**Node by node in networks, mathematician Edward R. Scheinerman models the motion along direct routes.**

The Internet…interstate highways…interpersonal relations… e-mails…cell phones…cellular growth…Google. Welcome to a world composed entirely of networks. Edward R. Scheinerman thinks that’s just fine. A professor of mathematics in the Whiting School of Engineering’s Department of Applied Mathematics and Statistics, Scheinerman loves to model and analyze networks—and he uses advanced mathematics to do just that across many engineering disciplines.

“Networks are everywhere,” Scheinerman explains, “from looking inside of a cell at how genes interact to finding your way from Baltimore to Colorado. My field is discrete mathematics, especially graph theory, partially ordered sets, random graphs, and combinatorics, with applications to robotics and networks. The explosion of networks in the world has also led to an explosion in discrete mathematics.

“MapQuest is a perfect example,” Scheinerman observes. “You want to get from point A to point B.” Instead of simply seeking the shortest distance, however, “you have to follow a network. When you come to a stoplight, you have to make a decision. There is no, ‘Well, maybe I’ll take a half left,’ because there is no road there.”

This fall, Scheinerman returned to campus after a year’s sabbatical, during which he completed the second edition of his popular textbook, Mathematics, a Discrete Introduction (now available from Brooks Cole). He also conducted applied mathematics research; began “and mostly completed,” he says, a new book on computer programming for mathematicians; and was a visitor at the Center for Computing Sciences in Bowie, Maryland.

The mathematician has served the Whiting School as chair of his department and recently as interim associate dean for academic affairs. He received his master’s (1981) and PhD (1984) degrees in mathematics from Princeton University, where he was first introduced to the charms of discrete mathematics and graph theory.

In his textbooks, Scheinerman provides a better understanding of discrete mathematics, graph theory, and ways that networks can explain human interaction systems. The first edition of his Mathematics: A Discrete Introduction was published in 2000. Invitation to Dynamical Systems (Prentice Hall) was released in 1996.

“When I started at Hopkins in the mid-’80s,” Scheinerman recalls, “there was no introductory-level course in discrete mathematics, and I thought it was important that there be one.” Much to his surprise, then-department chair Robert J. Serfling suggested he develop one. Out of this experience, Scheinerman produced Mathematics, A Discrete Introduction. “That’s one of the reasons I love Hopkins,” he continues. “It has afforded me the opportunity to make changes, to have some influence, to make a difference.”

Another area where the mathematician is making a difference is in helping computer science more directly address the needs of mathematicians. In fact, the book Scheinerman has almost completed seeks to raise the profile of the programming language C++ among this group. “There has long been a great need” for such a resource, he notes. “C++ is widely used in the computer engineering world, but it has not been widely embraced by the mathematics community.”

The mathematical problems that Scheinerman tackles are not merely abstract. “Some of my papers would perhaps be of interest only to mathematicians,” he says. “Others deal directly with specific application areas, such as robotics.” What does tie them all together is that they all draw on the relatively new field of mathematics known as “discrete.”

“In discrete mathematics, you’re either A or B. You are not something muddled in between.” Edward R. Scheinerman

**What Makes This Math “Discrete”? **

“Most people study continuous mathematics,” Scheinerman explains. “Calculus, the study of things in motion, is one of the crowning achievements of continuous mathematics.” Calculus developed to answer such questions as, “How does the Moon move around the Earth, and why?” For this reason, he says, “for a very long time, there was no clear distinction between mathematics and physics. To ask whether Newton was a mathematician or a physicist would not have made any sense. In his time, the two things were intermixed.”

Fast-forward to the 20th century, and a world of finite (as opposed to continuous) mathematical problems. “We have invented everything from computers to MapQuest, and Google to the Rubik’s Cube. Not one of these is readily analyzed by continuous mathematics,” Scheinerman says. “Likewise with computers, everything is digital. In discrete mathematics, you’re either A or B. You are not something muddled in between.”

Much of mathematics is application-driven, and the need for a mathematics that can deal effectively with motion through networks—as opposed to smooth and continuous motion—has grown up with the computer era. Using discrete mathematics to model and understand networks finds applications in many fields. “One of my colleagues at Hopkins, Carey Priebe [professor of Applied Mathematics and Statistics], analyzed the e-mail communications of Enron employees to better understand how the social networks differed from corporate networks,” Scheinerman explains. “Nothing happens in engineering these days that doesn’t have a mathematics connection.”

No network, it seems, can escape the discrete mathematician’s ability to capture it in a graph that can be studied to better understand the connections between individuals—human or otherwise.

“Graph” is really somewhat a misnomer, the mathematician says, because the graphs he makes bear little resemblance to a typical pie or bar chart. Scheinerman’s graphs model discrete connections between entities, or nodes. Depending on the network one hopes to graph, there can be hundreds—or thousands—of direct connections between various nodes. Consider for example, the e-mail communication paths within an organization, or the connections represented by the growth and changes inside of a cell. The graphs reveal patterns, and Scheinerman tests theorems against these patterns, looking for mathematical proofs that can be used to discover the “truths” lying hidden in the complex interconnections.

We have a Johns Hopkins mathematician to thank for this specific use of the word “graph.” J. J. Sylvester, the legendary 19th-century British-born mathematician, coined the term “graph” for the kind of representation used by Scheinerman. (Perhaps contributing to Sylvester’s legendary status is the fact that in 1877, as a condition for accepting the position as the University’s first mathematics professor, he stipulated that his annual salary of $5,000 be paid in gold.)

Indeed, language and logic are exceedingly important to mathematics. As the chapter titles in Mathematics: A Discrete Introduction attest, mathematics seems more closely connected with philosophy than science, with which mathematics is often linked. Such chapter titles as “Why?” “The Nature of Truth,” and “If-then,” would sound right at home on a shelf next to Kierkegaard, Bonhoeffer, and Kant.

“The whole idea of ‘definition, theorem, proof’ is key,” Scheinerman explains. “This is universal to all of mathematics, to a mathematical way of knowing.” As it turns out, this is the exact opposite of the scientific method of discovering truth, which is to prove by empiricism. In science, “when you make a statement that has a one-in-a-million exception, no one is going to criticize you,” he says, “especially in a biological or social science.”

In mathematics, however, “truth means 100 percent. When we say a theorem is true, it is true without exceptions. Period. You cannot achieve that by empirical methods,” Scheinerman says with the certainty of a man who speaks mathematical truth. In all probability, he can produce a graph to prove it.

**To learn more about Edward R. Scheinerman, visit www.mts.jhu.edu/~ers . To see a graph of how the Applied Mathematics and Statistics faculty members are networked in their research, visit www.ams.jhu.edu/ams/ research/general.html**