Controlling random light is a key enabling technology that pioneered statistical imaging methods like speckle microscopy. Such low-intensity illumination is especially useful for bio-medical applications where photobleaching is crucial. Since the Rayleigh intensity statistics of speckles do not always meet the requirements of applications, considerable effort has been dedicated to tailoring their intensity statistics. A special random light distribution that naturally comes with radically different intensity structures to speckles are caustic networks. Their intensity statistics support low intensities while allowing sample illumination with rare rouge-wave-like intensity spikes. However, the control over such light structures is often very limited, resulting in patterns with inadequate ratios of bright and dark areas. Here, we show how to generate light fields with desired intensity statistics based on caustic networks. We develop an algorithm to calculate initial phase fronts for light fields so that they smoothly evolve into caustic networks with the desired intensity statistics during propagation. In an experimental demonstration, we exemplarily realize various networks with a constant, linearly decreasing and mono-exponential probability density function.