I took this picture in Arizona, but never knew it's name. I tried to look it up on the net, but the closest thing I found on the net is mimosa.....but they never show red, only pink
I also thought that it was a bottle brush, but then I went to SuniBC's page looked at her pic. there is another one of that bush when I clicked on it and the clicked again it brought the picture MUCH bigger. My mom had bottle brush, but I don't remember them having little white somethings in the middle, but it may just be the season, but Bottle Brush would also be my guess.
But what a fine spirit will mark Christ’s rule! “Come to me, all you who are toiling and loaded down,” said Jesus, “and I will refresh you. Take my yoke upon you and learn from me, for I am mild-tempered and lowly in heart, and you will find refreshment for your souls. For my yoke is kindly and my load is light.”—Matt. 11:28-30.
I also went and looked. You were right when you said it expanded into a MUCH larger photograph. I'd be willing to bet good money that it is indeed a Bottle Brush. We are sort of looking straight at the flower and that makes it a little harder to identify. Here is a photo that shows the Bottle Brush flower a little bit better.
Not going, not coming, Rooted, deep and still. Not reaching out, not reaching in. Just resting. At the center, A single jewel, the flawless crystal drop. In the blaze of it's brilliance is The Way Beyond.
Looks like Bottle Brush to me too. There are a lot of varieties though and I don't know which one this is. If you're interested in seeing the different varieties do a Google image search for Callistemon and Melaleuca. There are several varieties of each.
Humankind has not woven the web of life. We are but one thread within it. Whatever we do to the web, we do to ourselves. All things are bound together. All things connect.
At first glance I thought bottle brush but its certainly not a bottle brush bush as the leaves are wrong and they dont have those white bits in them. Id say it is a mimosa probably one thats been grafted to have this colouring.
Here is a photo of Bottle Brush that I found on Flickr. I still say it is Bottle Brush. My mother used to have Bottle Brush in our backyard when I was a kid and several of my neighbors have it too.
This is a Calliandra which I though looked like your pic, but once again the leaves are hard to see. 
Red Powder Puff - Calliandra haematocephala (the leaves look the same too)