Harris appeared to be a psychopath, lacking empathy, Levin said. Harris wrote in his diary, according to the sheriff's department, "I'm full of hate and I love it." He wrote in his 1998 yearbook, on the pictures of almost every student, words including "worthless," "die" and "beat."

Still, people with mental problems generally are not automatically violent. A 2006 study in the American Journal of Psychiatry found that only 5% of all violent crimes in Sweden are committed by people with severe mental illness, which is likely to be similar in other countries such as the United States, the study said.

Russ Hanoman, a friend of Lanza's mother, described Lanza as "very withdrawn emotionally." He and other acquaintances reported Lanza's mother said that he had Asperger's syndrome, a diagnosis of high-functioning autism that CNN has not been able to confirm. There is no link between autism and planned violence, autism groups have noted.

Killers tend to blame others, not themselves, for their problems. Mass killers tend to target people whom they imagine would torment them, or whom they blame for their distress, Ash said. Why Lanza would target young children remains mysterious. Did he want to provoke outrage? Ash wonders. Is it because children seem happy and carefree?

"I believe he also sought to get even with society at large by killing the most cherished members: our children," Levin said.

Feeling alone, and with access to guns

Almost all the killers Levin has studied appeared to lack social support, having no one to turn to when in trouble. They reject their peers, and they are in turn rejected -- "alone in a psychological sense," Levin said.

Often the shooter has experienced a catastrophic loss, such as rejection from a girlfriend or getting fired from a job, Levin said. Parents may also inadvertently push teens and young adults over the edge by, for example, pressuring them to be successful.

Access to and training in the use of firearms is another major factor in executing a massacre like this, Levin said. The Bureau of Alcohol, Tobacco, Firearms and Explosives confirmed to CNN that Lanza and his mother frequented several gun ranges over the past several years. Those who knew the mother said she kept many weapons, including assault rifles and handguns, in her home. Investigators believe Lanza took his mother's guns to Sandy Hook Elementary School.

Although people have a constitutional right to gun ownership, they should not make them readily available to children who might be troubled or severely mentally ill, Levin said. School shootings in recent decades often happened with weapons taken from parents.