Generalized Gelation Theory describes Human Online Aggregation in support of Extremism
Pedro D. Manrique, Minzhang Zheng, Zhenfeng Cao, Neil F. Johnson
GGeneralized Gelation Theory describes Human OnlineAggregation in support of Extremism
Pedro D. Manrique, Minzhang Zheng, Zhenfeng Cao & Neil F. Johnson Physics Department, University of Miami, Coral Gables, Florida FL 33126, U.S.A.
Though many aggregation theories exist for physical, chemical and biologicalsystems, they do not account for the significant heterogeneity found, for example,in populations of living objects . This is unfortunate since understanding howheterogeneous individuals come together in support of an extremist cause, forexample, represents an urgent societal problem. Here we develop such a theoryand show that the intrinsic population heterogeneity can significantly delay thegel transition point and change the gel’s growth rate. We apply our theory toexamine how humans aggregate online in support of a particular extremist cause.We show that the theory provides an accurate description of the online extremistsupport for ISIS (so-called Islamic State) which started in late 2014.
Physical theories of aggregation rely on kinetic equations where two clusters of sizes i and j join to form a new cluster of size i + j at a rate K ij , also referred to as the kernel
6, 7, 9 .The conventional theory proposes a set of rate equations for the number of clusters of size s , n s ( t ), for s = 1 , , ... . This deterministic analysis is known as Smoluchowski theory .It is known that if the aggregation rate increases rapidly with the cluster sizes, the systemundergoes a large-scale transition where a non-negligible fraction of the total population1 a r X i v : . [ phy s i c s . s o c - ph ] D ec ggregates into the largest cluster, known as a gel . A kinetic model of polymerizationallows for analytical treatment for a few types of kernels. In particular, it has been shownthat a kernel of the form K ij ∼ ( ij ) ω yields a gel transition for ω > /
2, and the system’s sizedistribution at the point of transition follows a power-law (PL) with exponent τ = 3 / ω
16, 17 . Such aggregates may subsequently fragment, however our focus in this paper is onhow they initially emerge and grow, and the consequences of this. The theory that we nowdevelop shows that heterogeneous systems with aggregation rules based on objects’ mutualaffinity, can effectively delay the gel transition point and drastically alter its growth rate.We then apply our theory to analyze the formation and early dynamics of a collective humansupport network underlying extremism. Our study provides new insights into the potentialmechanisms underlying their online aggregation dynamics. More generally, our theory canbe applied to an aggregating population of any type in which there is significant individualheterogeneity.We incorporate the heterogeneity among objects by a single variable x that for simplic-ity we refer to as its ‘character’ and which is assigned to each individual
18, 19 . For simplicity,we consider x to be a real number between 0 and 1 given that any other one-dimensionalrange can be easily rescaled to this. In addition, we assume that x is static over time, thoughthis aspect could be changed in the future to account for experience or external influence.For a given population of N objects, a distribution of character values q ( x ) is used to ran-domly assign a unique x i to each object i , ( i = 1 , , , ..., N ). Interactions between objectsare described by means of their mutual affinity. We define the similarity S ij between indi-2idual i and individual j as S ij = 1 − | x i − x j | , so that individuals with alike character havea high similarity while individuals with unlike character have a low similarity. Hence weillustrate two contrasting aggregation mechanisms: Mechanism 1 (which we refer to as M1)forming alike clusters and Mechanism 2 (which we refer to as M2) forming unlike clusters.Figure 1(a) schematically illustrates these aggregation mechanisms that are quantified bythe coalescence probability C ( S ij ) between individuals i and j .Starting from an isolated population of N objects, clusters form over time by randomlyselecting two individuals i and j that merge into a new cluster with a probability C ( S ij ) orremain separated with a probability of 1 − C ( S ij ). A diagram of the aggregation processis shown in the left panel of Fig 1(b). This process can be traced analytically by meansof Smoluchowski theory where two clusters join at a rate proportional to the product oftheir sizes (i.e. product kernel) and is weighted by the mean-field aggregation probability F that accounts for the heterogeneity and formation mechanism. In general, F determines thelikelihood for any pair of elements i and j to merge into a new cluster at a given timestep t .The set of coupled differential equations for the dynamics of the number of clusters of size s , n s ( t ), is given by:˙ n s ( t ) = − F sn s N ∞ (cid:88) r =1 rn r + FN s (cid:88) r =1 rn r ( s − r ) n s − r , s ≥ n s ( t ) = − F n s N ∞ (cid:88) r =1 rn r , s = 1 . (2)The first term on the right-hand side of both Eqs. 1 and 2 represents the population ofclusters of size s that merge with other clusters, while the second term in Eq. 1 is the3opulation of smaller clusters that merge to form clusters of size s . Since K ij is a productkernel with exponent ω = 1, the system undergoes a gelation transition at some finitepoint in the dynamics , and follows a PL size distribution with exponent τ = 5 / t c . Its exact location is determined mathematically by a singularityin the second moment of the size distribution and found to be equal to t c = N/ F (seeSupplementary Information for the full derivation).The expression for the evolution of the gel cluster G can be obtained by means of theexponential generating function E ( y, t ) ≡ (cid:80) s ≥ sn s e ys whose partial time derivative takes theform of the inviscid Burgers equation which can be solved by the method of characteristics(see Ref. for the case of homogeneous systems). Above the gel transition point, theformalism determines that the size of G follows the following equation G ( t ) = N (cid:16) − e − FtN G ( t ) (cid:17) . (3)The solution of (3) can be expressed in terms of the W -Lambert function as G/N = 1 − W ( z exp z ) /z where z = − F t/N .The aggregation mechanism favoring joining similar individuals (i.e. M1) is definedsimply as C = S ij , while for aggregating dissimilar individuals (i.e. M2) it is defined as C = 1 − S ij . The limit of random aggregation is obtained by considering all charactervalues to be identical (i.e. q ( x ) = δ ( x − x )) which makes the process character-free with anaggregation probability C ( S ij ) = 1. For the case of interest of a heterogeneous population,the mean-field probability F depends on both the formation mechanisms and the character4istribution. For example for a uniform character distribution q ( x ), the probability densityfunction (PDF) of the similarity y = S ij for M1 is f ( y ) = 2 y and hence the mean-fieldaggregation probability F = (cid:82) yf ( y ) dy = 2 /
3. Similarly, for M2 the mean-field probabilityresults in F = 1 /
3. The homogeneous (i.e., character free) limit occurs when y = 1 and thecharacter distribution is a Dirac delta which yields F = 1.Figure 2(a) summarizes these parameters as well as illustrating a single simulationresult of the gel formation (before and after the theoretical transition point t c ) for each ofthe mechanisms. The disks represent the evolution of G while the rings are smaller clusterswhose radii are proportional to the square root of their respective size. Figure 2(b) comparesthe time evolution of G for each of the aggregation mechanisms calculated from the mean-field approach (solid lines) and the discrete simulations (dots) averaged over 500 realizations.They are in good agreement.Our model can be used to analyze the evolution of heterogeneous systems in the naturalworld as well as social domains. As an illustration, here we study the formation of onlineextreme clusters (henceforth referred to as ‘groups’ since they are online social media groups)each of which comprises a finite number of individual human users. Such online groupformation in support of Islamic State (ISIS) through VKontakte (VK, http://vk.com) hasbeen reported in previous studies
21, 22 . VKontakte is the largest European social mediaplatform and is based in Russia. As of September of 2017, VKontakte had 447 million usersworldwide and is known to have been heavily used to spread pro-ISIS propaganda as well as5SIS recruitment and financing
21, 23 . Figure 3(a) shows a snapshot of the pro-ISIS networkextracted on January 10th 2015: 59 different social media groups supporting ISIS werefound, with a total of 21,881 followers combined and 48,605 links (i.e. follows). As a resultof the extreme content shared in these groups, moderators are constantly chasing them andshutting them down . Unlike platforms like Facebook which shuts down these groupsalmost immediately, VK can takes weeks or even months to act. This allows us to studytheir rich dynamics. During the period between the end of 2014 and the beginning of 2015, asudden and roughly continuous growth in the number of added links (i.e. follows) within thewhole network occurred and lasted until mid-2015 where a decay process set in . The firstfew weeks of this sudden growth are particularly interesting since the number of shutdownevents was minimal and hence aggregation processes dominated the system dynamics. Themonitoring of the group size distribution over time revealed that, for three consecutive daysstarting December 28, 2014, a PL distribution with exponent near 5 / p ≈ . t c =December 30,2014). Figure 3(b) shows that our heterogeneous model compares well with the real data fora mean-field aggregation probability of F = 0 . ± . t c , during whichmultiple aggregation and fragmentation events would have taken place, it is understandablethat a certain level of noise is present within the data. We consider this background noise asthe floor from which the gel cluster arises at t = t c , as shown in Fig. 3(b). Our results suggestthat dissimilar follows collectively assemble to create the network of pro-ISIS support groupsin VK (i.e. pro-ISIS support is dominated by mechanism M2). This finding is consistent withthe fact that different online social media groups support pro-ISIS causes in complementaryways, such as financing, recruitment, spreading of propaganda among others
21, 23 .Going deeper into the group formation analysis, we now look into the dynamics ofindividual groups. From the wide ecology of groups found in 2015, we selected those withfeatures that resemble those of our heterogeneous model. In particular, we had to weed outgroups that were inactive and/or put their web setting as invisible for long periods of time.In addition, some groups experienced large dynamical increments in very short periods oftime which reveals abrupt changes akin to processes such as explosive percolation. Moreover,since the total number of potential follows varies over time, we restrict the modeling to thefirst few active weeks where the assumption of a constant subpopulation of follows (i.e. N )holds. We measured the goodness-of-fit of Eq. 3 with the real data and found a total of32 groups that give an r-squared higher than 80% during the initial growth period. Ourapproach supposes that each group is a gel cluster formed by a subpopulation of followsfrom a larger pool comprising the entire network. Our results are presented in Fig. 4.Since groups have the option to turn themselves invisible for any length of time, our data7ontain some gaps for those particular dates. However we can see that our gelation approachcaptures well the group growth at the early stage and associates each one with a particularvalue of F (and hence a formation mechanism) ranging from 1 / N that changes over time. A first approach to thischallenge is by considering small linear variations of N in the original equation. We explorethis aspect by adding a small linear increment to the size ( N ( t ) = N + kt ) and comparingto the original case (i.e. k = 0). Figure S3 illustrates that this variation resembles the groupdynamics for a larger time period than the original modeling. Interestingly there are nosignificant differences in the estimated parameter F , which for the static (i.e. k = 0) anddynamical (i.e. k (cid:54) = 0) versions of the model are F = 0 . ± .
022 and F = 0 . ± . Methods
The full mathematical derivation of Eq. 3 is given in the SI. The power-law analysis thatwe use for the distribution of group sizes follows strict testing procedures described in thereferences of the SI. The data extraction is explained in full detail in Ref. of the mainpaper.1. Medini, D., Covacci, A. & Donati, C. Protein homology network families reveal step-wisediversification of type III and type IV secretion systems. PLoS Computational Biology , 12, e173, 1543-1551 (2006).2. G. Bounova & O. de Weck. Overview of metrics and their correlation patterns formultiple-metric topology analysis on heterogeneous graph ensembles. Phys. Rev. E. ,916117, (2012)3. H. R. Pruppacher, J. D. Klett. Microphysics of clouds and precipitation, Reidel, Dor-drecht, 1978.4. S. N. Wall, G. E. A. Anniansson. Numerical calculations on the kinetics of stepwisemicelle association. J. Phys. Chem. Proc. R. Soc. Lond. A.
J. Chem. Phys. Science
Physica D , Topics in Current Aerosol Research , Vol. 3, G. M. Hidy and J. R. Brock,eds. (Pergamom Press, New York, 1972), Part 2.102. M. H. Ernst,
Fractals in Physics , L. Pietronero and E. Tosatti, eds. (North-Holland,Amsterdam, 1986), p.28913. E.M. Hendriks, M.H. Ernst, and R.M. Ziff. Coagulation equations with gelation.
J. Stat.Phys. , 3, (1983)14. A. A. Lushnikov. Gelation in coagulating systems. Physica D , , 37-53, (2006).15. P. L. Krapivsky, S. Redner and E. Ben-Naim A Kinetic View of Statistical Physics ,(Cambridge University Press, Cambridge, 2010).16. R. M. Ziff, E.M. Hendriks and M.H. Ernst. Critical Properties for Gelation: A KineticApproach.
Phys. Rev. Lett. , 8, (1982),17. E.M. Hendriks, M.H. Ernst and R.M. Ziff. Coagulation Equations with Gelation. J. Stat.Phys. , 3, (1983).18. N. F. Johnson, P. Manrique and P. M. Hui. J. Stat. Phys. , 395 (2013)19. P. D. Manrique, P. M. Hui and N. F. Johnson.
Phys. Rev. E , 062803 (2015)20. P. G. J. van Dongen and M. H. Ernst. J. Stat. Phys. , 889-926 (1987)21. N. F. Johnson, M. Zheng, Y. Vorobyeva, A. Gabriel, N. Velasquez, P. Manrique, D.Johnson, E. Restrepo, C. Song and S. Wuchty. Science , 6292, 1459-1463 (2016)22. M. Zheng, Z. Cao, Y. Vorobyeva, P. Manrique, C. Song, N.F. Johnson. Multiscale dy-namical network mechanisms underlying aging from birth to death. arXiv:1706.00667[physics.soc-ph], (2017) 113. BBC News. Russian students targeted as recruits by Islamic State. (July 24, 2015)24. BBC News. Islamic State web accounts to be blocked by new police team. Available at (June 22, 2015)25. J.M. Berger and H. Perez. The Islamic State’s Diminishing Returns on Twitter. GWProgram on Extremism (2016). Available at https://cchs.gwu.edu/sites/cchs.gwu.edu/files/downloads/Berger_Occasional%20Paper.pdf
26. J. Paraszczuk. Why Are Russian, Central Asian Militants Vanishing From Social Net-works? RadioFreeEurope November 05, 2015. Available at
Competing Interests
The authors declare that they have no competing financial interests.
Correspondence
Correspondence and requests for materials should be addressed to P.D.M. (email:[email protected]).
Acknowledgments
We are grateful to Yulia Vorobyeva for detailed help and translation con-cerning ISIS activity on Russian-language Vkontakte. The authors acknowledge funding from theNational Science Foundation grant CNS1522693 and Air Force (AFOSR) grant 16RT0367.
Author contributions
P.D.M. and N.F.J. worked on the mathematical model. P.D.M., M.Z.and Z.C. worked on the data and data analysis. P.D.M and N.F.J wrote the paper. All authorsrevised and approved the manuscript. igure 1 Our model of aggregation in a heterogeneous population (a) Individual hetero-geneity is modeled by introducing a unique hidden variable x to each agent. The value of x is randomlyassigned from a given distribution q ( x ) . The formation mechanism depends on the affinity among indi-viduals. Favoring high similarity in establishing a connection is mechanism 1 (which we refer to as M1)while favoring dissimilarity is mechanism 2 (which we refer to as M2). These mechanisms are quantifiedby the coalescence probability C ( S ij ) , which is a function of the similarity S ij = 1 − | x i − x j | of a givenpair of objects i and j . (b) The aggregation process and hence the formation of the gel cluster ( G )can be described by a discrete simulation (left panel) and analytically traced by a mean-field approach(right panel). Figure 2
Stochastic and mean-field approach results for the gel dynamics (a) Formationrules and parameters for each mechanism (free, M1 and M2) and a sample simulation of the dynamics.The ’free’ mechanism refers to a homogeneous system with all x values identical. Colored disks representthe evolution of G while the gray ones are smaller clusters. For all cases the radii grow proportional to s / and the time limit is set when G reaches % of N ( N = 10 agents). Dashed horizontal linesindicate the theoretical transition time t c . (b) Contrast between simulation (points) and mean-fieldapproach (lines) for the evolution of G for the proposed mechanisms. Figure 3
Rise of global pro-ISIS online support (a) Sample of the online network of groupsin support of Islamic State on the VKontakte (VK) platform for an example day: January 10, 2015.Black dots represent groups, white dots are users which are connected through blue links (i.e. follows).(b) Evolution of the total number of follows s tot , i.e. links in the bipartite network shown in (a)(black circles), compared to our analytical model of heterogeneous objects undergoing a gelation process(blue line). The transition point t c is found to be December 30, 2014 which is the time where the ize distribution of online groups is approximately / ( α = 2 . ). A fitting of Eq. 3 yields F =0 . ± . . Figure 4
Modeling single online extreme groups . Evolution of the size of individual onlinepro-ISIS support groups s group (circles) compared to individual gelation processes (curves) resulting indifferent values of mean-field coalescence probability F ranging from / up to (inset).(inset).