Spectrum of the Elimination of Loops and Multiple Arrows in Coupled Cell Networks

A uniform lift of a given network is a network with no loops and no multiple arrows that admits the first network as quotient. Given a regular network (in which all cells have the same type and receive the same number of inputs and all arrows have the same type) with loops or multiple arrows, we prove that it is always possible to construct a uniform lift whose adjacency matrix has only two possible eigenvalues, namely, 0 and −1, in addition to all eigenvalues of the initial network adjacency matrix. Moreover, this uniform lift has the minimal number of cells over all uniform lifts. We also prove that if a non-vanishing eigenvalue of the initial adjacency matrix is fixed then it is always possible to construct a uniform lift that preserves the number of eigenvalues with the same real part of that eigenvalue. Finally, for the eigenvalue zero we show that such a construction is not always possible proving that there are networks with multiple arrows whose uniform lifts all have the eigenvalue 0, in addition to all eigenvalues of the initial network adjacency matrix. Using the concept of ODE-equivalence, we prove then that it is always possible to study a degenerate bifurcation arising in a system whose regular network has multiple arrows as a bifurcation of a bigger system associated with a regular uniform network.