This month's topics:
Math, Models and Mystification
Paul Krugman wrote about "economics professors who are totally convinced that [people like PK] are ignorant, unwashed types who don't know anything about modern macroeconomics" in his New York Times column (October 4, 2010): "I believe that what we're looking at is people who know their math, but don't know what it means: they can grind through the equations of their models, but don't have any feel for what the equations really imply. ..."
Topology is not enough
to determine the vulnerability of electrical distribution grids. Writing in Chaos (20, 33122), Paul Hines, Eduardo Cotilla-Sanchez and Seth Blumsack state: "A number of articles have recently used topological (graph theoretic) models to assess vulnerability in electricity systems. In this article, we illustrate that under some circumstances these topological models can lead to provocative, but ultimately misleading conclusions."
A schematic example of the difference between a topological (nearest neighbor) model for system vulnerability, and one based on Kirchoff's laws. Suppose that node 2 fails in power grid a. In the simplest topological model (b.), the load from 2 is shared by its nearest neighbors, in this case 1 and 3. In a physically more realistic model (c.) the failure causes an open circuit: the load shifts to nodes 5 and 6, and node 3 has no power flow. Image adapted from Chaos, 20, 33122.
To justify this assertion, they consider three measures of the vulnerability of a network. Two of them are graph-theoretical in nature: characteristic path length (the average distance between node pairs) and connectivity loss, defined as the expected value over all nodes i of 1 - g(i)/G, where G is the number of generators in the network and g(i) is the number that can be reached from node i across unfailed links. The third one, which they devised, is blackout size, calculated using equations that "capture the effects of Ohm's and Kirkhoff's laws." They also consider different kinds of attack-vectors on the nodes of a network, including degree attack (nodes with the highest connectivity are removed first), maximum-traffic attack (nodes transporting the highest amounts of power are removed first) and betweenness attack (nodes traversed by the highest number of shortest paths are removed first). Their conclusion is that "the highest impact attack-vectors differ in each of the three vulnerability metrics. Thus, one would draw different conclusions about the greatest risks depending on the vulnerability measure used. From the path length metric, betweenness attacks appear to have the greatest impact. From connectivity loss, one would conclude that degree-based attacks are most dangerous. From the blackout model, max-traffic attacks appear to contribute most to vulnerability." So topological models need to be supplemented by "physics-based" models to avoid "misallocation of risk-mitigation resources." This article was picked up in the "Editor's Choice" section of Science, October 29, 2010.
Markov models for Bengalese finch songs
Sonograms (frequency spectrum plotted against time) of seven basic Bengalese finch song syllables. The frequency range is 1-10kHz; duration of sample is written above the sonogram. Image courtesy of Alexay A. Kozhevnikov.
According to Dave Mosher in Wired Science (November 16, 2010), "The notoriously complex song of the finch has finally succumbed to statistics. Physicists have developed a model that can map out and predict the notes birds sing in sequence." The physicists are Dezhe Lin and Alexay Kozhevnikov of Penn State; their article, "A compact statistical model of the song syntax in Bengalese finch," is on arXiv. They dissected Bengalese finch songs into seven syllables (ABCDEFG in the sonograms above) and set out to find a model that would predict the way finches string these syllables together in their songs. They observed that a Markov model (each syllable is associated with one state, and transition probabilities between the states do not depend on the state transition history) was not sufficient. In fact, "a state transition model that accurately describes the statistics of the syllable sequences includes adaptation of the self-transition probabilities when states are repeatedly revisited, and allows associations of more than one state to the same syllable." This is a POMMA, a partially observable Markov model with adaptation.
The POMMA constructed by Jin and Kozhevnikov to describe Bengalese finch song (the letters in the ovals correspond to the syllables illustrated above. "The pink oval represents the start state. Cyan indicates that the state has a finite probability of transitioning to the end state. The numbers indicate the transition probabilities."). It correctly predicts such sequences as E, DDFBBBGBAAAA, DDFBBGBAAAA, DDFBBGBAAAG, ECCC and ECCCDDDFBBGBAAAGG. Image courtesy of Alexay A. Kozhevnikov.
Their model has implications for songbird neurophysiology: "The success of the POMMA supports the branching chain network hypothesis of how syntax is controlled within the premotor song nucleus HVC [High Vocal Center], and suggests that adaptation and many-to-one mapping from neural substrates to syllables are important features of the neural control of complex song syntax."
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