In the past month alone, eight multiple-shooting incidents have occurred in the United States and Europe, leaving 70 dead, including the three Pittsburgh police officers killed April 4.
In the criminology trade, they are known as mass murders, and they usually share several common traits, experts say.
The police shootings here, and the actions of suspect Richard Poplawski, fit some of these patterns, they say, but in other ways, the Pittsburgh case and another incident in which four Oakland, Calif., police officers were killed are atypical.
A common factor in most mass murders, said criminologist Jack Levin of Northeastern University in Boston, is revenge.
"At the most basic level," Dr. Levin said, "the revenge is directed against family members," who are the main victims in about 30 percent of all mass killings.
"The next most likely target is the workplace, where an ex-worker who was fired or laid off comes back shooting, killing the boss and co-workers."
And finally, he said, there are mass killers who blame society in general for their problems and may walk into a mall and open fire, or target certain groups for destruction, including, occasionally, the police, because "the police are representatives of society."
Pittsburgh already had two infamous examples of mass killers who targeted certain groups, Dr. Levin said, in the twin rampages in early 2000 of Ronald Taylor, a black man who shot five white men, killing three, and Richard Baumhammers, a white man who shot six people who belonged to racial or ethnic minorities, killing five.
Besides an embittered sense of revenge, the experts said, another common thread in these killings is access to high-powered weapons, which is a particularly American phenomenon.
"You need to understand how our society permits easy access to lots and lots of guns if you're going to understand why these kinds of killings happen so much in the United States as opposed to somewhere else," said David Hemenway, a health policy professor at Harvard University.
"As far as I can tell, the psychological problems of these killers are not unique to the United States, but what is unique is that it's so easy for people in the U.S. to get access to weapons."
Daniel Nagin, a criminologist at Carnegie Mellon University, agreed. "It's technologically impossible to kill a lot of people very quickly without access to these assault weapons," he said.
Besides the revenge motive, mass murderers usually share certain other psychological or behavioral characteristics, the experts said.
Grant Duwe, a criminologist with the Minnesota Department of Corrections who has written a history of mass murders in America, cited five:
• The killer blames others for his problems.
• He is much more likely to have a mental illness, particularly paranoid schizophrenia, than homicide perpetrators in general.
• He is often a loner, with few friends or social connections.
• He carefully plans his attacks, taking days to months to get ready.
• He is much more likely to be suicidal than a typical killer. "Because the mass murderer considers his life no longer worth living, he will either kill himself or force the police to kill him," Dr. Duwe said.
For his doctoral dissertation, Dr. Duwe examined 909 cases of mass murder in America from 1900 through 1999 and found that out of 116 mass public shootings in that data set, 47 percent ended with the killer committing suicide, compared with less than 5 percent among homicide offenders in general.
During the incident April 4, Mr. Poplawski called friends during the shootout with police to say he was going to die that day, although he ended up surrendering to police.
That occasionally happens, said Eric Hickey, director of forensic studies at Alliant International University in San Diego, who has studied mass murders for several years.
"Most of these guys don't plan an exit strategy," he said, "and if they get caught, it's not by design -- it's more that they've vented and they're just out of ammo, and so they sit down and say come and get me."
While the Stanton Heights shootings shared some of the features cited by the experts, it also was unusual in certain key ways.
Franklin Zimring, a renowned criminal justice researcher at the University of California at Berkeley, said the fact that Mr. Poplawski's mother was spared is highly unusual.
In most mass murders, family members are often the first to be killed, especially if they are in conflict with the shooter.
After hearing details of the Pittsburgh incident, he said "mom is pretty strong in that relationship, and the significant factor here seems to be that the mother was going to get him thrown out of the house, and from his perspective this was almost like capital punishment, because it appears he was so dependent on her."
An incident March 21 in which a man killed four Oakland, Calif., police officers and then was shot to death himself also does not fit the usual mass killing pattern, Dr. Zimring said.
The suspect in that case was facing a return to prison when he was pulled over at a traffic stop and probably shot the first officer in a state of panic, "and all the other killings followed from that first one."
The researchers were divided on how much of a role the poor economy has played in the recent surge of multiple shootings.
"Is the economy a stressor?" asked Carnegie Mellon's Dr. Nagin. "Probably. But there are lots of stressors in life, so I'm quite skeptical that there's some systematic explanation for these really tragic events."
Northeastern's Dr. Levin said he doesn't believe the economy directly caused any of the killings but does think it could have been a critical trigger point.
"The economy is not the cause," he said, "but bad economic times create more catastrophic losses, and I can tell you in almost every one of the mass murder cases, there is some triggering event that takes the form of a catastrophic loss, such as a job loss, a loss in the stock market, a loss of a relationship -- some kind of loss that precedes the attack and kind of pushes the killer over the edge."
Dr. Hickey said in the recession of the early 1990s, the United States was seeing almost one multiple killing per week.
Gradually, that declined to about one per month, but "you knew that couldn't last and it started going back up a year ago or so, and in the past three or four months, there has been a spike, which could be an anomaly, but I don't think so."
"I would be shocked if we didn't continue on this track for a while."
While mass murders often focus attention on the kind of people who commit such crimes, it's important not to lose sight of larger issues such as gun regulations, said Harvard's Dr. Hemenway.
Dr. Hemenway drew the analogy of the accident investigation research he sometimes does. It is common in that field to find that many serious crashes occur at one intersection but hardly any at a nearby intersection.
"So what makes for the difference?" he said. "It's not the people who are driving through the intersection because they're the same at each spot, but instead, it's the way we set up the lights or crosswalks or whatever."
Or, as Dr. Hickey put it: "I'm not against guns, but I just think there are lot of people who shouldn't have guns, yet they do."
Mark Roth can be reached at firstname.lastname@example.org or 412-263-1130.