Complexity of a Microblogging Social Network in the Framework …downloads.hindawi.com/journals/complexity/2018/4732491.pdf · 2019. 7. 30. · ResearchArticle Complexity of a Microblogging

Research ArticleComplexity of a Microblogging Social Network inthe Framework of Modern Nonlinear Science

Andrey Dmitriev Vasily Kornilov and Svetlana Maltseva

School of Business Informatics National Research University Higher School of Economics Moscow 101000 Russia

Correspondence should be addressed to Andrey Dmitriev admitrievhseru

Received 9 August 2018 Accepted 11 November 2018 Published 2 December 2018

Guest Editor Piotr Brodka

Copyright copy 2018 Andrey Dmitriev et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Recent developments in nonlinear science have caused the formation of a new paradigm called the paradigm of complexity Theself-organized criticality theory constitutes the foundation of this paradigm To estimate the complexity of a microblogging socialnetwork we used one of the conceptual schemes of the paradigm namely the system of key signs of complexity of the externalmanifestations of the system irrespective of its internal structure Our research revealed all the key signs of complexity of the timeseries of a number ofmicropostsWe offer a newmodel of amicroblogging social network as a nonlinear random dynamical systemwith additive noise in three-dimensional phase space Implementations of this model in the adiabatic approximation possess all thekey signs of complexity making themodel a reasonable evolutionarymodel for amicroblogging social networkTheuse of adiabaticapproximation allows us to model a microblogging social network as a nonlinear random dynamical system with multiplicativenoise with the power-law in one-dimensional phase space

1 Introduction

Social networks have been studied longer than any othertype of networks It is remarkable that one of the signs ofnetwork complexitymdasha power law of nodesrsquo degree distri-bution [1]mdashwas first empirically formulated by D Price in1965 for social networks In 1999 A L Barabasi a physicistfrom the University of Notre Dame (USA) and his graduatestudent R Albert determined [2 3] that for many networksinstead of the expected Poisson probability distribution ofnodesrsquo degree (ie the number of connections a node has toother nodes) the distribution they obtained approximatelyfollowed a power law as all critical states do In many realnetworks a small number of nodes have a large number ofconnections whereas a large number of nodes have just a fewconnections Such networks are called scale-free networksThis name was not invented specifically for this type ofnetworks It came from the theory of critical phenomenawhere fluctuations in critical states also follow a power lawThe theory of scale-free networks is considered to be one ofthe scenarios complex systems follow when they come into acritical state As of late such networks are more often calledcomplex networks

Some other relevant works in this area are those of refs[4ndash8]

An extensive body of research on the modeling of thestructure and functioning of social networks is availabletoday This research has two directions The first directionrelates to the analysis of the social networks data (see oneof the latest reviews [9]) while the second concerns thedevelopment of models of the structure dynamics andevolution of social networks The distinction between thesetwo directions is somewhat arbitrary since in most casesthese directions overlap (see eg [10 11])

Starting from the second half of the 20th century theideas andmethods of physics have tended to infiltrate naturalsciences and traditional humanities Methods of physicalmodeling are often used in such areas of science as demo-graphics sociology and linguistics As a result sociophysicalmodels of social networks such as the Ising model [12ndash15] Bose-Einstein condensate model [3 16] Quantum walkmodel [17] Ground state and community detection [18 19]among others were developed

Despite having a variety of sociophysical modelsthe results and theories of nonlinear science with some

HindawiComplexityVolume 2018 Article ID 4732491 11 pageshttpsdoiorg10115520184732491

2 Complexity

exclusions (see eg [20 21]) are not used to model theevolution of social networks First of all we are talkingabout the complexity and self-organized criticality theorydescribing the mechanism of complexity [22ndash24] Mech-anisms of self-organized criticality in social knowledgecreation process are presented in the paper [25] It isnoteworthy that the key sign of complexity of a systemregardless of its internal structure ie one based solely on itsexternal characteristics was formulated in the framework ofthis theory According to this theory a system is consideredto be complex if it is able to generate unexpected andorextraordinary events (for instance bursts of values in timeseries) This motivated our research The purpose of theresearch is a nonlinear dynamical interpretation of thecomplexity of a microblogging network and the developmentof an appropriate network model that could explain itscomplexity using the third paradigm of nonlinear sciencecalled the complexity paradigm Another motivation for theresearch was the results presented in [26ndash31] where the timeseries of a number of microposts are characterized by themajority of key signs of the system complexity (a detaileddescription of the key signs of the system complexity ispresented in Section 2)

This paper is organized as follows Section 2 dealswith thekey signs of the system complexity according to the complex-ity paradigm Section 3 presents the results of the analysis ofan empiric time series of a number of microposts includingthe results of the calculation of the key signs of the complexitySection 4 presents amodel of amicroblogging social networkas a nonlinear deterministic dynamical system including itscapabilities and restrictions Section 5 presents a generalizedmodel of a microblogging social network modified by theconsideration of stochastic sources and a decrease in theorder parameter as well as the results of an analysis in theadiabatic approximation Section 6 contains the main resultsof the research and a discussion

2 Nonlinear DynamicalInterpretation of Complexity

The development of any branch of science leads to the for-mulation of paradigms namely initial conceptual schemesmodels of problem statements and solutions of the problemsAt this time three paradigms have been developed in non-linear science The first paradigm is that of self-organizationThe second is the paradigm of deterministic chaos The mostrecent development of nonlinear science is closely linked tothe third paradigm which could be defined as a paradigmof complexity that has the theory of self-organized criticalityas its foundation The paradigm of complexity lies at thejunction of the first two paradigms If the first two paradigmsdeal with order and chaos respectively the third is usuallydescribed as ldquolife on the edge of chaosrdquo [32]

Since it is impossible to rigorously define complexity ourresearch is limited to consideration of the key signs of systemcomplexity defined in the publications by Per Bak and co-authors [22ndash24] and their application to the interpretationof the complexity ofmicroblogging social networks As statedin the introduction first of all we consider the complexity of

external system manifestations regardless of internal struc-ture For the purposes of this research we define ldquoexternalsystemmanifestationsrdquo as signals (the time series of a numberof microposts) of a microblogging social network generatedas a result of nontrivial interactions within a very large poolof users

One of the key signs of complexity is its inclination tothe occurrence of catastrophic eventsmdasheither unexpected(ie nonpredictable) or extraordinary (ie prominent amongsimilar events) or both Importantly in either case we canconclude that the system that has generated such an event iscomplex From simple systems we could expect predictabilityand similarities in their behavior As for the signals of amicroblogging system such events qualitatively correspondto considerable bursts seen on a plot of value incrementsof the time series of a number of microposts One of thequantitative criteria of the existence of catastrophic eventsis the existence of power low of the probability densityfunction (PDF) for the values of the time series It is worthmentioning that in the majority of cases the occurrence ofsuch events on the network signal level corresponds to thequalitative restructuring of the system ie a transition fromapolycentric state to amonocentric state and vice versa (suchtransitions are thoroughly described in [33])

Another key sign of complexity is scale invariancemeaning that events or objects lack their own characteristicdimensions durations energies etc At the level of externalmanifestations of a microblogging network scale invariancemeans that the time series of a number of micropostsare fractal or multifractal time series (such time series aredescribed in detail in [34])

In a general case a power low for PDF is a statisticalexpression of scale invariance of the time series

119901 (119909) prop 119909minus120572 (1)

where usually 120572 isin (1 2] PDF (1) belongs to the class offat-tailed PDFs For statistical description of catastrophicevents PDF (1) is a rule with almost no exceptions PDF (1)differs from compact distributions (for example Gaussiandistribution) because the events corresponding to the tail ofthe distribution are not rare enough to be neglected PDF (1)reflects a strong interdependence of the events For examplesuch distributionmay be caused by an avalanche-like increaseof the number of microposts in the network as a result of aldquochain reactionrdquo caused by reposting

Another manifestation of the scale invariance of the timeseries is the existence of the power spectral density (PSD)specific for flicker noise

119878 (119891) prop 119891minus120573 (2)

where 120573 cong 1 The existence of PSD (2) means thata considerable part of the energy is linked to very slowprocesses For amicroblogging network the existence of PSD(2) means that it is impossible to predict the behavior of thetime series of a number of microposts without consideringglobal information exchange processes

The aforementioned features of PDF and PSD are notthe only criteria of scale invariance Besides PDF and PSD

Complexity 3

we used a fractal dimension and a Hurst exponent alongwith other quantitative measures and criteria It is importantto stress that the scale invariance and an inclination tocatastrophes are typical only for systems that are far fromequilibrium Therefore a nonequilibrium state of the systemand therefore a nonlinearity are the necessary conditions forthe complexity of the system

Lastly the third key sign that characterizes complexsystems is their integrity The integral properties of a systemusually are statistically described by power-law space andtime correlations These correlations are known as distantspace and time correlations The existence of distant timecorrelations or long memory in time series is characterizedby the autocorrelation function (ACF) in the following form[34]

119860119862119865120591 prop 120591minus120574 (3)

where 120574 isin (01) The existence of the relationship (3) impliesthe absence of characteristic times at which the informationabout the previous events could be lost A catastrophicbehavior and integrity are connected in the following way forthe catastrophic behavior part of the system should be ableto function in coordination For a microblogging networkan avalanche-like increase of the number of microposts ispossible when a user and his followers followers of thesefollowers etc are working in coordination Integrity ispossible in complex systems only due to the processes of self-organization Here we talk about coarse scale properties ofthe system since minor changes in system parameters do notaffect its integrity

Therefore a microblogging network is a complex systemwhen all the key signs of complexity listed above are satisfiedThis statement forms the foundation of our research and iskey to the construction of a model of microblogging networkevolution

3 Analysis of Empirical Data from Twitter

Empirical data used for our research is a sample of morethan 3 million microposts (tweets retweets and links) aboutthe first US presidential debates of 2016The sample includesmicroposts posted by more than 1 million users from 1345on September 26 2016 to 1100 on September 27 2016 with1-second increments

Figure 1 shows the total number of microposts vs time(Twitter time series 119872119875119905) It is easy to see that 119872119875119905 hasextraordinary events and unexpected events (bursts)

To estimate the correlation dimension (119863119862) and embed-ding dimension (119898) we used the GrassbergerndashProcacciaalgorithm [35] We obtained 119863119862 = 3032 for119898 le 6

Hence the process leading to the 119872119875119905 series is notrandom it depends on a limited number of key parameters[36] The 119872119875119905 series is not stochastic it is chaotic Forinstance for a stochastic series corresponding to Gaussiannoise 119863119862 = 9304 for 119898 le 13 and if the series correspondsto generalized Brownian then noise119863119862 = 8165 for119898 le 9

Using the RS analysis we obtained the Hurst exponent(119867) To calculate the fractal dimension of a time series (119863119865)we used the algorithm presented in [37] We obtained the

09-2

6 17

09-2

6 20

09-2

6 23

09-2

7 02

09-2

7 05

09-2

7 08

09-2

7 11

09-2

7 14

0280400600800

1000

Num

ber o

f mic

ropo

sts

Time (s)

Figure 1 Twitter time series

ndash200 ndash150 ndash100 ndash50 0 50 100 150 2000

001

002

003

004

005

Normal distributionEmpirical distribution

Figure 2 Normal and empirical PDFs

following results 119867 = 0801 119863119865 = 1199 Therefore119872119875119905 isa fractal time series (the fractal dimension is not an integerand exceeds the topological dimension of the time series)Moreover 119872119875119905 is persistent ie the time series is trend-resistant (05 lt 119867 lt 1) Such a time series has a long memoryand is inclined to follow trends [38]

Figure 2 shows PDF for the increments (returns) of119872119875119905time series and PDF for a normal distribution

Empirical probabilities lie outside the normal PDF in theintervals (minusinfinminus3120590] and [3120590 +infin) This means that heavytails exist DrsquoAgostinorsquos K-squared test [39] also confirms thepossibility to reject the null hypothesis about the normalityof the distribution at the significance level of 001 when thestatistics 1198702 = 641989 Another proof that heavy tails existis presented by the fact that the distribution follows a powerlaw of probability distribution Figure 3 shows PDF and thecomplimentary cumulative distribution function (CCDF)Both functions are well approximated by linear functions

Let us determine the type of noise (parameter 120573 inPSD 119878(119891) = 1119891120573) for 119872119875119905 To calculate 120573 we used thedetrended fluctuation analysis method (DFA) [40] After thecalculations we obtained the scaling exponent 120572 = 113 andthe PSD parameter 120573 = 2120572 minus 1 = 126 The 120573 value obtainedcorresponds rather to a flicker noise (120573 = 1) than to any othertype of noiseThevalue120573 = 126 obtained by theDFAmethod

4 Complexity

CCDFPower law fit CCDF Power law fit PDF

PDF

10ndash1

102

100

10ndash4

10ndash5

10ndash2

10ndash3

Figure 3 PDF and CCDF for empirical time series119872119875119905

ln(f)

lnS(

f)

Empirical noiseLinear approximation

100

101

102

103

1013

1011

109

107

105

103

Figure 4 PSD for empirical time series119872119875119905

coincides with the value obtained via the approximation ofthe time series PSD by a linear functionThe PSD obtained byapplying fast Fourier transform to119872119875119905 is shown in Figure 4on a log-log scale A linear fit yields 120573 = 129

The autocorrelation function (119860119862119865120591) for an 119872119875119905 timeseries is described by a decreasing power function (3) with theexponent 120574 = 002 Hence this function has long memory

Figure 5 presents 119860119862119865120591 and its linear approximation on alog-log scale A linear fit gives 120574 = 0024 Microblogging Social Network as aNonlinear Deterministic Dynamical System

41 Main Assumptions for the Model A social network is amacroscopic systemThe number of users for such a system isN ≫ 1This assumption is justified for Twitter since accord-ing to the existing estimations N sim108 In the proposedmodel out of all possible degrees of freedom we chooseand consider just a few macroscopic degrees of freedom(phase or dynamic variables corresponding to hydrodynamic

ln()

ln(ACF

)

Autocorrelation functionLinear approximation

100

100

101

Figure 5 Autocorrelation function for empirical time series119872119875119905

modes in physics) Such a reduction can be justified bythe synergetic subordination principle This principle statesthat during the evolution the hydrodynamicmodes suppressthe behavior of microscopic degrees of freedom and fullydetermine the systemrsquos self-organization As a result thecooperative behavior of a system is determined by severalhydrodynamic variables that represent the amplitudes ofhydrodynamic modes This way we do not need an infinitenumber of microscopic degrees of freedom and there isno need to thoroughly study the microscopic interactionsbetween the users of a social network

A social network is modeled as a point autonomousdynamical system This model was chosen because it ispossible to compare the results with empirical data providedby the Twitter time series Each user of a social networkcan be in one of the two possible states either passive (|119901⟩-state) or active (|119886⟩-state) A Twitter user in |119886⟩-state can sendmicroposts to other network users In this state a networkuser has enough information to send microposts If a user isin |119901⟩-state (the user does not have enough information) heor she cannot send microposts

A microblogging social network is an open nonequi-librium system A social network is capable of informa-tion exchange with the environment The incoming flow ofexternal (for the system) information comes into the systemfrom different sources for example from other mass mediaThis flow feeds the network with information and creates aninverse population of states of network users 119873|119886⟩ ≫ 119873|119901⟩where 119873|119886⟩ is the number of network users in |119886⟩-state and119873|119901⟩ is the number of network users in |119901⟩-state

The distribution of Twitter users can be represented withgood accuracy by a Boltzmann distribution

119873|119886⟩ = 119873|119901⟩ exp[minus(119868|119886⟩ minus 119868|119901⟩)120579 ] (4)

where 119868|119886⟩ is the amount of information the users in |119886⟩-state possess 119868|119901⟩ is the amount of information the usersin |119901⟩-state possess and 120579 is a parameter that describesthe average intensity of stochastic interactions between thenetwork users A simple analysis (4) allows us to define twomacroscopic network states a steady state and a nonequilib-rium state If 119868|119886⟩ minus 119868|119901⟩ ≫ 120579 then 119873|119886⟩ ≪ 119873|119901⟩ In this case

Complexity 5

the network is in a steady state If 119868|119886⟩ minus 119868|119901⟩ ≫ 120579 then119873|119886⟩ ≫119873|119901⟩ In this case the network is in a nonequilibrium stateSince a social network is constantly fed with information itis constantly in a nonequilibrium state creating an avalancheof microposts Because of the constant feed of information asteady state can almost never be reached It is very importantthat the existence of chaotic states is a fundamental propertyof open nonequilibrium systems

42 Phase Variables and Relationships between Them Let usdefine the phase variables of a dynamical system These vari-ableswill be used tomodel Twitter as an opennonequilibriumsystem These variables are as follows 120578119905 equiv 119872119875119905 minus 1198721198750is the deviation of the number of microposts (119872119875119905) fromthe corresponding equilibrium value (1198721198750 is the number ofmicroposts in the steady state) ℎ119905 equiv 119868119905 minus 1198680 is the deviationof aggregated intrasystem information (119868119905) the network userspossess from the corresponding equilibrium value (1198680 isthe aggregated intrasystem information the network userspossesswhenTwitter is in steady state) 119878119905 equiv 119873|119886⟩119905minus119873|119901⟩119905 is theinstantaneous difference at the moment of time 119905 between thenumbers of strategically oriented social network users (usersfollowing a particular strategy) in |119886⟩-and |119901⟩- states If Nis the difference between the total number of users in |119886⟩-and |119901⟩- states then N minus 119878119905 users act randomly (randomlyoriented users)

According to [41] business users and spam users can beconsidered as strategically oriented users

Business users follow marketing and business agendason Twitter The profile description strongly depicts theirmotive and a similar behavior can be observed in theirtweeting behavior Spammers mostly postmalicious tweets athigh rates Automated computer programs (bots) mostly runbehind a spam profile and randomly follow users expectinga few users to follow back

Personal users and professional users can be consideredrandomly oriented users Personal users are casual homeusers who create their Twitter profile for fun learning toget news etc These users neither strongly advocate any typeof business or product nor have profiles affiliated with anyorganization Generally they have a personal profile andshow a low to mild behavior in their social interactionProfessional users are home users with professional intent onTwitter They share useful information about specific topicsand involve in healthy discussion related to their area ofinterest and expertise

Let us determine relationships between the dynamicvariables and their rates of change

The rate of deviation of the number of microposts isdetermined by the relaxation of a social network into asteady state (minus120578119905) and the change in deviation of aggregatedintrasystem information from the equilibrium value (+119886120578ℎ119905)

120591120578 120578119905 = minus120578119905 + 119886120578ℎ119905 (5)

The term minus120578119905 in Eq (5) is due to the relaxation of thesocial network as a nonequilibrium system According to LeChateliers principle when a system deviates from the steadystate this generates ldquoforcesrdquo that try to restore the systemback

to the steady state As follows from Eq (5) without the term119886120578ℎ119905 the equation takes the following form

120578119905 = minus120578119905120591120578 (6)

The solution to Eq (6) is given by the function 120578119905 =119860exp(minus119905120591120578) Hence119872119875119905 997888rarr 1198721198750 when 119905 997888rarr infin (the socialnetwork tends to its steady state) In Eq (6) 120591120578 is the time ofrelaxation to the steady state

The term 119886120578ℎ119905 in Eq (5) can be easily explained as thedeviation of aggregated intrasystem information increasesthe rate of the deviation of the number of micropostsincreases as well

The rate of the deviation of the aggregated intrasysteminformation from the equilibrium value is determined by therelaxation of a social network towards a steady state (minusℎ119905) andthe product +119886ℎ120578119905119878119905

120591ℎℎ119905 = minusℎ119905 + 119886ℎ120578119905119878119905 (7)

The term minusℎ119905 in Eq (7) is also explained by Le Chatelierrsquosprinciple as in Eq (6) The term +119886ℎ120578119905119878119905 appears because theamount of information each user of a social network acquiresfrom a stream of microposts is proportional to the deviationof the number of microposts and depends on the state of theuser in the social network In other words the average contri-bution to the deviation of aggregated intrasystem informationis proportional to the product of the deviation of the numberof microposts and the difference between the numbers ofusers in |119906⟩- and |119897⟩- states

Finally the third equation describes the change in inver-sion of population of strategically oriented users and can bewritten as follows

120591119878 119878119905 = (1198780 minus 119878119905) minus 119886119878120578119905ℎ119905 (8)

where 120591119878 is the corresponding relaxation time and 1198780 isthe initial number of strategically oriented users (this valuereflects the intensity of information feeding into the socialnetwork) In other words 1198780 is the difference between thenumbers of strategically oriented users of a social networkwhich are in |119886⟩- and |119901⟩-states at the time 119905 = 0 The termminus119886119878120578119905ℎ119905 reflects the effective power that a stream of microp-osts applies to create aggregated intrasystem information in asocial network This power can be positive or negative

Thus the evolution of amicroblogging social network canbe described by the well-known Lorenz system of equations

120591120578 120578119905 = minus120578119905 + 119886120578ℎ119905120591ℎℎ119905 = minusℎ119905 + 119886ℎ120578119905119878119905120591119878 119878119905 = (1198780 minus 119878119905) minus 119886119878120578119905ℎ119905

(9)

43 Synergetic Interpretation of a Nonlinear Dynamical Sys-tem The system of self-consistent equations (9) is a well-known method to describe a self-organizing system TheLorenz synergetic model was first developed as a simplifi-cation of hydrodynamic equations describing the Rayleigh-Benard heat convection in the atmosphere it is now a

6 Complexity

classical model of chaotic dynamics Further research on theLorenz system presented in a series of publications provedthat the system provides an appropriate kinetic picture ofthe cooperative behavior of particles in any macroscopicdynamical system where the actualization of potential orderis possible Processes in such self-organizing complex systemsin nonequilibrium state lead to the selection of a smallnumber of parameters from the complete set of variablesthat describe the system all other degrees of freedom adjustto correspond to these selected parameters Following theterminology used in the synergy theory these parameters arethe order parameter (120578119905) conjugated field (ℎ119905) and controlparameter (119878119905) According to the Ruelle-Takens theoremwe can observe a nontrivial self-organization with strangeattractors if the number of selected degrees of freedom isthree or more

In the system of equations (9) 119886120578 is a coefficient andpositive constants 119886ℎ 119886119878 are measures of feedback in asocial network Functions 120578119905120591120578 ℎ119905120591ℎ (1198780 minus 119878119905)120591119878 describethe autonomous relaxation of the deviation of the numberof microposts deviations of aggregated information andinversion of population of strategically oriented users of asocial network to the stationary values 120578119905 = 0 ℎ119905 = 0 119878119905 = 1198780with relaxation times 120591120578 120591ℎ 120591119878

Eq (10) takes into account that in the autonomousregime the change in the aforementioned parameters ofa social network is dissipative In addition Le Chatelierrsquosprinciple is very important since the growth of the controlparameter 119878119905 is the reason for self-organization the values 120578119905and ℎ119905 must vary so as to prevent the growth of 119878119905 Formallythis fact could be explained as the existence of a feedbackbetween the order parameter 120578119905 and the conjugated field ℎ119905Lastly a positive feedback between the order parameter 120578119905and the control parameter 119878119905 leading to the growth of theconjugated field ℎ119905 is very important since this feedback isthe reason for self-organization

44 Capabilities and Restrictions of a Deterministic Model forthe Interpretation of a Social Networkrsquos Complexity First of allwe have to note that Eq (9) was first obtained by EdwardLorenz in 1963 as a result of some simplifications of theproblem of a liquid layer heated from below In this problemEq (9) is obtained when the flow velocity and temperatureof the initial hydrodynamic system are presented as two-dimensional truncated Fourier series and the Boussinesqapproximation is used For the problem of convection ina layer the Lorenz equations serve as a rough not veryaccurate approximation It is only adequate in the regionof regular modes where uniformly rotating convection cellsare observed The chaotic regime typical of Eq (9) doesnot describe the turbulent convection However the Lorenzequations became a suitable model for describing systemsand processes of various natures convection in a closed loopsingle-mode laser water wheel rotation financial marketstransportation flows dissipative oscillator with nonlinearexcitation and some others

How reliable is the model (9) for the description ofthe evolution of a microblogging social network We willconsider the model ldquoreliablerdquo if there is a good correlation

between theoretically predicted and empirically observed keysigns of complexity of the systemThe results of the compari-son of key signs of complexity for the theory-based deviationsof the number of microposts 120578119905 and the correspondingempirical data are presented below

As shown earlier (see Eq (4)) a steady state of thenetwork is almost impossible to achieve due to a constantinformation feed Theoretically a dynamical system (9) hasan asymptotically stable zero stationary point as a node for1198780 = 0 In this case 119872119875119905 asymp 1198721198750 119868119905 asymp 1198680 and 119873|119886⟩119905 asymp 119873|119901⟩119905as 119905 997888rarr infin However in practice a microblogging networkas an open nonequilibrium system always has a non-zerodifference between the numbers of strategically orientedusers that are in |119886⟩-state and in |119901⟩-state at the time 119905 = 0Therefore despite a theoretical feasibility of the steady statefor a social network this state cannot be achieved in practiceWhen the difference between the numbers of strategicallyoriented users that are in |119886⟩-state and in |119901⟩-state at thetime 119905 = 0 reaches some critical value 1198780119888 Eq (9) enters achaotic regime and a strange attractor appears A transitionalstate that corresponds to 1198780 isin (1 1198780119888) cannot be realized inpractice

We will consider 1198721198750 as constant for a long enoughperiod of time and compare differentmeasures for theoretical(120578119905 the solution of system (9) in chaotic regime) and empir-ical data 119872119875119905 The model of a social network presented inthe form (9) explains the fractal and chaotic nature of theobserved 119872119875119905 119863119862 = 2067 and 119863119865 = 1504 Howeverthe model (9) cannot explain the observed key signs ofcomplexity of a social network Theoretical119872119875119905 constitutes atime series without memory (119860119862119865120591 exponentially decreases)PSD is constant (white noise 120573 = 0) PDF is multi-modalwith ldquotruncated tailsrdquo (see Figure 6)

Compactness and multi-modality of the distribution aredetermined by the existence of three stationary points of thedynamical system (9)

Thus the Lorenz system (9) is not a reliable model forthe description of the evolution of a microblogging socialnetwork as a complex system

5 Microblogging Social Network as aNonlinear Random Dynamical System

As shown earlier the nonlinear dynamic model (9) explainsthe fractality and chaotic nature of empirical 119872119875119905 as well asthe dissipative nature of the system On the other hand Eq(9) cannot explain someother phenomena found in empiricaldata and first of all the key signs of complexity of a socialnetwork a power law of PDF 1119891-noise and long memoryLet us consider different ways of improving (generalizing) Eq(9) in order to adequately describe a microblogging socialnetwork

Since the correlation dimension and embedding dimen-sion of the empirical time series (119863119862 = 3032 119898 le 6)exceed the corresponding theoretical values (119863119862 = 2067119898 le 4) one of the ways to improve Eq (9) is to increasethe number of phase variables of the dynamic systemAnother approach to improving Eq (9) is to consider theself-consistent behavior of the order parameter conjugated

Complexity 7

ndash20 ndash15 ndash10 ndash5 0 5 10 150

100

200

300

400

500

Figure 6 Histogram for the time series119872119875119905

field and control parameter taking into account the noisefor each of those parameters Different generalizations of Eq(9) have been proposed and studied by Alexander Olemskyand collaborators [42ndash44] in particular in the context of itsapplications to the study of self-organization of continuumevolution of financial markets and economical structure ofsociety cooperative behavior of active particles and self-organized criticality

Taking into account stochastic terms and the fractionalityof the order parameter Eq (9) takes the following form

120591120578 120578119905 = minus120578120572119905 + 119886120578ℎ119905 + radic119868120578120585119905120591ℎℎ119905 = minusℎ119905 + 119886ℎ120578120572119905 119878119905 + radic119868ℎ120585119905120591119878 119878119905 = (1198780 minus 119878119905) minus 119886119878120578120572119905 ℎ119905 + radic119868119878120585119905

(10)

In Eq (10) 119868119894 are noise intensities for each phase variable 120585119905is white noise where ⟨120585119905⟩ = 0 ⟨1205851199051205851199051015840⟩ = 120575(119905 minus 1199051015840) 120572 isin (0 1]The random dynamic system (RDS) (10) is a generalizationof the deterministic dynamic system (9) where stochasticsources are added the feedback is weakened and the orderparameter is relaxedThe replacement of the order parameter120578119905 by a smaller value 120578120572119905 (120572 le 1) means that the process ofordering influences the self-consistent behavior of the systemto a lesser extent than it does in the ideal case of 120572 = 1

For the convenience of the analysis of Eq (10) we willtransform it into a dimensionless formThen time 119905 deviationof the number of microposts (120578119905) deviation of the aggregatedintrasystem information (ℎ119905) the difference between thenumbers of strategically oriented users in different states (119878119905)and corresponding noise intensities (119868119894) will be scaled asfollows

119905119888 equiv 120591120578 (119886120578119886ℎ)(120572minus1)(2120572) ℎ119888 equiv (1198862120578119886ℎ119886119878)minus12 120578119888 equiv (119886ℎ119886119878)minus1(2120572) 119878119888 equiv (119886120578119886ℎ)minus12

119868119888120578 equiv (119886ℎ119886119878)minus1120572 119868119888ℎ equiv (1198862120578119886ℎ119886119878)minus1 119868119888119878 equiv (119886120578119886ℎ)minus12

(11)

Now Eq (10) can be written down as follows

120578 = minus120578120572 + ℎ + radic119868120578120585120591ℎ119905119888 ℎ = minusℎ + 120578

120572119878 + radic119868ℎ120585120591119878119905119888 119878 = (1198780 minus 119878) minus 120578120572ℎ + radic119868119878120585

(12)

Let us analyze RDS (12) in adiabatic approximation when thecharacteristic relaxation time of the number of microposts ina network considerably exceeds the corresponding relaxationtimes of aggregated intrasystem information and the numberof strategically oriented users 120591120578 ≫ 120591ℎ 120591119878 This means thataggregated intrasystem information ℎ asymp ℎ(120578) and the numberof strategically oriented users 119878 asymp 119878(120578) follow the variation inthe deviation of the number of microposts (120578) When 120591120578 ≫120591ℎ 120591119878 the subordination principle allows us to set (120591ℎ119905119888)ℎ =(120591119878119905119888) 119878 = 0 in Eq (12) ie to disregard the fluctuations inℎ asymp ℎ(120578) and 119878 asymp 119878(120578)

For a microblogging social network functioning as anopen nonequilibrium system the adiabatic approximationmeans that when the external information feed tends to zerothe stream of microposts slowly decreases and at the sametime the aggregated intrasystem information and the numberof strategically oriented users in active state decrease as well

An adiabatic approximation is a necessary conditionfor the transformation of the three-dimensional RDS withadditive noise (12) into a one-dimensional RDS with multi-plicative noise of the following form

120591120578 120578 = 119891 (120578) + radic119868 (120578)120585 (13)

The terms of Eq (13) corresponding to the drift and diffusion(intensity of the chaotic source) have the following form

119891 (120578) equiv minus120578120572 + 1198780120578120572 (1 + 1205782120572) (14)

119868 (120578) equiv 119868120578 + (119868ℎ + 1198681198781205782120572) (1 + 1205782120572)2 (15)

The Langevin equation (13) has an infinite set of randomsolutions 120578Their probability distribution (119901(120578 119905)) is given bythe Fokker-Planck equation

120597119905119901 (120578 119905) = 120597120578 [minus119891 (120578) 119901 (120578 119905) + 120597120578 (119868 (120578) 119901 (120578 119905))] (16)

In the stationary case (120597119905119901(120578 119905) = 0) the distribution 119901(120578 119905) isgiven by the following relationship

119901 (120578) prop 119868minus1 (120578) exp [int 119891 (120578)119868 (120578) 119889120578] (17)

8 Complexity

As a result the stationary probability distribution densityof the deviation of the number of microposts from thecorresponding equilibrium value has the following form

119901 (120578) = 119885minus1 (1 + 1205782120572)minus2 exp[[int (1 + 1205782120572)minus2

120578120572 119889120578]] (18)

where 119885 is a normalization constantBefore we draw any conclusion about distribution (18)

let us direct our attention to one significant fact that distin-guishes theory from practice Distributions of real systemsand processes regardless of their nature cannot have aninfinite expected value or variance Therefore power-lawPDFs like 119901(119909) prop 119909minus2120572 (2120572 is chosen for the purposes ofanalysis of expression (18)) are approximate and not validfor large 119909 The exponential decrease of PDF correspondsto the intermediate asymptotics and in practice instead ofheavy tails we should have semi-heavy tails (see distributionin Figure 2)

119901 (120578) prop 120578minus2120572P( 120578120578119888) (19)

where the scaling functionP(120578120578c ) is approximately constantat 120578 cong 120578119888 and quickly decreases when 120578 997888rarr infin Here theldquoheaviness of the tailrdquo is shifted toward the intermediate rangeof 120578 values Thus the dimensionless deviation of the numberof microposts 120578 scaled for 120578119888 serves as a scaling variable 120578120578119888in (19) Since the integral in Eq (16) is regular at 120578 997888rarr 0 thePDF obtained has a power-law form

The power law for PDF of the deviation of the numberof microposts 120578119905 which is equivalent to119872119875119905 for large timeswas obtained and justified analytically However we couldnot obtain analytical expressions for PSD 119860119862119865120591 or thecorrelation and fractal dimensions Therefore we presentbelow the results of numerical calculations for a family ofrealizations of RDS (13) for 120572 = 05 based on algorithmsstudied and used earlier

Let us determine the type of noise typical for 120578119905 We usedthe DFA method to calculate 120573 We obtained the scalingexponent 120572 = 118 and 120573 = 2120572minus1 = 136The value obtainedfor 120573 corresponds rather to flicker noise (120573 = 1) than to anyother type of noise The value 120573 = 118 obtained by the DFAmethod is close to the value obtained through fitting PSDtime series by a linear function PSD obtained by applyingfast Fourier transform to 120578119905 is presented in Figure 7 in log-logscale A linear fit gives 120573 = 134119860119862119865120591 for the time series 120578119905 decreases by following thepower law (3) with the exponent 120574 = 004 and hence hasdouble memory Figure 8 shows 119860119862119865120591 for 120578119905 in log-log scale

To estimate the correlation dimension (119863119862) and embed-ding dimension (119898) we used the GrassbergerndashProcacciaalgorithm We obtained 119863119862 = 2852 for 119898 le 5 Hencethe process that leads to the series 120578119905 is not random it iscontrolled by a limited number of key parameters The series120578119905 is chaotic rather than stochastic

Using the results of RS analysis we determined the Hurstexponent (119867) To calculate the fractal dimension of the timeseries (119863119865) we used the algorithm described in [20] We

0100

0001

0010

100

1

10

0100005000100005 10500 5ln(f)

lnS(

f)

Figure 7 PSD for 120578119905 time series

0010

0100

1

0001

0100005000100005 10500 5ln()

ln(ACF

)

Figure 8 Autocorrelation function for 120578119905 time series

obtained 119867 = 0765 119863119865 = 1235 Hence 120578119905 is a persistentfractal time series Such time series has a long-term memoryand tends to follow trends

Therefore the generalized Lorenz system (12) adequatelymodels the evolution of a microblogging social networkas a complex system The characteristics of 120578119905-realizationsof RDS (12) are quantitatively close to the correspondingcharacteristics of empirical time series

6 Results and Discussion

For the convenience of further discussion Table 1 presentsthe results of calculations of key characteristics and propertiesof complex systems (ie systems that tend to have unexpectedandor extraordinary events)

The empirical time series of microposts has all the keyproperties of complexity a power-law PDF noise that is closeto flicker noise time correlationswith longmemory and scaleinvariance in a time series of microposts The existence ofbursts in time series of microposts (see Figure 1) allows us toconclude that a microblogging network is a complex systemand it is far from equilibriumThe time series ofmicroposts ischaracterized by scale invariance ie it is a fractal time seriesSuch time series in particular are characterized by power-law PDFs caused by an avalanche-like increase of the numberof microposts (see bursts in Figure 1) after a ldquochain reactionrdquoof reposting An avalanche-like increase of the number ofmicroposts is possible if a user coordinates his actions withhis followers followers of those followers and so on Thisdefines a connection between the catastrophic behavior andintegrity of a microblogging network

Complexity 9

Table 1 Characteristics of empirical and theoretical time series

Time Series PDF 120573 120574 119863119862 119898 119863119865 119867Empirical Power 129 002 3032 6 1199 0801

LawLorenz Compact 0 Exponential 2067 4 1504 0496Generalized Power 136 004 2852 5 1235 0765Lorenz Law

For a description of the evolution of a microbloggingnetwork the nonlinear dynamical system model (9) is arough not very accurate approximation First the modeldoes not predict the occurrence of catastrophic values ina time series of the number of microposts which wouldsignify the complexity of a microblogging network or theexistence of long memory or the time seriesrsquo tendency tofollow trends Despite this deficiency Eq (9) allows one tostudy social networks far from equilibrium (see distribution(4) and comments thereon) and it also explains the existenceof dynamical chaos in a time series as well as their fractality

The nonlinear random dynamical system (10) is a gener-alization of themodel (9) that accounts for external stochasticsources and the fractionality of the order parameter (weaken-ing of feedback and relaxation of the order parameter) Thismodel adequately describes the evolution of a microbloggingsystem

Quantitative characteristics of the model (10) in adiabaticapproximation are close to the corresponding characteristicsof the empirical time series of microposts (see Table 1)An adiabatic approximation allows us to reduce a three-dimensional random adiabatic system with additive noise(10) to a one-dimensional random dynamical system withexponential multiplicative noise (13)

7 Conclusions

The main results of this research were obtained by analyzinga single time series of microposts whose values howeverconstitute a representative sample Similar results of analysisof an empirical time series of a microblogging network arepresented in [24ndash29] We cannot claim that the time seriessamples studied by us or other researchers are representativewhich would be essential for a generalization of the resultsonto the entire general population In the framework of thisapproach it is necessary to analyze all the available data onmicroposts and users collected since the beginning of themicroblogging network However this step could be avoidedif we consider the scale invariance of social networks Thisallows us to extrapolate and interpolate the results of thenetwork analysis onto any large or small scale Hence thefractality of a sample predetermines the fractality of the entirenetwork A justification of the scale invariance for Twitteris presented in [45] Therefore the conclusion about thecomplexity of microblogging networks in the framework ofthe paradigm of complexity is justified

What follows from the fact that a microblogging networkis complex We can give two answers to this question Thefirst is connected with the possibility of second-order phase

transitions in a microblogging network the second concernsthe analysis and prediction of a time series of microposts Letus elaborate on each answer

It has been established that time series of microposts arecharacterized by long-range time correlations This is trueboth for empirical time series and for realizations of therandom dynamical system (13) Long-range correlations andother characteristics of time series discussed above are typicalof critical phenomena such as second order phase transitions

For simplicity let us consider the kinetics of a phasetransition in a microblogging network in the framework ofthe model (9) taking into account the stochasticity of thefeed (the difference in the initial number of strategicallyoriented users in active and passive states 1198780) In this caseit can be shown (a detailed proof lies outside the scope ofthis paper) that as 1198780 increases and exceeds a certain criticalvalue a microblogging network evolves according to thestrategy chosen by a relatively small number of strategicallyoriented users The aforementioned avalanche-like increaseof the number of microposts takes place The critical valueof 1198780 is determined by the geometric mean of the total andcritical values of the number of strategically oriented users Aformalism that leads to the above result is presented in [42]

The results obtained in this paper are valuable fromboth theoretical and practical points of view Firstly theyshow that the systems under consideration (in this case thenumber ofmicroposts) are deterministic despite having noisecomponents (ie they are not stochastic) This allows us touse the theory of dynamical systems and analyze the timeseries of microposts in a different way using the dimensiontheory and the theory of dynamic systems Secondly thevalues of invariants obtained can help solve the problem ofprediction For example the embedding dimension showshow many terms of a time series determine the next termwhereas the correlation entropy and the largest Lyapunovexponent allow us to estimate the time of predictability of thesystem

In conclusion we would like to mention that there existmany interesting problems that are not studied yet suchas critical phenomena of self-organization in microbloggingnetworks based on the analysis of the nonlinear randomdynamic system (10) This will be the subject of our futureresearch

Data Availability

Data was obtained by hydrating a list containing 3183202identifiers of tweets from the set of 12 lists of identifiersprovided by Harvard University This list is about the 2016

10 Complexity

USA presidential elections laquo2016 United States PresidentialElection Tweet Idsraquo (2016) The list was created by JustinLittman Laura Wrubel and Daniel Kerchner The authors ofthe list used SocialFeed service to gather data after the firstdebates Tweets on the subsequent debates were not includedin the sampleThe sample obtained has about 1 million emptyentries This happened because some users whose identifierswere in the initial list later removed their tweets or madethem private The resulting sample has the following charac-teristics a micropost can correspond to one hashtag or sev-eral hashtags (debate debates debatenight debate2016debates2016) the presence of the micropostrsquos author in thelist of followers of one or several users (CPD (debates)HillaryClinton (HillaryClinton) KDonald J Trump (real-DonaldTrump)) 2290855microposts 934656 users 76458time intervals one-second increments After the list of tweetswas received the information was extracted in idoriginal idformat Here id is a unique identifier of the user who madethe retweet the original id is a unique identifier of the userwho made the initial tweet If a tweet is not a retweet id andoriginal id coincide

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Thework was supported by the Russian Foundation for BasicResearch (grant 16-07-01027)

References

[1] D J De Solla Price ldquoNetworks of scientific papersrdquo Science vol149 no 3683 pp 510ndash515 1965

[2] A Barabasi and R Albert ldquoEmergence of scaling in randomnetworksrdquo Science vol 286 no 5439 pp 509ndash512 1999

[3] R Albert and A Barabasi ldquoStatistical mechanics of complexnetworksrdquo Reviews of Modern Physics vol 74 no 1 pp 47ndash972002

[4] C T Butts ldquoThe complexity of social networksTheoretical andempirical findingsrdquo Social Networks vol 23 no 1 pp 31ndash712001

[5] J Skvoretz ldquoComplexity theory and models for social net-worksrdquo Complexity vol 8 no 1 pp 47ndash55 (2003) 2002

[6] M G Everett ldquoRole similarity and complexity in social net-worksrdquo Social Networks An International Journal of StructuralAnalysis vol 7 no 4 pp 353ndash359 1985

[7] H Ebel J Davidsen and S Bornholdt ldquoDynamics of socialnetworksrdquo Complexity vol 8 no 2 pp 24ndash27 (2003) 2002

[8] S Boccaletti V Latora Y Moreno M Chavez and D WHwang ldquoComplex networks Structure and dynamicsrdquo PhysicsReports vol 424 no 4-5 pp 175ndash308 2006

[9] S Tabassum F S F Pereira S Fernandes and J Gama ldquoSocialNetworks Analysis An Overviewrdquo WIREs Data Mining andKnowledge Discovery pp 1ndash21 2018

[10] S Saganowski B Gliwa P Brodka A Zygmunt P Kazienkoand J Kozlak ldquoPredicting community evolution in socialnetworksrdquo Entropy vol 17 no 5 pp 3053ndash3096 2015

[11] P De Meo F Messina D Rosaci and G M L Sarne ldquoFormingtime-stable homogeneous groups intoOnline Social NetworksrdquoInformation Sciences vol 414 pp 117ndash132 2017

[12] A Grabowski and R A Kosinski ldquoIsing-based model ofopinion formation in a complex network of interpersonal inter-actionsrdquo Physica A Statistical Mechanics and its Applicationsvol 361 no 2 pp 651ndash664 2006

[13] S Dasgupta R K Pan and S Sinha ldquoPhase of Ising spins onmodular networks analogous to social polarizationrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 80no 2 Article ID 025101 2009

[14] G Bianconi ldquoMean Field Solution of the Ising Model on aBarabasi-Albert Networkrdquo Physics Letters A vol 303 no 2-3pp 166ndash168 2002

[15] C Li F Liu and P Li ldquoIsing model of user behavior decision innetwork rumor propagationrdquo Discrete Dynamics in Nature andSociety vol 2018 Article ID 5207475 2018

[16] J-L Guo Q Suo A-Z Shen and J Forrest ldquoThe evolutionof hyperedge cardinalities and bose-Einstein condensation inhypernetworksrdquo Scientific Reports vol 6 Article ID 33651 2016

[17] M Faccin T Johnson J Biamonte S Kais and P MigdałldquoDegree Distribution in Quantum Walks on Complex Net-worksrdquo Physical Review X vol 3 no 4 Article ID 041007 2013

[18] J Reichardt and S Bornholdt ldquoStatisticalmechanics of commu-nity detectionrdquoPhysical ReviewE Statistical Nonlinear and SoftMatter Physics vol 74 no 1 Article ID 016110 2006

[19] B PChamberlain J Levy-KramerCHumby andMPDeisen-roth ldquoReal-time community detection in full social networkson a laptoprdquo PLoS ONE vol 13 no 1 Article ID e0188702 2018

[20] Y Matsubara Y Sakurai B A Prakash L Li and C FaloutsosldquoNonlinear dynamics of information diffusion in social net-worksrdquo ACM Transactions on the Web (TWEB) vol 11 Article11 no 2 2017

[21] N Hegde L Massoulie and L Viennot ldquoSelf-organizing flowsin social networksrdquo Theoretical Computer Science vol 584 no13 pp 3ndash18 2015

[22] P Bak C Tang and K Wiesenfeld ldquoSelf-organized CriticalityAnExplanation of 1f-noiserdquoPhysical Review Letters vol 59 no4 pp 381ndash384 1987

[23] P Bak C Tang and K Wiesenfeld ldquoSelf-organized CriticalityrdquoPhysical Review A vol 38 no 1 pp 364ndash374 1988

[24] P Bak How Nature Works The Science of Self-organized Criti-cality Springer-Verlag 1996

[25] B Tadic M M Dankulov and R Melnik ldquoMechanisms of self-organized criticality in social processes of knowledge creationrdquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 96 no 3 Article ID 032307 2017

[26] MAguilera IMorer X Barandiaran andM Bedia ldquoQuantify-ing Political Self-Organization in Social Media Fractal patternsin the Spanish 15M movement on Twitterrdquo in Proceedings ofthe 12th European Conference on Artificial Life pp 395ndash402Michigan USA 2013

[27] K Lyudmyla B Vitalii and R Tamara ldquoFractal time seriesanalysis of social network activitiesrdquo in Proceedings of the2017 4th International Scientific-Practical Conference Problemsof Infocommunications Science and Technology (PIC SampT) pp456ndash459 IEEE Kharkov Ukraine October 2017

[28] T De Bie J Lijffijt C Mesnage and R Santos-RodriguezldquoDetecting trends in twitter time seriesrdquo in Proceedings of the2016 IEEE 26th InternationalWorkshop onMachine Learning forSignal Processing (MLSP) pp 1ndash6 Vietri sul Mare Salerno ItalySeptember 2016

Complexity 11

[29] A Mollgaard and J Mathiesen ldquoEmergent user behavior ontwitter modelled by a stochastic differential equationrdquo PLoSONE vol 10 no 5 pp 1ndash12 2015

[30] A Dmitriev V Dmitriev O Tsukanova and S Maltseva ldquoAnonlinear dynamical approach to the interpretation of mi-croblogging network complexityrdquo Studies in ComputationalIntelligence vol 689 pp 390ndash400 2017

[31] B Tadic V Gligorijevic M Mitrovic and M Suvakov ldquoCo-evolutionary mechanisms of emotional bursts in online socialdynamics and networksrdquo Entropy vol 15 no 12 pp 5084ndash51202013

[32] V V Waldrop Complexity The Emerging Science at the Edgeof Order and Chaos Touchstone New York USA 1993

[33] O A Tsukanova E P Vishnyakova and S VMaltseva ldquoModel-based monitoring and analysis of the network communitydynamics in a textured state spacerdquo in Proceedings of the 16thIEEE Conference on Business Informatics CBI 2014 pp 44ndash49Switzerland July 2014

[34] Ming Li ldquoFractal Time SeriesmdashA Tutorial ReviewrdquoMathemat-ical Problems in Engineering vol 2010 Article ID 157264 26pages 2010

[35] P Grassberger and I Procaccia ldquoMeasuring the strangeness ofstrange attractorsrdquo Physica D Nonlinear Phenomena vol 9 no1-2 pp 189ndash208 1983

[36] M Z Ding C Grebogi E Ott T Sauer and J A Yorke ldquoEsti-mating correlation dimension from a chaotic time series whendoes plateau onset occurrdquo Physica D Nonlinear Phenomenavol 69 no 3-4 pp 404ndash424 1993

[37] M M Dubovikov N V Starchenko and M S DubovikovldquoDimension of the minimal cover and fractal analysis of timeseriesrdquo Physica A Statistical Mechanics and its Applications vol339 no 3-4 pp 591ndash608 2004

[38] B B Mandelbrot and J W Van Ness ldquoFractional brownianmotions fractional noises and applicationsrdquo SIAM vol 10 no4 pp 422ndash437 1968

[39] R B DrsquoAgostino A Belanger and R B DrsquoAgostino ldquoA sugges-tion for using powerful and informative tests of normalityrdquoTheAmerican Statistician vol 44 no 4 pp 316ndash321 1990

[40] R B Govindan J D Wilson H Preil H Eswaran J Q Camp-bell and C L Lowery ldquoDetrended fluctuation analysis ofshort datasets an application to fetal cardiac datardquo Physica DNonlinear Phenomena vol 226 no 1 pp 23ndash31 2007

[41] M M Uddin M Imran and H Sajjad ldquoUnderstanding Typesof Users on Twitterrdquo in Proceedings of 6th ASE InternationalConference in Social Computing Stanford USA 2014

[42] A I Olemskoi A V Khomenko and D O Kharchenko ldquoSelf-organized criticality within fractional Lorenz schemerdquo PhysicaA Statistical Mechanics and its Applications vol 323 no 1-4 pp263ndash293 2003

[43] A I Olemskoı ldquoTheory of stochastic systems with singularmultiplicative noiserdquoPhysics-Uspekhi vol 41 no 3 pp 269ndash3011998

[44] A I Olemskoi andAV Khomenko ldquoThree-Parameter Kineticsof a Phase Transitionrdquo Journal of Theoretical and ExperimentalPhysics vol 81 no 6 pp 1180ndash1192 1996

[45] S Aparicio J Villazon-Terrazas and G Alvarez ldquoA Model forScale-Free Networks Application to Twitterrdquo Entropy vol 17no 12 pp 5848ndash5867 2015

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2 Complexity

exclusions (see eg [20 21]) are not used to model theevolution of social networks First of all we are talkingabout the complexity and self-organized criticality theorydescribing the mechanism of complexity [22ndash24] Mech-anisms of self-organized criticality in social knowledgecreation process are presented in the paper [25] It isnoteworthy that the key sign of complexity of a systemregardless of its internal structure ie one based solely on itsexternal characteristics was formulated in the framework ofthis theory According to this theory a system is consideredto be complex if it is able to generate unexpected andorextraordinary events (for instance bursts of values in timeseries) This motivated our research The purpose of theresearch is a nonlinear dynamical interpretation of thecomplexity of a microblogging network and the developmentof an appropriate network model that could explain itscomplexity using the third paradigm of nonlinear sciencecalled the complexity paradigm Another motivation for theresearch was the results presented in [26ndash31] where the timeseries of a number of microposts are characterized by themajority of key signs of the system complexity (a detaileddescription of the key signs of the system complexity ispresented in Section 2)

This paper is organized as follows Section 2 dealswith thekey signs of the system complexity according to the complex-ity paradigm Section 3 presents the results of the analysis ofan empiric time series of a number of microposts includingthe results of the calculation of the key signs of the complexitySection 4 presents amodel of amicroblogging social networkas a nonlinear deterministic dynamical system including itscapabilities and restrictions Section 5 presents a generalizedmodel of a microblogging social network modified by theconsideration of stochastic sources and a decrease in theorder parameter as well as the results of an analysis in theadiabatic approximation Section 6 contains the main resultsof the research and a discussion

2 Nonlinear DynamicalInterpretation of Complexity

The development of any branch of science leads to the for-mulation of paradigms namely initial conceptual schemesmodels of problem statements and solutions of the problemsAt this time three paradigms have been developed in non-linear science The first paradigm is that of self-organizationThe second is the paradigm of deterministic chaos The mostrecent development of nonlinear science is closely linked tothe third paradigm which could be defined as a paradigmof complexity that has the theory of self-organized criticalityas its foundation The paradigm of complexity lies at thejunction of the first two paradigms If the first two paradigmsdeal with order and chaos respectively the third is usuallydescribed as ldquolife on the edge of chaosrdquo [32]

Since it is impossible to rigorously define complexity ourresearch is limited to consideration of the key signs of systemcomplexity defined in the publications by Per Bak and co-authors [22ndash24] and their application to the interpretationof the complexity ofmicroblogging social networks As statedin the introduction first of all we consider the complexity of

external system manifestations regardless of internal struc-ture For the purposes of this research we define ldquoexternalsystemmanifestationsrdquo as signals (the time series of a numberof microposts) of a microblogging social network generatedas a result of nontrivial interactions within a very large poolof users

One of the key signs of complexity is its inclination tothe occurrence of catastrophic eventsmdasheither unexpected(ie nonpredictable) or extraordinary (ie prominent amongsimilar events) or both Importantly in either case we canconclude that the system that has generated such an event iscomplex From simple systems we could expect predictabilityand similarities in their behavior As for the signals of amicroblogging system such events qualitatively correspondto considerable bursts seen on a plot of value incrementsof the time series of a number of microposts One of thequantitative criteria of the existence of catastrophic eventsis the existence of power low of the probability densityfunction (PDF) for the values of the time series It is worthmentioning that in the majority of cases the occurrence ofsuch events on the network signal level corresponds to thequalitative restructuring of the system ie a transition fromapolycentric state to amonocentric state and vice versa (suchtransitions are thoroughly described in [33])

Another key sign of complexity is scale invariancemeaning that events or objects lack their own characteristicdimensions durations energies etc At the level of externalmanifestations of a microblogging network scale invariancemeans that the time series of a number of micropostsare fractal or multifractal time series (such time series aredescribed in detail in [34])

In a general case a power low for PDF is a statisticalexpression of scale invariance of the time series

119901 (119909) prop 119909minus120572 (1)

where usually 120572 isin (1 2] PDF (1) belongs to the class offat-tailed PDFs For statistical description of catastrophicevents PDF (1) is a rule with almost no exceptions PDF (1)differs from compact distributions (for example Gaussiandistribution) because the events corresponding to the tail ofthe distribution are not rare enough to be neglected PDF (1)reflects a strong interdependence of the events For examplesuch distributionmay be caused by an avalanche-like increaseof the number of microposts in the network as a result of aldquochain reactionrdquo caused by reposting

Another manifestation of the scale invariance of the timeseries is the existence of the power spectral density (PSD)specific for flicker noise

119878 (119891) prop 119891minus120573 (2)

where 120573 cong 1 The existence of PSD (2) means thata considerable part of the energy is linked to very slowprocesses For amicroblogging network the existence of PSD(2) means that it is impossible to predict the behavior of thetime series of a number of microposts without consideringglobal information exchange processes

The aforementioned features of PDF and PSD are notthe only criteria of scale invariance Besides PDF and PSD

Complexity 3



119860119862119865120591 prop 120591minus120574 (3)









09-2

6 17

09-2

6 20

09-2

6 23

09-2

7 02

09-2

7 05

09-2

7 08

09-2

7 11

09-2

7 14

0280400600800

1000

Num

ber o

f mic

ropo

sts

Time (s)



001

002

003

004

005







4 Complexity


PDF

10ndash1

102

100

10ndash4

10ndash5

10ndash2

10ndash3


ln(f)

lnS(

f)


100

101

102

103

1013

1011

109

107

105

103






ln()

ln(ACF

)


100

100

101






119873|119886⟩ = 119873|119901⟩ exp[minus(119868|119886⟩ minus 119868|119901⟩)120579 ] (4)


Complexity 5








120591120578 120578119905 = minus120578119905 + 119886120578ℎ119905 (5)



120578119905 = minus120578119905120591120578 (6)




120591ℎℎ119905 = minusℎ119905 + 119886ℎ120578119905119878119905 (7)



120591119878 119878119905 = (1198780 minus 119878119905) minus 119886119878120578119905ℎ119905 (8)




(9)


6 Complexity














Complexity 7


100

200

300

400

500





(10)





(11)




(12)




120591120578 120578 = 119891 (120578) + radic119868 (120578)120585 (13)


119891 (120578) equiv minus120578120572 + 1198780120578120572 (1 + 1205782120572) (14)

119868 (120578) equiv 119868120578 + (119868ℎ + 1198681198781205782120572) (1 + 1205782120572)2 (15)


120597119905119901 (120578 119905) = 120597120578 [minus119891 (120578) 119901 (120578 119905) + 120597120578 (119868 (120578) 119901 (120578 119905))] (16)



8 Complexity



120578120572 119889120578]] (18)



119901 (120578) prop 120578minus2120572P( 120578120578119888) (19)






0100

0001

0010

100

1

10

0100005000100005 10500 5ln(f)

lnS(

f)


0010

0100

1

0001

0100005000100005 10500 5ln()

ln(ACF

)







Complexity 9







7 Conclusions








Data Availability


10 Complexity




Acknowledgments


References





























Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3



119860119862119865120591 prop 120591minus120574 (3)









09-2

6 17

09-2

6 20

09-2

6 23

09-2

7 02

09-2

7 05

09-2

7 08

09-2

7 11

09-2

7 14

0280400600800

1000

Num

ber o

f mic

ropo

sts

Time (s)



001

002

003

004

005







4 Complexity


PDF

10ndash1

102

100

10ndash4

10ndash5

10ndash2

10ndash3


ln(f)

lnS(

f)


100

101

102

103

1013

1011

109

107

105

103






ln()

ln(ACF

)


100

100

101






119873|119886⟩ = 119873|119901⟩ exp[minus(119868|119886⟩ minus 119868|119901⟩)120579 ] (4)


Complexity 5








120591120578 120578119905 = minus120578119905 + 119886120578ℎ119905 (5)



120578119905 = minus120578119905120591120578 (6)




120591ℎℎ119905 = minusℎ119905 + 119886ℎ120578119905119878119905 (7)



120591119878 119878119905 = (1198780 minus 119878119905) minus 119886119878120578119905ℎ119905 (8)




(9)


6 Complexity














Complexity 7


100

200

300

400

500





(10)





(11)




(12)




120591120578 120578 = 119891 (120578) + radic119868 (120578)120585 (13)


119891 (120578) equiv minus120578120572 + 1198780120578120572 (1 + 1205782120572) (14)

119868 (120578) equiv 119868120578 + (119868ℎ + 1198681198781205782120572) (1 + 1205782120572)2 (15)


120597119905119901 (120578 119905) = 120597120578 [minus119891 (120578) 119901 (120578 119905) + 120597120578 (119868 (120578) 119901 (120578 119905))] (16)



8 Complexity



120578120572 119889120578]] (18)



119901 (120578) prop 120578minus2120572P( 120578120578119888) (19)






0100

0001

0010

100

1

10

0100005000100005 10500 5ln(f)

lnS(

f)


0010

0100

1

0001

0100005000100005 10500 5ln()

ln(ACF

)







Complexity 9







7 Conclusions








Data Availability


10 Complexity




Acknowledgments


References





























Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity


PDF

10ndash1

102

100

10ndash4

10ndash5

10ndash2

10ndash3


ln(f)

lnS(

f)


100

101

102

103

1013

1011

109

107

105

103






ln()

ln(ACF

)


100

100

101






119873|119886⟩ = 119873|119901⟩ exp[minus(119868|119886⟩ minus 119868|119901⟩)120579 ] (4)


Complexity 5








120591120578 120578119905 = minus120578119905 + 119886120578ℎ119905 (5)



120578119905 = minus120578119905120591120578 (6)




120591ℎℎ119905 = minusℎ119905 + 119886ℎ120578119905119878119905 (7)



120591119878 119878119905 = (1198780 minus 119878119905) minus 119886119878120578119905ℎ119905 (8)




(9)


6 Complexity














Complexity 7


100

200

300

400

500





(10)





(11)




(12)




120591120578 120578 = 119891 (120578) + radic119868 (120578)120585 (13)


119891 (120578) equiv minus120578120572 + 1198780120578120572 (1 + 1205782120572) (14)

119868 (120578) equiv 119868120578 + (119868ℎ + 1198681198781205782120572) (1 + 1205782120572)2 (15)


120597119905119901 (120578 119905) = 120597120578 [minus119891 (120578) 119901 (120578 119905) + 120597120578 (119868 (120578) 119901 (120578 119905))] (16)



8 Complexity



120578120572 119889120578]] (18)



119901 (120578) prop 120578minus2120572P( 120578120578119888) (19)






0100

0001

0010

100

1

10

0100005000100005 10500 5ln(f)

lnS(

f)


0010

0100

1

0001

0100005000100005 10500 5ln()

ln(ACF

)







Complexity 9







7 Conclusions








Data Availability


10 Complexity




Acknowledgments


References





























Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5








120591120578 120578119905 = minus120578119905 + 119886120578ℎ119905 (5)



120578119905 = minus120578119905120591120578 (6)




120591ℎℎ119905 = minusℎ119905 + 119886ℎ120578119905119878119905 (7)



120591119878 119878119905 = (1198780 minus 119878119905) minus 119886119878120578119905ℎ119905 (8)




(9)


6 Complexity














Complexity 7


100

200

300

400

500





(10)





(11)




(12)




120591120578 120578 = 119891 (120578) + radic119868 (120578)120585 (13)


119891 (120578) equiv minus120578120572 + 1198780120578120572 (1 + 1205782120572) (14)

119868 (120578) equiv 119868120578 + (119868ℎ + 1198681198781205782120572) (1 + 1205782120572)2 (15)


120597119905119901 (120578 119905) = 120597120578 [minus119891 (120578) 119901 (120578 119905) + 120597120578 (119868 (120578) 119901 (120578 119905))] (16)



8 Complexity



120578120572 119889120578]] (18)



119901 (120578) prop 120578minus2120572P( 120578120578119888) (19)






0100

0001

0010

100

1

10

0100005000100005 10500 5ln(f)

lnS(

f)


0010

0100

1

0001

0100005000100005 10500 5ln()

ln(ACF

)







Complexity 9







7 Conclusions








Data Availability


10 Complexity




Acknowledgments


References





























Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity














Complexity 7


100

200

300

400

500





(10)





(11)




(12)




120591120578 120578 = 119891 (120578) + radic119868 (120578)120585 (13)


119891 (120578) equiv minus120578120572 + 1198780120578120572 (1 + 1205782120572) (14)

119868 (120578) equiv 119868120578 + (119868ℎ + 1198681198781205782120572) (1 + 1205782120572)2 (15)


120597119905119901 (120578 119905) = 120597120578 [minus119891 (120578) 119901 (120578 119905) + 120597120578 (119868 (120578) 119901 (120578 119905))] (16)



8 Complexity



120578120572 119889120578]] (18)



119901 (120578) prop 120578minus2120572P( 120578120578119888) (19)






0100

0001

0010

100

1

10

0100005000100005 10500 5ln(f)

lnS(

f)


0010

0100

1

0001

0100005000100005 10500 5ln()

ln(ACF

)







Complexity 9







7 Conclusions








Data Availability


10 Complexity




Acknowledgments


References





























Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7


100

200

300

400

500





(10)





(11)




(12)




120591120578 120578 = 119891 (120578) + radic119868 (120578)120585 (13)


119891 (120578) equiv minus120578120572 + 1198780120578120572 (1 + 1205782120572) (14)

119868 (120578) equiv 119868120578 + (119868ℎ + 1198681198781205782120572) (1 + 1205782120572)2 (15)


120597119905119901 (120578 119905) = 120597120578 [minus119891 (120578) 119901 (120578 119905) + 120597120578 (119868 (120578) 119901 (120578 119905))] (16)



8 Complexity



120578120572 119889120578]] (18)



119901 (120578) prop 120578minus2120572P( 120578120578119888) (19)






0100

0001

0010

100

1

10

0100005000100005 10500 5ln(f)

lnS(

f)


0010

0100

1

0001

0100005000100005 10500 5ln()

ln(ACF

)







Complexity 9







7 Conclusions








Data Availability


10 Complexity




Acknowledgments


References





























Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity



120578120572 119889120578]] (18)



119901 (120578) prop 120578minus2120572P( 120578120578119888) (19)






0100

0001

0010

100

1

10

0100005000100005 10500 5ln(f)

lnS(

f)


0010

0100

1

0001

0100005000100005 10500 5ln()

ln(ACF

)







Complexity 9







7 Conclusions








Data Availability


10 Complexity




Acknowledgments


References





























Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9







7 Conclusions








Data Availability


10 Complexity




Acknowledgments


References





























Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity




Acknowledgments


References





























Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11

























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Complexity of a Microblogging Social Network in the Framework …downloads.hindawi.com/journals/complexity/2018/4732491.pdf · 2019. 7. 30. · ResearchArticle Complexity of a Microblogging