What you see could be related to a few different things, actually. You have to eliminate other possibilities before being sure it is the developer.
Edge effects have to do with two chemical effects that happen simultaneously. The first is the depletion of developer. This happens when developer is very dilute and there is a lack of agitation. In stagnant liquids, the diffusion rate is typically in the order of 1 cm/day (as opposed to 1 cm/sec for gases). That means that fresh developer molecules take a long time to diffuse to areas on the film where developer has become depleted due to more rapid development. When this diffusion takes places on the border between very bright and dark areas, the developing rate does something that is similar to unsharp mask in software sharpening. It drops a little on the dark side of the edge, causing that side to be slightly darker than the rest of the dark area as the developer gets "robbed" there, and it inceases slightly on the bright side of the edge as the fresh developer flows in, causing the edge to be slightly brighter than the rest of the bright area. This is visually similar to sharpening, and has the same effect if done correctly: it gives the image more acutance (but not more resolution!). The second effect is build-up of bromide, which is released when the silver bromide is reduced to metallic silver. The bromide acts as developer restrainer, i.e. the higher the concentration, the slower development becomes. Both effects combine to preserve highlights with dilute Rodinal (which is sensitive to bromide). Both effects can be effectively eliminated with constant or over-zealous agitation. Just like unsharp mask, the effect when exagerated can be garish and even objectionable. Dilutions of 1:50 to 1:100 should be okay with intermittent agitation, but note that Bob Schwalberg specifically warned against 1:100 dilution with films such as TriX and HP5.
Some other possibilities:
i) Lens flare or ghosting due to dirty or scratched elements or poor coatings will cause spillover of light into darker areas of the image. It is not easy to see the opposite, where lighter areas become darker. Generally, such images will also have poor local contrast and an overall muddy or foggy look. If it happens under normal lighting, with excellent lenses that are multi-coated, your problem is something else. Suspect the filter too, if you have one on.
ii) Some films have no or weak anti-halation backing, which causes light to bounce back from the carrier that the emulsion is coated on, into the emulsion to illuminate it a second time, but diffused. This is seen as halos around bright objects and ghosting inside the darker areas. It is fairly easy to find out whether your particular film has this characteristic, as it is well documented for all films that are still available. Lucky SHD100 is a good example, but I am also aware of earlier Kodak films, and also the Rollei Retro films.
iii) It may be that you see developer depletion, but it is not necessarily accompanied by visible acutance edge effects. This will be the case if you start with too little developer, that is locally exhausted before the film is fully developed. It will cause uneven and unpredictable development, and should be discouraged. You should not use less than 10 ml concentrate per standard roll, 7 ml as an absolute minimum. So calculate your volume after mixing from there. For 35 mm and 120 film, I always use 10 ml made up to 500 ml, regardless of whether I need less to cover the reel. Resist the temptation to develop two rolls of 120 on one reel in 500 ml of 1:50. As Parodinal is so cheap (it costs me less than 10 US cents per dose of 10 ml to make), there is no conceivable reason to skimp on a few ml when the film costs $5 to start with. Even commercial Adonal/Rodinal or whatever you use, should still be very cheap compared to the film itself, so using less than the recommended dose is penny wise pound foolish. As for 1:100 stand development, unless your highlights are exposed completely out of whack, I cannot really understand the need for this approach as a general one. The problem in my view is better solved with better exposure. Stand development comes with other potential problems, and will rarely produce an optimal tone curve for modern films. While it may solve a specific problem when encountered, I think it creates more problems than it solves for general use. I am talking from the perspective of printing negatives, not scanning them, of course.