Ah ha! The purple ones are Grape Hyacinth. I love them because they are practically unkillable. They are absolutely great in those sunny, dry areas. But they can also become a nuisance and grow quite big in rich soil that gets a little shade. You already noted how they can naturalize (spread like wildfire).
They produce seeds in papery capsules after the the flower are done and will grow from those seeds if scattered around. Small dug up bulbs will turn up elsewhere and will do their best to grow no matter where they end up.
There are other color flower cultivars including white, pink and white edged sky blue. Larger flowered vs. smaller. But the small purple ones are most common.
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