We see Billy heading to catch a plane out of Boston while he talks to Dignam on the phone. Now that he knows Costello has someone high up in Special Investigations, he knows he's going to get found out soon. The only way for him to survive is to find the police rat before the police rat finds him.
So Billy tells Dignam to send some false info through the department to help narrow down which person is passing stuff on to Costello.
Billy decides to stick around, and the next time we see him, he's having coffee with Madolyn. He asks her about her relationship, and she tells him. Then he asks what she'd say if he walked in and saw Madolyn and Colin together, and Madolyn says she'd lie.
Next time we see Madolyn, she's moving some of her stuff into Colin's apartment. He asks her why she works for the state when her degree could get her a way better job. She says she believes in public service.
Frank Costello calls to ask why Colin let his phone die. He seems concerned that Colin's concentration has been slipping since he's gotten serious with Madolyn. He ends the convo by giving Colin an ultimatum: find the rat in my operation, or something bad will happen to Madolyn.