p238
ネットワークは信頼できるか？ たいてい出来ない．
食物連鎖についてはいままで得られているデータはひどい単純化(gross oversimplification) である．たとえば，I. D. Hodkinson and S. J. Coulson, “Are high Arctic terrestrial food chains really that simple? ―The Bear Island food web revisited,” Oikos 106 , 427 (2004)．
Crudeness, if not incorrectness, of self similarity in omic networks
See, for example, R. May, ``Subnets of scale-free networks are not scale-free: Sampling properties of networks,'' PNAS 102, 4221 (2005) as to omic networks; R. May, ``Network structure and the biology of populations,'' Trends Eco. Evo. 21 , 394-399 (2006) as to ecology. May writes, ``many of the observed degree distributions for intracellular signaling, although roughly linear on a log-log plot (i.e. a power law), are in fact better fit with an exponential or other degree distribution.''
As to protein-protein interaction network, N. Przlj, D. G. Corneil and Jurisica, ``Modeling interactome: scale-free or geometric?'' Bioinformatics 20 , 3508 (2005) demonstrates that high precision interaction network for yeast is definitely not scale-free as will be discussed later.
Is physiological 3/4-power law meaningful?
For physiology, see Glazier, ``The 3/4-Power law is not universal: evolution of isometric, ontogenetic metabolic scaling in pelagic animals,'' BioScience 56 , 325 (2006): For pelagic life style the linear law holds robustly. Several models have been proposed that appear to provide a strong theoretical basis for the law (due to West et al. or Banavar et al.). However, empirical work suggests that the 3/4-power law is not universal, and should at most be regarded as a statistical rule or a trend, rather than as an inviolable law. A metaanalysis result is in WHITE, ``ALLOMETRIC EXPONENTS DO NOT SUPPORT A UNIVERSAL METABOLIC ALLOMETRY,'' Ecology, 88 , 315 (2007):
The debate about the value of the allometric scaling often revolves around a dichotomous distinction between the 3/4-power exponent predicted by recent models. Such an approach does not allow for the possibility that there is no single `true' exponent. Significant differences between scaling exponents were also identified between ectotherms and endotherms, as well as between metabolic states (e.g., rest, field, and exercise). That is, there is no universal metab olic allometry.
The nongenericity of this scaling is confirmed in the following paper, which tried to correct the power law taking the finite size effect into account: Savage et al., ``Sizing Up Allometric Scaling Theory,'' PLoS Comp. Biol. 4 , e1000171 (2008). The WBE model result only holds in the limit of infinite network size (body mass) and that the actual exponent predicted by the model depends on the sizes of the organisms being studied. When accounting for corrections over a size range spanning the eight orders of magnitude observed in mammals, the WBE model predicts a scaling exponent of 0.81, seemingly at odds with data. The current canonical model may need amendments. This paper frankly admits the data Glazier compiled cannot be explained by the scaling idea.
Failure of 3/4-power law has a reason
Glazier, ``Effects of metabolic level on the body size scaling of metabolic rate in birds and mammals,'' Proc. Roy. Soc. 275 , 1405 (2008). Significant deviations from the `3/4-power lawﾕ have been observed. Therefore, the author proposed a new model, the metabolic-level boundaries (MLB) hypothesis. According to the MLB hypothesis, the scaling slope b should vary between two extreme boundary limits: 2/3 as a result of surface-related constraints on fluxes of resources, wastes and heat, and 1 as a result of mass (volume) constraints on energy use or power production.
＊As predicted, in both of these independently evolved endothermic taxa, the scaling slope approaches 1 at the lowest and highest metabolic levels (as observed during torpor and strenuous exercise, respectively), whereas it is near 2/3 at intermediate resting andcold-inducedmetabolic levels. Remarkably, both taxa show similar, approximately U-shaped relationships between the scaling slope and the metabolic (activity) level.
＊Variation of the scaling slope is not merely noise obscuring the signal of a universal scaling law, but rather is the result of multiple physical constraints whose relative influence depends on the metabolic state of the organisms beinganalyzed.
Food webs incompleteness must be the rule
I.¥ D.¥ Hodkinson and S.¥ J.¥ Coulson, ``Are high Arctic terrestrial food chains really that simple? ---The Bear Island food web revisited,'' Oikos 106 , 427 (2004):
The work supports the increasingly accepted view that many food webs presented in theliterature are gross oversimplifications and that analysis of their structure can produce misleading conclu sions.
Parasites are really significant in ecosystems
Kuris et al., ``Ecosystem energetic implications of parasite and free-living biomass in three estuaries,'' Nature 454 , 515 (2008). The biomass is estimated for free-living and parasitic species in three estuaries on the Pacific coast of California and Baja California. Parasites have substantial biomass, which exceeded that of top predators. The biomass of trematodes was particularly high, being comparable to that of the abundant birds, fishes, burrowing shrimps and polychaetes.
Mayは工学的な例ならスケーリングの成り立つネットワークもいいかもしれないといったが，そういうことさえない．以下はすでに上で引用したものと同じである．
Garbage In, Gospel Out
W Willinger, D Alderson, and J C. Doyle, ``Mathematics and the Internet: A Source of Enormous Confusion and Great Potential,'' Notices AMS {¥bf 56}, 586 (2009) advocates measurement based internet research (i.e., phenomenology).
``We illustrate why and how in the case of the Internet, scale-free network models of the preferential attachment type have become a classic lesson in how errors of various forms occur and can add up to produce results and claims that create excitement among non-networking researchers, but quickly collapse under scrutiny with real data or when examined by domain experts.''
``We motivate here the development of a novel modeling approach for Internet-like systems that (1) respects the highly designed nature of the network; (2) reflects the engineering intuition that exists about a great many of its parts; (3) is fully consistent with a wide range of measurements; and (4) outlines a mathematical agenda that is more challenging, more relevant, and ultimately more rewarding than the type of mathematics motivated by an alluring but largely misguided approach to Internet modeling based on scale-free graphs of the preferential attachment type.''
``{¥em Ask not what mathematics can do for [the Internet]; ask what [the Internet] can do for mathematics.}'' (Ulam)
Foremost among these issues are the dangers of taking available data ``at face value'' without a deeper understanding of the idiosyncracies and ambiguities resulting from domain-specific collection and measurement techniques. No amount of number crunching or mathematical sophistication can extract knowledge we can trust from low-quality data sets, whether they are of petabyte scale or not.
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