Using of aggregate indexes in the process of oil export modeling on an example of the Republic of Tatarstan before the 2009 crisis

The degree of relevance of this research is defined by the need of expansion and strengthening of the foreign economic regional relations as an instrument of increase of the economic status of the modern region (Republic of Tatarstan) through the interregional and foreign trade linkages. In respect of the foreign trade of the region there is a set of approaches for the decision of these tasks. Thus basic data which could be used for the analysis of current cooperation linkages on a separate period of time for the purpose of seeking the ways of increasing their efficiency due to its contents are limited mainly by the volumes of the purchases/sales which could be expressed in any of the trade, logistic or cost units. The index method chosen for the modeling of foreign economic relations allows us to define the general index of efficiency of trade i.e. the index number with the changed and constant (fixed) weights, an index of structural changes, to calculate the absolute variation of the 2009 to 2007 result and to show which part of this increment (decrease) is received from the changes of the size of basic cost, from the productivity of trade with the certain countries or from the changes of structure of basic cost during the processes of trade. So, application of the aggregated indexes even on the limited massive of basic data allows to carry out the factorial analysis of the most essential parts of the analysed phenomenon. Therefore, the index analysis due to principal limitations of statistical data in the modern Russia conditions concerned with the famous social and economic reasons could be very productive instrument in the analysis of the chosen objects of research. * Corresponding author. Tel.: 79274489346. E-mail address: iraahmetova@mail.ru. 2 Author name / Procedia Economics and Finance 00 (2012) 000–000

Introduction
As we noted above for the creation of any model is especially important the choice of the most adequate formalized tools (mathematical apparatus) according to statistical information at the time of research. Accordingly it could be offered an algorithm of the solution of this task in a conditions when in the analysed group of the exported (imported) goods from the territory of the certain region (on the example Republic of Tatarstan) by the separate commodities the suppliers are constrained by the strict liabilities which aren't concerned with the other commodities trade of analysed group.

Choice of the mathematical model
As it is known the efficiency of creation of social -economic models is mainly determined by the character of basic data and adequate to them procedures of the formalized processing. Even the superficial review of applied methods testifies the considerable opportunities of the correlation and regression analysis taking a noticeable place among other methods of the analysis and modeling. However its application shows to basic data on a studied problem serious requirements which in some cases can't be fully realized (deficiency of information of the necessary sort and its available contents, existence of available information in various scales, etc.). At the same time in economic statistics are successfully applied the methods which have been initially focused for aprioristic deficiency of the information to which without noticeable reservations it is possible to carry so-called index method focused on calculations of relative statistics, which characterize the phenomena in time, in space and on the relation with the general plan see Boiko and Karmanov, 1999. Also known feature of an index method is its orientation for the minimum statistical information, however divided for two strict periodsbasic and reporting. Indexes can be divided on individual and the general which in turn can be constructed as the averages from individual and as aggregate. Especially the aggregate indexes as formalization tools probably, can be somehow adapted for the solution of the problem above. Here could be interested not only separate aggregate indexes, but also such set of indexes and their relationship as indexes of the constant weights, the fixed weights and an index of structural changes.
In the classical statement of an index task of this kind at substantial level it is necessary and enough to have in the basic and reporting periods some abstract "productivities" (y 0 , y 1for the basic and reporting periods) and their initial "opportunities" (P 0 , P 1also for the basic and reporting periods). Then their products with the corresponding indexes (y • P) can be interpreted as some "results" which we will show on the following example (see table 2) where as an initial statistical information could be used the levels of dynamic ranks (table 1) offered to the subsequent consideration. On the basis of contents of table 1, displaying dynamic ranks with the levels of studied parameter given in the table (the amount of realization of oil from RT, mln. of US dollars), having assumed as a basis 2007 as a basic year, it is possible to create the modified statistical data both for basic and for reporting periods if we will use the following rules.

Adaptation of initial information for the method of index modelling
The monetary volume of sales of the previous year taken as a basic (in mln. US dollars) as a basic parameter for the next index analysis will be estimated as an "opportunity" whereas its "productivity" could be estimated in relation to the next year as the relation of monetary volume from oil sale of the next year (2008) to considered year (2007). Then, if to accept a matrix given in table 1 in the form of a data file in an interval scale C = {Cij}, where i=1, N = 11; j = 1, m = 3 (2007,2008,2009), "productivity" for the basic period of y0i could be estimated as the relation of export sales of the following (2008) to previous (2007) year: Сi,j+1 у0i = --. (1)

Cij
By the same rule for "productivity" of y1i for the reporting period there will be equally fair the ratio similar to expression (1): Сi,j+2 у1i = --. (2) Ci,j+1 The calculated on expressions (1) and (2) y0i and y1i values are given in tab. 2 in the P0 and P1 columns. It is quite obvious that dimensions of counted parameters in expressions (1) and (2) needs some specification. Formally they are obliged to be dimensionless coefficients (mln. US dollars/mln. US dollars). At the same time, according to the contents as "productivity" it has to comprise information in so-called "the removed look". As (1) and (2) reflect the attitudes of the previous cost indexes towards the subsequent in time their denominators reflect the so-called initial cost (IC of mln. US dollars), whereas numerators reflect the new created cost (NCC of mln. US dollars). Then dimension of "productivity" in both expressions could be expressed as [to NCC mln. US dollars/ IC of mln. US dollars]. If as the basic "opportunity" logically accept the statistical set of oil sales of the table 1 for 2007 as P 0istatistical set of C2007, so as the reporting "opportunity" it is possible to consider P 1i -set of C2008 from the table 1. Then taking into account (1), (2) and given above results the table 2 will be the following. According to the accepted expressions (1) and (2) given in table 2 in the names of columns as operands with the corresponding mathematical indexes for carrying out the index analysis have the following dimensions: "y" are measured in units [NCC mln. US dollars/ IC of mln. US dollars], "(P)" -in units [IC of mln. US dollars], products of "yP" are measured in (NCC mln. US dollars/ IC of mln. US dollars) • (IC of mln.US dollars) = [by NCC of mln. US dollars]. Therefore, from the available as the statistical data short dynamic ranks of table 1 due to the formulated rule in an adaptation of basic statistical data to the index analysis we accept as the basic period 2007, and as the reporting -2008, as it is reflected in working table 2. Further, using basic data in the form of y 0i , y 1i , P 0i and P 1i , it is easy to find products necessary for calculations look like (yP) and their sums up with the corresponding interlinear symbols: ∑P 0 = 17701,46 IC mln. US dollars; ∑P 1 = 16938,78 IC mln.US dollars; ∑y 0 P 0 = 16941,31 NCC mln.US dollars; ∑y 1 P 1 = 15481,55 NCC mln. US dollars; ∑y 0 P 1 = 52487,94 NCC mln. US dollars. It is clear that the sums of "productivity" ∑ y 0 dimension [NCC mln. US dollars/ IC of mln. US dollars] and ∑y 1 dimension [NCC mln. US dollars/ IC of mln.US dollars] don't have physical sense and therefore don't pay off. As the basic period (2007) and reporting (2008) are created by the rules declared here it must be kept in mind that circumstance that "result", for example, look like ∑y 1 P 1 will be already belong to the following, 2009 owing to specifics of adaptation of initial information (table 1) to modified initial information (table 2).

Modelling on the basis of modified initial information
Chosen for modelling of foreign economic relations famous index method as provided in Gromuko, 1999 allows us to define the general index of productivity of trade -the variable and fixed (constant)  The received results generally mean very insignificant decrease, however is important the identification not only the maintenance of probably revealed relationships of cause and effect of the received preliminary result as consequences but also an assessment the sizes of influence of the possible factors revealed at modelling.
For this purpose (identification of factors of decrease for 4,5% of overall effectiveness of trade that made losses of 0,043 mln. US dollars) it is necessary to find the indexes of productivity of trade of the fixed structure and an index of structural changes.
2. We find an index of trade productivity of the fixed structure eliminating influence of change of initial costs structure (IC, mln. US dollars) for the basic and reporting periods on a dynamics of average efficiency: This means that from the change of trade productivity with the certain countries (which could be seen from the table 1) average efficiency for the reporting period decreased in relative expression on Δу f.c. % = 29,5% -100% = -70,5% or, in absolute expression, on Δу f.c. % = 0,914 -3,099 = -2,185 (NCC mln. US dollars/ IC of mln. US dollars).
3. Further we find an index of the structure reflecting the structural changes of basic data according to tab. 1 over all countries on dynamics of average productivity of trade for the considered period: This means that from the change of initial opportunities structure (commodity volume turnovers over the countries) average productivity of trade increased in relative expression on Δу str. % = 323,77% -100% = 223,77% or, in absolute expression, on Δу str. = 3,099 -0,957 = + 2,1416 (NCC mln. US dollars/ IC of mln. US dollars). Thus, the general decrease in average efficiency of trade (on -0,043 NCC mln. US dollars/ IC of mln. US dollars; in another words decreased the speed of creation of new cost in relation to the initial cost) which occurred from 2007 to 2008 is caused by action of two factors: decrease in productivity of trade over the certain countries (on -2,185 NCC mln. US dollars/ IC of mln. dollars) and change of structure of IC of the certain countries (on + 2,141 NCC mln. US dollars/ IC of mln. US dollars). Then indexes of the fixed structure and structural shifts in the absolute expressions (-2,185 and + 2,142) are the explaining factors of decrease in average productivity of trade as a whole (-0,043), and the balance of ratios can be presented as below: -0,043 = -2,185 + 2,142 (NCC mln. US dollars/ IС mln. US dollars) (3) In particular using a ratio (3), it is easy to estimate the weights of the revealed factors influencing throughout the studied period on average productivity of trade in relation to the considered countries. As a result of simple ratios it is possible to conclude that to the share of the productivity of trade with the certain countries (negatively operating factor) it takes 50,5% whereas the factor of structure of the trade (positively operating factor) takes 49,5% (as a result it is about 50 to 50). Thus, the general insignificant decrease in the general efficiency (efficiency) of trade (only on 43 thousand dollars) is caused upon really operating with two mutually counterbalancing each other (in this case) by processes: on the one handobvious decrease in efficiency of use of earlier saved up experience and the possible developed traditions of trade with the certain countries, with anotherin the Republic of Tatarstan for the studied period and within this model quite convincing counterbalance in the form of creation of such successful structure of interaction (in this modelin the form of oil sales volumes in 2007 and 2008) is shown. As identification of the real reasons (purposefully from state bodies of RT, business owners or it is spontaneous) so successful balance presented in expression (3) in a research objective doesn't enter. At this investigation phase expedient it will be simple to estimate structures of initial opportunitiesvolumes of commodity turnovers (on our terminology) the basic and reporting periods (see tab. 3), having used the primary (initial) statistical information provided in tab. 1 (weight in the % reflecting structure of trade by years in tab. 3, are provided through a virgule). As we could see from the table 3, the structure of oil supply over the far abroad and neighbouring countries in the reporting and basic periods differs very significantly. In the revealed structure of trade practically for 2007 -2008 period there is no stable supply of oil for any of the countries whereas the overall picture testifies the unstable nature of studied process unless it is possible to carry as "quasicontinuous" trade with known convention such countries as the Netherlands, Poland, Belarus and France. For other countries given in table 3 instability is the main characteristic, however, the same is looked and for 2009 (see table 1). Nevertheless such reorientation in supply of oil that is reflected in structure of trade for RT for the studied period was factually by the result as a compensating factor, despite decrease in trade as a whole: in 2007on 17701,46 mln. US dollars, whereas in 2008on 16938,78 mln. US dollars.
Therefore, having rather limited set of basic data (see table 1 4. For the solution of these objectives it is necessary to determine firstly in absolute expression the change of total "results" of oil sale as multiply of "productivity" and "opportunities" (Δop.b. ) in reporting 2008 in comparison with basic 2007: Δ = ∑у 1 P 1 -∑у 0 P 0 = 15481,55 -16941,31 = -1459,76 (NCC mln. US dollars).
So, for this period (2007 -2008) took place the reduction of total result of oil sale on 1459,76 mln. US dollars or ≈ 1,46 NCC bln. US dollars that make considerable volume new, half-received cost for RT. In this connection also as well as in a case with the productivity of oil sale is interested the former question of identification as the maintenance of the reasons of this circumstance and an assessment of their level of influence on total (summary) result. For the identification of that and another factor we will make the following calculations.
Here the productivity is taken of different kinds for 2008 and for 2007, but in relation to opportunities of 2008 that is nothing as a difference between the numerator and the denominator of an index of productivity of the fixed structure (weight) in an aggregate look.
7. Assessment of acquisitions/losses at the expense of changes of initial opportunities structure (see table  3  2) on -37006,39 NCC mln. US dollars gross collecting decreased from the decrease in productivity (efficiency) of export to the certain countries; 3) on + 36276,56 NCC mln. US dollars the result increased from the change of structure of initial volumes of oil sales in the basic and reporting periods.
Weight indicators of the operating factors listed here (it is revealed three of them) unlike the factors influencing to the productivity of trade (them it was revealed two) will be others. To the share of a factor "reductions from the reduction of the size of initial opportunities" (owing to the operating objective / subjective reasonshereinafter it isn't investigated) it is 0,99% fall, the factor of "decrease in productivity of sales" falls to the share of 50,0%, to a factor share of "change of structure of initial volumes" it is 49,01% (in total 100%).
The weight of two last factors listed above differ from earlier given scales relating to an assessment of action of factors on the share of productivity of trade with the certain countries (also negatively operating factor) takes 50,5%, and for the factor of trade structure (also positively operating factor) it takes 49,5%. Therefore the factor "the size of initial opportunities" in the here accepted terminology has no noticeable impact.

Conclusion
So, application of the aggregated indexes even on the limited massive of basic data allows carrying out the factorial analysis of the most essential parts of the analysed phenomenon. Therefore, the index analysis due to principal limitations of statistical data in the modern Russia conditions concerned with the famous social and economic reasons could be very productive instrument in the analysis of the chosen objects of research. It is enough to have only as it was already shown some statistical data, and after that it is possible to use the scheme: "Efficiency (y)" multiply to "Opportunities (P)" = "Result (yP)".


extfieldsofstudy:
Computer Science
Medicine

text:
How Do Online Social Networks Grow?

Online social networks such as Facebook, Twitter and Gowalla allow people to communicate and interact across borders. In past years online social networks have become increasingly important for studying the behavior of individuals, group formation, and the emergence of online societies. Here we focus on the characterization of the average growth of online social networks and try to understand which are possible processes behind seemingly long-range temporal correlated collective behavior. In agreement with recent findings, but in contrast to Gibrat's law of proportionate growth, we find scaling in the average growth rate and its standard deviation. In contrast, Renren and Twitter deviate, however, in certain important aspects significantly from those found in many social and economic systems. Whereas independent methods suggest no significance for temporally long-range correlated behavior for Renren and Twitter, a scaling analysis of the standard deviation does suggest long-range temporal correlated growth in Gowalla. However, we demonstrate that seemingly long-range temporal correlations in the growth of online social networks, such as in Gowalla, can be explained by a decomposition into temporally and spatially independent growth processes with a large variety of entry rates. Our analysis thus suggests that temporally or spatially correlated behavior does not play a major role in the growth of online social networks.

Introduction
Online social networks (OSNs) have become increasingly important as they allow us to interact across any geographical scale. Communication networks, transport networks and OSNs are often interconnected and interdependent. This opens up great economic and social opportunities but can also involve considerable risks such as cascading breakdowns [1]. The study of OSNs is of importance for understanding the behavior of individuals, groups and societies. Hence, particular types of growth in social, economic and other networked systems have attracted a lot of attraction in the past years [2][3][4][5][6][7][8].
Gibrat's law states that both the average growth rate and the standard deviation of the growth rate of a given observable are constant and independent of the specific value of the observable [9]. However, this empirical law, originally observed in economic systems, has been challenged by many socio-economic studies [10,11], notably very recently [4,5,8].
Any social growth dynamics is expected to depend on social factors such as gender, age, social status and so forth. Unfortunately, available datasets that comprise such information are typically too small to investigate emergent scaling or large-scale collective behavior. In this paper, we focus on the population growth dynamics of three large OSNs. Our datasets do not resolve individual social factors but their size allows for studying scaling and long-range correlations, both temporally and spatially.
We find evidence for certain scaling laws in the growth rate and the variance, although for Renren and Twitter the exponents characterizing fluctuations are found to deviate from those that have been reported previously for social and economic systems.
These deviations carry important information about the growth of online social systems. In particular, we find that the relative number of registered users increases almost temporally and spatially independently of each other. This contrasts the behavior of offline growth in many social and economic systems where growth is a long-range correlated process and thus a collective phenomenon. Even for Gowalla where scaling indicates longrange correlated growth a decomposition into independent growth processes unravels the seemingly long-range collective behavior to be a mere artifact of the large variability of entry rates [12].

Data
We analyze three OSN datasets. The first OSN Renren (rr), often referred to as the ''Chinese Facebook'', is one of the largest online social networks in China. The dataset covers about 1,000,000 users in the time period of January 2006 to December 2010 (60 months) with online interactions from over 10,000 registered locations.
The second OSN data set, comes from a subset of Twitter (tw), a microblogging online social service sited in the United States. It covers more than 250,000 members between August 2006 and September 2010 (50 months) from about 9,000 locations.
The third OSN, Gowalla (gw), was an online check in social service launched in 2007 and closed in 2012 in the United States. Users were able to check in at certain locations, referred to as Spots, either through a mobile phone application or Gowalla's mobile website. Among other things, checking-in allowed for the dropping or swapping of virtual items. The dataset covers 21 months (from February 2009 to October 2010) with around 200,000 members from about 5,000 locations.
We acquired the first two datasets by crawling user profiles in the web sites from Renren.com [13] and Twitter.com [14] through their APIs. We only crawled the user profiles which are publicly available. The Gowalla dataset is obtained from a shared data source [15] by other researchers. Due to the tremendous size of OSNs, we only acquired a sampled subset of each OSN. To eliminate sample bias we deployed the Breadth First Search (BFS) bias correction procedure by Kurant et al. [16].
For these three datasets we define a population at a location l (an integer number ID) at time t as the set of all users with home location l. The spatial resolution of the location refers to as a city code, associated with the administrative area (i.e. the city name) of the user's home location. For Renren and Twitter we assume the registered location of the user as the user's home location. For Gowalla we assume the most visited location as the home location. For the spatial analysis we used an assignment of GPS coordinates (via Google Maps Application Programming Interfaces (API)) to the location l, and calculated the distance between two locations via their GPS coordinates. The estimated GPS coordinates of a user's home location may thus be incorrect for a certain fraction of a given population. This, however, may not alter any of the conclusions made in this article.

Results
Here, we investigate the mean growth rate and its fluctuation in OSN populations and ask the question how these observables depend on the initial population size.
We denote the population size, i.e. the number of users with home location index 1ƒlƒl max at time 0vtƒT, by S l (t). Following Refs. [17][18][19] we define the logarithmic growth rate r between time t 0 and t 1 (t 0 vt 1 ƒT) as where S 0~S l (t 0 ) and S 1~S l (t 1 ) are the population size at a location l but at different time points t 0 and t 1 [5].
To characterize fluctuations, we study the average growth rate r(S 0 ) and the standard deviation as a function of the initial population size S 0 , see Figure 1. In other words, the average growth rate Sr(S 0 )T corresponds to only those online populations with size at least S 0 until time t 0 . The conditional standard deviation of the growth rate s(rDS 0 ) for those populations expresses the statistical spread or fluctuation of growth among populations with S 0 . Both quantities show a power law dependence on the initial population size, The range of b for Renren and Twitter is in agreement with those previously reported exponents for b. However, in contrast to the previous work mentioned above our analysis (employing maximal likelihood estimation (MLE) and bootstrapping) do not suggest significant deviations from b~0:5 which would indicate uncorrelated growth. In contrast, for Gowalla we find b significantly smaller than b~0:5 (Pv0:05).
Second, we find the range of exponents for the average growth rate for all studied online social networks significantly above a~0 (Pv0:05) indicating a violation of Gibrat's proportionate growth, which is in agreement with social and economic systems [4,5,10,[20][21][22].
The average growth rate Sr(S 0 )T and the conditional standard deviation of the growth rate s(rDS 0 ) allow for direct comparison with the literature for other social systems and Gibrat's law but are only averages. As suggested by studies of certain assets in economical systems the distribution of the variance can often exposes important information that cannot be seen in averages [23].
Since for a given S 0 there is only a single value of s(rDS 0 ) (see Fig. 1), we ask what is the relative variation of s(rDS 0 ) across all values of s(rDS 0 0 ) that occur in a given dataset. We thus focus on the relative fluctuation function (rff), as a function of S 0 . Specifically, we study the complementary cumulative relative fluctuation function (ccrff), which is given by the complement of the integrated rff, We chose the ccrff representation because it shows (if exists) a clearer scaling than the rff and thus better exposes different (scaling) regimes. The ccrff is obtained by collecting all locations with a given value of S 0 using exponential binning (see Fig. 2 and Methods). In contrast to Renren and Twitter where we find no significant bimodality, for Gowalla the ccrff as a function of S 0 exhibits a remarkable bimodal behavior.  For Gowalla, Figure 2C suggests a bimodal distribution of standard deviations, characterized by an exponential decay that is followed by a power law S Ã~9 3 for Gowalla marks the crossover point (determined from MLE, see Methods). MLE suggests that the power law decay is characterized by the exponent n gw~0 :13 (R 2~0 :966).

Gowalla: Correlations in the growth rate
The above findings suggest to consider two groups of locations: one group with initial population size S 0 vS Ã , and the other one with initial population size S 0 §S Ã . We study the monthly population growth rates for each location and calculate their autocorrelation function (ACF) [24][25][26]. For S 0 vS Ã the ensemble averaged ACF exhibits an exponential decay, C(t)*e {ct , indicating that the population growth is short-term correlated, see Figure 3A. We obtain the exponent c~0:13 (R 2~0 :992 from MLE), which is equivalent to a correlation time constant of about two weeks.
In contrast, for S 0 §S Ã the ACF is well described by a power law, C(t)*t {d with power law exponent d~0:73 (R 2~0 :955 from MLE), see Figure 3B. This is consistent with long-term correlations characterized by P ? t~{? C(t)~?, see [27] and references therein.

Superposition model
Seemingly long-range correlation can often be explained by a finite set of independent processes whose superposition accounts for the algebraic decay in the ACF, and the divergence of its infinite sum. In 1979 van der Ziel established that any ensemble of uncoupled short-range correlated stochastic oscillators is sufficient for explaining long-range correlations in their superposition, if and only if the time constants of the mixed processes are sufficiently broadly distributed [12]. More recently, it has been shown that a superposition of Poisson processes, together with circadian activity, very likely account for many scaling laws of human activity patterns [28]. Here, as the growth rates are broadly distributed, we follow this spirit by considering a superposition of populations and surrogate time series from these, see Methods.
Gowalla's population growth of the superposition ensemble obtained from a random selection of population with S 0 vS Ã results in the occurrence of seemingly long-term correlations for This suggests that the seemingly long-term correlated population growth found for locations with S 0 §S Ã results from superpositions of short-term correlated growing populations.

Spatial dependence
To study geographical factors we investigate correlations of the populations growth rates r i and r j between different places [29,30].
We therefore study the Pearson's correlation coefficient where s i,j~ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi S(r i,j {Sr i,j T) 2 q is the standard deviation of r i and r j , respectively.
We investigate the monthly population growth rates and Pearson's correlation coefficient between a pair of locations as a function of the geographic distance of the users. Figure 5 shows the Pearson's correlation coefficients for the three data sets. The average correlation ScT is found at a level of about 0:7{0:8, effectively independent of the geographic distance. The high value of the cross correlation agrees well with the plausible assumption that individuals join online social networks collectively but independently of the geographic distance to each other.

Discussion
We find scaling in the population growth rate and variance in online social networks. Our results suggest that the population growth in online social networks is neither significantly determined by population size [31] nor by spatial factors. The results deviate from Gibrat's law as previously found in many social and economic systems. The seemingly long-term correlated growth behavior for Gowalla suggested by scaling in the standard deviation is explained by a simple decomposition into short-term correlated population growth with broadly distributed growth rates. Our method may help interpreting (seemingly) long-range correlations in the growth of large heterogenous (online) social and economic systems. Seemingly collective behavior in online social systems may result from the high variability of loners' actions and not from correlated collective behavior.

Ethics statement
We use the APIs that provided by Renren.com and Twitter.com for data collection from these two websites. The acquirement of Renren and Twitter datasets is in accordance with the websites' terms of service.

Data availability statement
We use three datasets in this article. The Renren and Twitter datasets can be obtained upon the request, which is ''data available on request''. The request can be send to the Computer Networks Group at University of Göttingen via email (net@cs.unigoettingen.de). The Gowalla dataset is obtained from a shared data source [15] by other researchers. The requester can download it from snap.stanford.edu, which is ''data available from online''.

Exponential binning
Fitting of average, standard deviation and the ccrff is performed by exponential binning, by which the bins are evenly distributed on a logarithmic scale. Specifically, the beginning of each bin is b j~t cR j s, exponentially increasing in j, with constants c and Rw1, so that bins have size b jz1 {b j &b j (R{1). We use exponential binning for both Figure 1 (with c~1, R~2) and  window. Following the methodology of studies in human population growth in the real world [4] and the human interaction activities in OSNs [5], we determined t 0 as a result from time when the number of locations with growing populations reaches the peak. That is, t 0 :~35 for Renren, t 0 :~36 for Twitter and t 0 :~14 for Gowalla, respectively, see Figure 6.

Determination of S Ã
To determine the best value for S Ã , we fit the distribution of standard deviation with respect to S 0 ranging from 0 to 300 by using MLE. For each S 0 , we calculate R 2 for exponential and power-law fitting, denoted as R 2 exp and R 2 pow , respectively. To characterize the overall fitting quality (FQ) we use where we use log-log-scaling for determining the coefficient of determination for the power law, and log-linear-scaling for the exponential. We choose S Ã~a rgmax(FQ(S 0 )) where FQ takes its maximum at the value of S Ã~9 3, as shown in Figure 7.

Spatially resolved monthly growth rates
For each location with integer ID l, we extract a time series from t 0 to t 1 of the monthly population growth rate according to equation (1) as r t~l n Stz1 St , t 0 ƒtvt 1 being the tth month. We calculate the autocorrelation function (ACF) from the time series r t as where t is the time lag and s r is the standard deviation of r t .

Superposition construction
To study superpositions we select all populations at locations with S 0 vS Ã . The randomized surrogate data set is created by shuffling these entries, and creating a time series from these shuffled entries as follows.
(1) From the set of populations with S 0 vS Ã we select randomly a population and add up its initial population size S 0 , irrespective of its location. (2) We repeat (1) until the sum exceeds S Ã . This results in a set of locations whose total initial population size equals or slightly exceeds S Ã . We call this set of locations one realization of a superposition. (3) For each realization we study the temporal development with respect to total populations size of the thereafter fixed selected locations. For each superposition we construct a time series, that is, the population growth rates in monthly resolution, from t 0 to t 1 . For this set of time series we obtain the ensemble averaged ACF.
Author Contributions Figure 7. The selection of S Ã for Gowalla. The fitting quality as function of S 0 . S Ã is defined as the position (argmax) of the maximum, which is S Ã~9 3 for Gowalla. doi:10.1371/journal.pone.0100023.g007