This is a tough one!
I don't have any children so therefore all I can only attempt to put myself in the (teens) shoes. Oddly enough parents seem to erase their own childhoods when it comes to attempting to understand their children.
Generally speaking there are two reasons why teens and children runaway.
1. They consider themselves to be "adults" and they resent the rules and discipline established in the household. Essentially they want FREEDOM!
2. They feel trapped and depressed. This could be tied to abuse in the home, being bullied at school, not liking how they look, feeling hopeless. Essentially they too want to ESCAPE.
Generally speaking adults and teens live in parallel universes that rarely intersect. Parents are very busy providing the essentials for the children and their household in addition to maintaining their own relationship and friendships.
Teens on the other hand believe they are "grown" and rarely seek guidance from their parents. If they're having problems they'd rather talk it over with their friends or peers. If they go to an adult with the problem it affirms that they themselves are not adults!
This is the last thing they want to admit.
Another reason why teens don't consult with their parents is because they believe they already know what their parents will say and it's not what they want to hear!
Bottom line is happy teens living in happy and loving homes don't run away.
Therefore the first step as a parent is to assume they left because of you or issues at home. The next step is put yourself in their shoes.
Go over the previous conversations and interactions you had leading up to them running away. Were you spending time together or was she being distant and secretive? Has she talked about friends you've never met?
This will let you know if her actions were "impulsive" or "calculated".
The only way I believe a teen goes back home is if (they feel being home is better than running in the streets). If you really thought you were going to catch hell when you got home you'd avoid going back!
On the other hand if a parent reached out and said:
"I just want you home safe and sound, to sit down with you and talk to understand what (we) can do to make life better for you. Whenever you're ready to come home call me day or night. Never forget you are loved more than you know and you do have a place to call home. Love, Mom."
Afterwards you just let it go. Don't initiate anymore calls or texts.
The ball is her court.