There is no single profile that fits a child molester or a pedophile or a sexual abuser.  But there are a number of discrete profiles that can differentiate pedophiles or men who have committed child sexual abuse from people who don't sexualize children.
 

The Sexually and Emotionally Aroused

There are those pedophiles who are primarily sexually and emotionally aroused at children in the absence of any sexual arousal to adults.  These are people who are drawn to children occupationally, emotionally, psychologically, and who live their lives in relation to children and not in relation to adults.
 

The Impulsive and Exploitative

There are child molesters who are more impulsive and exploitative, who take advantage of the vulnerability of children, of the fact that children will not refuse or resist, and the fact that children can be easily manipulated.  The exploitative child molester is characterized by problems in impulse control, problems in antisocial behavior.  That kind of person generally not only has sexual offenses against children, but may also have other sexual offenses such as exhibitionism or voyeurism, and they also have general antisocial problems.
 

The Sexually Compulsive

Another child molester profile coincides with those people who have major mental illness who either have sexual compulsions, and the major mental illness may be as a result of organic conditions, or have loss of control due to alcohol or to brain damage or to senility.

All of those profiles can define a discrete group of men who are sexually attracted to children.