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w ideas — none of which biology. city, the more the aver- produces and consumes, ources or ideas4. On aver- reases, per capita uantities such as er of patents pro- er of educational utions all increase 5% more than the rowth4. There is, de: negative met- me, traffic conges- of certain diseases ing the same 15% e bad and the ugly ated, predictable, ws that, despite are approximately one another (see and Tokyo are, to redictable degree, -up versions of San Fran- assumption upon which modern cities and economies are based. Sustaining that growth with limited resources requires that major innovations — such as those historically asso- ciated with iron, coal and digital technology — be made at a continuously accelerating rate. The time between the ‘Computer Age’ and the ‘Information and Digital Age’ was some undesirable consequence in developed and develo be viewed as experimen designed and measured the creation of an integrat ory and a new science of p planning. Examples of increasingly com poster children su Spain or Curitiba i of new initiatives in don. Ideally, by cou (such as lower car actionable policie indicators of socia cesses and failures corrected for, guid theory and creatin Cities are the c civilization, the potential disaste of the solution to lems. It is therefo understand their and evolution in a dictable, quantita ference between ‘policy a Data from 360 US metropolitan areas show that metrics such as wages and crime scale in the same way with population size. −2.0 −1.5 −1.0 −0.5 0 0.5 1.0 1.5 2.0 log (city population/city population average) log (metric/metric average) −2.0 −1.5 −1.0 −0.5 0 0.5 1.0 1.5 2.0 2.5 Income Crime METRIC: Patents GDP L. Bettencourt, G. West, “A uniﬁed theory of urban living”, Nature 2010
and extraordinary geographic variability, we have shown cities belonging to the same urban system obey pervasive ng relations with population size, characterizing rates of vation, wealth creation, patterns of consumption and human vior as well as properties of urban infrastructure. Most of e indicators deal with temporal processes associated with the l dimension of cities as spaces for intense interaction across spectrum of human activities. It is remarkable that it is ipally in terms of these rhythms that cities are self-similar nizations, indicating a universality of human social dynam- despite enormous variability in urban form. These findings ide quantitative underpinnings for social theories of ‘‘ur- sm as a way of life’’ (12). Our primary analytical focus here was concerned with consequences of population size on a variety of urban metric this sense, we have not addressed the issue of location (49–5 a determinant of form and size of human settlements. We however, shed some light on associated ideas of urban hierarch urban dominance (14, 51): increasing rates of innovation, w creation, crime, and so on, per capita suggest flows of quantities from places where they are created faster (source those where they are produced more slowly (sinks) along an u hierarchy of cities dictated, on average, by population size. A related point deals with limits to urban population gro Although population increases are ultimately limited by impac Table 2. Classiﬁcation of scaling exponents for urban properties and their implications for growth Scaling exponent Driving force Organization Growth ␤ Ͻ1 Optimization, efﬁciency Biological Sigmoidal: long-term population limit ␤ Ͼ1 Creation of information, wealth and resources Sociological Boom/collapse: ﬁnite-time singularity/unbounded growth; accelerating growth rates/discontinuities ␤ ϭ 1 Individual maintenance Individual Exponential ncourt et al. PNAS ͉ April 24, 2007 ͉ vol. 104 ͉ no. 17 ͉ Reguła kciuka
might predict (rank 88th in income, 184th in GMP), not very inventive (178th in patents) and quite safe (267th in violent crime). San Francisco is the most exceptional large city, being rich (11th in income), creative (19th in patents) and fairly safe (181th in violent crime). The truly exceptional MSAs are smaller, such as technologically d period. The cha auto-correlation (see Materials an spectrum Pi(v)~ definitions). Thei
Economies”, http://arxiv.org/abs/1102.4101 5e+04 1e+05 2e+05 5e+05 1e+06 2e+06 5e+06 1e+07 2e+07 0 20000 40000 60000 80000 MSA population Gross product per capita (2001 dollars/person−year) Fig. 2. Horizontal axis: population, as in Figure 1, log scale. Vertical axis: gross product per capita, but on a linear and not a logarithmic scale. The two largest values are 7.8 × 104 dollars/person-year (in Bridgeport-Stamford-Norwalk, CT, a center for hedge funds and other ﬁnancial ﬁrms) and 7.7 × 104 dollars/person-year (in San Jose-Sunnyvale-Santa Clara, CA, i.e., Silicon Valley), and the smallest are 1.5 × 104 dollars/person-year (in McAllen-Edinburg- (ale ma fajnego bloga)
for navian and a few other research units, including 4 U.K. universities. The er e y al al n d e y h n. i- d es o al logarithm of the number of support staff, N(Support), is plotted versus the logarithm of number of academic staff, N(Academic). This indicates a power-law scaling relationship between these 2 groups, N(Support) ϭ C ϫ N(Academic)␤, where C is a constant (Ϸ0.07) and the exponent is ␤ Ϸ 1.30 Ϯ 0.03. The Danish data fall above the linear fit to all of the data while the Swedish data fall below. Swedish universities have Ϸ10% less support staff than Norwegian and Danish universities with a similar number of total staff. The observed power-law behavior, which covers a range of Ͼ3 orders of magnitude (decades), reflects that the larger the units are, the higher the number of support staff is relative to the academic staff. Data points for subunits within Norwegian universities follow the same power-law trend and point to a hierarchical organization of the larger research units. The scale- free (power-law) relationship suggests that units of all sizes may be equally well (or poorly) organized to promote the research and education production with available resources. In principle, units given the hierarchical structure of most European univer- sities. Fig. 2 shows a simple hierarchical model that results in scaling behavior similar to that observed. The basic organiza- tional unit in the hierarchy is composed of 3 ‘‘basic research units.’’ Each research unit is composed of 1 ‘‘support staff’’ and Fig. 1. Diagram showing the distribution of academic and support staff for Scandinavian and a few other research units, including 4 U.K. universities. The data in this log-log plot were ﬁtted to a power law, using an orthogonal distance regression. We used simple bootstrap sampling with repetition to seek the spread in the data and found a close to Gaussian distribution in the values for the exponents. Conﬁdence intervals on the slopes were estimated using bootstrapping. Slopes for individual data sets (with conﬁdence inter- vals): Norway, 1.32 (1.24–1.40); Denmark, 1.34 (1.19–1.39); Sweden, 1.39 (1.07–1.51). The analyses were carried out in Past, version 1.89, Hammer et al. (6). Sources of data: Norwegian data (from around 1990) can all be found in the database for the statistics of higher education in Norway (http:// dbh.nsd.uib.no/dbhvev/ansatte/tilsatterapport.cfm). Data for the University of Oslo from before 1990 are available from annual reports. Data from Swedish, Danish, and other research units are found at the individual univer- sity’s web pages, usually with references to the respective annual reports. The National Health Service (NHS) data are published at http://www.ic.nhs.uk/ pubs/nhsstaff. No data have been excluded from the diagram. All data are provided in Tables S1 and S2. Fig. 2. A simple deterministic hierarchical model for the structure of a research unit. Three levels of the self-similar hierarchical organization are shown. The basic organizational unit (BOU) consists of 1 central administrator (pink hexagon) and 3 administrators (green square) dealing directly with the academic staff (blue circles). The iterative step in the construction of the hierarchy is to assume that the growth of the administration follows the pattern of the basic unit. Hence we obtain 1 central administrative unit, which has contact with 3 basic research units (BRU). The exponent for this construc- tion is log(4)/log(3) ϳ 1.26. In general, the exponent is given by log(X1)/ log(X1 Ϫ X2), where ‘‘X1’’ is the total number of support staff and ‘‘X2’’ is the number of ‘‘central support staff.’’ A general feature of these models is that the central support becomes increasingly distant from the basic research units, B. Jamtveita, E. Jettestuena, J. Mathiesena, “Scaling properties of European research units”, doi:10.1073/pnas.0903190106 C ⇡ 0.07
dependency of research quality on group size”, arXiv:006.0928 N in a region around the breakpoint, to illustrate the maximal gradient there. Without breakpoint, the steepest gradient condition would lead to supporting only the biggest among the large groups so that only one group remains at the end. The resulting instability is described in Harrison (2009) in the case of the former Soviet Union, where the scenarios oscillated between this and the alternative one. At the end of Section The relationship between quality and quantity, we mentioned that the opposite causal mechanism of increasing quality driving increasing group size, may be dismissed as the dominant mechanism on the basis of empirical evidence. Such a model 0 250 500 750 0 5 10 15 20 25 S N applied mathematics Fig. 5 The absolute strenght S = sN for applied mathematics near the breakpoint Nc = 12.5, where the gradient is maximised 538 R. Kenna, B. Berche -50 -25 0 25 50 S- <S> institution Z A applied mathematics Fig. 6 The renormalised quality s - hsi for applied mathematics, where hsi is the quality predicted by the model. The tighter bunching of the data compared to the raw plot of Fig. 1b (the two ﬁgures are to the same scale) illustrates the validity of the model and that the averaged individualised performances are better than hitherto realised Critical mass and the dependency of research quality 539 ncy of the expected quality s = S/N of research teams may therefore be measurement of the relative importance of cooperation or collaboration iscipline. Beyond Nc , two-way communication between all team members dominant driver of research quality leading to a milder dependency of s on enon is akin to phase transitions in physics (Kenna and Berche 2010). The akpoint point Nc may be considered a demarkation point between ‘‘small/ large’’ groups in a given discipline and a measurement of the number of whom a given individual can collaborate in a meaningful sense. to capture the essence of this behaviour, and to measure Nc for different ecewise linear regression analysis is applied to data sets corresponding to mic disciplines. We ﬁt to the form s ¼ a1 þ b1 N if N Nc a2 þ b2 N if N ! Nc; ð8Þ and b2 are related to the parameters appearing in (7). Note in particular that at the slope to the right of the breakpoint should be small for disciplines point values. Breakpoint values Nc are also estimated through the ﬁtting R. Kenna, B. Berche 0 25 50 institution QMUL Brighton Liverpool Plymouth UWE 0 25 50 75 100 0 10 20 30 40 50 60 70 80 90 100 s N QMUL Warwick (b) applied mathematics Cambridge Brighton Liverpool Oxford Plymouth UWE esearch quality s for each of 45 UK applied mathe- earch groups (a) arranged alphabetically and (b) as of group size N. (The sample institutions include ry, University of London (QMUL) and the Univer- West of England Bristol (UWE).) In (b) the solid piecewise linear regression best-ﬁts to the data and jth individuals, the group N(N−1)/2 i,j =1 bgi,j = N¯ ag + N resents the link between no erage strength of interaction molecular ﬁeld theory , w strength of such complete si S ∝ ¯ aN + where ¯ a and ¯ b are the mean strength averaged over all g In fact, since two-way com ried out eﬀectively between may be further expected th is increased, a transition po beyond which the clusterin creases. If a given node ca at most Nc others, the gro subgroups, of mean size αNc interactions can take place.
50 100 150 s N (a) physics 0 25 50 75 100 0 30 60 90 s N (b) Earth sciences 0 25 50 75 100 0 50 100 150 200 250 s N (c) biology ig. 3: Success rate or research quality s as a function of group size N for (a) physics, (b) Earth science and (c) biology. A Fig.1(b), the solid lines are piecewise linear regression best-ﬁts to the data and the dashed curves represent 95% condidenc tervals for these ﬁts. roup is measured quantitatively. As mentioned, the UK’s cademic communities provide suitable testing arenas as he quality of research groups in many disciplines has been easured through RAE. A piecewise linear regression analysis is applied to data ts corresponding to measured quality of research groups, tting to the form (3). In Fig. 3 the data for physics which includes experimental physics), Earth sciences and ology are presented. For these, as in Fig. 1(b) for ap- ied mathematics, the positive correlation between qual- y and team size reduces beyond a discipline-dependent that local cooperation is less signiﬁcant in pure mathemat ics, where the work pattern is more individualized. Thi is consistent with experience: papers in pure mathemat ics tend to be authored by one or two individials, rathe than by larger collaborations. The results for chemistr are presented in Fig. 4(b), where there is also an absenc of small groups leading to a relatively large error in th critical mass estimate Nk = 18 ± 7. The above work is based upon the measures of researc quality as determined in the UK. To check its broade generality, we compare the results of the RAE with thos 0 25 50 75 100 0 20 40 60 s N (a) pure mathematics 100 R. Kenna, B. Berche, “The extensive nature of group quality”, arXiv:1004.3155