The spectrum of the electron spin-spin interactions largely determines which mechanism is responsible for the growth of the nuclear spin polarization in dynamic nuclear polarization (DNP). When electron spin-spin interactions are weak and their spectrum is narrow, the solid effect (SE) dominates the process. When they are stronger, the cross effect (CE) and thermal mixing (TM) come into play. Then a narrow spectrum favours the CE - that is an exchange of electron Zeeman energy with the nuclear spins - and a broad spectrum also TM - that is an exchange of electron spin-spin interaction energy with the nuclear spins. Moreover, the spectrum of the electron spin-spin interactions critically determines the rate of spectral diffusion of electron spin polarization across the electron spin resonance (ESR) line, and the associated conversion of electron Zeeman energy into electron spin-spin interaction energy. This way electron spin-spin interactions indirectly influence the DNP process. The present work describes Monte Carlo simulations of the spectrum of these interactions for approximately spherical radicals in glasses and analytical approximations of the simulation results. As an example application expressions for the relative strengths of the energy flows due to the CE and TM are derived.