Here it is worth recalling that the "empty" IGM, including the voids, is permeated by photons, by which we mean the EBL at all photon energies. Particles with energy EP 1018 eV interact with other particles and photons, so that all of space becomes a (passive) particle physics laboratory. This fact is partially illustrated in Figure 13. At some rate, galaxy-supplied particles and magnetic fields will diffuse out of the walls and filaments into the voids. The cosmic voids are also the place to look for a pre-galactic, or primordial field. However at this point, the detection of magnetic fields in cosmic voids still lies mostly in the realm of Gedanken experiments.
For VHE particles the time of arrival, deflection, energy, and composition can all be measured in principle, and calculated. Figure 13, is based on ideas originally advanced by R. Plaga , and is a concept illustration of reactions and deflections in a intergalactic magnetic field. It is based on the existence of a burst event producing - rays (-ray burster), neutrinos, and/or hadrons, all at very high energies. The first VHE -rays could be products of original VHE hadrons.
Figure 13. Propagation path 1 applies to photons of order ~ 1 Tev that propagate directly through the IGM. Photons of 100 TeV initiate a chain of reactions involving i.r. photons of the EBL, e+, e-, and, at lower energies, the CMB photons to produce a photon-particle cascade. Case 3 occurs if BIGM is sufficiently high to prevent the e+, e- from continuing the cascade along the line of sight to us.
If the electron Larmor radii for BIGM 10-12 G are sufficiently large, we might see a coherent photon cascade, where lower energy photons in path 1 mark the un-delayed arrival time. For BIGM 10-12 G we would observe a photon halo around the source of the bursting event. In this case (3 in Fig. 13), the halo size and spectrum would lead to an estimate of the magnetic field strength in the intergalactic vicinity of the source. Depending on the parameter space, the radiation could be pair annihilation radiation at 0.5 MeV, or electron/positron synchrotron radiation. Time delays, and energy losses for intergalactic protons propagating in an intergalactic magnetic field have been calculated and discussed by Stanev et al. . Observable effects for proton propagation here refer to a higher regime of BIGM, in the vicinity of 10-9 G. The reader could consult the interesting paper by Stanev et al.  for an informative discussion of loss mechanisms (GZK, Bethe-Heitler, etc.), and other UHECR propagation effects.
I wish to thank my many co-workers for their contributions to the material I have presented here. I thank the organizers for their invitation to participate in this very interesting meeting, and Dr Henry Glass for his invaluable assistance in the preparation of this manuscript.
This work supported by the U.S. Department of Energy, to LANL, and NSERC (Canada).