On the Randomness of Eigenvectors Generated from Networks with Random Topologies

Jack W. Silverstein
1979 SIAM Journal on Applied Mathematics  
A model for the generation of neural connections at birth led to the study of W, a random, symmetric, nonnegative definite" linear operator defined on a finite, but very large, dimensional Euclidean space ]. A limit law, as the dimension increases, on the eigenvalue spectrum of W was proven, implying that realizations of W (being identified with organisms in a species) appear totally different on the microscopic level and yet have almost identical spectral densities. The present paper considers
more » ... the eigenvectors of W. Evidence is given to support the conjecture that, contrary to the deterministic aspect of the eigenvalues, the eigenvectors behave in a completely chaotic manner, which is described in terms of the normalized uniform (Haar) measure on the group of orthogonal transformations on a finite dimensional space. The validity of the conjecture would imply a tabula rasa property on the ensemble ("species") of all realizations of W.
doi:10.1137/0137014 fatcat:poa6y3o6v5dhnehxujalj72ciu