The Measures of Rank or Status: A Reformulation and Reinterpretation

Abstract

This article investigates methods in social network analysis to identify the most important or the most prominent actors in a social network by ranking them appropriately. Although much has been done since Seeley's (1949) seminal work, several questions remain: How does scaling an adjacency matrix of a social network affect its eigenvalues and their corresponding eigenvectors? How can the differences between the left and the right eigenvectors be reconciled? Do both depict the two sides of the same coin? Can they be “merged” objectively to yield a single eigenvector, and how? Under what conditions does an adjacency matrix characterize, as a first-order approximation, the dynamics of its corresponding social network? Most importantly, how can a general dynamic model be derived to make clear the interrelationships of existing models?

In answering these questions, we derive a general model for the evolving rank (which we define as a general measure of prominence) of a (fixed) set of actors in a social network from a system of nonlinear equations; show that, under a rank equivalence condition, an adjacency matrix characterizes the dynamics of its corresponding social network as a first-order approximation; put some existing models within a single framework; and give a solution to the vexing problem of specifying the reference values of the rank.

Keywords:

• Boolean matrix
• equivalence condition
• matrix scaling
• measures
• models
• rank of the receiver
• rank of the sender
• rank prestige

Acknowledgments

We are most grateful to Drs. Michael Davies and Tim Pattison of Defence Science and Technology Organisation (DSTO) for their valuable comments on an earlier draft of this work, and to two anonymous reviewers for their insightful comments, which significantly improve its quality.

Notes

¹In the mathematical literature, Perron eigenvector is referred to as the eigenvector corresponding to the maximal eigenvalue. If A corresponds to a strongly connected graph, the Perron-Frobenius Theorem can also be invoked here to show that the right and left eigenvectors associated with this eigenvalue have nonnegative entries.

Note. The out-degrees are the choices given, the relative out-degrees are the relative choices given, the in-degrees are the choices received, and the relative in-degrees are the relative choices received. The relative out-degrees and the relative in-degrees of matrix A ^k as the kth successive approximation to the left and the right Perron eigenvectors v _L  = (1/3, 2/7, 5/21, 1/7) and v _R  = (1/4, 1/4, 1/4, 1/4) ^T of matrix A that correspond to the Perron eigenvalue 2 of the four eigenvalues λ = {2, (−1)^2/3, –(−1)^1/3, −1}.

Note. Matrices A ^T A, AA ^T and J are symmetric and have the identical right and left eigenvectors, and, in general, the right and left singular vectors V _A and U _A of a square matrix A differ from its right and left eigenvectors S _A and , i.e., V _A  ≠ S _A and .

²In general, neither the row sums nor the column sums of matrix A are unity. Forcing them to be so is strongly discouraged.

Note. The comments refer to whether both eigenvectors have the same number of nonzero entries.

Note. EIES = Electronic Information Exchange System. The five assumptions about D _p|q in Eq.(44) are (1) no scaling: B = C = I; (2) scaled by rows: the ith diagonal entry of matrix B is (∑ _j a _ij )⁻¹ and C = I; (3) scaled by columns: B = I and the jth diagonal entry of matrix C is (∑ _i a _ij )⁻¹; (4) scaled by rows and columns: the ith diagonal entry of matrix B is (∑ _j a _ij )⁻¹ and the jth diagonal entry of matrix C is (∑ _i a _ij )⁻¹; and (5) scaled by rows and columns: the entries B1 and C1 of diagonal matrices B and C are chosen, such that matrix BAC or (BAC) ^T is doubly stochastic. In a Perron eigenvector, the integer ranks the preceding real number.

* = Perron eigenvalues.

Note. EIES = Electronic Information Exchange System. The five assumptions about D _p|q in Eq. (44) are (1) no scaling: B = C = I; (2) scaled by rows: the ith diagonal entry of matrix B is (∑ _j a _ij )⁻¹ and C = I; (3) scaled by columns: B = I and the jth diagonal entry of matrix C is (∑ _i a _ij )⁻¹; (4) scaled by rows and columns: the ith diagonal entry of matrix B is (∑ _j a _ij )⁻¹ and the jth diagonal entry of matrix C is (∑ _i a _ij )⁻¹; and (5) scaled by rows and columns: the entries B1 and C1 of diagonal matrices B and C are chosen, such that matrix BAC or (BAC) ^T is doubly stochastic. In a singular vector, the integer ranks the preceding real number; , .

Note. , and matrices A ^T A, AA ^T and J are all symmetric and have the identical right and left eigenvectors.