Atypical scaling behavior persists in real world interaction networks

A TYPICAL SCALING BEHA VIOR PERSISTS IN REAL WORLD INTERACTION NETWORKS HARR Y CRANE AND W AL TER DEMPSEY A bstract . Scale-free power law structure describes complex networks derived from a wide range of r eal world pr ocesses. The extensive literature focuses almost exclusively on networks with power law exponent strictly larger than 2, which can be explained by constant vertex growth and prefer ential attachment. The complementary scale-free behavior in the range between 1 and 2 has been mostly neglected as atypical because there is no known generating mechanism to explain how networks with this property form. However , empirical observations reveal that scaling in this range is an inherent feature of real world networks obtained from repeated interactions within a population, as in social, communication, and collaboration networks. A generative model explains the observed phenomenon through the r ealistic dynamics of constant edge growth and a positive feedback mechanism. Our investigation, ther efore, yields a novel empirical observation grounded in a str ong theoretical basis for its occurrence. Self-organizing dynamics of many pr ocesses produce a common heter o- geneous structur e characterized by power law degr ee distributions, which have been discover ed in the W orld W ide W eb [1 – 3], social networks [4], telecommunications networks [5], biological networks [6], and many oth- ers [7 – 10]. A network exhibits power law degree distribution with exponent γ > 1 if the pr oportion p k of vertices with degr ee k satisfies p k ∼ k − γ for large k . Figur e 1 plots the degree distributions of four well known networks: the actors collaboration network [1], Enron email network [11], W ikipedia voting network [12], and Facebook social cir cles network [13, 14]. The power law exponent in each of these networks is between 1 and 2, behavior that cannot be explained by preferential attachment models. Prefer ential attachment dynamics provide an intuitive description of networks that undergo constant vertex growth and exhibit power law greater than 2. These pr operties do not accurately describe the networks in Figure 1: (A) In the actor collaboration network, vertices correspond to actors and edges repr esent that two actors wer e cast in the same movie. Her e the data permits multiple edges between vertices if the actors were cast together more than once. Thus, the network gr ows as a consequence Date : July 13, 2015. H. Crane is partially supported by NSF grant DMS-1308899 and NSA grant H98230-13-1- 0299. 1 2 HARR Y CRANE AND W AL TER DEMPSEY 0 1 2 3 4 5 6 7 −12 −8 −4 actors log degree log propor tion * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 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* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * γ = 1.75 (A) 0 1 2 3 4 5 6 7 −10 −6 −4 −2 enron log degree log propor tion * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 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* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * γ = 1.85 (B) 0 1 2 3 4 5 6 7 −8 −6 −4 −2 wikipedia log degree log propor tion * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 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* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * γ = 1.15 (C) 0 1 2 3 4 5 6 7 −8 −6 −4 facebook log degree log propor tion * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 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* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * γ = 1.3 (D) F igure 1. Empirical degree distributions for (A) Actors col- laboration network, (B) Enr on e-mail network, (C) W ikipedia link network, and (D) Facebook social cir cles network. In each panel, the slope of the dashed line is − γ , where γ is the estimated power law exponent. Fitting the two-parameter generative model in (1) to the data, we obtain estimates (A) ( α actor , θ actor ) = (0 . 75 , 1 . 14), (B) ( α enron , θ enron ) = (0 . 85 , − 0 . 63), (C) ( α wiki , θ wiki ) = (0 . 15 , 350), and (D) ( α fb , θ fb ) = (0 . 30 , 168). A TYPICAL SCALING BEHA VIOR PERSISTS IN REAL WORLD INTERACTION NETWORKS 3 of movie pr oduction, i.e., edge formation, rather than the influx of new actors. (B) In the Enron email network, vertices correspond to employees at the Enron corporation and an edge r epresents that an email has been exchanged between those employees. As emails are exchanged, new edges form without any requir ement that new vertices be added. (C) The W ikipedia voting network r epresents voting behavior for elec- tions to the administrator role in W ikipedia. V ertices are W ikipedia users and a directed edge points from i to j if user i voted for user j . The network grows when elections are held, i.e., new edges are formed. (D) The Facebook social cir cles network repr esents friendships among Facebook users. The network grows by the formation of new friend- ships, i.e., edges, which usually result from social interactions among users. Each of the above networks grows by the addition of edges that connect according to a positive feedback mechanism, whereby past interactions reinfor ce future behavior . For example, an email sent from employee A to employee B is likely to be reciprocated by a reply from B to A; actors cast together in one movie likely play complementary r oles which may be suitable in future movies; and so on. While positive feedback exhibits obvious similarities to preferential at- tachment, it di ff ers in that edge formation need not be accompanied by the addition of new vertices. Furthermore, the range of the power law exponent implies additional gr owth properties about the network. Power law exponent γ > 2 implies that the expected vertex degree grows at rate P n k = 1 k · k − γ ≈ R n 1 x − γ + 1 dx ∼ O (1) as a function of the number of vertices, making the total number of edges grow at the rate n · O (1) = O ( n ). Ther efore, prefer ential attachment models implicitly assume a network for which the number of edges gr ows linearly with the number of vertices. On the other hand, for 1 < γ < 2, the expected degree grows at rate P n k = 1 k · k − γ ∼ O ( n 2 − γ ) as a function of the number of vertices n , indicating total edge growth at rate n · O ( n 2 − γ ) = O ( n 3 − γ ) in the intermediate range between sparsity O ( n ) and density O ( n 2 ). Some r ecent progr ess in the mathematical literature demonstrates the fundamentally di ff erent structural pr operties of sparse and dense networks [15 – 17]. Figure 1 suggests that power law exponent between 1 and 2 is also of scientific interest, and understanding this intermediate range should provide important insights into the structure of real world networks. W e replicate these featur es in the following generative model, which produces scale-free networks with exponent 1 < γ < 2 and closely resembles how networks (A)-(D) form. W e generate a network with n edges by sequentially adding one edge at each time t = 1 , 2 , . . . , n . Our model is determined by two parameters α and θ in the range 0 < α < 1 and θ > − α . Befor e time t , 4 HARR Y CRANE AND W AL TER DEMPSEY the network has t − 1 edges and a random number of vertices N t , with the initial condition N 1 = 0. W e label these vertices i = 1 , . . . , N t and write D ( i , t ) to denote the total degr ee of vertex i before the t th edge is added. (Note that each self-loop fr om a vertex to itself contributes 2 to its degr ee.) When the t th edge arrives, its two incident vertices v 1 ( t ) , v 2 ( t ) ar e chosen randomly among vertices 1 , . . . , N t and a new vertex N t + 1 as follows. W ith N 1 t = N t , we first choose v 1 ( t ) randomly with probability (1) pr( v 1 ( t ) = i ) ∝ ( D ( i , t ) − α, i = 1 , . . . , N 1 t θ + α N 1 t , i = N 1 t + 1 . After choosing v 1 ( t ), we define N 2 t according to whether or not v 1 ( t ) is a newly observed vertex: if v 1 ( t ) = N 1 t + 1, then we define N 2 t = N 1 t + 1; otherwise, we put N 2 t = N 1 t . W e then choose v 2 ( t ) as in (1) with N 1 t replaced by N 2 t . When generating a network with directed edges, we orient edges to point from v 1 ( t ) to v 2 ( t ); in the undirected case, the edge between v 1 ( t ) and v 2 ( t ) has no orientation. W e write G n to denote the network generated after n steps of this procedur e. The above generative model produces a sequence of networks ( G n ) n = 1 , 2 ,... , where G n has n edges and a random number of vertices N n . For k = 1 , 2 , . . . , we write N n ( k ) to denote the number of vertices in G n with degree k , so that N n = P k ≥ 1 N n ( k ). From properties of the generating mechanism in (1) [18], the empirical degr ee distributions p n ( k ) = N n ( k ) / N n converge to α · k − ( α + 1) / Γ (1 − α ), where Γ ( t ) = R ∞ 0 x t − 1 e − x dx is the gamma function. The simulation results in Figur e 2 verify this pr operty of our model. Moreover , the expected number of vertices satisfies (2) E ( N n ) ∼ Γ ( θ + 1) α · Γ ( θ + α ) (2 n ) α , as n → ∞ . Given an observed power law exponent 1 < γ < 2, we can use these two properties to estimate the model parameters α and θ by setting α = γ − 1 and choosing the value of θ so that Equation (2) is satisfied by the observed network. The estimated parameters in the caption of Figur e 1 wer e obtained by this method. Remarkably , we can express the pr obability distribution of the random network G n in closed form by: (3) pr( G n = G ) = α # V ( G ) ( θ/α ) ↑ # V ( G ) θ ↑ (2 n ) Y v : deg( v ) > 1 (1 − α ) ↑ (deg( v ) − 1) where G is any network with n edges that can be generated by (1) , deg ( v ) is the degree of vertex v in G , # V ( G ) is the number of vertices in G , and x ↑ j = x ( x + 1) · · · ( x + j − 1) is the ascending factorial function. A further important property of G n is that its distribution (3) is independent of the order in which edges arrive during network formation. As this information is typically unavailable for network data, viable statistical models should be A TYPICAL SCALING BEHA VIOR PERSISTS IN REAL WORLD INTERACTION NETWORKS 5 agnostic to it. Nevertheless, many network models, including prefer ential attachment models, do depend on the order of arrival, severely limiting the scope of statistical infer ences [19]. Under our model, this lack of information has no adverse consequences. Therefor e, we expect that the discovery of (3) and its intuitive explanation of network formation should lead to significant progr ess in statistical network analysis. Our generating mechanism allows for self-loops and multiple edges be- tween vertices, features common in many of the interaction networks we consider . For the Enr on and actors networks, respectively , self-loops corre- spond to emailing oneself and acting in the same movies as oneself, while multiple edges r eflect an exchange of multiple emails between individuals and a casting of the same actors in multiple movies. Although these features may be pr esent in the underlying real world phenomenon, network datasets are often simplified by r educing multiple edges to a single edge. In fact, of the four networks in Figure 1, only the actor collaboration network dataset recor ds multiple edges. Thus, Figur e 1 suggests that atypical scaling is not only pr esent in interaction networks with multiple edges but also in their projection to a simple network by r educing multiple edges to a single edge. Figure 2 demonstrates that our model preserves the same scaling under this operation. The parameters of our model have a clear interpr etation in terms of the network generating mechanism. In (1) , we see that α controls the rate at which a vertex accumulates edges, leading to the explicit relationship between α and the power law exponent γ = α + 1. Given the value of α , θ controls the growth of vertices, with large values corresponding to faster growth. The θ parameter exhibits its biggest influence at the beginning of network formation. High estimates of θ for the W ikipedia and Facebook networks support the conclusion that most votes in W ikipedia elections involve users who did not participate in previous elections and the formation of Facebook social cir cles begins with rapid addition of new individuals. The moderate estimate of θ for the actors network supports the opposite conclusion; indeed, a core of the same movie actors ar e cast repeatedly while the majority of actors struggle for r oles. The negative estimate of θ for the Enron network reflects the tendency for communication within a closed group to outpace the rate at which new team members ar e introduced. The occurrence of power law exponent between 1 and 2 in several common network datasets brings forth a previously undetected feature of r eal world network evolution. While our sequential constr uction in (1) and pr eferential attachment dynamics both grow the network in a size-biased manner — higher degree vertices accumulate edges at a faster rate—network gr owth under our model is driven by the addition of edges, which accurately reflects the dynamics of the underlying network. Prefer ential attachment models, on the other hand, achieve the complementary power law behavior by sequential addition of vertices, behavior not reflective of networks (A)-(D). Even with state of the art methods [20, 21], statistical network models are 6 HARR Y CRANE AND W AL TER DEMPSEY 0 1 2 3 4 5 −6 −5 −4 −3 −2 log degree log propor tion * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * γ = 1.67 (A2) 0 1 2 3 4 5 −11 −9 −8 −7 log degree log propor tion * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * γ = 1.67 (A1) 0 2 4 6 −8 −6 −4 −2 log degree log propor tion * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * γ = 1.25 (B2) 0 2 4 6 −10 −8 −6 −4 log degree log propor tion * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * γ = 1.25 (B1) F igure 2. Simulation r esults showing degree distribution of networks and their projection to a simple network by remov- ing multiple edges. (A1) Network generated from model with parameters ( α, θ ) = (0 . 67 , 1), (B1) Network generated from model with parameters ( α, θ ) = (0 . 25 , 1), (A2) Simple network obtained by reducing multiple edges to single edge in (A1) network, (B2) Simple network obtained by r educ- ing multiple edges to single edge in (B1) network. Results suggest that the generated network and its induced simple network both exhibit power law of similar degree. A TYPICAL SCALING BEHA VIOR PERSISTS IN REAL WORLD INTERACTION NETWORKS 7 not su ffi ciently robust to answer many questions of practical inter est [19]. Our model also possesses fundamental statistical properties that lead to straightforward estimation of the parameters α and θ and, hence, the power law exponent. Explicit calculation of the distribution (3) opens the door to much more detailed statistical analyses by likelihood-based and Bayesian techniques. Although power law exponent in this range has not received much attention, we expect that it is widespr ead in r eal world interaction networks. Our framework should lay the foundation for futur e investigations, both scientific and mathematical. R eferences [1] Barabási, A.-L. & Albert, R. Emergence of scaling in random networks. Science 286 , 509–512 (1999). URL http://dx.doi.org/10.1126/science.286.5439.509 . [2] Faloutsos, M., Faloutsos, P . & Faloutsos, C. On power -law relationships of the internet topology . ACM Comp. Comm. Review 29 (1999). [3] Kumar , R., Raghavan, P ., Rajagopalan, S. & T omkins, A. T rawling the web for emerging cyber communities. Proceedings of the 8th W orld Wide Web Conference (1999). [4] W atts, D. & Strogatz, S. Collective dynamics of ‘small-world’ networks. Nature 393 , 440–442 (1998). [5] Abello, J., Buchsbaum, A. & W estbrook, J. A functional approach to external graph algorithms. Proceedings of the 6th European Symposium on Algorithms 332–343 (1998). [6] Jeong, H., Mason, S., Barabási, A.-L. & Oltvai, Z. Lethality and centrality in protein networks. Nature 411 , 41 (2001). [7] Chung, F . & Lu, L. Complex graphs and networks , vol. 107 of CBMS Regional Confer ence Series in Mathematics (Published for the Conference Board of the Mathematical Sciences, W ashington, DC, 2006). [8] Dorogovtsev , S. N. & Mendes, J. F . F . Evolution of networks (Oxford University Press, Oxford, 2003). URL http://dx.doi.org/10.1093/acprof:oso/9780198515906.001. 0001 . From biological nets to the Internet and WWW . [9] Durrett, R. Some features of the spread of epidemics and information on a random graph. Proceedings of the National Academy of Sciences 107 , 4491–4498. [10] Newman, M. E. J. The structure and function of complex networks. SIAM Rev . 45 , 167–256 (electronic) (2003). URL http://dx.doi.org/10.1137/S003614450342480 . [11] McAuley , J. & Leskovec, J. Learning to discover social circles in ego networks. NIPS (2012). [12] Leskovec, J., Huttenlocher , D. & Kleinberg, J. Signed networks in social media. CHI (2010). [13] Klimt, B. & Y ang, Y . Introducing the enron corpus. CEAS (2004). [14] Leskovec, J., Lang, K., Dasgupta, A. & Mahoney , M. Community structure in large networks: Natural cluster sizes and the absence of large well-defined clusters. Internet Mathematics 5 , 29–123 (2009). [15] Borgs, C., Chayes, J. T ., Cohn, H. & Zhao, Y . An L p theory of sparse graph conver- gence I: limits, sparse random graph models, and power law distributions. accessed at arXiv:1401.2906 (2014). [16] Borgs, C., Chayes, J. T ., Cohn, H. & Zhao, Y . An L p theory of sparse graph convergence II: LD convergence, quotients, and right convergence. accessed at arXiv:1408.0744 (2014). [17] Lovász, L. & Szegedy , B. Limits of dense graph sequences. J. Comb. Th. B 96 , 933–957 (2006). [18] Pitman, J. Combinatorial stochastic pr ocesses , vol. 1875 of Lecture Notes in Mathematics . 8 HARR Y CRANE AND W AL TER DEMPSEY [19] McCullagh, P . What is a statistical model? Ann. Statist. 30 , 1225–1310 (2002). URL http://dx.doi.org/10.1214/aos/1035844977 . W ith comments and a r ejoinder by the author . [20] Bickel, P . & Chen, A. A nonparametric view of network models and Newman–Girvan and other modularities. Proceedings of the National Academy of Sciences of the United States of America 106 , 21068–21073 (2009). [21] Bickel, P ., Chen, A. & Levina, E. The method of moments and degree distributions for network models. Ann. Statist. (2011). D ep ar tment of S t a tistics & B iost a tistics , R utgers U niversity , 110 F relinghuysen A venue , P isca t a w a y , NJ 08854, USA E-mail address : hcrane@stat.rutgers.edu URL : http://stat.rutgers.edu/home/hcrane D ep ar tment of S t a tistics , U niversity of M ichigan , 1085 S. U niversity A ve , A nn A rbor , MI 48109, USA E-mail address : wdem@umich.edu