MULTIFRACTAL NETWORK GENERATORS

Transcription

1 MULTIFRACTAL NETWORK GENERATORS AUSTIN R. BENSON, CARLOS RIQUELME, SVEN P. SCHMIT (0) Abstract. Generating ranom graphs to moel networks has a rich history. In this paper, we explore a recent generative moel, the multifractal network generator, through both a theoretical an empirical lens. We erive concise analytic formulas for the moments of graph properties, incluing number of eges, number of weges, an number of triangles. Our empirical experiments show that the multifractal network generator can simultaneously moel a few graph properties of real-worl networks to high accuracy using a very fast algorithm. Furthermore, we give evience that fitting these local properties carries over well to global properties, such as egree istribution. 1. Generative graph moels 1.1. Prior work. Methos for generating ranom graphs that can realistically moel the structure in networks we see in the real-worl have receive much attention. The olest an best unerstoo ranom graph moel is ue to Erős an Rényi [1], but it fails to resemble real-worl networks. Subsequently, more complicate moels have arisen, such as the Kleinberg s small-worl moel [4] an the Watts-Strogatz moel [13] Stochastic Kronecker graphs. More recently, Leskovec et al. introuce Stochastic Kronecker graphs (SKG) [5], an the compact generative moel has effectively reprouce real-worl graph properties such as heavy-taile egree istributions an small iameter. SKGs [5, 6] constitute the most wiely use moel that is similar in structure to the MFNG. Given a graph G, the KronFit algorithm [5] is a maximum likelihoo estimation technique for creating an SKG that generates graphs similar to G. Similar work has applie SKG to graph completion [3]. More theoretical rigor has been evelope for SKG, incluing the estimation of graph properties [, 11] Multifractal network generators. The multifractal network generator (MFNG) was introuce by Palla, Lovász, an Vicsek [9], an follows a recursive structure that resembles SKG. An avantage of MFNG is that several graph parameters can be easily estimate from the small set of generative parameters (see Section 3). In this project, we explore how easy it is to control these parameters to generate realistic networks. In particular, given a network, we want to reprouce similar networks via MFNG. The theoretical analysis for SKG is quite technical, an the simple characterization of graph properties from MFNG is attractive to our problem. Recent work by Palla [10] has extene the MFNG framework, but fining MFNG that moel many real-worl network properties has not been explore. 1.. Contributions. This paper erives several analytic expressions for moments of graph properties in MFNG (see Section 3). In particular, we provie expressions for: The expectation an variance of the number of eges. The expecte number of -stars. This inclues the expecte number of weges, which are -stars. The expecte number of t-cliques. This inclues the expecte number of triangles, which 3-cliques. The egree istribution. Date: December 10,

2 The central theoretical result is Theorem, which allows us to compute probabilities of properties by only looking at the original MFNG parameters, which is both appealing from both a theoretical an computational stanpoint, especially for large networks. This constitutes a significant improvement over expressions for first moments of some graph properties in [9]. Futhermore, our framework makes it easier to evelop more theoretical results. We then use these new methos to fit MFNG parameters to real-worl networks in Section 4. Our experiments show that MFNG is a promising approach to moelling networks.. Preliminaries.1. Multifractal network generators. The MFNG is a recursive generative moel base on a generating measure, W k. The measure W k consists of a set of probabilities p ij, p ij = p ji, 1 i, j m along with a set of m lengths l 1,..., l m with m l i = 1. In this paper, we will refer to the intervals as categories. An unirecte graph G is istribute accoring to W k if it is generate by the following proceure: (1) Partition [0, 1] by the lengths l i. For each interval, recursively partition the interval k times into m intervals, using the relative lengths l i. This creates m k intervals l i1,...,i k of length k x=1 l i x such that l i1,...,i k = 1. i 1,...,i k () Sample N points uniformly from [0, 1] an create noes x 1,..., x N. Each noe x i lans in some interval of length l i1,...,i k. We can ientify x i by the sequence (i 1,..., i k ). In other wors, x i is ientifie by a k-tuple of categories. (3) For every pair of noes x i an x j ientifie by (i 1,..., i k ) an (j 1,..., j k ), we a ege (x i, x j ) to G with probability k x=1 p i xj x. While the generation is intricate, MFNG amits a geometric interpretation. Consier first k = 1. We partition the unit square into rectangles accoring to the lengths l i. The point (x i, x j ) [0, 1] [0, 1] lans in the unit square, insie a rectangle of area l i1 l j1. The ege survives to the next roun with probability p i1,j 1. In the next roun, we recursively create the unit square partition, scale to the rectangle. The relative positions of x i an x j lan the point in the rectangle of size l i l j. This process repeats k times. The iea is illustrate in Figure 1. x j p 1 x i p p P ((x i,x j ) G) =p p 31 p 1 Figure 1. Geometric interpretation of MFNG with k = m = 3. At each recursive level, the position of the noes are relative to the position in the previous rectangle.

3 3 u 1 u 1 3 u 1 u Figure. -stars with center noe u The benefit of the multifractal network generator is that certain graph properties can be analytically compute given the generating measure an interval lengths. In [9], the authors provie simple, ineficient computations for a few graph properties. These formulas are given with respect to the extene m k m k generating measure that is, consiering m k interval. However, as we show in Section 3, we can efficiently compute many graph properties. In our approach the formulas are base on the inepenence of certain events happening at the k ifferent levels of the process. In many cases, the expressions for MFNG are much simpler than the analogous ones for SKG. However, MFNG has not yet been shown to moel real-worl networks as well as SKG... Overview of graph properties. In this section, we review the efinitions an notation of graph properties consiere in this paper. Given a graph G = (V, E), we use the following efinitions: A wege is a pair of eges that share a single noe, entoe {(u, v), (u, w)}, where u, v, an w are all istinct. A triangle is a complete subgraph of three noes, enote (u, v, w). Note that the clustering coefficient of a graph is three times the number of triangles ivie by the number of eges. A -star is a noe u along with eges (u, v 1 ),..., (u, v ) for u v i V istinct. Note that a -star is the same as a wege. Figure illustrates -stars for {, 3, 4, 5}. 3. Theoretical results In this section we state the main theoretical result of our work (Theorem ). First, we give an interpretation an iscuss why this is significant. Then we state the main theorem, followe by some corollaries. Proofs of all results are in Appenix A Interpretation an significance. The main theorem gives simple formulas for several graph relate properties, where there is no nee to expan the measure an use recursion. Expansion is use in [9], where they consier calculating properties after expaning the measure. This leas to a probability matrix of size m k m k, enote by P, compare to our metho, which only nees to access the probabilities on the square, enote by p. Note that p is m m, an that k is implicitly a function of the number of noes (in our experiments in Section 4, k is between log N/ an log N ). Hence, using the theorem, we are able to scale these computations to graphs with arbitrary number of noes. 3.. Notation. Let W k enote the istribution of a ranom graph generate by measure W with k levels. We call each of the m initial intervals a category, an we enote by c i the category corresponing to the interval of length l i, 1 i m. C enotes the inices of all categories, i.e., C = {1,..., m}. In this section, all graphs have the same set of noes V an V = n. A given noe u belongs to k categories (one at each level), an the categories of noe u are enote by c u = (c 1,..., c k ). In particular, we enote the i-th category, by c u i. Furthermore, the length of a category c is enote by l c. 3

4 3.3. Main results. Lemma 1. Let W be a given generating measure. Let graphs H 1,..., H k W 1, inepeently rawn, an also enote H i = (V, E Hi ). Consier graph G = (V, E G ) where, for each u, v V, (u, v) E G (u, v) E Hi for every i = 1,..., k, i.e., G is the graph intersection of the H i. Then, G W k. The main result is the following theorem. Theorem (Main theorem). Consier a multifractal graph G = (V, E), generate accoring to measure W k. For any event A on G that can be written as A = {S E}, where S {(i, j) : i, j {1,..., n}, i < j}, P Wk (A) = P W1 (A) k. In other wors, the probability that a subset of the eges exists if the graph is rawn from W k is the k-th power of the probability that these eges exist if the graph is raw from W 1. The conition that A can be written as A = {S E}, where S {(i, j) : i, j {1,..., n}, i < j} is very subtle. Let us therefore spen some time explaining its consequences. It states that the above theorem hols if we can specify a subset of the eges that must be present the graph. We can also be inifferent about certain eges, but the theorem oes not allow us to specify that an ege is not present. As an example, consier a graph with three noes, u, v an w. We can use the above theorem to calculate the probability that it contains the wege {(u, v), (v, w)} E (an then use basic probability to calculate whether the graph contains a wege). Furthermore, we can use the theorem to calculate the probability that a graph contains the triangle (u, v, w) E. However, we cannot use the theorem to calculate the probability that the wege {(u, v), (v, w)} is not a triangle, i.e. aitionaly require (u, w) / E. Let us look at one more anti-example. Define µ(g) to be the chromatic number of G. Then we cannot use the theorem to compute P(µ(G) < 10), as that woul give us P(µ(G) < 10) = P(µ(H 1 ) < 10) k. But clearly, P(µ(G) < 10) P(µ(H 1 ) < 10) since taking the intersection of graphs can only reuce the chromatic number. The following corrolary summarizes how we can efficiently compute several useful graph properties using Theorem. Corollary 3. Let G be a multifractal graph with n noes generate accoring to measure W k = (m, k, l, p ij ). (1) The expecte number of eges E in G, is an its variance is given by ( ( n Var( E ) = )s k 1 where we efine i,j C E[ E ] = ( ) n )s k + n ( ) n s k, ( n 1 ) ω + ( n s : p ij l i l j, ω : p ij p it l i l j l t. () The expecte number of weges Λ in G is given by ( ) n 1 E[Λ] = n ω. (3) The expecte number of t cliques C t in G is given by E[C t ] = ( ) n s k t, where s t : t 4 i 1,...,i t C j,q [t] j<q p iji q )( n ) s k, l i 1 l i l it.

5 In particular, the expecte number of triangles is ( ) n E[ ] = s k 3 3, where s 3 : p ij p it p jt l i l j l t. (4) We enote the expecte number of noes with egree by E, which satisfies n ( ) i E[E ] = E[X ] E[E i ], i=+1 where X is the number of stars in the graph an its expectation is given by ( ) k n E[X ] = n p i1i j l ij. i 1,...,i +1 C In orer to be able to solve the recurrences, we also have that E[E n 1 ] = E[X n 1 ]. Note that, for a given measure W k, we coul empirically compute the value of E[C t ] for each t until we fin E[C t ] 1 > E[C t +1], which is a goo inicator of the expecte clique number of G size of the largest inuce complete subgraph in G, while a concentration result is still neee to claim that the clique number will be in a small neighborhoo of t with high probability. j= j=1 4. Experimental results Given a graph G, we are intereste in fining a generating measure W k such that the MFNG has similar properties. Since MFNGs generate ranom graphs, we seek to match the expecte value of several graph properties to the true graph properties. Since we can easily compute the moments of several graph properties from the generating measure, we use an optimization framework Fitting graphs to MFNG. Let P = {P j } enote a set of graph properties that we want to match. For example, P j (G) coul be the number of eges or the number of triangles in the graph. For a generating measure W, we will interpret P j (W) to be a ranom variable. We will also efine a istance function D that measures the istance between a graph G an a generating measure W, i.e., D(W, G) R. D shoul reflect the graph properties that we want to match. For this report, we will restrict D to be of the following form: D(W, G) = P (E[P j (W)] P j (G)) (P j (G)) We coul also inclue ifferent weights w j for ifferent properties P j. However, for simplicity, we equally weight each property. Suppose that k is fixe an m is upper boune by m u. Then we are intereste solving the following optimization problem minimize p,l D(W, G) (4.1) subject to 0 p ij 1, 1 i j m u 0 l i 1, 1 i m u m u l i = 1 The objective function is nonlinear, an the constraints are linear. The number of variables is 1 m u(m u + 1) + m u. Each variable has a lower an upper boun, an there is a single equality constraints (the lengths must sum to one), so there are m u (m u 1) + 4m u + 1 constraints. Note that, in this framework, m can effectively take any value in {1,..., m u }. The effective value of m is the number of non-zero l i. 5

6 Graph property MFNG SKG E[E] n(n 1) ( m m j=1 l il j p ij ) k (α + β + γ) r (α + γ) r ( m m ) k m E[Λ] n(n 1)(n 1) j=1 h=1 l il j l h p ij p ih ((α + β) + (β + γ) ) r (α (α + β) + γ (γ + β)) r ( m m ) k m 6E[ ] n(n 1)(n ) j=1 h=1 l il j l h p ij p ih p jh ( α + β + γ ) r + ( α + γ ) r (α 3 + 3β (α + γ) + γ 3) r 3 ( α ( α + β ) + γ ( β + γ )) r + ( α 3 + γ 3) r Table 1. Analytic expressions for the expectation of graph properties for SKG an MFNG in terms of moel parameters, where n is the number of noes in the graph. The results for SKG are from []. As a consequence of Theorem, we can compute E[P j (W)] for several P j in O(m 3 u) time. This allows us to quickly evaluate D(W, G) an efficiently use optimization software. We will use SKG as a comparison against MFNG. Again, we will use D(Θ, G) to enote a istance function between a SKG with an initiator matrix Θ an again interpret P j (Θ) as a ranom variable. Furthermore, we restrict Θ to be a matrix: ( ) α β Θ =. β γ The optimization problem for SKG is: (4.) minimize α,β,γ D(Θ, G) subject to 0 α 1 0 β 1 0 γ 1 There are three variables an six constraints. More sophisticate techniques to solving (4.) exist []; however, we since our focus is on SKG, we use this simple formulation. 4.. Fitting eges, weges, an triangles. For our experiments, we attempt to match the number of eges (E), weges (Λ), an triangles ( ) of real-worl graphs. These properties are easy to compute for SKG [], which we will use for comparison. Table 1 summarizes the expecte value of these properties for MFNG an SKG. We solve (4.1) an (4.) using Matlab s active set nonlinear optimization solver. For each optimization problem, we use 000 trials with a ranom starting vector an take the best result over all trials. Note that the graph properties for MFNG epen on k, the number of recursive layers. We sample k uniformly from { log (N) /,..., log (N) } an solve the optimization problem with k fixe. A isavantage of SKG is that the exponent r in Table 1 is fixe at log (N). For large r, this makes the function evaluations quite sensitive to changes in parameters. We also use KronFit to generate a maximum-likelihoo SKG initiator matrix. The approach of KronFit is funamentally ifferent from the non-linear optimization approach, an we provie the results for comparison. The following notation is use in the results: Let P 1 be the number of eges in the graph an efine E err = (E[P 1 (W)] P 1 (G)) / (P 1 (G)) to be the error in the number of eges. Let P be the number of weges in the graph an efine Λ err = (E[P (W)] P (G)) / (P (G)). Let P 3 be the number of triangles in the graph an efine err = (E[P 3 (W)] P 3 (G)) / (P 3 (G)). 6

7 Network n E Λ Pokec 1.63e e07.09e09 3.6e07 cit-hepph 3.45e04 4.0e05.63e07 1.8e06 roanet-ca 1.96e06.77e e06 1.0e05 Table. Summary of graph properties of the three networks use in our experiments. n is the number of noes, E the number of eges, Λ the number of weges, an the number of triangles. Network Metho E err Λ err err (p 11, p 1, p ) (l 1, l ) k (α, β, γ) Pokec cit-hepph roanet-ca MFNG.87e e-1 1.6e-1 (0.91, 0.09, 0.70) (0.57, 0.43) 14 SKG.87e e (0.5, 0.41, 1.00) SKG (KF) (0.8, 0.55, 0.37) MFNG 6.08e e e-11 (0.68, 0.18, 0.98) (0.36, 0.64) 13 SKG 1.38e (0.3, 0.56, 1.00) SKG (KF) 1.94e (1.00, 0.50, 0.35) MFNG e-11 (1.00, 0.04, 0.76) (0.43, 0.57) 17 SKG (0.96, 0.08, 1.00) SKG (KF) (0.67, 0.41, 0.57) Table 3. Results of fitting MFNG an SKG to real-worl graphs using (4.1) an (4.), as well as the results for KronFit for comparison. MFNG reprouces the expecte number of eges, weges, an triangles to high accuracy, while SKG has ifficulty matching the number of triangles. Our test graphs consist of three networks representing vastly ifferent ata. The first is the Pokec social network [1], which contains ata about all links between users in Slovakia. This network is especially interesting because it contains the full graph of a social network, which therefore suffers less from selection bias as ego networks. Although this network is irecte, we transform it an treat it as an unirecte network. The secon network is a citation network in the high energy physics community (cit-hepph) [7]. Apart from the two social networks above, we also consier a non-social network, the California roa network (roanet-ca) [8]. For a backgroun an analysis of the networks, we refer to the cite papers. Table summarizes the relevant graph propreties of these networks. The results of the optimization proceure an KronFit are in Table 3. We inclue the values of the probability matrix of MFNG (p 11, p 1, p ), the lengths of the intervals of MFNG (l 1, l ), an the SKG parameters (α, β, γ). Note that KronFit is not trying to match the graph properties, an the results are given for comparison. The results show that MFNG accurately matches the expecte number of eges, weges, an triangles of the networks. On the other han, SKG has ifficulty matching the expecte number of triangles. KronFit oes a reasonable job at matching the number of eges an weges, even though the algorithm is not irectly optimizing for these graph properties. The MFNG fit shows that properties along the iagonal (p 11 an p ) ten to be very large. This is expecte. In clustere networks, noes falling into the same category in MFNG shoul be more likely to connect, an this is reflecte by the large iagonal probabilities. In all cases, the lengths (l 1 an l ) for MFNG are not skewe. Heuristically, this tells us that the fit is realistic an not too extreme. 7

8 Network S = Λ S 3 S 4 S 5 Pokec.08e e1.83e e18 cit-hepph.63e e e e1 roanet-ca 6.00e06.95e e05.95e04 Table 4. Number of -stars in the test netwroks for {, 3, 4, 5}. The number of -stars is the same as the number of weges. Network E err Λ err err S 3 err S 4 err S 5 err (p 11, p 1, p ) (l 1, l ) k Pokec (1.00, 0.71, 0.4) (0.19, 0.81) 13 cit-hepph e (0.3, 0.18, 1.00) (0.57, 0.43) 7 roanet-ca e (0.91, 0.03, 0.99) (0.5, 0.48) 0 Table 5. Fit of MFNG to real-worl networks while trying to match several graph properties. MFNG can match the citation network but has ifficulty with the Pokec an roas networks Aitional fitting of -stars in MFNG. One property that we have ignore so far is egree istribution. While there are formulas for egree istribution in [9], they are approximations base on asymptotics of the binomial istribution. Instea, we will count -stars as a proxy for egree istributin. Let S be the number of -stars in a graph. In aition to fitting eges, weges, an triangles, we now also try to fit S, for {3, 4, 5}. Table 4 lists the number of -stars in our test networks. For the Pokec an citation network, S < S 3 < S 4 < S 5 while for the roa network, S > S 3 > S 4 > S 5. We solve (4.1) while fitting these six graph properties. Again, there were 000 calls to the optimization solver with ranom starting points. Table 5 lists the results of the best solution. For the Pokec network, MFNG fit all properties except for the number of triangles, an for the citation network, MFNG fit all properties. However, MFNG ha a ifficult time fitting the roas network. The ifficulty with roas is most likely a result of the fact that S k < S k+1 for k =, 3, Generating MFNGs efficiently. One of the ownsies of the MFNG graphs is that it is computationally intensive to generate them. A naive metho might calculate the P matrix, which is m k m k, an then loop over every pair of noes (u, v) an throw a properly (un)balance coin. For reasonably size graphs, k often is too large to compute this matrix explicitly. We have implemente an approximation algorithm inspire by the ball-ropping algorithm use to generate SGKs [5]. Our algorithm consists of two steps. First, we group noes that belong to the same set of categories, i.e., we group all noes u by c u = (c 1,..., c k ). The secon part is more intricate. We consier D possible eges. For each ege, fin a box at the lowest recursive layer of the MFNG via a ball-ropping proceure, using both the probability matrix as the lenghts of intervals. In particular, the probability of the ball being roppe into a box at each level is proportional to p ll, where l is the vector of initial lengths. After repeating this proceure k times, we have reache a box at the lowest level of the MFNG. For now, assume that the box contains at least one caniate ege, i.e., there is a pair of noes x i, x j [0, 1] such that (x i, x j ) is in the box. From the set of all ege caniates, choose one uniformly at ranom an connect them. At face value, there is a central flaw in this approach. Since the noes are generate uniformly on [0, 1], some category groups may have have a isproportionally small or large number of ege caniates. To account for this, we instea consier λd possible eges, where λ > 1. When arriving at a box, we calculate the expecte number of ege caniates in the box, n p, an the actual number of ege caniates in the box, n a. We select a ranom ege from the box, an a it to the graph with probability na λn p. This corrects the iscrepancy in the number of ege caniates. 8

9 Figure 3. Hops plot for the Physics citation network (left) an the MFNG fitte graph (right). We still have to pick λ. On one han, we want λ to be as small as possible, as the larger λ, the more work we have to o. On the other han, if λ too small, the quantity na λn p coul be bigger than 1, resulting in artifacts in the graph. In practice, we fin that λ between an 3 works well. Selecting D is another challenge. In the fast generation of SKGs, people often use D = E[ E ], but this is rather artificial. However, fining the istribution of E is not straightforwar. We opt for a compromise by computing E[ E ] an Var E, an noting that since every ege is a Bernoulli trial, E is tightly concentrate aroun E[ E ]. Therefore, we sample D N (E[ E ], Var[ E ]). Emperically, this proves to be a very accurate approximation. The last issue is empty boxes. We o not have a proper solution to this, except for avoiing them; to o this, we have to boun k, ensuring it is small enough to avoi empty boxes with high probability. While it woul be esirable to get ri of this restriction, we can still generate realistic graphs with boune k, as we see in the following section Simulating large graphs. The result is a metho that can generate reasonably large graphs, that coul also be parallelize for further spee-up. In this section, we compare the Physics citation network to a MFNG generate graph where the parameters are fitte to match eges, weges, triangles, 3-stars, 4-stars an 5-stars. In particular, because we are only able to fit local parameters using the fast technique we escribe, it is both important an interesting to see how well this translates to global properties. Figures 3, 4 an 5 show that we achieve mixe results. The hop plot (Figure 3) is very accurate, but the egree istribution (Figure 4) shows the same staircase effect as the (non-stochastic) Kronecker graphs [5]. The clustering coefficient (Figure 5) plot shows that, even though we can match the number of eges an weges, that oes not mean that the clustering coefficient per noe is well matche. 5. Conclusion an future work This paper investigate both the theoretical unerlying of MFNGs as well as the practical implications by using it to fit real-worl graphs. In particular, we have shown that it is possible to compute several important graph properties using very simple computations that only epen on the initial measure W, an for which the running time is inepenent of k. It is remarkable that we are able to fit the MFNG moel to large-scale graphs in negligible time, inee it is much faster than calculating the properties of the graph we inten to fit. Furthermore, empirical results show that fitting local structures such as -stars an triangles also leas to the fitting of global structures such as egree istribution. There are several areas for future work. First, we woul like to provie more rigorous results on fast generation of MFNG. This is three-fol. First, we have to upperboun k to eal with empty boxes, which 9

10 Figure 4. Degree istribution plot for the Physics citation network (left) an the MFNG fitte graph (right). Figure 5. Clustering coefficient plot for the Physics citation network (left) an the MFNG fitte graph (right). is restrictive. Secon, the fast graph generator is still too slow to generate graphs with more than a few hunre thousan noes. It woul be very interesting to scale the metho up such that it can be use to generate graphs with more than, say, a million noes. We note that apart from improving the current algorithm, it is also possible to parallelize the current algorithm an generate larger graphs that way. Thir, we notice the awkwar artifacts in the egree istribution that are similar to the non-stochastic Kronecker graphs as note in [5] when generating MFNG graphs. How to get ri of these artifacts is not exactly clear an woul be esirable. The optimization algorithm we use is fast but slightly a hoc. More application-specific optimization routines can be explore. We also note that many ifferent measures lea to an equally goo fit of the moel. This coul point to room for aitional properties to be taken into account, thereby ifferentiating between these moels. Furthermore, there is room for more theoretical results of graph properties. Can Theorem be extene such that we can eal with more general events or use to calculate other properties of such graphs? Lastly, it is also not clear that fitting the expecte value of several graph properties is the best approach. We have shown some encouraging evience that it might be, but further theoretical an empirical evience is neee. 10

11 References [1] P. Erős an A. Rényi. On ranom graphs. Publicationes Mathematicae Debrecen, 6:90 97, [] D. F. Gleich an A. B. Owen. Moment-base estimation of stochastic kronecker graph parameters. Internet Mathematics, 8(3):3 56, 01. [3] M. Kim an J. Leskovec. The network completion problem: Inferring missing noes an eges in networks. In SDM, pages 47 58, 011. [4] J. Kleinberg. The small-worl phenomenon: an algorithm perspective. In Proceeings of the thirtysecon annual ACM symposium on Theory of computing, pages ACM, 000. [5] J. Leskovec, D. Chakrabarti, J. Kleinberg, C. Faloutsos, an Z. Ghahramani. Kronecker graphs: An approach to moeling networks. The Journal of Machine Learning Research, 11: , 010. [6] J. Leskovec an C. Faloutsos. Scalable moeling of real graphs using kronecker multiplication. In Proceeings of the 4th international conference on Machine learning, pages ACM, 007. [7] J. Leskovec, J. Kleinberg, an C. Faloutsos. Graph evolution: Densification an shrinking iameters. ACM Transactions on Knowlege Discovery from Data (TKDD), 1(1):, 007. [8] J. Leskovec, K. J. Lang, A. Dasgupta, an M. W. Mahoney. Community structure in large networks: Natural cluster sizes an the absence of large well-efine clusters. Internet Mathematics, 6(1):9 13, 009. [9] G. Palla, L. Lovász, an T. Vicsek. Multifractal network generator. Proceeings of the National Acaemy of Sciences, 107(17): , 010. [10] G. Palla, P. Pollner, an T. Vicsek. Rotate multifractal network generator. Journal of Statistical Mechanics: Theory an Experiment, 011(0):P0003, 011. [11] C. Seshahri, A. Pinar, an T. G. Kola. An in-epth analysis of stochastic kronecker graphs. Journal of the ACM (JACM), 60():13, 013. [1] L. Takac an M. Zabovsky. Data analysis in public social networks. [13] D. J. Watts an S. H. Strogatz. Collective ynamics of small-worl networks. Nature, 393(6684):440 44, Appenix A. Proofs of main results This appenix contains the proofs Lemma 1 an Theorem along with calculations an proofs of the formulae in Corollary 3. A.1. Proof of Lemma 1. Proof. We prove the lemma by showing that a graph generate using the graphs H 1,..., H k satisfies the construction of a graph whose istribution follows W k. The proof consists of two parts: (1) First, we show that () Then, we show that P(c u = (c 1,..., c k )) = l ci. P((u, v) E G c u = (c u 1,..., c u k), c v = (c v 1,... c v k)) = 11 p c u i c v, i

12 inepenently of other eges. Since the graphs H i are inepenent, it follows that the components of c u = (c u 1,..., c u k ) are inepenent. Note also that the categories are inepenent of other noes ue to construction of {H i }. Moreover, we have that P(c u = (c 1,..., c k )) = P(c u i = c i ) = l ci, which completes the proof of (1). For (), we note that the inepenence between eges given the categories follows irectly from the efinition of {H i }. Furthermore, P ((u, v) E G c u, c v ) = P ((u, v) E Hi i c u, c v ) = P ((u, v) E Hi c u i, c v i ) = which completes the proof. A.. Proof of Theorem. Proof. This is a straightforwar consequence of Lemma 1. Let A be an event that can be written as A = {S E}, where S {(i, j) : i, j {1,..., n}, i < j}. Then we have A.3. Proof of Corollary 3. P Wk (A) = P Wk (s E G, s S) = P k(s E (W1) H i, s S, i {1,..., k}) = P (W1) k(s E H i, s S) = P (W1) k(s E H 1, s S) k = P W1 (A) k. A.3.1. Expecte number of eges. Let u an v be two ranom noes of G. Let A enote the event A = {(u, v) E}, an we efine A (i) to enote the analogous event restricte to level i in the multifractal generator. By Theorem, we have that P(A) = P(A (i) ) = P(A (1) ) k. Now, we see that if we are only concerne with the first original level, then P(A (1) ) P(A (1) c u 1 = i, c v 1 = j) P(c u 1 = i, c v 1 = j) We conclue that i,j C p ij P(c u 1 = i)p(c v 1 = j) i,j C p ij l i l j := s. i,j C (A.1) P(A) = P((u, v) E) = s k. The expecte number of eges is then given by ( ) n (A.) E[ E ] = s k. 1 p c u i,c v i

13 A.3.. Expecte number of triangles. Let u, v an w be three ranom noes of the graph. We efine E u,v,w to be the event that there is a triangle between u, v, w an, similarly, let E u,v,w (i) be the event that there is a triangle between u, v, w in level i [k]. Then, by Theorem, (A.3) P(E u,v,w ) = P(E (i) u,v,w) = P(E (1) u,v,w) k. Now, we compute the probability of a triangle happening between three ranom noes at the first level, that is, accoring to W 1. We see that P(E u,v,w) (1) = P((u, v), (u, w), (v, w) E) P((u, v), (u, w), (v, w) E c u 1 = i, c v 1 = j, c w 1 = t) P(c u 1 = i, c v 1 = j, c w 1 = t) By (A.3), we conclue that P((u, v), (u, w), (v, w) E c u 1 = i, c v 1 = j, c w 1 = t) P(c u 1 = i)p(c v 1 = j)p(c w 1 = t) P((u, v) E c u 1 = i, c v 1 = j)p((u, w) E c u 1 = i, c w 1 = t)p((v, w) E c v 1 = j, c w 1 = t) l i l j l t p ij p it p jt l i l j l t =: s 3. (A.4) P(E u,v,w ) = s k 3. We can now compute the expecte number of triangles in G as (A.5) E[ ] ( ) ( ) n n 1(E S ) = P(E u,v,w ) = s k S V S =3 A.3.3. Expecte number of t cliques. Accoring to the notation we use in the previous section, given t ranom noes S = {u 1,..., u t } an letting E S be the event that they form a t-clique, we have that Also, it follows that P(E S ) = P(E (1) S ) i 1,...,i t C P(E (i) S ) = P(E(1) S )k. j,q [t] j q p iji q l i 1 l i l it := s t Finally, let C t be the expecte number of t cliques in G. We conclue that ( ) n (A.6) E[C t ] = s k t. t A.3.4. Expecte number of weges. Let u, v, w be three istinct noes of G. We efine A to be the event that there is a wege centere at u in G, that is, A = {(u, v), (u, w) E(G)}. Similarly, as in previous sections, we efine A (i) to be the event restricte to level i. By Theorem, we see that P(A) = P(A (i) ) = P(A (1) ) k. 13

14 Now, by consiering only the first level, we fin that P(A (1) ) = P((u, v), (u, w) E) = P((u, v), (u, w) E c u 1 = i, c v 1 = j, c w 1 = t)p(c u 1 = i, c v 1 = j, c w 1 = t) P((u, v) E c u 1 = i, c v 1 = j)p((u, w) E c u 1 = i, c w 1 p ij p it l i l j l t =: ω. It follows that the expecte number of weges Λ in G is given by ( ) n 1 (A.7) E[Λ] = n ω. = t) l i l j l t A.3.5. Variance of the number of eges. Let X ij be the inicator ranom variable of the event (v i, v j ) E for i j. We also efine X i j X ij, the total number of eges. Let us compute the secon moment of X, E[X ] = E X ij X ij i<j i<j = E i<j X ij = E[X] + E[Λ] + + E X ij X ik + E i,j k i k,j z,k z E [X ij ] E [X kz ] i k,j z,k z ( )( ) n n = E[X] + E[Λ] + P ((v i, v j ) E) ( )( ) n n = E[X] + E[Λ] + s k. X ij X kz Hence, we see that Var(X) = E[X ] E[X] ( )( ) n n = E[X] + E[Λ] + s k E[X] ( ) ( )( ) n 1 n n = E[X](1 E[X]) + n ω + s k ( ( ( ) ( ) ( )( ) n n n 1 n n = )s k 1 )s k + n ω + s k. A stars an Degree Distribution. A star centere at noe u is a graph containing + 1 vertices whose eges go from u to each of the other vertices in the graph. We start by noting the following key an simple fact: the number of vertices with egree in a graph G equals the number of copies of stars in G that are not part of any ( + 1) star in G. Let us efine X to be the ranom variable that counts the number of stars in G, for any [n 1]. Let > an suppose that vertex u has egree. Then, u will contribute with ( ) stars to X. We efine V to be the ranom variable that counts the number of noes with egree. Similarly, we enote by E the number of noes with egree. We see that E = V V +1, 14

15 which irectly implies E[E ] = E[V ] E[V +1 ]. Our goal is to write V as a function of X an X +1. We see that E n 1 = X n 1 an n ( ) i E = X E i. Taking expectations, we conclue that E[E ] = E[X ] i=+1 n i=+1 ( ) i E[E i ]. We can compute the expecte number of stars in G. Let u 1,..., u +1 be +1 istinct noes, an let S be the event that there is a star centere at u 1 in G. In other wors, S = {(u 1, u ), (u 1, u 3 ),..., (u 1, u +1 ) E}. By Theorem, P(S) = P(S i ) = P(S 1 ) k. Now, let us compute P(S 1 ). P(S 1 ) = P((u 1, u ), (u 1, u 3 ),..., (u 1, u +1 ) E) = P((u 1, u ), (u 1, u 3 ),..., (u 1, u +1 ) E c uj 1 = i j j) P(c uj 1 = i j j) = = i 1,...,i +1 C i 1,...,i +1 C i 1,...,i +1 C P((u 1, u ) E c u1 1 = i 1, c u 1 = i ) P((u 1, u +1 ) E c u1 1 = i 1, c u +1 1 = i +1 ) +1 j= p i1i j +1 l ij. j=1 +1 l ij j=1 We can now compute the expecte number of stars in G: ( ) ( ) n 1 n 1 E[X ] = n P(S) = n P(S 1 ) k ( ) n 1 +1 = n p i1i j i 1,...,i +1 C j= +1 l ij j=1 k. 15