Individual heterogeneity generates explosive system network dynamics
IIndividual heterogeneity generates explosive system network dynamics
Pedro D. Manrique and Neil F. Johnson
Physics Department, University of Miami, Coral Gables, Florida FL 33126, U.S.A. (Dated: July 24, 2018)Individual heterogeneity is a key characteristic of many real-world systems, from organisms tohumans. However its role in determining the system’s collective dynamics is typically not wellunderstood. Here we study how individual heterogeneity impacts the system network dynamicsby comparing linking mechanisms that favor similar or dissimilar individuals. We find that thisheterogeneity-based evolution can drive explosive network behavior and dictates how a polarizedpopulation moves toward consensus. Our model shows good agreement with data from both biolog-ical and social science domains. We conclude that individual heterogeneity likely plays a key role inthe collective development of real-world networks and communities, and cannot be ignored.
I. INTRODUCTION
The mechanisms by which single entities (e.g.molecules, cells, people) come together as groups under-lie myriad physical, biological and social processes [1–9]. Though multiple models have been proposed in orderto explain the different aspects of real-world groupingbehavior, the feature of heterogeneity among individu-als has hardly been considered. Indeed, the universal-ity of some collective processes across contrasting sys-tems seems to have overshadowed the reality of intrinsi-cally heterogeneous populations, and more importantlysystem-level evolution driven by individual heterogene-ity. Percolation refers to the dynamical transition tolarge-scale connectivity by the addition of individual links[10, 11]. Before the transition, all clusters are of negli-gible size compared to that of the whole system. Fromthe transition point onward, the largest cluster gathersa finite non-negligible fraction of the total number of in-dividuals. The simplest way to form and grow clustersis by the classic Erd´os-R´enyi (ER) model [12, 13] whichat each timestep a new link is established between tworandomly selected nodes. This model reaches the perco-lation transition when the number of added links reacheshalf of the total number of nodes. At this point, thesecond moment associated with the cluster size distribu-tion presents a singularity and the system passes to a gelphase [14–16].Careful recent work has been dedicated to exploringthe dynamical control of percolation, particularly thetransition point [17–19]. Depending on the system, re-searchers look for ways to either delay or acceleratethe appearance of the percolation transition. This ismade possible by introducing the element of competitionamong the nodes to be associated together. Instead ofrandomly selecting one pair of nodes to be linked, threeor more nodes are selected to compete for link addition[18, 19]. The rules for addition vary from model to model,however the direct aspect that is being explored is howthe different potential new links might affect the size ofthe resultant new cluster. Thus, rules have been pro-posed so that only small clusters are formed, which delaysthe transition, or large clusters join resulting in the con- M1 M2 free 𝑘𝑘 = 20 𝑘𝑘 = 10 𝑘𝑘 = 30 M1 M2 free M2 M1 𝑘𝑘 =20 ER (free) 𝑘𝑘 =30 𝑘𝑘 =20 𝑘𝑘 =30 M2 M1 E D C
1 2 3 4 k Model 𝐶𝐶 ( 𝑆𝑆 𝑖𝑖𝑖𝑖 ) low M1
0 1 𝐶𝐶 ( 𝑆𝑆 𝑖𝑖𝑖𝑖 ) high 𝐶𝐶 ( 𝑆𝑆 𝑖𝑖𝑖𝑖 ) low M2 𝑥𝑥 A B 𝐶𝐶 ( 𝑆𝑆 𝑖𝑖𝑖𝑖 ) high Heterogeneity of nodes
A new link is established between the pair that maximizes 𝐶𝐶 ( 𝑆𝑆 𝑖𝑖𝑖𝑖 ) randomly select 𝑘𝑘 unconnected nodes FIG. 1.
Modeling heterogeneous grouping
A. The het-erogeneity of nodes is modeled by the character { x } whichis a real number defined in the [0,1] interval and randomlyassigned to each node from a distribution q ( x ). Our modelof interacting nodes comprises two mechanisms: M1 whichfavors linking similar nodes (i.e. close in character) and M2favoring dissimilar nodes (i.e. distant in character). B. Themodel follows a two steps cycle where it first randomly selectsa sample of k unconnected nodes and second selects the pairthat maximizes the coalescence function C ( S i,j ) that links to-gether. C. Dynamics of the largest cluster G as a function ofthe number of links for both heterogeneous formation mecha-nism as well as the random case (i.e. free) and different valuesof sampling size k . The total population is N = 10 nodes. D.Variation of the transition point for M1, M2 and free modelwith the sampling size k . E. Dependence of the largest gapin the growth of G for both mechanisms (shadows) and freemodel, and several values of sampling. Lines are guides to theeye. For D and E we show an average over 500 realizations. trary effect. These new models of large-scale connectivityshow abrupt variations in the size of the largest clusterand hence are known as explosive percolation. Examplesfrom the biological and social domain have been shownto have features akin to explosive percolation [20–22].However, it is not obvious why these models should beable to ignore individuals’ heterogeneity when determin-ing which clusters are formed and when. Even in our a r X i v : . [ phy s i c s . s o c - ph ] D ec everyday lives, we do not arbitrarily form clusters with-out some underlying factor determined by the character-istics of the individuals involved – for example in socialgatherings, we often join with family members who bydefinition have similar genes to us, while in a sport wejoin those having complementary skills in order to forma strong team. Our findings suggest that heterogeneity-based cluster formation and hence node-to-node affinityinteractions can play a crucial role in generating explo-sive phenomena in real-world systems, and influence thepoint of transition. As a result, our work provides freshinsights into how the diversity of individuals could affectthe overall dynamics of the system. Moreover, the outputof our models closely captures the dynamics of real-worldsystems such as online extremism and protein homologynetworks. II. MODEL
The heterogeneity is introduced as a hidden variable x i that we call the ‘character’ and is assigned randomly toeach node i from a distribution q ( x ) [23, 24]. For simplic-ity we consider all { x } to be real numbers between 0 and1 since larger ranges can be easily rescaled to be withinthis interval. Also, we consider that the character valuesare constant in time though this can be modified to in-clude variations over time to account for experience or ex-ternal influence. The mechanisms of link addition followdirectly from the relationship among the { x } values asso-ciated to the nodes to be linked. This is quantified by thesimilarity S ij between node i and node j which is definedas S ij = 1 − | x i − x j | . Thus highly similar nodes are closeto each other in the x spectra and otherwise for highlydissimilar nodes. The mechanisms of link addition rest inthe definition of the coalescence function C ( S ij ). We con-sider two complementary mechanisms for link addition:mechanism 1 (M1) favoring similar nodes and mechanism2 (M2) favoring dissimilar nodes (see Fig.1A). A systemfollowing M1 tends to form groups of alike individuals(e.g. kin) while M2 tends to form groups with unlike orcomplementary individuals (e.g. teams). Hence, for M1a coalescence function is defined as C ( S ij ) = S ij while forM2, C ( S ij ) = 1 − S ij .Our model starts with all N nodes unconnected. Ateach timestep a sample from the system is randomly se-lected and a new link is established between the pairof nodes that maximizes the coalescence function C ( S ij )(see Fig. 1B). Thus, the distribution q ( x ), the samplingmethod and size, and the specific mechanism of link addi-tion (i.e. M1 or M2), dictate the evolution of the system.Though the sampling can be either of nodes or links, theevolution of the network present similar properties. Sam-pling of nodes can be related to individuals that ran intoeach other in the course of a time period (e.g. a singleday) and among all the interactions only a few connec-tions are established based on their mutual affinity. Herewe will consider sampling of nodes for simplicity, with the link version in the Supplemental Material (SM). Thesmallest sampling size refers to two randomly selectednodes which sets the limit of a random graph model (ERmodel). Note that this is independent of the link additionmechanism and the distribution q ( x ). In a similar wayconsidering a distribution q ( x ) to be a Dirac Delta, im-plies that all the nodes are identical (i.e., character free)and hence the limit of a random graph model is also foundindependently of the remaining parameters. The formerand latter observations talk about the competitive anddiverse aspects of the network formation process, respec-tively. Unless both aspects are present the system willfollow an ER process.The evolution of the system is typically described bymeans of the size of the largest cluster that we call G .All present clusters sizes are denoted G i , where i is thesize rank starting from the largest (i.e., i = 1). Figure1C shows the evolution of G for several sampling sizes k , and mechanisms M1 and M2 when the character val-ues are assigned randomly from a uniform distribution q ( x ). The aggregation mechanisms and sampling sizeshave contrasting macroscopic effects in the evolution of G . For example, for M1 the percolation transition shiftsfrom the random case (character free) point at L/N = 0 . L/N and exhibits macroscopic fea-tures akin to explosive percolation such as large jumps.On the other hand, for M2 the transition shifts to smallervalues of
L/N than the random case and the growth tendsto be smoother. In all cases the shift tends to be moresevere as the sampling size increases as shown in Fig. 1D.These features can be explained by means of the forma-tion mechanisms. M1 links pairs of nodes whose x valuesare the closest to each other. The probability densityfunction (PDF) f ( y ) of the similarity y = S i,j associatedto the uniform distribution is f ( y ) = 2 y for y ∈ [0 , y = 1, there is alarge number of pairs with high similarity that could lieat any point in the character spectra. Hence, the forma-tion of small and medium size clusters in all x regions isexpected. This explains the appearance of jumps in theevolution of the largest cluster resulting from the aggre-gation of small and medium-size clusters to G . On theother hand, M2 tends to link nodes that lie far from eachother in the character spectra. The PDF tells us thatthe number of optimal pairs for M2 process (i.e. smallor zero S ij and hence y ) is low. As a consequence, forlarge samples it is very likely that either of these opti-mal nodes already belong to the largest component andtherefore the formation of medium size clusters is lessprobable. Thus, the growth of G for M2 becomes grad-ual and smoother than that of M1.Figure 1E illustrates this fact by examining the size ofthe largest gap ∆ G max in the evolution of G (∆ G max =max { G ( L + 1) − G ( L ) } ) due to the addition of a sin-gle link. It clearly shows that the gaps associated to M2are far smaller than those of M1. Interestingly we findthat the gap scales algebraically with the system size asshown in other explosive percolation models [17, 18] and B D L / N α = 1 L / N F α = 1/2 𝑥𝑥 𝑥𝑥 α = 1/5 C L / N E α = 5 𝐺𝐺 𝐺𝐺 𝐺𝐺 𝐺𝐺 𝐺𝐺 𝐺𝐺 𝐺𝐺 α = 2 L / N L / N 𝑥𝑥 α = 1 α = 5 α = 2 α = 1/2 α = 1/5 A 𝑥𝑥 𝑖𝑖 P D F single-parameter Beta distribution FIG. 2.
Effect of diversity in grouping . A. Probabilitydensity function of Beta distribution for several values of theparameter α for the case when α = β . B-F. Grouping evolu-tion as a function of the average character of the largest sevenclusters (colored bubbles). The size of the bubble is propor-tional to the square root of the size and each panel shows adifferent α value as shown in each panel. The total populationis N = 10 nodes, sampling k = 10 and M1 aggregation rule. this behavior is independent of the link addition mecha-nism. However, it is found that there is an upper boundin the largest gap for M1 around 1 / (cid:104)| x i − x j |(cid:105) ) between any two nodes is 1 /
3. Thismeans that, for a large sampling size, the optimal pairwould carry a character value difference smaller than oraround 1 /
3. Thus, linking nodes that are separated formore than 1 / / q ( x ). To this end, here we consider q ( x ) to be equal to a single parameter Beta distribution( β = α ) which is symmetric around x = 0 . model data 𝐺𝐺 / 𝑁𝑁 𝐺𝐺 / 𝑁𝑁𝐺𝐺 / 𝑁𝑁 𝐺𝐺 / 𝑁𝑁𝐺𝐺 / 𝑁𝑁 𝐺𝐺 / 𝑁𝑁 fraction of added links, ( 𝑝𝑝 ) Human protein homology network formation: { 𝐺𝐺 , 𝐺𝐺 , 𝐺𝐺 , 𝐺𝐺 } 𝑝𝑝 = 0.5 𝑝𝑝 = 1.0 𝐺𝐺 𝑗𝑗 / 𝑁𝑁 𝑝𝑝 = 0.5 𝑝𝑝 = 1.0 BA FIG. 3.
Heterogeneity in the Human Protein Homol-ogy Network
A. Snapshots of the human protein homologynetwork as links are added. The parameter p represents thefraction of added links. Top panel is the early stage for p = 0 . p = 1. Each panelshows the largest four clusters (i.e. G i , for i = 1 , , , k = 8. populations but with the same average character value.The uniform distribution previously presented is foundfor the special case of α = 1. Polarized populations canbe represented by 0 ≤ α <
1, where the severity in thepolarization increases as alpha approaches zero. In thelimit of α = 0, the system is maximally polarized withhalf of the nodes having character equal to 1 and halfequal to 0. This binary system follows two independentrandom graph formation processes that will join togetheronly after both graphs are fully formed (see SM). Bycontrast, unpolarized populations result for α > α → ∞ corresponds to the random graphprocess is found since the character values of all nodesare identical.Figures 2(B-F) illustrate single simulation results ofthe evolution of the seven largest clusters (colored bub-bles) vertically positioned at their average character valuewhen the population follows a single parameter Beta dis-tribution with different α values. The size of the bub-ble is proportional to the square root of the number ofnodes within the cluster and the colors represent the rankfrom one to seven according to their size (see legend atFig. 2F). Here we look at the grouping dynamics for thecase of M1 while M2 will be presented in the SM. For α <
1, the distribution is highly polarized and groupsare formed at opposite extremes of the character spectra.Depending on the severity of the polarization the sys-tem would require more or less links to reach consensusaround x = 0 .
5. This is because as α →
0, the numberof nodes at the center of the x -spectra (e.g. moderates)is smaller and links between groups at opposite poles be-comes less likely (see Fig. 2B). A less severe polarization(e.g. α = 1 /
2) allows for a certain number of moderateswhich serve as bridges between the groups in the polesand help reach consensus (see Fig. 2C). Populations withcomparable number of nodes along each portion of the x spectra (i.e. α = 1) form small and medium size clustersat different x points that subsequently join together intothe largest component around x = 0 .
5. Interestingly, af-ter consensus is attained, some extreme groups tend toappear at both poles (see Fig. 2D). This behavior is alsopresent for a distribution with a larger amount of moder-ates than extremists as depicted in Fig. 2E for the caseof α = 2. Finally, for α = 5 we find that the extremesare formed around the edges of the consensus group asshown in Fig. 2F. These results can provide some in-sights concerning how to create consensus (e.g. political)in a diverse and even polarized population of interactingindividuals, but warn of the possibility of leaving resid-ual isolated pockets of individuals with rather extremeaverage values of x (i.e. away from 0 . III. REAL-WORLD NETWORKS
We now explore two different real network systemsfrom the biological and social science domains that expe-rience explosive grouping behavior.
A. Protein Homology Network
Networks of proteins can be generated by identifyinghomology relationships, i.e., commonalities in the aminoacid sequences of a pair [22, 25, 26], and connecting themaccordingly. Here proteins are viewed as nodes that arelinked to each other through weighted edges. In princi-ple, the network can be comprised of all deduced proteins.For simplicity, here we look into the subset of human pro-teins which have shown features akin to explosive perco-lation [20]. The link’s weight is determined by the ho-mology relationship of a given pair. Highly homologousproteins have a greater weight than heterogeneous pro-teins. The homology between a given pair and hence theweight of their connecting link is measured by the align-ment score s ij while the score accuracy is determinedby the Expectation ( E ) value [27, 28]. The smaller the E -value the more reliable the score s ij becomes [27, 28].According to UniProt [29], a total of 159 ,
522 human pro-teins are deduced which can be divided in 20 ,
214 thathave been reviewed manually against 139 ,
338 that awaitrevision. Here we analyze the scores among the subsetof reviewed human proteins provided by the SimilarityMatrix of Proteins project (SIMAP) [26] and we haveused links with E -values up to 10 − . For simplicity, weuse the score ratio (SR) as a weight measure which isdefined as: SR = s ij / max { s ii , s jj } , where s ii is the self- homology score. Note that SR is defined within the [0 , G i , for i = 1 , , ,
4) fora mid-point evolution stage (top panel) where half of thelinks have been added, and the final stage (bottom panel)where the last link has been added illustrating their con-trasting topologies. Fig. 3B shows that the size of thelargest cluster (red circles) tends to show explosive dy-namics as new links are added. This behavior is capturedby our heterogeneous grouping model (pink rings) usinga uniform distribution q ( x ), M1 formation and samplingsize of k = 8. Both systems experience explosive groupingbehavior with comparable rates and gaps. Interestingly,our model also simultaneously reproduces some of thefeatures in the dynamics of the second and third largestclusters (squares and triangles, respectively). The agree-ment tends to be higher after the percolation transitionthan earlier. We attribute this to the strong homologyin some protein communities where single nodes can actas hubs gathering many individuals and rising to a non-negligible size. In our model, all nodes are equally likelyto be sampled for potential addition. Moreover, the re-strictions inflicted by the E -value leave many potentiallinks absent which explains why at the last stage thenetwork is not fully unified (see Fig.3A). Despite thesecomplications in the alignment parameters, our model isstill in reasonable greement with several features of theevolution of the protein network. This is an indication ofthe wide flexibility that a heterogeneity framework bringsto the grouping behavior which makes it adaptable to dif-ferent dynamical systems. B. Online grouping
We next consider the online social group formationin support of Islamic State (ISIS) whose data was col-lected in Ref. [30]. This occurs on Europe’s largest so-cial media platform based in Russia, VKontakte (VK,https://vk.com). As of December 2017, this site countswith 460 millions of users worldwide and has been usedby the extremists to spread propaganda and to recruitsympathizers. A snapshot of the pro-ISIS network is pre-sented in Fig. 4A for January 10, 2015 where 59 differ-ent extreme groups where active with a total of 21 , ,
605 connections (i.e.follows). This platform has become ideal for extremegroups in part because similar networking services suchas Facebook shut down these type of online groups almostimmediately, while VK takes more time to act. Duringthat active time the online groups attract followers andgrow in particular ways. The methodology of the data groupuserlink (follow) pro-ISIS network sample at VK January 10, 2015
A B sample of pro-ISIS online groups at VK g r oup s i ze (f o ll o w s ) modeldatamodeldatamodeldata g r oup s i ze (f o ll o w s ) C FIG. 4.
Explosive grouping in pro-ISIS online groups
A. Snapshot of online pro-ISIS support groups on the VKplatform on January 10, 2015. B. Evolution of a sample ofsharkfin pro-ISIS groups from first detection to shutdown. C.Examples of explosive behavior in pro-ISIS groups as com-pared to our heterogeneous model for k = 5 (left panel) and k = 10 (middle and right panel). For each case N is set to bethe respective group population at the moment of shutdown. extraction is presented in Ref [30] where different evolu-tionary adaptations have also been uncovered. For ex-ample, groups can change names, restart a new groupafter a shut down or switch between visible and invisiblepreferences in order to avoid moderators and being shutdown, among other.Figure 4B shows the evolution of a sample of extremeonline groups from the time of their earliest detectionup to the moment where they were shut down, at dailyresolution. The size of the groups is determined by thenumber of users that decide to follow them. We note thatin this particular sample the size of the group passes fromzero to an average size of 200 follows with a maximum ofnearly 1000 follows in a single day. In addition, we seethat this irregular growth is repeated at several sectionsof the formation process. These irregular jumps in thesize at the start or during the evolution of a given group g , could be the consequence of a group (or groups) g being shut down and hence all its (or their) former mem-bers coordinate to either join group g likely due to affin-ity with their message and generating a jump in its size,or to open a new group g and hence creating a jump atthe start of its evolution. Note that the latter processindicates that group g is a continuation of group g andtherefore they are essentially the same group, while theformer is a cluster aggregation process. In both casesthe changes in the sizes are abrupt and an associationwith a random aggregation process is less accurate. We therefore propose that a heterogeneous percolation modelwith group formation M1 cannot be discarded as a poten-tial mechanism for the creation and subsequent growthof these particular groups.Figure 4C strengthens our proposal by capturing keyfeatures of the formation of extreme online groups. Thepanels show how our heterogeneous model compares withthree of the extreme groups shown in Fig. 4B (color indi-cates the specific modeled group). The remaining groupsare compared with the model in Fig. S6 of the SM. Themodel interprets the VK system as a collection of severalsub-systems, each with a specific sub-population of po-tential follows that aggregate over time and whose largestcomponent grows to become the extreme group. Notethat the sub-population is not of users but ‘follows’ sinceeach user can follow several groups simultaneously. Po-tential followers explore groups daily and decide whetherto join or not, arguably based on affinity. This can beconsidered a competitive process where only some users(either isolated or from former shut down groups) addto the extreme group’s population. Hence we implementa competitive modeling where on each timestep k nodescompete for addition. Our results of Fig. 4C as well asFig. S6 show that values of k between 2 and 10 capturegeneral growth trends as well as some key features such asthe size jumps. Due to the bipartite nature of the groupevolution, the formation process considers inter-link ad-ditions only. This framework allows us to estimate thestart of the online activity even when the group was in-visible or not yet sufficiently extreme, and hence did notappear in the data collection radar. Also, this modelingopens the gate for exploration of different interventionstrategies to mitigate the spreading of the group by at-tacking it, for example, at its earliest stage. We note thatnot all the groups identified can be modeled by this ex-plosive percolation framework for technical reasons, e.g.because of missing data. IV. SUMMARY
We have shown that a heterogeneous population ofinteracting individuals can generate explosive groupingbehavior. In addition, our model provides a frameworkto study the impacts of new links on polarized popula-tions. Linking individuals can result in the formation ofnew residual clusters at the extremes. We also testedour model against two different heterogeneous real-worlddatasets capturing specific features of the formation pro-cess and showing that heterogeneity plays a decisive partin the system’s network evolution.
V. ACKNOWLEDGMENTS
We thank Chaoming Song for discussions regarding themodel, Minzhang Zheng and Yulia Vorobyeva for assis-tance with the pro-ISIS data and Thomas Rattei for pro-viding the protein data. N. F. J. is grateful to the Na-tional Science Foundation (NSF) grant CNS1522693 andAir Force (AFOSR) grant FA9550-16-1-0247. The views and conclusions contained herein are solely those of theauthors and do not represent official policies or endorse-ments by any of the entities named in this paper. [1] M. Anghel, Z. Toroczkai, K.E. Bassler and G. Korniss,Phys. Rev. Lett. , 058701 (2004)[2] A. Soulier and T. Halpin-Healy, Phys. Rev. Lett. ,258103 (2003)[3] B. Goncalves and N. Perra, Social Phenomena: Data An-alytics and Modeling (Springer, Berlin, 2015)[4] G. Palla, A.L. Barabasi, and T. Vicsek, Nature , 664(2007)[5] E. Estrada, Phys. Rev. E , 042811 (2013)[6] C. Song, S. Havlin and H. Makse, Nature Phys. , 275(2006)[7] G. Caldarelli, Scale-Free Networks: Complex Webs inNature and Technology (Oxford University Press, Oxford,2007)[8] A.L. Barabasi and H.E. Stanley,
Fractal Concepts in Sur-face Growth (Cambridge University Press, 1995)[9] F. Radicchi and S. Fortunato, Phys. Rev. E , 036110(2010)[10] Stauffer, D. & Aharony, A. Introduction to PercolationTheory (Taylor & Francis, 1994)[11] Sahimi, M.
Applications of Percolation Theory (Taylor &Francis, 1994)[12] Erd¨os, P. & R´enyi, A. On random graphs I. Math.
De-brecen , 290-297 (1959)[13] Erd¨os, P. & R´enyi, A. On the evolution of random graphs Publ. Math. Inst. Hungar. Acad. Sci. , 17-61 (1960).[14] E.M. Hendriks, M.H. Ernst, and R.M. Ziff. Coagulationequations with gelation. J. Stat. Phys. , 3, (1983)[15] A. A. Lushnikov. Gelation in coagulating systems. Phys-ica D , , 37-53, (2006).[16] P.L. Krapivsky, S. Redner and E. Ben-Naim, A KineticView of Statistical Physics (Cambridge University Press,Cambridge, 2010)[17] D’Souza R. M. & Nagler, J. Anomalous critical and su-percritical phenomena in explosion percolation.
NaturePhysics , 531-538 (2014).[18] Nagler, J., Levina, A. & Timme M. Impact of single linksin competitive percolation. Nature Physics , 265-270(2011)[19] Achlioptas, D., D’Souza, R. M. & Spencer, J. Explosivepercolation in random networks. Science , 5920, 1453- 1455 (2009).[20] Rozenfeld, H. D., Gallos, L. K. & Makse, H. A. Explosivepercolation in the human protein homology network.
Eur.Phys. J. B Phys. Rev. E. ,016117, (2012)[22] D. Medini, A. Covacci, & C. Donati. Protein homologynetwork families reveal step-wise diversification of typeIII and type IV secretion systems. PLoS ComputationalBiology , 12, e173, 1543-1551 (2006).[23] N. F. Johnson, P. Manrique and P. M. Hui. J. Stat. Phys. , 395 (2013)[24] P. D. Manrique, P. M. Hui and N. F. Johnson.
Phys. Rev.E , 062803 (2015)[25] P.F. Jonsson, T. Cavanna, D. Zicha & P.A. Bates. Clus-ter analysis of networks generated through homology: au-tomatic identification of important protein communitiesinvolved in cancer metastasis. BMC Bioinformatics , , 2(2006)[26] Thomas Rattei, Roland Arnold, Patrick Tischler, Do-minik Lindner,Volker St´’umpflen & H. Werner Mewes.SIMAP: the similarity matrix of proteins, Nucleic AcidsResearch , , D252-D256 (2006).[27] S.F. Altschul, W. Gish, W. Miller, G. Myers, and D.J.Lipman. A basic local alignment search tool. J. Mol.Biol. , , 403-410 (1990)[28] W.R. Pearson. Flexible sequence similarity searchingwith the FASTA3 program package. Methods Mol. Biol. , , 185-219 (2000)[29] The UniProt Consortium. UniProt: the universal proteinknowledgebase. Nucleic Acids Res. D158-D169 (2017)[30] Johnson, N. F., Zheng, M., Vorobyeva, Y., Gabriel, A.,Qi, H., Velasquez, N., Manrique, P., Johnson, D., Re-strepo, E., Song, C. & Wuchty, S. New online ecologyof adversarial aggregates: ISIS and beyond.
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