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Tony Phillips' Take on Math in the Media A monthly survey of math news |
Steven Strogatz's series of mathematics columns, "The Elements of Math," ran once a week, January 10 through May 9, 2010 in the Opinion Pages of the New York Times (all of which are indexed). Now he has just completed "Me, Myself and Math," a new six-part series focusing "on how the subject I love--math--relates to the subject we all love--ourselves."
A report in the September 27, 2012 Nature, by Fragkiskos Papadopoulos, Dmitri Krioukov and collaborators, examines the emergence of scaling in growing networks from a new perspective. In "Popularity versus similarity in growing networks," the authors review various hypotheses for explaining how "the distribution of the number of connections possessed by nodes follows power laws, as observed in many real networks." The most common explanation ("preferential attachment") is that new connections go preferentially to more popular nodes. But the authors marshall evidence to show that "popularity is just one dimension of attractiveness; another dimension is similarity." They construct a geometric model for this two-dimensional analysis. Nodes are added to a network one by one, with each new one linking to the $m$ most attractive nodes present ($m$ depends on the model). "The simplest proxy for popularity is node birth time" (because older nodes will have had more chances to be linked to) whereas similarity is measured by assigning to node number $s$ a spot $\theta_s$ on the circle: the most similar nodes will be those at smallest angular distance. In their model, node number $t$ will link to the $m$ pre-existing nodes $s$ minimizing $s\theta_{st}$ where $s$ is the birth rank and $\theta_{st} = \theta_{ts}$ is the angular distance between nodes $s$ and $t$.
How similarity attracts; in these images, $t=3$ and $m=1$. Here node $1$ is the most popular node, but in (a), node $3$ connects to node $2$ which has a smaller $s\theta_{st}$, since $2\theta_{23}=2\pi/3<1\theta_{13}=5\pi/6$. On the other hand, in (b), since $1\theta_{13}=2\pi/3<2\theta_{23}= \pi$, node $3$ connects to node $1$. Images courtesy of Dmitri Krioukov.
The authors discovered that in a planar representation, if polar coordinates are used to place node $s$ at $(r_s,\theta_s)$, where $r_s = \ln s$ and $\theta_s$ is as above, then minimizing $s\theta_{st}$ is well approximated by minimizing the distance between node $t$ and node $s$ in the hyperbolic metric.
"An optimization-driven network with $m=3$ is simulated for up to $20$ nodes. The radial (popularity) coordinate of new node $t=20$ is $r_t=\ln t$. ... This node connects to the three hyperbolically closest nodes." Image adapted from Nature 489 537.
The authors observe: "As opposed to preferential attachment, our optimization framework accurately describes the large-scale evolution of technological (the Internet), social (trust relationships between people) and biological (Escherichia coli metabolic) networks, predicting the probability of new links with high precision." In particular, their model predicts clustering ("the probability that two nodes connected to the same node are also connected to each other") much better than preferential attachment, where "clustering is asymptotically zero, whereas it is strong in many real networks."
The New Criterion for October 2012 ran an excerpt from Edward Frenkel's forthcoming book, Love and Math. The Berkeley professor describes what it was like to be a young, brilliant, Jewish math student in the Soviet Union. In particular he takes us with him to Moscow in 1984, when he was a 16-year-old high-school senior applying for admission to the Mechanics/Mathematics Department at Moscow State University. He had been warned "Do you know that Jews are not accepted to Moscow University?" but he applied nonetheless. He performed perfectly on the written exam including the fifth problem, traditionally "deadly" and "unsolvable." Next was the the oral examination. He came and drew the two questions he would have to answer. "(1) With a circle inscribed in a triangle, what is the formula for the area of the triangle in terms of the radius [and the perimeter]?, and (2) What is the derivative of the ratio of two functions (formula only)? Easy questions, which I could answer in my sleep." He raised his hand to show he was ready but "bizarrely, [the examiners] ignored me, as if I did not exist. They looked right through me. I was sitting like this, with my hand raised, for a while. No response." Finally two middle-aged men entered the room and were pointed to Frenkel. These would be his special interrogators. When he started his explanation, one interrupted: "What's the definition of a circle?" "The set of points in the plane equidistant from a given point." "Wrong! It's the set of all points in the plane equidistant from a given point." And so it went. He was officially informed that he had failed. A final bizarre touch: in the elevator after the test, when he was alone with one of the examiners, the man congratulated him on his "really impressive performance," suggested that he apply to the Moscow Institute of Oil and Gas ("They take students like you there") and wished him good luck.
An Editor's note at the end of the article lets us know that Frenkel did find some unofficial support, finished his education and started original research. His mathematical discoveries earned him an invitation to Harvard as Visiting Professor at age 21, and, at 28, his appointment as Full Professor at Berkeley.
Tony Phillips
Stony Brook University
tony at math.sunysb.edu