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.

Google’s PageRank and the Katz centrality are variants of the eigenvector centrality.

**Using the adjacency matrix to find eigenvector centrality**

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. 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. 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.

`def` `eigenvector_centrality(G, max_iter` `=` `100` `, tol` `=` `1.0e` `-` `6` `, nstart` `=` `None` `, ` ` ` `weight` `=` `'weight'` `): ` ` ` `"""Compute the eigenvector centrality for the graph G. ` ` ` ` ` `Eigenvector centrality computes the centrality for a node based on the ` ` ` `centrality of its neighbors. The eigenvector centrality for node `i` is ` ` ` ` ` `.. math:: ` ` ` ` ` `\mathbf{Ax} = \lambda \mathbf{x} ` ` ` ` ` `where `A` is the adjacency matrix of the graph G with eigenvalue `\lambda`. ` ` ` `By virtue of the Perron–Frobenius theorem, there is a unique and positive ` ` ` `solution if `\lambda` is the largest eigenvalue associated with the ` ` ` `eigenvector of the adjacency matrix `A` ([2]_). ` ` ` ` ` `Parameters ` ` ` `---------- ` ` ` `G : graph ` ` ` `A networkx graph ` ` ` ` ` `max_iter : integer, optional ` ` ` `Maximum number of iterations in power method. ` ` ` ` ` `tol : float, optional ` ` ` `Error tolerance used to check convergence in power method iteration. ` ` ` ` ` `nstart : dictionary, optional ` ` ` `Starting value of eigenvector iteration for each node. ` ` ` ` ` `weight : None or string, optional ` ` ` `If None, all edge weights are considered equal. ` ` ` `Otherwise holds the name of the edge attribute used as weight. ` ` ` ` ` `Returns ` ` ` `------- ` ` ` `nodes : dictionary ` ` ` `Dictionary of nodes with eigenvector centrality as the value. ` ` ` ` ` ` ` `Notes ` ` ` `------ ` ` ` `The eigenvector calculation is done by the power iteration method and has ` ` ` `no guarantee of convergence. The iteration will stop after ``max_iter`` ` ` ` `iterations or an error tolerance of ``number_of_nodes(G)*tol`` has been ` ` ` `reached. ` ` ` ` ` `For directed graphs this is "left" eigenvector centrality which corresponds ` ` ` `to the in-edges in the graph. For out-edges eigenvector centrality ` ` ` `first reverse the graph with ``G.reverse()``. ` ` ` ` ` ` ` `"""` ` ` `from` `math ` `import` `sqrt ` ` ` `if` `type` `(G) ` `=` `=` `nx.MultiGraph ` `or` `type` `(G) ` `=` `=` `nx.MultiDiGraph: ` ` ` `raise` `nx.NetworkXException(` `"Not defined for multigraphs."` `) ` ` ` ` ` `if` `len` `(G) ` `=` `=` `0` `: ` ` ` `raise` `nx.NetworkXException(` `"Empty graph."` `) ` ` ` ` ` `if` `nstart ` `is` `None` `: ` ` ` ` ` `# choose starting vector with entries of 1/len(G) ` ` ` `x ` `=` `dict` `([(n,` `1.0` `/` `len` `(G)) ` `for` `n ` `in` `G]) ` ` ` `else` `: ` ` ` `x ` `=` `nstart ` ` ` ` ` `# normalize starting vector ` ` ` `s ` `=` `1.0` `/` `sum` `(x.values()) ` ` ` `for` `k ` `in` `x: ` ` ` `x[k] ` `*` `=` `s ` ` ` `nnodes ` `=` `G.number_of_nodes() ` ` ` ` ` `# make up to max_iter iterations ` ` ` `for` `i ` `in` `range` `(max_iter): ` ` ` `xlast ` `=` `x ` ` ` `x ` `=` `dict` `.fromkeys(xlast, ` `0` `) ` ` ` ` ` `# do the multiplication y^T = x^T A ` ` ` `for` `n ` `in` `x: ` ` ` `for` `nbr ` `in` `G[n]: ` ` ` `x[nbr] ` `+` `=` `xlast[n] ` `*` `G[n][nbr].get(weight, ` `1` `) ` ` ` ` ` `# normalize vector ` ` ` `try` `: ` ` ` `s ` `=` `1.0` `/` `sqrt(` `sum` `(v` `*` `*` `2` `for` `v ` `in` `x.values())) ` ` ` ` ` `# this should never be zero? ` ` ` `except` `ZeroDivisionError: ` ` ` `s ` `=` `1.0` ` ` `for` `n ` `in` `x: ` ` ` `x[n] ` `*` `=` `s ` ` ` ` ` `# check convergence ` ` ` `err ` `=` `sum` `([` `abs` `(x[n]` `-` `xlast[n]) ` `for` `n ` `in` `x]) ` ` ` `if` `err < nnodes` `*` `tol: ` ` ` `return` `x ` ` ` ` ` `raise` `nx.NetworkXError(` `"""eigenvector_centrality(): ` `power iteration failed to converge in %d iterations."%(i+1))"""` `) ` |

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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.

`>>> ` `import` `networkx as nx ` `>>> G ` `=` `nx.path_graph(` `4` `) ` `>>> centrality ` `=` `nx.eigenvector_centrality(G) ` `>>> ` `print` `([` `'%s %0.2f'` `%` `(node,centrality[node]) ` `for` `node ` `in` `centrality]) ` |

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The output of the above code is:

`[` `'0 0.37'` `, ` `'1 0.60'` `, ` `'2 0.60'` `, ` `'3 0.37'` `] ` |

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The above result is a dictionary depicting the value of eigen vector centrality of each node. The above is an extension of my article series on the centrality measures. Keep networking!!!

**References**

You can read more about the same at

https://en.wikipedia.org/wiki/Eigenvector_centrality

http://networkx.readthedocs.io/en/networkx-1.10/index.html

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