Weighting Eigenvector Centrality in a Minimum Spanning Tree Portfolio

Summary

The document defines eigenvector centrality for a node in a minimum spanning tree using the adjacency matrix and its largest eigenvalue. It then asks how to combine the centralities of multiple nodes with a portfolio weighting vector, framing centrality as a possible input to portfolio analysis.

No method for calculating the weighted result is supplied: the text is a question and gives no worked example, evidence, or recommendation. A reader would need to specify what the weights represent and how they should be normalized or applied before the requested portfolio measure is well defined. The discussion also does not explain how the tree or adjacency matrix is constructed, or what financial interpretation to assign to a high-centrality asset. Treat it as a prompt about combining a graph statistic with portfolio weights, rather than a complete weighting procedure.

Key ideas

• Eigenvector centrality assigns each node a score based on the scores of its adjacent nodes.
• The score is defined using the adjacency matrix and its largest eigenvalue.
• The document asks how to combine node centralities with portfolio weights but does not provide a calculation.
• Interpreting a weighted centrality measure requires defining the role and normalization of the portfolio weights.

Tags

# Calculating the eigenvector centrality of a portfolio described by a minimum spanning tree


# Calculating the eigenvector centrality of a portfolio described by a minimum spanning tree












I am working with the eigenvector centrality of a minimum spanning tree, which can be calculated as:

`v(i) = lambda^-1 * sum[Omega(i,j)*v(j)] `

where:

- `v(i)` is the eigenvector centrality of the node i-th

- `Omega` is an adjacency matrix (square)

- `lambda` is the biggest eigenvalue of the Omega matrix

- `v(j)` is the eigenvector centrality of each j-th neighbor node

Let's assume that I want to calculate the eigenvector centrality of a portfolio, i.e. I want to weight each i-th eigenvector centrality v(i) by means of a weighting vector `w`. How can I calculate the weighted centrality of n nodes?

Thank you for your help.

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