web's Haystack". AMS.
5. M. E. J. Newman. "The mathematics of networks" (PDF). Retrieved 2006-11-09.{{cite journal}}: Cite journal requires |journal= (help)
6. Fletcher, Jack McKay and Wennekers, Thomas (2017). "From Structure to Activity: Using Centrality Measures to Predict Neuronal Activity". International Journal of Neural Systems. 28 (2): 1750013. doi: 10.1142/S0129065717500137 . hdl: 10026.1/9713 . PMID   28076982.
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8. Holme, Peter (15 April 2019). "Firsts in network science" . Retrieved 17 April 2019.
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10. Palacios-Huerta, Ignacio; Volij, Oscar (2004). "The Measurement of Intellectual Influence" (PDF). Econometrica. 72 (3). The Econometric Society: 963–977. doi:10.1111/j.1468-0262.2004.00519.x. hdl: 10419/80143 . ISSN   0012-9682.
11. Solá, Luis; Romance, Miguel; Criado, Regino; Flores, Julio; García del Amo, Alejandro; Boccaletti, Stefano (2013). "Eigenvector centrality of nodes in multiplex networks". Chaos: An Interdisciplinary Journal of Nonlinear Science. 23 (3): 033131. arXiv: 1305.7445 . Bibcode:2013Chaos..23c3131S. doi:10.1063/1.4818544. ISSN   1054-1500. PMID   24089967. S2CID   14556381.
12. De Domenico, Manlio; Solè-Ribalta, ALbert; Omodei, Elisa; Gòmez, Sergio; Arenas, Alex (2015). "Ranking in interconnected multilayer networks reveals versatile nodes". Nature Communications. 6: 6868. arXiv: 1305.7445 . doi:10.1063/1.4818544. ISSN   2041-1723. PMID   25904405. S2CID   14556381.
13. Cruz, Cesi; Labonne, Julien; Querubin, Pablo (2017). "Politician Family Networks and Electoral Outcomes: Evidence from the Philippines". American Economic Review. 107 (10). University of Chicago Press: 3006–37. doi: 10.1257/aer.20150343 .
14. Jackson, Matthew O. (2010-11-01). Social and Economic Networks. Princeton University Press. doi:10.2307/j.ctvcm4gh1. ISBN   978-1-4008-3399-3. JSTOR   j.ctvcm4gh1.
15. Elliott, Matthew; Golub, Benjamin (2019). "A Network Approach to Public Goods". Journal of Political Economy. 127 (2). University of Chicago Press: 730–776. doi:10.1086/701032. ISSN   0022-3808. S2CID   158834906.