Irreversible Aggregation and Network Renormalization
Seung-Woo Son, Golnoosh Bizhani, Claire Christensen, Peter Grassberger, Maya Paczuski
aa r X i v : . [ phy s i c s . d a t a - a n ] M a r Irreversible Aggregation and Network Renormalization
Seung-Woo Son, Golnoosh Bizhani, Claire Christensen, Peter Grassberger,
1, 2 and Maya Paczuski Complexity Science Group, University of Calgary, Calgary T2N 1N4, Canada FZ J¨ulich, D-52425 J¨ulich, Germany (Dated: October 16, 2018)Irreversible aggregation is revisited in view of recent work on renormalization of complex networks.Its scaling laws and phase transitions are related to percolation transitions seen in the latter. Weillustrate our points by giving the complete solution for the probability to find any given state inan aggregation process ( k + 1) X → X , given a fixed number of unit mass particles in the initialstate. Exactly the same probability distributions and scaling are found in one dimensional systems(a trivial network) and well-mixed solutions. This reveals that scaling laws found in renormalizationof complex networks do not prove that they are self-similar. PACS numbers: 89.75.Hc, 02.10.Ox, 05.70.Ln
Droplets beget rain, goblets coagulate to make butteror cream, and dust particles stick together to form ag-gregates that can eventually coalesce into planets. Atthe microscopic level, irreversible aggregation of atomsand molecules creates many familiar forms of mattersuch as aerosols, colloids, gels, suspensions, clusters andsolids [1]. Almost a century ago, Smoluchowski proposeda theory based on rate equations to describe processesgoverned by diffusion, collision and irreversible mergingof aggregates [2]. The theory predicts how many smalland large clusters exist at any given time and yields amass distribution that depends on certain details suchas the initial conditions, reactions present, relative rates,the presence or absence of spatial structure, etc. A keyinterest to physicists has been to derive scaling laws thatcharacterize different universality classes [3, and refer-ences therein].By contrast, wide interest in complex networks [4–7] has emerged recently. Vast applications to physics,computer science, biology, and sociology [8–10, and ref-erences therein] continue to be vigorously investigated.An important question is whether or not complex net-works exhibit self-similarity at different length scales andif they can be grouped into universality classes on thatbasis. Renormalization schemes for networks were pro-posed [11–14] to address this question. Scaling of themass or degree distribution of the renormalized nodeswas used to argue that many complex networks are self-similar. The semi-sequential renormalization group (RG)flow underlying the box covering of [11–14] was studiedcarefully in [15, 16], where it was found that scaling lawsmay be related to an “RG fixed point” which was ob-served for a wide variety of networks. A convenient, fullysequential scheme called random sequential renormaliza-tion (RSR) was introduced [17]. At each RSR step, onenode is selected at random, and all nodes within a fixeddistance ℓ of it are replaced by a single super-node.We point out a simple mapping between RSR and irre-versible aggregation on any graph. Hence any conclusiondrawn for one process holds also for the other. Indeed, a local coarse-graining step to produce a new super-noderepresents one aggregation event, where a ‘molecule’ ag-gregates with all its neighbors within distance ℓ to pro-duce a new cluster. Exact analysis in one dimensionreveals that even this trivial network exhibits scalinglaws for the cluster mass distribution under RSR – withexponents that depend on ℓ . Consequently, and some-what counter-intuitively, self-similarity observed in RSRand similar network renormalization schemes cannot beused to prove that complex networks are themselves self-similar. Instead scaling laws arise due to a percolationtransition in irreversible aggregation.The correspondence between aggregation and renor-malization is relevant for any model with stochasticcoarse-graining of a network. For instance, the theoryof space and time “Graphity” [18, 19], based on loopquantum gravity, involves a stochastic coarsening simi-lar (albeit more structured) to RSR. Hence the criticalpoint of aggregation may also be relevant in that and re-lated cases. The breakdown of conventional universality,where critical exponents depend on the microscopic scaleof coarse-graining, ℓ , seems to present a dilemma for the-ories based on stochastic coarse-graining of a network toarrive at e.g. a universal large scale theory of gravity.In order to demonstrate these points, here we considerirreversible aggregation ( k +1) X → X , where a randomlypicked cluster coalesces with k neighbors. For even k =2 ℓ this corresponds precisely to RSR on a 1-d chain withcoarsening range ℓ . The mass of the newly formed clusteris the sum of the ( k + 1) masses. We assume that the‘target’ cluster is picked with uniform probability fromall clusters. Other choices will be discussed in [20].Let us start with the model defined on a ring, i.e., withperiodic boundary conditions. Initially, N sites labelledby i ∈ [1 , ...N ] are each occupied by a particle of mass m = 1. Time can be either discrete or continuous, butwe demand that two events never happen simultaneously.Hence events, ranked by increasing time, are denoted bypositive integer values t . For each event, particles coag-ulate to form clusters of mass m >
1. More precisely, an ii + 1 i + m − FIG. 1: (Color online) Illustration of aggregation on a ringwith k = 1, N = 24, and N = 5. The tree in color corre-sponds to a cluster of mass m = 5. It has five leaves (blue)and four internal nodes (red). Its leaves start at site i andend at site i + m −
1. The numbers beside internal nodescorrespond to the time when coalescence occurs. event consists of picking a random cluster with uniformprobability and joining it with k clusters to its imme-diate right, using periodic boundary conditions. For k even, the same results are found if we aggregate clusterssymmetrically. After t events, N t = N − kt clusters exist.Our main result is the probability to find any sequenceof adjacent cluster masses p N N t ( m , m . . . m N t ) – wherea cluster of mass m is followed by a cluster of mass m ,etc., moving clockwise (see Fig. 1). we start with thesingle cluster mass probability.Cluster masses are restricted to m ≡ k ). Defin-ing m − ks , the integer s is the number of eventsneeded to make the cluster of mass m . As depicted inFig. 1, we can represent any realization of the processby a forest of N t rooted trees with N leaves and t in-ternal nodes. Each tree α has s α internal nodes, with P α s α = t . We simplify the notation by N for N t .Let π N N ( m ) denote, for fixed k (the dependence on k is not written explicitly in the following), the probabilitythat a cluster of mass m has its left-most member at site i ∈ [1 , N ] after t events. The probability that any of the N clusters picked at random has mass m is then p N N ( m ) = N N π N N ( m ) , (1)because there are N choices for i and the chance topick that particular cluster, given that it exists, is 1 /N .Since events occur completely at random, each history occurs with equal probability. The term ‘history’ refersto a fixed forest, which includes a fixed temporal orderof events. Thus π N N ( m ) is equal to the number of histo-ries leading to a final configuration with a cluster of mass m starting at position i , divided by all possible historiesleading to N clusters. The latter is equal to n hist , tot = N × ( N − k ) × . . . ( N + k ) , (2) where each of the t factors equals the number of choicesfor the next event. Using Pochhammer k − symbols or,equivalently, generalized rising factorials [21–24], this canbe written as n hist , tot = ( N + k ) t,k . Similarly, the numberof histories leading to a cluster of size m starting at afixed position i is n hist , cluster = ( m − k )( m − k ) × . . . s,k (3)and the number of histories for the remaining N − n hist , rest = ( N − m − k )( N − m − k ) × . . . ( N − N − t − s,k . (4)So far we have not included the number of choices associ-ated with different time orderings for the s events in thecluster and ( t − s ) events in the rest of the forest. Thenumber of different time orderings is given by n orderings = (cid:18) ts (cid:19) . (5)Combining Eqs. (1) to (5), we obtain p N N ( m ) = N N (cid:18) ts (cid:19) ( N − t − s,k (1) s,k ( N + k ) t,k = (cid:18) ts (cid:19) ( N − t − s,k (1) s,k ( N ) t,k . (6)This result can be further simplified into beta functionsor, more conveniently, k -beta functions (see e.g. [22]), B k ( x, y ) = 1 k B ( xk , yk ) , giving a remarkably simple final result p N N ( m ) = (cid:18) ts (cid:19) B k ( N − m, m ) B k ( N − , . (7)We make a number of observations: (1) For k = 1the process maps to bond percolation on a ring. For N = 2, the mass distribution is uniform over the en-tire range m ∈ [1 , N − N >
2, the distribu-tion is proportional to the ( N − nd factorial power(( N − m − N − m − · · · ( N − m − N + 2)). (2)For N = 2 and any k ≥ p N N ( m ) is symmetric underthe exchange m ↔ N − m . (3) For N = 2 and k = 2we obtain an equation formally identical to Spitzer’s dis-crete arcsine law for fluctuations of random walks [25].(4) Asymptotic power laws for N → ∞ can be deter-mined using Stirling’s formula. If N is fixed and both m and ( N − m ) → ∞ , p N N ( m ) ∼ ( t − s ) N − k − s − k . (8) -4 -3 -2 -1 N=2 exact solution P N (m) m N=2 exact solution P N (m) m N=2 N=3 N=4 N=5 exact solutions P N N ( m ) m FIG. 2: (Color online) Cluster size distributions after t =50 events for k = 2, for different values of N averaged over10 realizations compared to exact results. The large sizebehavior changes from increasing to decreasing power law at N = k + 1. Inset: The discrete arcsine law found for N = 2. For small masses, this gives a decreasing power law withexponent − /k . For N = k + 1, the power law p N N ( m ) ∼ s − /k holds up to the largest possible value, m = N − N + 1, and the cutoff is a step function. For m/N → N = k + 1,and the sign of the exponent changes at N = k + 1. For N < k + 1, the distribution has a peak at m/N → N > k + 1. These scaling lawsare illustrated for k = 2 in Fig. 2. (5) The scaling lawsfound for m ≪ N are identical to those obtained byKrapivsky [26] for the well mixed case. However, the be-havior for m/N → p N N ( m ) satisfies a numberof recursion relations: p N N ( m + k ) = m ( N − m − N + 1)( m + k − N − m − k ) p N N ( m ) ,p N N + k ( m ) = N ( N − m − N + 1)( N − N − N ) p N N ( m ) . A third nonlinear recursion relation is given later.Joint distributions for masses of adjacent clusters canalso be found. We denote by p N N ( m , m ) the probabilityto find a cluster of mass m followed immediately to theright by a cluster of mass m . This is non-zero only if m = 1 + s k and m = 1 + s k , where s α is the numberof events needed to form a cluster of mass m α . By thesame arguments that led to Eq. (6) we get p N N ( m , m ) = (cid:18) ts , s , s (cid:19) ( N − s ,k (1) s ,k (1) s ,k ( N ) t,k , where s = t − P αβ =1 s β and the first factor is the multino-mial coefficient instead of the binomial coefficient. When α = 2, it is a trinomial coefficient that counts the numberof ways in which the three sequences of events – for thetwo clusters considered, and for all ( N −
2) other clusters– can be interleaved in a single history.For any 1 ≤ α ≤ N − α consecutive, adjacent clusters is a productof a multinomial coefficient and α + 1 Pochhammer k -symbols, divided by the Pochhammer k -symbol relatedto the total number of possible histories given N initialparticles. Defining again s as the number of events in allclusters except the first α ones, we can write the resultcompactly as p N N ( m , . . . m α ) = (cid:18) ts , . . . s α (cid:19) ( N − α ) s ,k Q αβ =1 (1) s β ,k ( N ) t,k . (9)In particular, this can also be done for the joint distri-bution for all N masses by setting α = N −
1. The re-sulting expression is then manifestly invariant under anypermutations of N numbers ( m , . . . m N ). Hence the N − cluster probability is independent of the spatial orderingof the clusters. While there are obvious correlations be-tween the mass values (the sum of all cluster masses mustbe N ), there are no spatial correlations .We now consider a line of N particles with openboundaries. Again, aggregation events consist of a ran-dom choice of a cluster, followed by its amalgamationwith its k nearest neighbors to the right. The targetcluster must be at least k steps away from the right-mostboundary. Following the same arguments leads immedi-ately to Eq. (9) for α = N −
1, showing that the twomodels lead to precisely the same statistics.The absence of spatial correlations indicates that thesame dynamics might result for the well-mixed case. Nowwe start with a bucket containing N balls, each of unitmass. An event consists of taking k + 1 balls out of thebucket, merging them together, and returning the newball to the bucket. The k + 1 balls are chosen completelyat random, independent of their masses.The single cluster mass distribution for the well-mixedmodel can be obtained using the same strategy as be-fore, but the details are quite different. Consider thetotal number of histories. Since events now correspondto choosing any k + 1 balls out of N − kt balls, we have,instead of the Pochhammer k -symbol, a product of bino-mial coefficients, n hist , tot = (cid:18) N k + 1 (cid:19)(cid:18) N − kk + 1 (cid:19) . . . (cid:18) N + kk + 1 (cid:19) . (10)The expressions for n hist , cluster and n hist , rest are analo-gous, with the factors ( m − jk ) (resp. ( N − m − jk )) inEq. (3) (resp. (4)) replaced by binomial coefficients. Thenumber of time orderings n orderings is exactly the sameas before, but the first factor N /N in Eq. (6) has tobe replaced by N (cid:0) N m (cid:1) . Putting all these things together,many cancellations take place, leading exactly to Eq. (7).This argument can be similarly extended to get the full N -particle distribution function, obtaining exactly thesame result as before, for any k .The time-reversed process of aggregation is fragmen-tation. When considering the fragmentation process as-sociated with any of these models, we have to care-fully evaluate fragmentation rates. Assuming uniformrates would not lead to all time-reversed histories hav-ing the same probability. Indeed the fraction of allmergers associated with making a cluster of mass m ′ is( s ′ /t ) = ( m ′ − / ( N − N ), which must equal the proba-bility that an existing cluster of mass m ′ will fragment atthe next step in the time-reversed process. If it does, thenfor consistency its fragmentation products must have amass distribution given by p m ′ k +1 ( m ). A quadratic recur-sion relation for p N N + k ( m ) can then be obtained by con-sidering the likelihood of all fragmentation events in aconfiguration of N clusters, with m being the mass ofone of the resulting k + 1 fragmentation products. Therelation is p N N + k ( m ) = N − N +1 X ′ m ′ = m + k N ( m ′ − p N N ( m ′ ) p m ′ k +1 ( m ) N − N , where the prime on the summation symbol indicates that m ′ must increase in steps of k .In summary, we derived complete solutions for theprobability to find any given state in three models – well-mixed solutions, particles on a ring reacting with their k nearest neighbors, and the same reaction for particles ona line with open boundaries – and show that these solu-tions are precisely the same. The fact that we could solveexactly a one dimensional model without detailed balancemight seem surprising since such models are in generalnot solvable. It stems from the fact that spatial correla-tions, although a priori not excluded, are in fact absent.Related to this is our finding that the well-mixed mod-els have exactly the same solutions. Our method can beused to solve the model where the target cluster is pickedwith a probability proportional to its mass [20]. Perhapsgeneralizations of these observations hold true for morecomplicated models, in which case weighted path inte-grals would replace sums over histories.We have pointed out a direct mapping between irre-versible aggregation and RSR. The latter was motivatedby claims that one can define finite fractal dimensions forreal networks [11], using similar but more complicatedand ambiguous schemes. Results for RSR with ℓ = 1on various graphs (critical trees [17], Erd¨os-Renyi andBarabasi-Albert networks [27], and regular lattices [28])concur with our present conclusions for k = 2. Apartfrom studying a system that is sufficiently simple to beexactly solvable and that is obviously not fractal, here wepresented results for ℓ >
1, showing that scaling laws de-pend in a non-trivial way on ℓ . Results for the elementary network (a one dimensional line) examined analyticallyhere proves that scaling under stochastic network renor-malization arises from an underlying percolation transi-tion in aggregation and does not prove fractality or self-similarity of the underlying graph.Our mapping suggests that the critical behavior ofaggregation may also turn up in “Graphity” [18, 19]or related models, where geometry, gravity, and matteremerge through an aggregation process of an underlyinggraph. “Geometrogenesis” is the complementary processof infinite cluster formation in irreversible aggregation.In that case, scaling that depends on the microscopiccoarse-graining scale ℓ seems to add further obstacles tothe persistent and challenging problem to derive a largescale theory of gravity from microscopic graph models. [1] A. Zangwill, Nature , 651 (2001).[2] M. von Smoluchowski, Z. Phys. Chem. , 129 (1917).[3] F. Leyvraz, Phys. Rep. , 95 (2003).[4] S. H. Strogatz, Nature , 268 (2001).[5] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys. , 47(2002).[6] S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. ,1079 (2002).[7] M. E. J. Newman, SIAM Rev. , 167 (2003).[8] S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes,Rev. Mod. Phys. , 1275 (2008).[9] A.-L. Barab´asi, Science , 412 (2009).[10] A.-L. Barab´asi, N. Gulbahce, and J. Loscalzo, NatureRev. Genet. , 56 (2011).[11] C. Song, S. Havlin, and H. Makse, Nature , 392(2005).[12] K.-I. Goh, G. Salvi, B. Kahng, and D. Kim, Phys. Rev.Lett. , 018701 (2006).[13] J. S. Kim et al., Phys. Rev. E , 16110 (2007).[14] H. Rozenfeld, C. Song, and H. Makse, Phys. Rev. Lett. , 25701 (2010).[15] F. Radicchi, J. J. Ramasco, A. Barrat, and S. Fortunato,Phys. Rev. Lett. , 148701 (2008).[16] F. Radicchi, A. Barrat, S. Fortunato, and J. J. Ramasco,Phys. Rev. E , 26104 (2009).[17] G. Bizhani, V. Sood, M. Paczuski, and P. Grassberger, e-print arXiv:1009.3955, to appear in Phys. Rev. E (2010).[18] T. Konopka, F. Markopoulou, and L. Smolin, e-printarXiv:hep-th/0611197v1 (2006).[19] A. Hamma et al., Phys. Rev. D , 104032 (2010).[20] S.-W. Son et al., in preparation (2011).[21] J. Normand, J. Phys. A , 5737 (2004).[22] R. D´ıaz and E. Pariguan, Divulg. Mat. , 179 (2007).[23] J. Pitman and J. Picard, Combinatorial stochastic pro-cesses (Springer, 2006).[24] J. Kingman, J. Appl. Prob. , 27 (1982).[25] F. Spitzer, Principles of random walk (Springer Verlag,2001).[26] P. Krapivsky, J. Phys. A24