Eigenvector centrality

In graph theory, eigenvector centrality (also called eigencentrality) 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. [1] [2]

Google's PageRank and the Katz centrality are variants of the eigenvector centrality.[3]

Using the adjacency matrix to find eigenvector centralityEdit

For a given graph   with   vertices let   be the adjacency matrix, i.e.   if vertex   is linked to vertex  , and   otherwise. The relative centrality,  , score of vertex   can be defined as:


where   is a set of the neighbors of   and   is a constant. 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 Perron–Frobenius theorem) that only the greatest eigenvalue results in the desired centrality measure.[4] 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.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.[3] Furthermore, this can be generalized so that the entries in A can be real numbers representing connection strengths, as in a stochastic matrix.


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.[5]

In neuroscience, the eigenvector centrality of a neuron in a model neural network has been found to correlate with its relative firing rate.[5]

The earliest use of eigenvector centrality is by Edmund Landau in an 1895 paper on scoring chess tournaments.[6][7]

See alsoEdit


  1. ^ M. E. J. Newman. "The mathematics of networks" (PDF). Retrieved 2006-11-09. Cite journal requires |journal= (help)
  2. ^ 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. doi:10.1073/pnas.1810452115. PMC 6310864. PMID 30530700.CS1 maint: multiple names: authors list (link)
  3. ^ a b David Austin. "How Google Finds Your Needle in the Web's Haystack". AMS.
  4. ^ M. E. J. Newman. "The mathematics of networks" (PDF). Retrieved 2006-11-09. Cite journal requires |journal= (help)
  5. ^ a b cite paper | author = Fletcher, Jack McKay and Wennekers, Thomas| title = From Structure to Activity: Using Centrality Measures to Predict Neuronal Activity| journal = International Journal of Neural Systems| volume = 0| number = 0| pages = 1750013| year = 2017| doi = 10.1142/S0129065717500137| url = http://www.worldscientific.com/doi/abs/10.1142/S0129065717500137 }}
  6. ^ Endmund Landau (1895). "Zur relativen Wertbemessung der Turnierresultate". Deutsches Wochenschach (11): 366–369. doi:10.1007/978-1-4615-4819-5_23.
  7. ^ Holme, Peter (15 April 2019). "Firsts in network science". Retrieved 17 April 2019.