CCentral limit theorems for local network statistics
P-A. Maugis, University College London
Abstract
Subgraph counts—in particular the number of occurrences of smallshapes such as triangles—characterize properties of random networks, andas a result have seen wide use as network summary statistics. However,subgraphs are typically counted globally, and existing approaches failto describe vertex-specific characteristics. On the other hand, rootedsubgraph counts—counts focusing on any given vertex’s neighborhood—are fundamental descriptors of local network properties. We derive theasymptotic joint distribution of rooted subgraph counts in inhomogeneousrandom graphs, a model which generalizes many popular statistical networkmodels. This result enables a shift in the statistical analysis of largegraphs, from estimating network summaries, to estimating models linkinglocal network structure and vertex-specific covariates. As an example,we consider a school friendship network and show that local friendshippatterns are significant predictors of gender and race.
We contribute the results—and associated statistical inference methods—toenable the study of large networks at the level of each vertex. To do so wefirst derive the limit joint distribution of vertex-specific subgraph counts ininhomogeneous random graphs. Then, we exhibit the usefulness of these countsby studying a network of school friendships and showing that local counts aresignificant predictors of gender and race.A graph, or network, is composed of vertices connected by edges. Forinstance, students connected by friendship ties in a school. Such a graph maybe studied at two different scales: one global, the other local. In our schoolfriendship example, the former would consider friendship patterns throughoutthe school. The latter would study potential links between local patternssurrounding a student and other known features (such as gender or race). Theformer is well studied, and powerful tools exist to study graphs at the globalscale [4, 8, 9, 24, 34, 35, 51, 52, 55, 60, 63]. We focus on the latter.To study graphs at the local scale we introduce rooted subgraph counts. Sucha count is calculated from a graph by counting the occurrences of a particularshape attached to a given vertex. The simplest such count is a vertex’s degree:1 a r X i v : . [ m a t h . S T ] J un a) Five edges (b) Five 2-paths (c) Five triangles(d) Seven edges (e) Six 2-paths (f) Five trianglesFigure 1: Diagrams illustrating rooted subgraph counts. In each subfigure weconsider the rooted subgraph count of the central vertex, highlighted in grey.The rooted subgraphs counted are, in the first to third column, an edge, a pathof length 2, and a triangle.the number of edges attached to a vertex (Fig. 1.(a) and (d)). The second mostcommon rooted count is the number of triangles attached to a vertex (Fig. 1.(c)and (f)). Rooted counts provide vertex-specific information: for instance, in ourschool friendship example, depending on how many edges and triangles a studentis connected to, this student will play a different role in a contagion process [30].Effectively, rooted counts recover powerful fundamental characterizers of localgraph properties [39, restricted homomorphism], and are key features used tocompare and classify biological networks [1, 3, 50, graphlet degree].To undertake vertex-specific inference we must quantify the variations ofrooted counts expected under a null. Specifically, a description of the ranges oflocal behaviors that can be explained by known global network features, such assparsity, heavy tailed degree distributions and community structure. The nullof inhomogeneous random graphs [13], which subsumes most models used inthe statistical literature [2, 8, 9, 24, 28, 29, 34, 35, 51, 52, 55, 59, 60, 62, 63], isrecognized as sufficient to model these three global features. The question weaddress then is: “Are observed local behaviors consistent with this model?”Surprisingly, we observe that while inhomogeneous random graphs do notexplicitly model local behavior—something transparent in our results below—the null is sufficient to account for local behavior in several social networks.Specifically, while the null can be rejected as a model for local behavior across thewhole network at once, it cannot be rejected as a model to describe local behaviorwithin covariate-informed subnetworks; in our school friendship example, wefind that the local behavior within grades is accounted for, but not betweengrades. Further studies, that our tools enable, could look into determining theintermediate scale at which the null starts to fail, and characterizing how localbehavior departs from the null.Underpinning this finding are two main contributions. The first is to derivethe asymptotic joint distribution of rooted counts under the null. Surprisingly,no standard results or proof techniques exist to study rooted counts—proof2echniques for global counts rely on the overlap between copies being limited [7,9, 19, 46], while this does not hold for rooted copies—and very little is knownabout rooted counts even in the Erd˝os-R´enyi model [31, 54, 58]. To prove ourresult we rely on the intuition of the historically oldest moment-based methodsfor global counts [11, 23, 53]: that moments of subgraph counts are tied togetherby an algebraic structure. To uncover such a structure between rooted counts,we follow the proof progression of [37] and first present a quasi homomorphismtying together products of rooted counts. This then enables us to determinehow rooted copies are most likely to overlap, elicit the sought-after algebraicstructure, and eventually deduce all the moments of rooted counts. We find thatodd central moments are negligible, while even central moments follow a doublefactorial progression, which ultimately allows us to resolve the rooted counts’limiting distribution as Normal.Our second main result is to prove that the sample of rooted counts—the arrayof rooted counts across vertices—can be treated, for the purpose of statisticalinference, as an independent and identically distributed (i.i.d.) sample; e.g.,one can perform maximum likelihood estimation, obtain the Fisher informationmatrix and produce confidence intervals in the standard way (see Remark 1).This enables fully non-parametric estimation of models linking rooted countsand vertex-specific covariates using generic statistical tools. This is key forapplications, as existing methods are hard to implement, do not enable the useof covariates, or scale so well to large graphs. A large literature exists on efficientcounting of rooted graphs; e.g., [1, 3, 50].Both results are the best of their kind: we obtain convergence in moments forany rooted count, but find that uniform convergence does not hold; we obtainuniform convergence for averages of rooted counts, but find that convergence tobe arbitrarily slow. In both cases it is the strength of the dependence that limitsthe convergence. This leads to additional care being needed when working ondata (e.g., Appendix C.3.) Outline
Section 2 introduces the null model and the definitions needed to treatrooted counts. Section 3 presents two new central limit results for rooted counts.Section 4 presents: a goodness-of-fit test that identifies the vertices causing thelack of fit; and a regression, with associated confidence regions, linking rootedcounts and vertex-specific covariates. Section 5 concludes. Proofs can be foundin Appendix A and B. Simulation experiments and algorithms can be found inAppendix C.
We now introduce our null model and our statistic. Our null model is thatof generalized, or kernel-based, random graphs [13]. Our statistics are rootedsubgraph densities, and are related to the statistics used in [3, 50, 54, 58].As in [9, 24, 63], to allow for sparsity, community structure and power-lawdegree distribution, we use inhomogeneous random graphs [13]:3 efinition 1 (Inhomogeneous random graphs sequence G ( ρ, κ )) . Let κ ∈ L ∞ ([0 , ) be a k ernel (i.e., symmetric, positive, bounded by , and suchthat (cid:82) [0 , κ ( x, · ) dx > ) and ρ = ( ρ n ) n> be a sequence in (0 , . We call G ( ρ, κ ) = ( G n ) n> the sequence of random graphs such that for each n , G n is the random graph on [ n ] where to each vertex i is associated independently x i ∼ Uniform([0 , and such that independently across pairs { i, j } ⊂ [ n ] , I { ij ∈ G n } | x i , x j ∼ Bernoulli (cid:0) ρ n κ ( x i , x j ) (cid:1) . Definition 1 subsumes, among others, blockmodels, latent space models [28,29] and the random dot-product models [5, 45, 59, 62]. Such models assign toeach vertex a position in a space, say z i ∈ X ⊂ R k , and the probability of anedge is tied to a function φ of the z i -s; i.e., P (cid:0) ij ∈ G | z i , z j (cid:1) = φ ( z i , z j ). Thisis equivalent to assuming that κ may be written as κ ( x i , x j ) = φ ( ψ ( x i ) , ψ ( x j )),with ψ : [0 , → X . Note that by Borel isomorphism theorem, ψ can be bijective,and that setting the x i -s in [0 ,
1] in Definition 1 is not restrictive.In the following we focus on asymptotic results in the limit of large n .Especially, we shall consider the sparse regimes where ρ n →
0. To address thisproblem, we define our statistic as the normalize rooted count, thereby yieldingstatistics that remains order one in the limit:
Definition 2 (Rooted graph [25]) . A rooted graph F = [ F, v ] is a labeled graph F in which a vertex v has been singled out. Two rooted graphs F = [ F, v ] and F (cid:48) = [ F (cid:48) , v (cid:48) ] are isomorphic, and we write F ≡ F (cid:48) , if there exists an adjacencypreserving isomorphism that maps F onto F (cid:48) and v onto v (cid:48) . Definition 3 (Rooted subgraph density) . Fix a rooted graph F , a simple graph G of order n , and a vertex i ∈ G . Denoting K n the complete graph on n verticesand e ( H ) the number of edges in a graph H , we write s i ( F, G ) = (cid:18) e ( G ) e ( K n ) (cid:19) − e ( F ) { F (cid:48) ⊂ G : [ F (cid:48) , i ] ≡ F } { F (cid:48) ⊂ K n : [ F (cid:48) , i ] ≡ F } for the rooted density of F at vertex i ∈ G . Definition 3 is standard—it may be understood as the normalized probabilityof | F | − F rooted at i —andrelates directly to others; e.g., Janson and Ruci´nski [31, rooted graph]; Spencer[58, graph extension]; Lov´asz and Szegedy [39, restricted homomorphism]; Aliet al. [3], Prˇzulj [50, graphlet]; Cunningham et al. [21], Moustafa et al. [42,graph in vertex’s ego-network]; Karrer and Newman [32], Miller [41], Newman[44, role sequence]. The global subgraph density, which is more common inthe literature, is up to a constant factor, the sum across vertices of the rooteddensities [8, 9, 13, 38].With this definition, we will show in Section 3 that as long as the averagedegree grows with n —i.e., if nρ n → ∞ , which is the minimal assumption forstatistical inference under our null model [9, 24, 63]—then for F a tree, the4 i ( F, G n ) are asymptotically Normal. Therefore, surprisingly, there is no gapbetween the minimal assumption necessary for local and global inferences ininhomogeneous random graphs. We now present our two main results. The first characterizes the limit distributionof rooted densities at any vertex. The second is a central limit result for theaverage, taken across vertices, of functions of rooted densities. In Section 4we will show through an example that these results lead to new methods forstatistical inference on network data.
Our proofs rely on the method of moments: we prove moment convergence toobtain convergence in distribution. Erd¨os and R´enyi [23], Bollob´as [11], andultimately Ruci´nski [53], exhibited the power of this proof technique to studyglobal subgraph counts. Especially, we find that although U -statistics andPoisson convergence based methods are easier to implement, only the method ofmoments handles the strong dependence that may exist between rooted copies,and yields sufficient and necessary conditions for any subgraph F and densityregime ρ .The power of the moment method, when applied to subgraph counts, restscrucially upon the very simple asymptotic behavior of the expected number ofrooted copies. Specifically, if ( G n ) ∼ G ( ρ, κ ), then for any rooted graph F andvertex i we have (see Proposition A.1) E { F (cid:48) ⊂ G n : [ F (cid:48) , i ] ≡ F } = Θ (cid:0) n | F |− ρ e ( F ) n (cid:1) . Thus, the simplest features of F —its number of vertices and edges—characterizethe first order behavior of its rooted count; e.g., by Markov’s inequality, thereare asymptotically no copies of F rooted at i if nρ e ( F ) / ( | F |− n → H F , the set of rooted graphsthat can be built from two copies of F sharing their root and at least one othervertex: for the rooted triangle (root vertex in grey), H = { , } ; for therooted cherry , H = { , , , , } . Then, we find that the variance ofthe number of rooted copies is of the same order of magnitude as the sum of thenumber of copies of elements of H F .We then show that the 2 k -th moment of the number of rooted copies is alsodriven by H F . This higher order moment is equal, up to a negligible term, tothe total number of copies of all the subgraphs that can be built from k elementsof H F sharing only their root. This enables us to show that 2 k -th moment isasymptotically equal to ( k − k -th power, and then—invoking the Hausdorff moment method—that the s i ( F, G n ) are asymptoticallyNormal (proof to be found in Appendix A):5 heorem 1 (Limit of rooted densities) . Set ( G n ) ∼ G ( ρ, κ ) | x i = x anddefine the parameter m = max { e ( H ) / ( | H | −
1) : v ∈ H ⊂ F, | H | > } . Then,asymptotically in n , s i ( F, G n ) → p if nρ mn → ,s x ( F, κ ) := E (cid:104) (cid:81) pq ∈ F κ ( x p , x q ) (cid:12)(cid:12)(cid:12) x i = x (cid:105) if nρ mn → ∞ . Furthermore, in the later case, and if s x ( F, κ ) > , s i ( F, G n ) − s x ( F, κ ) (cid:112) Var s i ( F, G n ) L −−→ Normal(0 , . The result extends to the multivariate setting: a vector of standardized rootedsubgraph densities is asymptotically multivariate Normal.
Theorem 1 generalizes Ruci´nski and Vince [54, Theorem 5] which considerssums of rooted counts of balanced subgraphs, and Spencer [58, Theorem 5] whichconsiders the rate at which rooted counts concentrate, both under a constantkernel. A formula for Var s i ( F, G n ) is provided in Remark A.1.The rate at which Var s i ( F, G n ) shrinks, and therefore the rate of convergencetoward the Normal distribution, changes with ρ . This parallels the behaviorof global subgraph counts in the Erd˝os-R´enyi model [53]. If nρ mn divergesslowly, copies of F are likely to substantially overlap, and the rate dependsin complex way on the structure of F . However, if nρ mn diverges fast enough,copies tend to only overlap over one edge, and the rate is √ nρ n . Finally, if nρ mn = Θ(1), although all moments converge, in general no limit distributioncan be elicited [53]. More details are provided in Appendix A. Theorem 1 shows that, conditionally on x i , s i ( F, G n ) is an unbiassed andasymptotically Normal estimator of s x i ( F, κ ). However, Theorem 1 leaves thejoint behavior of the rooted densities unexplained. We now consider to whatextent it is possible to use the s i ( F, G n ) to estimate distributional propertiesof s x i ( F, κ ). The end goal is, for a covariate y i in some set Y and observed ateach vertex i , to use the s i ( F, G n ) to estimate first Cov (cid:0) s x i ( F, κ ) , y i (cid:1) , and thena regression, or some other model, linking s x i ( F, κ ) to y i .To link ( s x i ( F, κ )) i ∈ [ n ] to Y = ( y i ) i ∈ [ n ] , we consider the average acrossvertices of functions of rooted counts; i.e., for some map f , estimators of theform n − (cid:80) i f ( s i ( F, G n ) , y i ). While linking y i directly to x i could seem better,that would not make an explicit link between covariates and observed features,making such an approach hard to interpret.Counterintuitively, the limit distribution of such estimators does not followimmediately from Theorem 1, even for a simple map f . Indeed, as for any F we have Cov (cid:0) s i ( F, G n ) , s j ( F, G n ) (cid:1) = Ω (cid:0) n − (cid:1) , the dependence between thesummands is too strong for a central limit theorem (CLT) to immediately hold;the level of dependence between the s i ( F, G n ) violates what are presented as6he best possible sufficient assumption for an exchangeable sequence to verify aCLT (Blum et al. [10, p227].)Furthermore, methods to address such dependent sums fail: CLT for U -statistics, as in [9], apply only if f is a projection; martingale results for ex-changeable arrays, such as Weber [61], fail because the dependence betweenthe s i ( F, G n ) is too strong; CLT for weakly dependent stationary sequences(e.g., Newman and Wright [43]), require a specific dependence structure betweenthe s i ( F, G n ) which need not apply for general κ .Nonetheless, results such as [9] and [19], show that for f a projection, theestimator is well behaved. Therefore, the covariance between the s i ( F, G ) issuch that some concentration can occur. Two observations clarify necessaryconditions on f . First, for the estimator to be consistent, f must be continuous,and since the s x i ( F, κ ) live in a compact, f can be understood to be uniformlycontinuous. Then, for Cov (cid:0) f ( s i ( F, G ) , y i ) , f ( s j ( F, G ) , y j ) (cid:1) to commensurate withCov (cid:0) s i ( F, G ) , s j ( F, G ) (cid:1) , f must have an almost everywhere bounded derivative(e.g., see Cuadras [20]). Therefore, f Lipschitz in its first argument is a necessaryassumption, which we find to be sufficient (see Appendix B.)
Theorem 2 (Central limit result for rooted densities) . Let (cid:0) G n , Y (cid:1) be such that ( G n ) ∼ G ( ρ, κ ) and ( x i , y i ) is i.i.d. D for some distribution D over [0 , × Y . Setthe parameter γ = max { e ( H ) / ( | H |−
1) : v (cid:54)∈ H ⊂ F, | H | > } and f : [0 , + ∞ [ ×Y (cid:55)→ R d , Lipschitz in its first argument. Then, if nρ γn → ∞ , Var s x ( F, κ ) > , and f ( s x ( F, κ ) , y ) | ( x, y ) ∼ D has second moments, √ n n (cid:88) i ∈ [ n ] f (cid:0) s i ( F, G n ) , y i (cid:1) − E f (cid:0) s i ( F, G n ) , y i (cid:1) L −−−−→ n →∞ Normal(0 , Σ) , where Σ = Cov ( x,y ) ∼ D f (cid:0) s x ( F, κ ) , y (cid:1) . The result still holds if s i ( F, G n ) is replacedby a vector of rooted densities, and admits a natural Berry-Esseen type extension(Proposition G.4.) Fixing f to be the identity function, and omitting Y , we observe thatTheorem 2 generalizes Ruci´nski [53, Theorem 2], Bickel et al. [9, Theorem 1],Bordenave et al. [15, Propositions 35–36] and Coulson et al. [19, Corollary 4.1].Theorem 2 also specifies these results by writing the limiting covariance matrixΣ as a function of the kernel κ and the subgraph F (see Remark A.1). TheBerri-Essen extension recovers the same rates as presented in Privault and Serafin[49, Theorem 4.2]. See Appendix G for comparison with [6].The assumption underlying Theorem 2 is stronger than the one behindTheorem 1. Furthermore, following [53, Theorem 2] and [9, Theorem 1], it isnecessary for √ n -consistency even for f a projection. Also, for some F , includingtrees and cycles, γ = m (see the notion of balanced graphs [23, 54]).Theorem 2 enables many tests for dependence between rooted counts andvertex covariates. We now turn to use this observation in Section 4, where wewill uncover the link between rooted counts and gender in a social network. Remark 1.
Theorem 2 shows that when using ( y i , s i ( F, G )) i to fit a probabilis-tic model, statistical inference proceeds as if one were using the i.i.d. sample y i , s x i ( F, κ )) i . To see this, fix a smooth log-likelihood function l θ ( · , · ) , andset ˆ θ G = argmax θ (cid:80) i l θ ( Y i , s i ( F, G )) , ˆ θ κ = argmax θ (cid:80) i l θ ( Y i , s x i ( F, κ )) . Then,Theorem 2 shows that ˆ θ G and ˆ θ κ converge at the same rate and toward thesame value. Further, the asymptotic variance of ˆ θ G may be computed in thestandard way, using the Fisher information matrix. Other approaches, such as,but not limited to, least square optimization and generalized method of moment,would also apply without adaptation. See Appendix F for a discussion of theassumptions on l θ necessary for this claim to hold. Remark 2.
Theorem 2 allows for non-parametric estimation of κ . Specifically,as kernels that are fully determined by a finite number of subgraph densities—alsotermed finitely forcible kernels [37] (e.g., blockmodels)—admit global subgraphdensities as sufficient statistics [4, 9], they also admit rooted densities as sufficientstatistics. Further, following [39], we conjecture that rooted densities are in factsufficient statistics for all kernels of finite rank. We study a social network through two new methods for statistical inference onnetwork data. The first method uses Theorem 1 to build a goodness-of-fit testable to locate which vertices cause the lack of fit. The second uses Theorem 2to estimate a regression model linking vertex covariates and rooted densities,and test for significant parameters. Simulations describing the small sampleconvergence of the proposed methods are presented in Appendix C.
The network we study, henceforth G , is taken from the US National LongitudinalStudy of Adolescent Health, specifically school 44 in [26] which was studiedby [47]. Each vertex in G is a student, and a pair of students is connected ifat least one of them nominated the other as friend. For almost all students,we have three categorical covariates: gender, race and school grade. There are n = 1117 students in G , comprising 559 males, 556 females, 429 blacks, 525whites, 153 other races. Furthermore, there are 209 , , , , ,
131 grade7–12 students respectively. We found no statistically significant correlationsbetween any pair of these covariates. The maximal degree is 32, while the averagedegree is 10 and the median degree is 9.To determine whether the local structure of G is a predictor of genderand race, we test whether students’ rooted counts are significant predictorsof the covariates available. We focus on and , and thus use as predictor (cid:0) s i ( G ) (cid:1) i ∈ G = (cid:0) s i ( , G ) , s i ( , G ) (cid:1) i ∈ G ∈ R n × . We choose and as: usingthe smaller subgraphs presented little power; and are not subgraphs ofeach other, which we observed reduces the correlation of the counts; they areprominently used to characterize local network properties [3, 50].8 - -6 -3 - - -6 -3 -6 -3 Figure 2: Goodness-of-fit within grades. Crosses are the standardized rooteddensity vectors ˆ t i ( G g ), where from top to bottom, and left to right, g ranges7–12. The red circle is the confidence region at the 10% level (see Appendix Cfor a description of our algorithm to estimate the radius of this circle). Only G is rejected, with one outlier vertex. Before analyzing G at the level of each student—through using the s i ( G ) topredict the covariates—we first test for G to be a realization of an inhomogeneousrandom graph. Indeed, while G ’s global features of sparsity, community structureand degree distribution have successfully been modeled by our null model [47],our results will now allow us to test if our null is also sufficient to account forlocal features of G . We do so by testing independently for each i ∈ G that s i ( G ) is consistent with a kernel estimate. We choose such a vertex-specificapproach, instead of a global one relying on averages over vertices, for two keyreasons: first, a global test would be less powerful, as averages of rooted count areasymptotically normal under other, more general, null models [19, 48]; second,this will enable us to determine which vertices cause the lack of fit.Out goodness-of-fit test takes places in three steps: i) the kernel and latentvariates are estimated using maximum likelihood, yielding ˆ κ and (ˆ x i ) i ; ii) weuse Monte-Carlo integration methods with ˆ κ and (ˆ x i ) i to compute the mean ˆ µ i and variance ˆ σ i of each s i ( G ) under the null; iii) we test across i that the ˆ t i =ˆ σ − / i ( s i ( G ) − ˆ µ i ) are realizations of the bivariate standard Normal distribution.Note that the approach only makes sense since we know from Theorem 1 thatall the t i tend to the same distribution. Further details, especially on theestimation of κ , are provided in Appendix C. Two key points are that: the plug-in estimators ˆ µ i and ˆ σ i may be used for standardization because the variationof the rooted densities is at least as large as that of ˆ κ and (ˆ x i ) i [16, 35, 55]; thedependence between the s i ( G ) is such that using Bonferroni correction yields atoo conservative test, and we present an alternative correction in Appendix C.Using this method we reject the hypothesis that G is a realization of aninhomogeneous random graph: 43 t i -s are too far from the origin. However, wenote that the vertices causing the lack of fit have more edges spanning grades9han most. Thus, we refine our analysis to the graphs formed by students in thesame grade; the G g for g ∈ { , . . . , } . There, using the same method we rejectthe null only for G (see Fig. 2). We now use the the rooted densities to predict gender and race. Specifically,following the outcome of the tests, we consider as predictor (cid:0) s i ( G g ( i ) ) (cid:1) , where i ranges the students in grade less than 11, and g ( i ) is the grade of student i .Then, we predict the gender and race covariate via a binomial linear regressionswith a logit link function. Following Remark 1, this model may be estimated bymaximum likelihood and analyzed in the standard fashion. There, we can makethe following observations (always at the 10% level): i) While white students donot have an average degree larger than black students, they present significantlylarger rooted triangle and square densities; ii) By conditioning on race, in bothcases we find that the likelihood of a student being female increases with thetriangle density and shrinks with the squares density, while for male students, itis the reverse; iii) The triangle and square densities do not evolve with grade.These findings are robust to removing the vertices causing the lack of fit in G .We note that when using (cid:0) s i ( G g ( i ) ) (cid:1) in a regression model, we do not relyon ˆ κ and (ˆ x i ) i . This makes the approach independent of any kernel estimate,vertex clustering, or group label assignment, but also requiring only negligiblecomputation beyond that of evaluating the rooted counts.To conclude, we have shown that: i) the whole network cannot be modeledby our null; ii) yet, local network structure changes with vertex covariate in aprovably predictable way. Expanding our analysis to more social networks wouldenable rich and general inference regarding the scale at which conditional inde-pendence is a relevant null, as well as the link between local network structuresand gender and race, ultimately allowing to explore the global consequence ofgroup specific behavior on the social graph. We contribute the results, and associated methods, that enable the study oflarge networks at the level of each vertex. This in the exact same range ofdensity regimes as group level inference is possible. The combinatorial prooftechniques introduced constitute a basis opening new venues for the study oflocal properties of random graphs. We expect that our proofs extend to: directedor weighted graphs, unbounded kernels as long as s U ( F, κ ) exists, as well as torooted subgraphs with multiple roots.Following our findings, it appears that inhomogeneous random graphs—whichare recognized as sufficiently flexible to fully account for the three key large scalefeatures of real-world networks that are sparsity, community structure and heavytailed degree distribution [47]—do not induce rich local behaviors. Indeed, afterstandardization, rooted densities are almost i.i.d normal, both across vertices10 egree-corrected blockmodel (BM)Degree-based model (DM) Mixed-membership (MM)Rooted edges(counts of 3, 5) Rooted triangles(counts of 2, 3) R e l a t i v e f r e q u e n c y o f o cc u rr e n ce N o r m a li ze dd e n s i t i e s Figure 3: Three different kernels that lead to similar rooted edge and triangledensities. We consider three networks ensuing from three kernels and havingsubstantially different community structures: no communities, sharply sepa-rated communities, and overlapping communities. After normalizing the rooteddensities to account for their different scales, we observe that the six obtaineddistributions are very similar across kernels and densities.and subgraphs. This observation parallels the results presented in [17, 56]. Weexplore the idea further in Fig. 3, where we show that rooted densities density,and community structure can be, to a large extent, dissociated under our nullmodel.This makes our methods especially useful, because they therefore uncovera new layer of information to be extracted from network data. However, italso strains the modeling capabilities of the null model. The random graphmodels discussed in [14, 32, 41, 44] generate richer local features. However, thesemodels all take local counts as input, and thus cannot be used to infer the linksbetween local counts and vertex covariates. Therefore, the problem of proposinga framework able to induce rich local behavior remains open.
APPENDIX A: Proof of Theorem 1
APPENDIX A.1: Number of rooted copies
In a first instance, the coreobject of our study will be the number of copies of a rooted graph within alarger graph. Therefore, for a fixed rooted graph F = [ F, v ] and a graph G , wefirst introduce the number of rooted isomorphic copies of F with root at a fixedvertex in G , and provide examples in Fig. A.1. Definition A.1 (Number of rooted isomorphic copies) . Fix a rooted graph i Figure A.1: Example of rooted graph counting. Call G the plotted graph,then we have: X ( G, i ) = 1, X ( G, i ) = 8, X ( G, i ) = 0, X ( G, i ) = 2, X ( G, j ) = 2, X ( G, j ) = 9, X ( G, j ) = 1, X ( G, j ) = 2. F = [ F, v ] and a graph G . We denote X F ( G, i ) the number of rooted isomorphiccopies of F rooted at i ∈ G ; i.e, X F ( G, i ) = (cid:8) F (cid:48) ⊂ G : i ∈ F (cid:48) and [ F (cid:48) , i ] ≡ F (cid:9) , with [ F (cid:48) , i ] ≡ F if there is an adjacency preserving isomorphism from F (cid:48) to F that maps i to v , and F (cid:48) ⊂ G spans the non-automorphic subgraphs of G . As we count non-automorphic subgraphs of G , we have that X F ( K n , i ) =( n − | F |− / aut( F ), with K n the complete graph, ( n ) k = n ( n − · · · ( n − k + 1)the falling factorial, and aut( F ) the order of the automorphism group of F (under ≡ .) The overall number of copies of F in G , usually denoted X F ( G ), isthen proportional to the sum over i ∈ G of the X F ( G, i ).To begin our analysis we consider how the product of the number of rootedisomorphic copies of two rooted graphs F and F behaves. Therefore, weevaluate X F ( G, i ) X F ( G, i ) and write X F ( G, i ) X F ( G, i ) = (cid:88) i ∈ F (cid:48) ⊂ G I { [ F (cid:48) , i ] ≡ F } (cid:88) i ∈ F (cid:48) ⊂ G I { [ F (cid:48) , i ] ≡ F } = (cid:88) i ∈ F (cid:48) ⊂ Gi ∈ F (cid:48) ⊂ G I { [ F (cid:48) , i ] ≡ F } I { [ F (cid:48) , i ] ≡ F } . (A.1)To reformulate this sum in a more concise way, we index all the unlabeled rootedgraphs that can be formed by two graphs F (cid:48) and F (cid:48) containing i and such that[ F (cid:48) , i ] ≡ F and [ F (cid:48) , i ] ≡ F . This yields the following definition and subsequentlemma. Definition A.2 (Overlapping copies [53]) . For two rooted graphs F and F ,we write H F ,F the set of unlabeled rooted graphs that can be formed by two , = { , , } ; c ( , ) = 2 , c ( , ) = 2 ,c ( , ) = 1 . H , = { , , , , , } ; c ( , ) = 2 , c ( , ) = 2 ,c ( , ) = 2 , c ( , ) = 2 ,c ( , ) = 2 , c ( , ) = 1 . H , = { , , , } ; c ( , ) = 1 , c ( , ) = 2 ,c ( , ) = 1 , c ( , ) = 2 . Figure A.2: Examples of H F ,F and c H . We add only within this figure thedependence of c H in F , F . The first element of each H F ,F is F F (seeDefinition D.3). The c H are obtained by counting the number of ways H may be covered by one rooted copy of F and F each. The c H may alsobe obtained inductively by using the implicit preorder in H F ,F as follows:i) write F ⊂ F if X F ( F ) >
0; ii) for the smallest elements of the poset( H F ,F , ⊂ ), say the H t , we have c H t = X F ( H t ) X F ( H t ) by Lemma A.1;iii) for the smallest elements of ( H F ,F \ { H t } t , ⊂ ), say the H t , we have c H t = X F ( H t ) X F ( H t ) − (cid:80) s c H s X H s ( H t ) again by Lemma A.1; and so onuntil termination. copies of F and F , and c H the number of ways a given H = [ H, v ] ∈ H F ,F can be built from copies of F and F : H F ,F = [ H, v ] ⊂ K | F | + | F | : ∃ [ F (cid:48) , v ] , [ F (cid:48) , v ] ⊂ K | F | + | F | s.t., [ F (cid:48) , v ] ≡ F , [ F (cid:48) , v ] ≡ F ,and H = F (cid:48) ∪ F (cid:48) / ≡ c H = (cid:26)(cid:0) [ F (cid:48) , v ] , [ F (cid:48) , v ] (cid:1) ⊂ [ H, v ] : [ F (cid:48) , v ] ≡ F , [ F (cid:48) , v ] ≡ F and H = F (cid:48) ∪ F (cid:48) (cid:27) . If H is obtained from copies of F and F overlapping only at the root, then c H = aut( H ) / aut( F )aut( F ) [53, p4]. No formula exists in the general case; seeFig. A.2. Lemma A.1 (Rooted copies pairwise interaction) . Fix two rooted graphs F , F and a graph G . Then, for any i ∈ G , X F ( G, i ) X F ( G, i ) = (cid:88) H ∈H F ,F c H X H ( G, i ) . roof. This proofs consists in rewriting (A.1) using Definition A.2. We first notethat for each F (cid:48) , F (cid:48) inf (A.1), I { [ F (cid:48) , i ] ≡ F } I { [ F (cid:48) , i ] ≡ F } = 1 if and only ifthere exists a rooted graph H ∈ H F ,F such that [ F (cid:48) ∪ F (cid:48) , i ] ≡ H . Thereforewe can reindex the sum in (A.1) as follows: X F ( G, i ) X F ( G, i ) = (cid:88) i ∈ F (cid:48) ,F (cid:48) ⊂ G I {∃ H ∈ H F ,F : [ F (cid:48) ∪ F (cid:48) , i ] ≡ H } = (cid:88) H ∈H F ,F (cid:88) i ∈ F (cid:48) ,F (cid:48) ⊂ G I { [ F (cid:48) ∪ F (cid:48) , i ] ≡ H } . We now note that by definition of c H , for each copy of H in G rooted at i ,there will be c H pairs F (cid:48) , F (cid:48) of copies of F and F rooted at i in G such that[ F (cid:48) ∪ F (cid:48) , i ] = H . Therefore, we can simplify the sum above to obtain, X F ( G, i ) X F ( G, i ) = (cid:88) H ∈H F ,F c H (cid:88) i ∈ H (cid:48) ⊂ G I { [ H (cid:48) , i ] ≡ H } = (cid:88) H ∈H F ,F c H X H ( G, i ) , yielding the sought-after result.A key consequence of Lemma A.1 is that the quantities X F ( G, i ) for all F aresufficient to express products of the form X F ( G, i ) X F ( G, i ). Indeed, althoughno simple formula exists to compute H F ,F [53], it is a functions only of F and F , but not of G . Then, through an induction argument, the quantities X F ( G, i ) for all F are sufficient to express higher order products of the form X F ( G, i ) · · · X F m ( G, i ) for some m >
1. As we shall see below, in the case where G is a random graph, a similar phenomena occurs: describing the first momentof X F ( G, i ) for all F is sufficient to describe the higher order moments of therooted counts, and ultimately the joint distribution of the rooted graph counts. APPENDIX A.2: Rooted copies in inhomogeneous random graphs
Wenow consider X F ( G n , i ) in the setting where G n is a sequence of growing inho-mogeneous random graphs (see Definition 1). Our focus will be the limitingdistribution of X F = X F ( G n , i ).In what follows, we first uncover the limiting distribution of X F conditionallyon x i = x for some x ∈ [0 , G n rooted at some vertex i , and ( G n ) as being a sequence of inhomogeneousrandom graphs where the latent variable x i is fixed to be equal to a given value x ∈ [0 , G n ) ∼ G ( ρ, κ ) | x i = x . Furthermore, we will omit, unless it isnecessary, the dependence in n of both G n and ρ n .We begin by presenting the first order properties of X F , the proof of whichis deferred to the Appendix D, where we also present slightly improved bounds: Proposition A.1.
With F a rooted graph and G ∼ G ( ρ, κ ) | x i = x , E X F = s x ( F, κ ) ρ e ( F ) X F ( K n , i ) , X F = (cid:0) O (1 /n ) (cid:1) (cid:80) H ∈H F,F \{ F } c H E X H , where s x ( F, κ ) = E (cid:2) (cid:81) pq ∈ F κ ( x p , x q ) | x i = x (cid:3) , and F is the rooted graphobtained by taking the disjoint union of two copies of F and identifying theirroots. Then, with m ( F ) = max { e ( H ) / ( | H | −
1) : v ∈ H ⊂ F, | H | > } , if nρ m ( F ) = ω (1) , we have that X F = E X F (cid:16) O p (cid:0) /nρ m ( F ) (cid:1)(cid:17) . (A.2) Furthermore, for F , F two labelled rooted graphs sharing their root, we have X F ∪ F = O p (cid:0) E X F E X F / E X F ∩ F (cid:1) . (A.3) Remark A.1 (Computing the variance) . Although it is less amenable to thesubsequent derivation, we note that:
Var X F = (cid:0) O (1 /n ) (cid:1) (cid:88) H ∈H F,F \{ F } s x ( H, κ ) ρ e ( H ) ( n ) | H |− aut( H ) . We now show how Proposition A.1, with some additional manipulations, maybe augmented to provide a description of any central moment of X F . In turns—through method of moment arguments—we obtain a more precise understandingof the joint behavior of the X F in the limit of large graphs. Theorem A.1.
Fix k rooted graphs F , . . . , F k . Let G ∼ G ( ρ, κ ) | x i = x andassume that (cid:15) = 1 /nρ max t ∈ [ k ] m ( F t ) = o (1) and that s X ( F t , κ ) > for t ∈ [ k ] .Then, if k is even, E k (cid:89) t =1 (cid:0) X F t − E X F t (cid:1) = (cid:0) O ( (cid:15) ) (cid:1) (cid:88) s ∈ [ k ] (2) (cid:89) { p,q }∈ s Cov (cid:16) X F p , X F q (cid:17) , where [ k ] (2) is the set of all partitions of [ k ] in subsets of size . On the otherhand, if k is odd, E (cid:81) t (cid:0) X F t − E X F t (cid:1) = O (cid:0) (cid:15) (cid:81) t Var( X F t ) / (cid:1) .Proof. This proof proceeds in three steps. First we develop a more tractable ex-pression of E (cid:81) t ( X F t − E X F t ) in the form of a sum mirroring that of Lemma A.1.Then, we determine the leading term in that sum, by determining how the copiesof the F t in G predominantly overlap. Finally, we conclude in the same fashionas in the proof of Proposition A.1. Expanding E (cid:81) t ( X F t − E X F t ) : Let F t , . . . , F tM t be the M t = X F t ( K n , i )labeled rooted copies of F t in K n . Then, X F t = (cid:80) M t m =1 I { F tm ⊂ G } , so that k (cid:89) t =1 (cid:0) X F t − E X F t (cid:1) = k (cid:89) t =1 (cid:32) M t (cid:88) m =1 I { F tm ⊂ G } − M t (cid:88) m =1 P [ F tm ⊂ G ] (cid:33) k (cid:89) t =1 (cid:32) M t (cid:88) m =1 ( I { F tm ⊂ G } − P [ F tm ⊂ G ]) (cid:33) = (cid:88) m ∈ [ M ] (cid:88) m ∈ [ M ] · · · (cid:88) m k ∈ [ M k ] k (cid:89) t =1 (cid:0) I { F tm t ⊂ G } − P [ F t ⊂ G ] (cid:1) , (A.4)where the last step follows form expanding a product of sum into a sum ofproducts, and using that for all t and m , P [ F tm ⊂ G ] = P [ F t ⊂ G ].We now distinguish between k -tuples ( m , . . . , m k ) according to the unlabelledrooted graph induced by ∪ t F tm t . Therefore, we call H the set of unlabelledrooted graphs that can be formed by overlapping rooted copies of the rootedgraphs F , . . . , F k ; i.e., generalizing Definition A.2 we write H = (cid:110) [ H, v ] ⊂ K (cid:80) t | F t | : ∃ [ F (cid:48) t , v ] s . t . ∀ t, [ F (cid:48) t , v ] ≡ F t and H = ∪ t F (cid:48) t (cid:111) / ≡ . Furthermore, to simplify notation, for H ∈ H we write˜ X H = (cid:88) m ∈ [ M ] (cid:88) m ∈ [ M ] · · · (cid:88) m k ∈ [ M k ] I {∪ t F tm t ≡ H } k (cid:89) t =1 (cid:0) I { F tm t ⊂ G } − P [ F t ⊂ G ] (cid:1) (A.5)Resuming from (A.4) we may therefore write: E k (cid:89) t =1 (cid:0) X F t − E X F t (cid:1) = (cid:88) H ∈H E ˜ X H . (A.6)Note that the F tm t forming H may overlap in varied ways. As we eventuallybound E ˜ X H by E X H , only the resulting shape H = ∪ t F tm t matters to ouranalysis, and how they overlap does not affect the following arguments. Thisobservation is also why we can choose this notation. Order of magnitude of E ˜ X H : We begin by controlling the order of magni-tude of each term in the sum of (A.6). For each H = ∪ t F tm t , this is done bydiscussing how many vertices and edges of the F tm t overlap.First we note that if there exists p such that | F pm p ∩ ( ∪ t (cid:54) = p F tm t ) | ≤
1, then wemust have that F pm p ∩ ( ∪ t (cid:54) = p F tm t ) = ( { i } , ∅ ) and F pm p has all its vertices exceptits root disjoint from the F tm t . Since vertex disjoint edges occur independentlyin G ( n, ρκ ) | x i = x , we have E k (cid:89) t =1 (cid:0) I { F tm t ⊂ G } − P [ F t ⊂ G ] (cid:1) = E (cid:89) t (cid:54) = p (cid:0) I { F tm t ⊂ G } − P [ F t ⊂ G ] (cid:1) E (cid:16) I { F pm p ⊂ G } − P [ F p ⊂ G ] (cid:17) = 0 , (A.7)16s E I { F pm p ⊂ G } = P [ F p ⊂ G ] by definition. Therefore, we need only consider H = ∪ t F tm t such that each F tm t overlap with at least one other F tm t . In thiscase, expanding the product using inclusion exclusion, we obtain E k (cid:89) t =1 (cid:0) I { F tm t ⊂ G } − P [ F t ⊂ G ] (cid:1) = (cid:88) S ⊂ [ k ] ( − | S | E (cid:2) I {∪ t (cid:54)∈ S F tm t ⊂ G } (cid:3) (cid:89) t ∈ S P [ F t ⊂ G ] . (A.8)From Lemma D.2, we have that E I {∪ t (cid:54)∈ S F tm t ⊂ G } = O ( ρ e ( ∪ t (cid:54)∈ S F tmt ) ) and (cid:81) t ∈ S P [ F t ⊂ G ] = O ( ρ (cid:80) t ∈ S e ( F t ) ). As, by assumption, we have | ( ∪ t (cid:54)∈ S F tm t ) ∩ ( ∪ t ∈ S F tm t ) | >
1, it must be that e ( ∪ t (cid:54)∈ S F tm t ) + (cid:80) t ∈ S e ( F t ) ≥ e ( ∪ t F tm t ), andsince ρ <
1, we therefore have
E I {∪ t (cid:54)∈ S F tm t ⊂ G } (cid:89) t ∈ S P [ F t ⊂ G ] = O (cid:16) ρ e ( ∪ t F tmt ) (cid:17) = O (cid:16) ρ e ( H ) (cid:17) . As a result, from (A.5), (A.8) and Lemma D.2, we obtain E ˜ X H = O (cid:16) ρ e ( H ) (cid:8) ( m , . . . , m k ) : ∪ t F tm t ≡ H (cid:9)(cid:17) = O (cid:16) ρ e ( H ) X H ( K n , i ) (cid:17) = O ( E X H ) . (A.9) Controlling the variance:
Before we proceed to bound E X H , we first controlthe standard deviation Var( X F t ) / . There, we note that from Proposition A.1,Var X F t = (cid:0) O (1 /n ) (cid:1) (cid:88) U ∈H Ft,Ft \{ F t } c U E X U . (A.10)Then, recall from Definition A.2 that for each U in (A.10) there are two overlap-ping copies of F t , say U and U , such that U = U ∪ U . Thus, we may applyProposition A.1 again to each E X U in the sum above, and reindex the sum bythe overlap J = U ∩ U , and as s x ( F t , κ ) > s x ( J, κ ) > E X J >
0, we obtainVar X F t = O (cid:88) J ⊂ F t : | J | > (cid:0) E X F t (cid:1) E X J , and, as only a finite number of terms is in the sum and therefore can dominateit, we get Var( X F t ) / = O (cid:88) J ⊂ F t : | J | > E X F t √ E X J . Furthermore, for every non-zero term in the right hand side of (A.10), we havefrom Proposition A.1 that E X U = Θ(( E X F t ) / E X J ). Therefore that for any J ⊂ F we have:Ω (cid:18) E X F t √ E X J (cid:19) = Var( X F t ) / = O (cid:88) J ⊂ F t : | J | > E X F t √ E X J . (A.11)17 egligible terms in (A.6) : We now show that if any F pm p overlaps withmore than one of the { F tm t } t (cid:54) = p , then E X H is a O (cid:0) (cid:15) (cid:81) t Var( X F t ) / (cid:1) . To doso, we proceed by induction on k . In the following, we write that { F tm t } t ∈ [ l ] verifies: ( ∗ ) if all F tm t overlap with at least another F pm p t (cid:54) = p ; ( ∗∗ ) if all F tm t overlap with at exactly one other F pm p for p (cid:54) = q . With these definitions, ourgoal is to prove that E X ∪ t F tmt = O (cid:16) (cid:15) (cid:81) t Var( X F t ) / (cid:17) if { F tm t } verifies ( ∗ ) but not ( ∗∗ ), O (cid:16)(cid:81) t Var( X F t ) / (cid:17) if { F tm t } verifies ( ∗∗ ). (A.12)We can now start our induction. For k ≤
2, there is nothing to do. Now fix k ≥ l ≤ k − • First, if { F tm t } t ∈ [ k ] verifies ( ∗∗ ). Assume without loss of generality that F m overlaps with F m . As vertex disjoint graphs occur independently, E X ∪ t F tmt = E X ∪ t> F tmt E X F m ∪ F m . Then, by Proposition A.1, andwith J = F m ∩ F m , we get E X F m ∪ F m = O (cid:0) ( E X F m / (cid:112) E X J )( E F m / (cid:112) E X J ) (cid:1) . Then, the induction assumption and (A.11), yield the result. • Let us now assume that { F tm t } t ∈ [ k ] verifies ( ∗ ) but not ( ∗∗ ). Then, fol-lowing [53, p7], there is p ∈ [ k ] such that { F tm t } t ∈ [ k ] \{ p } verifies ( ∗ ).Echoing Ruci´nski [53]’s argument, to see this imagine the hypergraphwhose vertices are the vertices of ∪ t F tm t and edges are the vertex-sets V ( F tm t ), t ∈ [ k ]. By assumption, there must be at least one connected com-ponent with at least three edges, and one of the edges in these componentsis V ( F pm p ).This way we have defined a map from k -tuples verifying ( ∗ ) but not ( ∗∗ ),and ( k − ∗ ). Furthermore, with J = F pm p ∩ (cid:0) ∪ t (cid:54) = p F tm t (cid:1) ,by this mapping every ( k − O ( n | F p |−| J | ) k -tuples.Then, using our induction assumption, (A.11), and recalling that E X J =Ω(1 /(cid:15) ) by construction, we obtain E X ∪ t F tmt = O (cid:16) n | F p |−| J | ρ e ( F p ) − e ( J ) E X ∪ t (cid:54) = p F tmt (cid:17) = O ( E X J ) − / E X F p √ E X J (cid:89) t (cid:54) = p Var( X F t ) / = O (cid:32) (cid:15) (cid:89) t Var( X F t ) / (cid:33) , which is the sought after result. 18 e-factorizing (A.6) : Recall that the aim of the above discussion is to de-termine the leading terms in (A.6). Equations (A.7) and (A.12) show that wecan ignore E ˜ X H in (A.6) if H = ∪ t F tm t is such that at least one F tm t overlapswith none or two others. (Recall that ˜ X H was defined in (A.5).) This yields theresult if k is odd, since then the F tm t cannot be organized in pairs, and we musthave E (cid:81) kt =1 (cid:0) X F t − E X F t (cid:1) = O (cid:0) (cid:15) (cid:81) kt =1 Var( X F t ) / (cid:1) .We now consider the case where k = 2 r is even. In that case, any termof interest must be built from a partition of [ k ] into sets of size 2 describingwhich pair of F tm t overlap. Let s = ( s , . . . , s r ) be one such partition, andwrite s t = { p t , q t } for t ∈ [ r ]. Further, for t ∈ [ r ], let F p t F q t be the rootedgraph obtained by taking the disjoint union of F p t and F q t and identifying theirroots. With these notations, we have that the only subgraphs H to consider arethose obtained by taking the disjoint union of H , . . . , H r , where each H t is in H F pt ,F qt \ { F p t F q t } , and identifying their root. In that setting, for one such H ,using as in (A.7) that vertex disjoint subgraphs occur independently, we have E ˜ X H = r (cid:89) t =1 E ˜ X H t . Therefore, with H s t = H F pt ,F qt \ { F p t F q t } we can write E k (cid:89) t =1 (cid:0) X F t − E X F t (cid:1) = (cid:0) O ( (cid:15) ) (cid:1) (cid:88) ( s ,...,s r ) ∈ [ k ] (2) (cid:88) H ∈H s · · · (cid:88) H r ∈H sr r (cid:89) t =1 E ˜ X H t . There, we recognize the sum of products as the expansion of a product of sums,which factorizes as follows: E k (cid:89) t =1 (cid:0) X F t − E X F t (cid:1) = (cid:0) O ( (cid:15) ) (cid:1) (cid:88) ( s ,...,s r ) ∈ [ k ] (2) r (cid:89) t =1 (cid:88) H ∈H st E ˜ X H . (A.13)To proceed, let us consider (A.13) in the special case where k = 2. Directlyreplacing in (A.13), and as (2) = 1, we obtain, E (cid:2)(cid:0) X F − E X F (cid:1)(cid:0) X F − E X F (cid:1)(cid:3) = (cid:0) O ( (cid:15) ) (cid:1) (cid:88) H ∈H { , } E ˜ X H . (A.14)Therefore, we can replace in (A.13) each sums of the form (cid:80) H ∈H st E ˜ X H bythe covariance Cov( X F pt , X F qt ) = E [( X F pt − E X F pt )( X F qt − E X F qt )], whichcompletes the proof.We finally describe the limiting behavior of the rooted subgraph densitiesintroduced in Theorem 1. We reproduce here the statement of the theorem forthe reader’s convenience. 19 heorem 1 (Limit of rooted densities) . Set ( G n ) ∼ G ( ρ, κ ) | x i = x anddefine the parameter m = max { e ( H ) / ( | H | −
1) : v ∈ H ⊂ F, | H | > } . Then,asymptotically in n , s i ( F, G n ) → p if nρ mn → ,s x ( F, κ ) := E (cid:104) (cid:81) pq ∈ F κ ( x p , x q ) (cid:12)(cid:12)(cid:12) x i = x (cid:105) if nρ mn → ∞ . Furthermore, in the later case, and if s x ( F, κ ) > , s i ( F, G n ) − s x ( F, κ ) (cid:112) Var s i ( F, G n ) L −−→ Normal(0 , . The result extends to the multivariate setting: a vector of standardized rootedsubgraph densities is asymptotically multivariate Normal.Proof.
We first address the limit in probability before turning to the limit indistribution. In both cases we use Bickel et al. [9, Theorem 1] showing that if nρ = Ω(1), and with ˆ ρ = e ( G ) /e ( K n ), thenˆ ρ = ρ (cid:16) O p (cid:0) n − / (cid:1)(cid:17) , (A.15)which enables use to replace ˆ ρ by ρ (as nρ = Ω( nρ m ( F ) ) for any F ). Limits in probability:
By definition, s i ( F, G ) = X F ( G, i ) / ˆ ρ e ( F ) X F ( K n , i ).Then, if nρ m ( F ) = o (1), from Proposition A.1, E X F ( G, i ) and Var X F ( G, i )are o (1), so that P [ X F ( G, i ) >
0] = o (1). Therefore, X F ( G, i ) = 0 with highprobability in the limit, so that P [ s i ( F, G ) >
0] = o (1), and we obtain our firstclaimed result.If nρ m ( F ) = ω (1), from Proposition A.1, X F ( G, i ) → p E X F ( G, i ) and E X F ( G, i ) = s x ( F, G ) ρ e ( F ) X F ( K n , i ). It follows that we have that X F ( G, i ) → p s x ( F, G ) ρ e ( F ) X F ( K n , i ); i.e., ˜ s i ( F, G ) → p s x ( F, κ ). Thus s i ( F, G ) → p s x ( F, κ )from (A.15), which is the result.
Limits in distribution: The univariate case:
To obtain the limits indistribution we show convergence in moments, and conclude by using that theNormal distribution is uniquely determined by its moment sequence—see forinstance [33, Theorem 2.a]. Thus, we consider the k -th standardized moment of s i ( F, G ). By homogeneity of rescaling and (A.15), we have E s i ( F, G ) k = E (cid:32) X F − E X F (cid:112) Var X F (cid:33) k (cid:16) O p (cid:0) n − / (cid:1)(cid:17) . Therefore, we can work directly with the standardized moments of X F . Now,from Theorem A.1, setting F t = F for each t ∈ [ k ], we obtain E (cid:32) X F − E X F (cid:112) Var X F (cid:33) k → (cid:40) k ] (2) if k is even,0 otherwise. (A.16)20urthermore, if k is even, [ k ] (2) = ( k − k − · · · k − k -th moment of a standardized Normal. Therefore, asymptotically in n , s i ( F, G ) has the same moments as a standardized Normal distribution, andby [33] we obtain the result. The proof in the multivariate case proceeds similarlyand is addressed in Appendix E.
APPENDIX B: Proof of Theorem 2
Up to this point we have only considered rooted densities at a fixed vertex in G . We now consider properties of the array ( s i ( F, G )) i ∈ [ n ] , and show that theconcentration results of Proposition A.1, as well as some uniform bounds oververtices, permit us to prove Theorem 2.We first observe that ( s i ( F, G )) i ∈ [ n ] is a row exchangeable triangular array: un-der our model, for any fixed n , ( s i ( F, G )) i ∈ [ n ] is a finitely exchangeable sequence;i.e., for any permutation ς of [ n ], we have ( s i ( F, G )) i ∈ [ n ] ∼ ( s ς ( i ) ( F, G )) i ∈ [ n ] . Oneconsequence of exchangeability is that for each n all the s i ( F, G ) are identicallydistributed across i ∈ [ n ].However, exchangeability does not imply independence, and for instancethe empirical mean of a row-wise exchangeable triangular array need not beasymptotically Normal [10, 18, 61]. Therefore, in the proof that follows, the keyingredient are upper bounds for Cov (cid:0) s i ( F, G ) , s j ( F, G ) (cid:1) . Theorem 2 (Central limit result for rooted densities) . Let (cid:0) G n , Y (cid:1) be such that ( G n ) ∼ G ( ρ, κ ) and ( x i , y i ) is i.i.d. D for some distribution D over [0 , × Y . Setthe parameter γ = max { e ( H ) / ( | H |−
1) : v (cid:54)∈ H ⊂ F, | H | > } and f : [0 , + ∞ [ ×Y (cid:55)→ R d , Lipschitz in its first argument. Then, if nρ γn → ∞ , Var s x ( F, κ ) > , and f ( s x ( F, κ ) , y ) | ( x, y ) ∼ D has second moments, √ n n (cid:88) i ∈ [ n ] f (cid:0) s i ( F, G n ) , y i (cid:1) − E f (cid:0) s i ( F, G n ) , y i (cid:1) L −−−−→ n →∞ Normal(0 , Σ) , where Σ = Cov ( x,y ) ∼ D f (cid:0) s x ( F, κ ) , y (cid:1) . The result still holds if s i ( F, G n ) is replacedby a vector of rooted densities, and admits a natural Berry-Esseen type extension(Proposition G.4.)Proof. The proof proceeds by approximating f ( s i ( F, G n ) , y i ) by f ( s x i ( F, κ ) , y i )and E f ( s i ( F, G n ) , y i ) by E ( x,y ) ∼D f ( s x ( F, κ ) , y ), and then showing that theerrors are negligible, allowing us to conclude. To simplify notation, we willwrite s i for s i ( F, G n ), s x i for s x i ( F, κ ), and x for { x i } i ∈ [ n ] , therefore making thedependence in n implicit. Further, let K be the Lipschitz constant for all f ( · , y ), y ∈ Y .We first approximate f ( s i , y i ) by f ( s x i , y i ). To this end we observe that the s i − s x i form an exchangeable array, and therefore are identically distributed.This prompts us to introduce the δ i -s, independent copies of the s i − s x i , so21hat the δ i -s are i.i.d., and the s x i + δ i are i.i.d. and of the same (marginal)distribution as the s i -s. This allows us to write:1 n (cid:88) i ∈ [ n ] f ( s i , y i ) − E f ( s i , y i ) =1 n (cid:88) i ∈ [ n ] f ( s x i + δ i , y i ) − E f ( s i , y i ) (cid:124) (cid:123)(cid:122) (cid:125) A n + 1 n (cid:88) i ∈ [ n ] f ( s i , y i ) − f ( s x i + δ i , y i ) (cid:124) (cid:123)(cid:122) (cid:125) B n . (B.1)We observe that, under our assumptions, the random variable A n is the sumof centered independent and identically distributed random variables with finitesecond moments. Therefore, we can apply the Lindeberg-L´evy central limittheorem (as Var s x > √ nA n L −−−−→ n →∞ Normal(0 , Σ) . (B.2)We now show that √ nB n → p
0, which will complete the proof. To do sowe consider increasingly complex maps f : first we consider f the projection onits first argument, then f Lipschitz but independent of y i , before treating thegeneral case with s i being multivariate. As we did in the proof of Theorem 1,we note here that ˆ ρ = ( e ( G n ) /e ( K n )) = (1 + O (1 /n )) ρ from [9, Theorem 1], andin most cases we may replace ˆ ρ by ρ without impacting our bounds. Sum of rooted subgraph counts:
In this case, we set f ( s i , y i ) = s i , sothat B n = n (cid:80) i ( s i − s x i − δ i ). By construction and Theorem 1 (which appliesas γ > m ), the δ i -s are i.i.d., of mean 0, and second moments tending to 0.Therefore we can apply the Lindeberg-L´evy central limit theorem, and obtainthat √ n (cid:80) i δ i → p n (cid:80) i ( s i − s x i ) → p
0. We use the law oftotal variance to obtain:Var (cid:32) √ n (cid:88) i s i − s x i (cid:33) = E Var (cid:32) √ n (cid:88) i s i − s x i (cid:12)(cid:12)(cid:12) x (cid:33) + Var (cid:32) √ n (cid:88) i E (cid:2) s i − s x i | x (cid:3)(cid:33) (B.3)We now consider each term of (B.3) in succession.For the first term in (B.3) we use that the s x i are constant conditionally on x and that the s i form an exchangeable array to write for any i (cid:54) = j ∈ [ n ] E Var (cid:32) √ n (cid:88) i s i − s x i (cid:12)(cid:12)(cid:12) x (cid:33) = E Var (cid:0) s i | x (cid:1) + n E Cov (cid:0) s i , s j | x (cid:1) . (B.4)22y Theorem 1 (which applies as γ > m ), the first term, the variance, tends to0. Remains the covariance. There, by the same steps we used to prove firstLemma A.1, and then Proposition A.1, this covariance is driven by the numberof overlapping copies of F rooted at i and j respectively. Here, because of theconditioning, edges form independently, and all terms that do not overlap at leastover one edge are independent. Thus, similarly as in (A.11), and as in Bickelet al. [9, Eq. 6.3] and Nowicki and Wierman [46, pp. 308–309], we have E Cov (cid:0) s i , s j | x (cid:1) = O (cid:88) H ⊂ [ F,v ] , v (cid:54)∈ H, | e ( H ) |≥ n −| H | ρ − e ( H ) . Since nρ γ = ω (1), we have that n −| H | ρ − e ( H ) = O (cid:0) /n ( nρ γ ) (cid:1) = o ( n − ) for all H in the sum. As the sum is finite, we have E Var (cid:32) √ n (cid:88) i s i − s x i (cid:12)(cid:12)(cid:12) x (cid:33) = o (1) . (B.5)We now consider the second term in (B.3), (cid:80) i E [ s i − s x i | x ] / √ n . As E [ s i − s x i | x ] = E [ s i | x ] − s x i (since s x i = E [ (cid:81) pq ∈ F κ ( x p , x q ) | x i = x ] is constantconditionally on x ,) we have to two terms to manipulate: (cid:80) i E [ s i | x ] and (cid:80) i s x i .To proceed we now show that n − (cid:80) i E [ s i | x ] is a U -statistic, and that the s x i are it’s Hajek projections [22, Definition 15.2], which will yield the result by [22,Theorem 15.1]. To see this, we observe that from Definition 3 and (A.15), wehave s i = (1 + O p (1 /n )) ρ − e ( F ) X F ( G, i ) /X F ( K n , i )= (1 + O p (1 /n )) ρ − e ( F ) aut( F ) (cid:0) ( n − | F |− (cid:1) − (cid:88) H ⊂ K n : [ H,i ] ≡ F I { H ⊂ G } . As for any H in the sum, E [ I { H ⊂ G } | x ] = ρ e ( F ) (cid:81) pq ∈ H κ ( x p , x q ), summingover i , we obtain: n − (cid:88) i E [ s i | x ] = n − (1 + O p (1 /n )) (cid:0) ( n − | F |− (cid:1) − (cid:88) i ,...,i | F | ∈ [ n ] (cid:89) pq ∈ F κ ( x i p , x i q )= (1 + O p (1 /n )) (cid:18) n | F | (cid:19) − (cid:88) i <... √ n (cid:32) n (cid:88) i E [ s i | x ] − n (cid:88) i s x i (cid:33) → . (B.6)23ogether, (B.3), (B.5), and (B.6) yield that √ nB n = o p (1) , (B.7)which is the sought-after result. The univariate case:
Here we consider the case where f ( s i , X i ) does notdepend on Y i but is K -Lipschitz in its first argument. Then, by the triangularinequality, (cid:12)(cid:12) √ nB n (cid:12)(cid:12) ≤ √ n (cid:88) i ∈ [ n ] (cid:12)(cid:12) f ( s i ) − f ( s x i + δ i ) (cid:12)(cid:12) ≤ K √ n (cid:88) i ∈ [ n ] (cid:12)(cid:12) s i − s x i − δ i (cid:12)(cid:12) . (B.8)There, as E B n = 0 by construction, and writing ∆ i = s i − s x i − δ i , parallelingthe previous segment of this proof, we have: (cid:12)(cid:12) √ nB n (cid:12)(cid:12) = O p (cid:16)(cid:113) Var | ∆ i | + n Cov( | ∆ i | , | ∆ j | ) (cid:17) . Now, in the previous section we have shown that (cid:80) i ∆ i / √ n → p
0, and thereforewe also have proved that Var ∆ i + n Cov(∆ i , ∆ j ) →
0. As Var | ∆ i | = O (Var ∆ i )and Cov( | ∆ i | , | ∆ j | ) = O (cid:0) Cov(∆ i , ∆ j ) (cid:1) we obtain that (cid:12)(cid:12) √ nB n (cid:12)(cid:12) ≤ K √ n (cid:88) i ∈ [ n ] (cid:12)(cid:12) s i − s x i − δ i (cid:12)(cid:12) = o p (1) , (B.9)which is the sought after result. For the covariance bound, while the result isintuitive in the sense that covariations are constrained by the absolute value,it can also be seen as an application of Cuadras [20, Theorem 1], where it isshown that f : R → R with a bounded derivative and for two real valued randomvariables U , V such that (cid:0) f ( U ) , f ( V ) (cid:1) has a bounded second moment, we havethat Cov( f ( U ) , f ( V )) < O (cid:0) Cov(
U, V ) (cid:1) . The general case:
In this case we have F = ( F , . . . , F k ) for k ≥
1, and wenaturally expand s i , s x i to being k -variate vectors. Using the subadditivity ofthe square root, we observe that (cid:107)√ nB n (cid:107) ≤ K √ n (cid:88) i ∈ [ n ] (cid:107) s i − s x i − δ i (cid:107) ≤ K (cid:88) t ∈ [ k ] √ n (cid:88) i ∈ [ n ] (cid:12)(cid:12) s i ( F t , G ) − s x i ( F t , κ ) − δ it (cid:12)(cid:12) . Then, (B.9) yields that √ nB n = o p (1) in that case as well. Remark B.2. If Var s x = 0 , we have √ nA n → p and the leading term is B n .Therefore, we do not converge to a Normal, but rather towards a mass at 0.However, [22, Theorem 15.6] applies on the U -statistic component of B n , andwe conjecture that nB n tends in distribution to a mixture of χ distributions.Hladk´y et al. [27, Theorem 1.1.c] shows that this conjecture holds at least forcliques. Proving this conjecture would require tighter bounds in (B.5) specific tothis regime. PPENDIX C: Material for Section 4
APPENDIX C.1: Estimating κ We use maximum likelihood to estimate κ . As doing so requires to perform an optimization over the space of measurepreserving transformations of [0 , . Furthermore, in using [47], we made thefollowing augmentations: i) the approach relies on a rule of thumb to determinethe number of blocks ˆ b to use, we considered all number of blocks in [ . b, . b ]; ii)for each considered number of blocks, as the algorithm is randomized, we performthe estimation 10 times and kept the estimate with the maximal likelihood; iii)when comparing likelihoods from estimates with different number of blocks, asthe model is then parametric, we use information criterions to correct for thedifference in number of parameters (here AIC). Algorithm 1:
Block-kernel estimator.
Input:
A graph G Output:
A kernel estimate ˆ κ and block assignments { P b } b ∈ [ k ] Use Louvain algorithm to organize G ’s vertices in blocks. Call k theensuing number of blocks. For each b ∈ [ k ], let P b be the set of vertices inblock labeled b ; Set ˆ B ∈ [0 , k × k , such that ∀ ( b, b (cid:48) ) ∈ [ k ] , ˆ B bb (cid:48) = { ij ∈ G : i ∈ P b , j ∈ P b (cid:48) } / | P b | ( | P b (cid:48) | − I { b = b (cid:48) } ); Set ˆ σ : (0 , → [ k ], such that ∀ u ∈ (0 , , ˆ σ ( u ) = (cid:80) b ∈ [ k ] I (cid:8) u > (cid:80) b (cid:48) ≤ b | P b (cid:48) | /n (cid:9) ; Set ˆ κ : (0 , → [0 , ∀ u, v ∈ (0 , , ˆ κ ( u, v ) = ˆ B ˆ σ ( u ) , ˆ σ ( v ) ; APPENDIX C.2: Producing the ˆ t i -s From Theorem 1 we know that if( G n ) n> ∼ G ( ρ, κ ), and vertex i has latent position x , then there is a sequenceΣ x ( n, κ ) such that t i ( G n , x, κ ) = Σ x ( n, κ ) − / (cid:0) s i ( , G n ) − s x ( , κ ) , s i ( , G n ) − s x ( , κ ) (cid:1) is asymptotically bivariate standard normal. We estimate s x ( , κ ), s x ( , κ ),and Σ x ( n, κ ) using bootstrap. In doing so, we observe that since ˆ κ is a step-function, the densities are step-function themselves, and we may average ourestimates across blocks, as well as index them by block label rather than x . SeeAlgorithm 2. We call ˆ t i ( G ) the estimator of t i resulting from using Appendix C.1and Algorithm 2. We observe through simulation that the ˆ t i -s are already closeto normal for n = 5000 (see Fig. C.3.) Using Olhede and Wolfe [47]’s code: github.com/p-wolfe/network-histogram-code . emark C.3. The parametric bootstrap is computationally expensive, as itrequires to count subgraphs in each replicate. However, as Σ x ( n, ρκ ) depends onthe decay rate of ρ , computing Σ x ( n, ρκ ) directly would require assuming onesuch rate. The parametric bootstrap avoids this assumption. However, averagingacross vertices in each block, as in Algorithm 2, improves precision by a factor oforder / √ n , and only few replicates are sufficient. An alternative is to evaluatethe moments directly though H F and associated s x ( H, κ ) , as in Remark A.1and [40]. In that case, given H F , the computational complexity will be that ofevaluating k | F |− integrals over [0 , | F | . Algorithm 2:
Estimator of rooted count distribution parameter.
Input:
A blockmodel κ , a network size n , and a number of replicates N Output: ˆ s b ( , κ ), ˆ s b ( , κ ) and (cid:98) Σ b ( n, κ ) for b ∈ [ k ] Set k to be the number of blocks in κ ; For i ∈ [ n ], set x i = i/ ( n + 1), and for b ∈ [ k ] let P b be the set of verticesin block b ; Generate G , . . . , G N , N independent realizations of G ( n, κ ) |{ x i } i ∈ [ n ] ; Compute the s i ( , G t ) and s i ( , G t ) for i ∈ [ n ] and t ∈ [ N ]; Compute for each b ∈ [ k ] ˆ s b ( , ˆ κ ) = N (cid:80) Nt =1 1 | P b | (cid:80) i ∈ P b s i ( , G t ) , ˆ s b ( , ˆ κ ) = N (cid:80) Nt =1 1 | P b | (cid:80) i ∈ P b s i ( , G t ) , (cid:98) Σ b ( n, ˆ κ ) = N − (cid:80) Nt =1 1 | P b | (cid:80) i ∈ P b (cid:0) s i ( , G t ) − ˆ s b ( , ˆ κ ) (cid:1)(cid:0) s i ( , G t ) − ˆ s b ( , ˆ κ ) (cid:1) ; APPENDIX C.3: Critical value
Our proofs imply that the convergence ofTheorem 1 is not uniform: higher order moments, while converging at the samerate, exhibit exponentially larger constant factors. Thus, extremal quantilescannot be expected to have converged when performing analysis, which weconfirm through simulation in Fig. C.3. We note here that [57] suggests thatconvergence may become uniform for dense enough random graph sequences G .We therefore adapt the critical value of the test, which we do throughbootstrap. Specifically, while for a given level α and using Bonferroni correctionthe asymptotic critical value is F − χ (1 − α/n ), we instead use (1 − α )-th quantileof max i ∈ [ n ] ,r ∈ [ R ] { ˆ t i ( G r ) } (where the { G r } r ∈ [ R ] are random graph replicates.)We present in Fig. C.3 an example of how the critical value is augmented. Finally,using simulation we confirm that doing so preserves both the level and power ofthe test. APPENDIX D: Proof of Proposition A.1
In what follows, we deconstruct Proposition A.1 in three results that we provesuccessively. This allows us to put forward one key intermediate result: an26 Theoretical Quantiles E m p i r i c a l Q uan t il e s -10 -5 - - Triangle-Square First Component T r i ang l e - S qua r e S e c ond C o m ponen t -10 -5 - - Triangle-Square First Component T r i ang l e - S qua r e S e c ond C o m ponen t Figure C.3: Simulation experiments.
Left:
QQ-Plot of realized distributionof standardized rooted counts against theoretical distribution. We simulate { G l } l ∈ [200] , each of order n = 5000 from κ a block model with 3 blocks of equalsizes and sparsity ρ = n − / . From { G l } l ∈ [200] we produce the { t ( G l ) } l ∈ [200] .We present the QQ-plot of {(cid:107) ˆ t ( G l ) (cid:107) } l against the χ (2) distribution. We do notobserve significant departure from the χ (2) distribution. Center:
Exemplarmovement of critical value. We use the same set-up as in the leftmost plot, butwith n = 200. Crosses are the { ˆ t i ( G ) } i ∈ [200] . The green circle is the asymptoticcritical value ( F − χ (2)(1 − α/n )), in red is the estimated one. With the estimatedcritical value, we recover the selected level of the test; i.e., we perform the teston G to G , and we cannot reject the null that the rejection probability is α . Right:
Triadic closure and test power. We use the same set-up as in thecentral plot. Then, we enforce 5% triadic closure on G (we closed 5% of theopened triangles attached to 5% of the vertices.) The plot is structured as thecentral one, and we observe two points outside the confidence zone. The observedrejection rate across graph replicates is not significantly different from 1.homomorphism property for rooted counts. This property mirrors results thathave proved central to the study of subgraph counts at the global scale (see Lov´asz[36, Equation 2, p964], Bollob´as and Riordan [12, Equation 1, p217] and Lov´asz[37, Equation 7.7, p117].) Lemma D.2.
Fix a rooted graph F and let G ∼ G ( n, ρκ ) | x i = x . Then E X F = s x ( F, κ ) ρ e ( F ) X F ( K n , i ) , (C.1) X F = O p (cid:16) n | F |− ρ e ( F ) (cid:17) . (C.2) where s x ( F, κ ) = E (cid:2) (cid:81) pq ∈ F κ ( x p , x q ) | x i = x (cid:3) . The s x ( F, κ ) notation in consistent with the literature: it uses the ‘ s ’ of [12]to denote an embedding density, and the ‘ x ’ subscript of [39] to denote thedensity of a partially labelled graph. Proof of Lemma D.2.
We prove both statements in succession. We start byevaluating the mean. To do so, we write the number of copies of F rooted at i G as follows: X F = { F (cid:48) ⊂ G : i ∈ F (cid:48) , [ F (cid:48) , i ] ≡ F } = (cid:88) i ∈ F (cid:48) ⊂ K n [ F (cid:48) ,i ] ≡ F I { F (cid:48) ⊂ G } . (C.3)We now exploit the conditional independence of edge occurrences, through thelaw of iterated expectation, to compute E I { F (cid:48) ⊂ G } . We obtain that for any[ F (cid:48) , i ] ≡ F , E I { F (cid:48) ⊂ G } = E (cid:89) pq ∈ F (cid:48) I { pq ∈ G } = E E (cid:89) pq ∈ F (cid:48) I { pq ∈ G } (cid:12)(cid:12)(cid:12) { x t } t ∈ F (cid:48) \ i = E (cid:89) pq ∈ F (cid:48) ρκ ( x p , x q ) (cid:12)(cid:12)(cid:12) x i = x = s x ( F, κ ) ρ e ( F ) . (C.4)In this last expression, we observe that E I { F (cid:48) ⊂ G } does not depend on thechosen copy F (cid:48) , but is a function of x , F , κ and ρ . Therefore, from (C.3)and (C.4), using the linearity of the expectation operator we obtain that E X F = s x ( F, κ ) ρ e ( F ) X F ( K n , i ) , which recovers (C.1), and therefore the first half of the sought-after result.We now consider the bound in probability of (C.2). First, denoting aut( F )the cardinality of the root preserving automorphism group of F as in Definition 3,we have X F ( K n , i ) = ( n ) | F |− aut( F ) = Θ (cid:16) n | F |− (cid:17) , as aut( F ) is fixed and greater than 0. Then, as s x ( F, κ ) > E X F = Θ (cid:0) n | F |− ρ e ( F ) (cid:1) , and we may fix M > n such that ∀ n ≥ n , E X F ≤ M n | F |− ρ e ( F ) . Finally, since 0 ≤ X F for all n , we can use Markov’s inequality and obtain thatfor any (cid:15) < ∀ n ≥ n , P (cid:34) X F n | F |− ρ e ( F ) ≥ (cid:15) − M (cid:35) ≤ E X F ( G, i ) /n | F |− ρ e ( F ) (cid:15) − M ≤ (cid:15), which is equivalent to X F = O p (cid:0) n | F |− ρ e ( F ) (cid:1) , the sought-after result.The key consequence of Lemma D.2 is that the order of magnitude of X F is driven solely by the number of vertices and edges in F , paralleling resultspresented [23] and [53] for the Erd˝os-R´enyi model. This translates into setoperations on rooted graphs presenting good properties in terms of order ormagnitude of number of rooted copies. We present two such properties in thefollowing propositions. 28 roposition D.2. Set F , F to be two rooted graphs sharing their root andlet G ∼ G ( n, ρκ ) | x i = x . Then, X F ∪ F = O p (cid:0) E X F E X F / E X F ∩ F (cid:1) , and if F ⊂ F , P [ X F >
0] = O (cid:0) E X F (cid:1) .Proof. We prove both results in succession. Let us begin with the first assertion,and write using Lemma D.2 and the classical equality | F ∪ F | = | F | + | F | −| F ∩ F | , X F ∪ F = O p (cid:16) n ( | F | + | F |−| F ∩ F | ) − ρ e ( F )+ e ( F ) − e ( F ∩ F ) (cid:17) = O p (cid:32) n | F |− ρ e ( F ) n | F |− ρ e ( F ) n | F ∩ F |− ρ e ( F ∩ F ) (cid:33) = O p (cid:18) E X F E X F E X F ∩ F (cid:19) , which is the sought-after result.We now turn to the second assertion and first write that P [ X F >
0] = P [ X F ≥ F ⊂ F , if there is a copy of F , there must also beone of F , so that P [ X F ≥ ≤ P [ X F ≥ P [ X F > ≤ P [ X F ≥ ≤ E X F , which is sufficient to obtain thesought-after result.Proposition D.2 shows that if any rooted subgraph of F has a vanishingexpected number of rooted copies, then X F = 0 with high probability (gener-alizing Ruci´nski and Vince [54, Lemma 1]). Conversely, Proposition D.2 alsoshows that if all subgraphs of both F and F have diverging expected numberof rooted copies, then X F ∪ F = o p ( E X F E X F ) when | F ∩ F | > m ( F ): m ( F ) = max { e ( H ) / | H | − v ∈ H ⊂ F, | H | > } . Now observe that for each v ∈ H ⊂ F , n | H |− ρ e ( H ) = ω (1) if and only if nρ e ( H ) / ( | H |− = ω (1). Furthermore, as ρ ≤
1, we have that nρ e ( H ) / ( | H |− ≥ nρ m ( F ) . Therefore, similarly as in Ruci´nski and Vince [54, Lemma 1], we observea type of zero-one law: if nρ m ( F ) = o (1), then X F is almost surely equal to zero,while if nρ max( m ( F ) ,m ( F )) = ω (1), then the number of rooted copies of elementsof H F ,F is dominated by E X F E X F .These results, in conjunction with Lemma A.1, will now allow us to provethat rooted graph counts verify a partial homomorphism property (the productof rooted subgraph counts approximately maps to the rooted subgraph countof their gluing product; see definition just below and Fig. A.2 for examples.)Ultimately, this implies the concentration of each count to its expectation for ρ large enough. This result generalizes Lov´asz [36, Equation 2, p964], Bollob´asand Riordan [12, Equation 1, p217] and Lov´asz [37, Equation 7.7, p117].29 efinition D.3 (Gluing Product [37, Section 4.2]) . We call gluing product oftwo rooted graphs F , F —and write F F —the rooted graph obtained by takingthe disjoint union of F and F , and then merging the two root vertices. Wewrite F k for the k -fold gluing product of F with itself. Proposition D.3.
Fix two rooted graphs F , F . Set G ∼ G ( n, ρκ ) | x i = x and (cid:15) F = 1 / min (cid:8) n | H |− ρ e ( H ) : v ∈ H ⊂ F (cid:9) . Then, (cid:15) F = O (cid:0) /nρ m ( F ) (cid:1) and (cid:40) X F X F = c F F X F F (cid:0) O p ( (cid:15) F F ) (cid:1) if nρ m ( F F ) = ω (1) , X F / E X F = 1 + O p ( (cid:15) F ) if nρ m ( F ) = ω (1) .Proof. We expand X F X F and then obtain the two claimed statements by iden-tifying the leading terms in the obtained expression. We start from Lemma A.1,that we recall for the reader’s convenience, X F X F = (cid:88) H ∈H F ,F c H X H . Then, we use Definition D.3 to split the sum in two terms as follows: X F X F = c F F X F F + (cid:88) H ∈H F ,F \{ F F } c H X H . (C.5)We now use (C.5) to prove that X F X F = c F F X F F (cid:0) O p ( (cid:15) F F ) (cid:1) .To control the sum on the right, we first fix H ∈ H F ,F \ { F F } and tworooted graphs F (cid:48) and F (cid:48) such that H = F (cid:48) ∪ F (cid:48) . Then, we note that byconstruction max( m ( F ) , m ( F )) = m ( F F ). Therefore, the assumption that nρ m ( F F ) = ω (1) along with Proposition D.2 directly implies that X H = O p (cid:32) E X F (cid:48) E X F (cid:48) E X F (cid:48) ∩ F (cid:48) (cid:33) = O p (cid:0) (cid:15) F F E X F E X F (cid:1) . (C.6)Now, recall that X F ( K n ) = ( n ) | F |− / aut( F ), with ( n ) k = n ( n − · · · ( n − k + 1)the falling factorial, so that from Lemma D.2 we obtain E X F E X F = (cid:32) ( n ) | F |− aut( F ) ρ e ( F ) s x ( F , κ ) (cid:33)(cid:32) ( n ) | F |− aut( F ) ρ e ( F ) s x ( F , κ ) (cid:33) = n | F | + | F |− aut( F ) aut( F ) ρ e ( F )+ e ( F ) (cid:16) O (cid:16) n − (cid:17)(cid:17) E (cid:89) pq ∈ F κ ( x p , x q ) (cid:12)(cid:12)(cid:12) x i = x E (cid:89) pq ∈ F κ ( x p , x q ) (cid:12)(cid:12)(cid:12) x i = x . We now use the definition of the gluing product—especially that by construction | F F | = | F | + | F | − e ( F F ) = e ( F ) + e ( F )—to factorize the twoexpectations, and obtain E X F E X F = n | F F |− aut( F ) aut( F ) ρ e ( F F ) (cid:16) O (cid:16) n − (cid:17)(cid:17) (cid:89) pq ∈ F F κ ( x p , x q ) (cid:12)(cid:12)(cid:12) x i = x = aut( F F )aut( F ) aut( F ) ( n ) | F F |− aut( F F ) ρ e ( F F ) s x ( F F , κ ) (cid:16) O (cid:16) n − (cid:17)(cid:17) = aut( F F )aut( F ) aut( F ) s x ( F F , κ ) ρ e ( F F ) X F ( K n ) (cid:16) O (cid:16) n − (cid:17)(cid:17) = c F F E X F F (cid:16) O (cid:16) n − (cid:17)(cid:17) . (C.7)Together (C.5), (C.6) and (C.7) imply that X F X F = c F F X F F (cid:0) O p ( (cid:15) F F ) (cid:1) , (C.8)which is the sought-after result.We now use (C.8) to prove that X F = E X F (cid:0) O p ( (cid:15) F ) (cid:1) . We do so usingthe second moment method. We begin by replacing F by F in both (C.7)and (C.8), to obtain E X F E X F = c F F E X F F (cid:0) O ( n − ) (cid:1) X F X F = c F F X F F (cid:0) O p ( (cid:15) F ) (cid:1) ⇒ (cid:40)(cid:0) E X F (cid:1) = c F F E X F (cid:0) O ( n − ) (cid:1) X F = c F F X F (cid:0) O p ( (cid:15) F ) (cid:1) . Therefore, we haveVar (cid:0) X F (cid:1) = E X F − (cid:0) E X F (cid:1) = O (cid:16) (cid:15) F (cid:0) E X F (cid:1) (cid:17) . (C.9)Then, since by Lemma D.2 E X F >
0, we can use Chebyshev’s inequality toobtain that for any (cid:15) > P (cid:20)(cid:12)(cid:12)(cid:12)(cid:12) X F E X F − (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:15) (cid:21) ≤ (cid:15) − Var (cid:0) X F (cid:1)(cid:0) E X F (cid:1) = O (cid:0) E (cid:15) F (cid:1) = O (cid:0) /nρ m ( F ) (cid:1) = o (1) . Therefore X F / E X F → p
1, which is the sought-after result.Proposition D.3 fully describes the distributional properties of rooted countsin the limit of large inhomogeneous random graphs: Rooted counts concentrateto their means, and their interactions concentrate to the corresponding gluingproduct. Further, these concentrations take place uniformly in x , and at leastat rate nρ m ( F ) . This parallels known results for the total, unrooted count inErd˝os-R´enyi random graphs [53]. APPENDIX E: Proof of Theorem 1
Here we address the multivariate case, resuming from the end of the proofof Theorem 1 in Appendix A. We consider F = ( F , . . . , F k ) a k -tuple of rooted31raphs and still call s i ( F, G ) the corresponding k -variate vector of densities.From (A.16) we know that each entry of s i ( F, G ) is asymptotically Normal.Remains to show that the joint distribution is also Normal. To do so, we willshow convergence in moments, since it is here also sufficient to conclude [33].Fix r = ( r , . . . , r k ) ∈ N k . By the same arguments as above, and cruciallyby Theorem A.1, we have with Σ the correlation matrix of s i ( F, G ) (which existsbecause s i ( F, G ) is bounded with high probability) and [ rk ] the multiset whereeach t ∈ [ k ] is repeated r t times, if (cid:80) t r t is even, E (cid:81) kt =1 (cid:20) s i ( F,G ) − E s i ( F,G ) √ Var( s i ( F,G )) (cid:21) r t → (cid:80) s ∈ [ rk ] (2) (cid:81) { p,q }∈ s Σ pq , otherwise, E (cid:81) kt =1 (cid:20) s i ( F,G ) − E s i ( F,G ) √ Var( s i ( F,G )) (cid:21) r t → . Thus, s i ( F, G ) converges in moments to Normal(0 , Σ), which completes the proof.
APPENDIX F: Application of Theorem 2
In Remark 1 we claim that:
Theorem 2 shows that when using ( y i , s i ( F, G )) i to fit a probabilistic model,statistical inference proceeds as if one were using the i.i.d. sample ( y i , s x i ( F, κ )) i .To see this, fix a smooth log-likelihood function l θ ( · , · ) , and set ˆ θ G = argmax θ (cid:88) i l θ ( Y i , s i ( F, G )) , ˆ θ κ = argmax θ (cid:88) i l θ ( Y i , s x i ( F, κ )) . Then, Theorem 2 shows that ˆ θ G and ˆ θ κ converge at the same rate and towardthe same value. Further, the asymptotic variance of ˆ θ G may be computed in thestandard way, using the Fisher information matrix. These claim hold if Theorem 2 holds uniformly in θ , or at least in a neigh-borhood of some θ , where θ is the true parameter. As is transparent from theproof of Theorem 2, the bound depend only on n and K . Therefore, for thisconvergence to occur we need there to be a finite constant K such that for all θ ,the bivariate maps l θ ( · , · ) and ∂l θ /∂θ (cid:0) · , · ) are K -Lipschitz in the neighborhoodof θ . APPENDIX G: Extensions to Theorem 2
Here we present a natural extension to Theorem 2, showing uniform convergenceto the normal distribution for the sum of subgraph counts. We will only presentthe result and proof for the univariate case, and the counts themselves, but anextension to the same setting as Theorem 2 would follow using the same prooftechnique.This following result is surprising in that one would have expected a standardBerry-Esseen theorem to hold, with concentration at the speed of √ n , while32e observe another, larger, error term in (cid:15) n . The cause of this additional termis the dependence between rooted counts. The reasons why this terms is solarge is because the intuition that higher order dependence terms—terms of theform E X p Y q for two variables X and Y and p, q > (cid:15) n , as they remain for all order explained bythe overlap between two copies. Proposition G.4 (Berry-Esseen type result for sum of rooted densities) . With F a rooted graph, G ∼ G ( ρ, κ ) such that E s x ( F, κ ) > and S n = (cid:80) i ∈ [ n ] ( X F ( G n , i ) − E X F ) . Then, if (cid:15) n = max (cid:8) /n | H |− ρ e ( H ) : H ⊂ F (cid:9) = o (1) , we have that: sup t (cid:12)(cid:12)(cid:12) F n − / S n ( t ) − F Normal (0 ,σ ) ( t ) (cid:12)(cid:12)(cid:12) = O (cid:0) √ (cid:15) n (cid:1) . Proof.
As in the proof of Theorem 2, we proceed by splitting S n into twocomponents. However, in this case, it is more convenient to do so implicitlythrough factorising the characteristic function. Here we note that we shall relyon elements of proofs of earlier results in this document, and therefore will notprovide as much details as for the earlier proofs.Let us first write the characteristic function of S n / √ n , which we call φ n : φ n ( t ) = E e iS n t/ √ n = (cid:88) k ≥ ( it/ √ n ) k k ! E S kn . Quite naturally following our earlier proofs, we focus on the moments, and write: E S kn = (cid:88) ≤ k ≤ ... ≤ k n ≤ k ; (cid:80) i k i = k E (cid:89) i ∈ [ n ] (cid:0) X F ( G n , i ) − E X F (cid:1) k i . We now focus on how the dependence between the X F ( G n , i ) inflate these terms,and at what speed this increase in dependence shrinks with n (for this reason weshall not keep dependence in k in the asymptotic analysis to follow.) There weobserve that by the proof of Theorem A.1, (cid:0) X F ( G n , i ) − E X F (cid:1) k i can be thoughof as the count of k i overlapping, copies of F rooted at i . Then, the probabilitythat any of the k i copies of F rooted at i overlaps with any of the k j copies of F rooted at any j (cid:54) = i is of order k i ( (cid:80) j (cid:54) = i k j ) (cid:15) n /n (using the same logic as in theparagraph after (B.4) in the proof of Theorem 2; note that k i ( (cid:80) j (cid:54) = i k j ) (cid:15) n → n i , we obtain that if ∀ i, k i (cid:54) = 1 E (cid:89) i ∈ [ n ] (cid:0) X F ( G n , i ) − E X F (cid:1) k i = (cid:0) O ( (cid:15) n /n ) (cid:1) (cid:89) i ∈ [ n ] E (cid:0) X F ( G n , i ) − E X F (cid:1) k i , else, since by Proposition D.3, X F ( G n , i ) = (cid:0) O p ( (cid:15) n /n ) (cid:1) E X F , we have that( (cid:15) n /n ) − / (cid:0) X F ( G n , i ) − E X F (cid:1) = O (1)33nd therefore E (cid:89) i ∈ [ n ] (cid:0) X F ( G n , i ) − E X F (cid:1) k i = O (cid:0) ( (cid:15) n /n ) k/ (cid:1) . It follows that with S (cid:48) n distributed at the sum of n independent and centredcopies of X F , and since there is order n k terms where at least one k i equals 1,we have: φ n ( t ) = (cid:88) k> ( it/ √ n ) k k ! (cid:16)(cid:0) O ( (cid:15) n /n ) (cid:1) E S (cid:48) kn + O (cid:0) ( n(cid:15) n ) k/ (cid:1)(cid:17) . We therefore recover three terms, one which is the standard one of a sum of i.i.d. random variables, one bounded by (cid:15) n /n , and one bounded by √ (cid:15) n . Then,expressing the distribution function as the Fourier transform of the characteristicfunction, addressing the first term using the Berry-Esseen theorem, we recover:sup t (cid:12)(cid:12)(cid:12) F n − / S n ( t ) − F Normal (0 ,σ ) ( t ) (cid:12)(cid:12)(cid:12) = O (cid:16) n − / + √ (cid:15) n (cid:17) . We obtain the result notting that for any ρ , n − / = O ( (cid:15) n ). Remark G.4 (Donsker extension to Theorem 2) . Proposition G.4 impliesthat in large sparse kernel based random graphs the empirical process of oursstatistic is not tight, so that a Donsker type extensions (as in convergence ofthe interpolated partial sums to a Brownian bridge) does not hold. For densegraphs, as in ρ → c ∈ (0 , , then (cid:15) n is of order n − , and the sequence is tight,so that a Donsker extensions would hold. This is consistent with the intuitionthat the dependence between rooted counts increases with sparsity. However, it isinteresting to observe that any degree of sparsity breaks this convergence. Remark G.5 (Relation to [6]) . Barbour and R¨ollin [6] presents a result relatedto Theorem 2. In [6] averages across vertices of functions of vertex’s neighborhoodare found to be normal in a special case of our null (degree based models, whichcorresponds to rank one kernels) and in the case of a very sparse graph ( nρ → c > . ) There we observe that in such a sparse setting, all but a finite numberof neighborhoods will be trees, and that for F a rooted tree, Theorem 2 holds onlyfor sequences ρ such that nρ → ∞ .Thus, [6] shows that the behavior Theorem 2persists even in sparser regimes. From [9, Theorem 1] we knew the global countto be Normal in the very sparse regime of nρ → c > (with an inflated variancecompared to the denser cases.) Acknowledgements
I gratefully acknowledge that the authors of [47] kindly shared the dataset used inthis work, and provided comments on early versions of this document. Discussions34ith members of UCL Statistical Science Department and C. J. Priebe helpedimprove this document. I thank anonymous referees for their constructivecomments, the Isaac Newton Institute for Mathematical Sciences, Cambridge,for support and hospitality during the program Theoretical Foundations forStatistical Network Analysis (EPSRC grant no. EP/K032208/1) where part ofthis work was undertaken. This work was supported in part by RELx, througha partnership with the UCL Big Data Institute, and by the NCSML PDRAChallenge Best Article prize. Any remaining mistakes are mine alone.
References [1] N. K. Ahmed, J. Neville, R. A. Rossi, N. G. Duffield, and T. L. Willke.Graphlet decomposition: framework, algorithms, and applications.
Knowl.Inf. Syst. , 50(3):689–722, 2017.[2] E. M. Airoldi, D. M. Blei, S. E. Fienberg, and E. P. Xing. Mixed membershipstochastic blockmodels.
J. Mach. Learn. Res. , 9:1981–2014, 2008.[3] T. Ali, T. Rito, G. Reinert, F. Sun, and C. M. Deane. Alignment-freeprotein interaction network comparison.
Bioinformatics , 30:i430–i437, 2014.[4] C. Ambroise and C. Matias. New consistent and asymptotically normalparameter estimates for random graph mixture models.
J. R. Stat. Soc.Ser. B. Stat. Methodol. , 74(1):3–35, 2011.[5] A. Athreya, C. E. Priebe, M. Tang, V. Lyzinski, D. J. Marchette, and D. L.Sussman. A limit theorem for scaled eigenvectors of random dot productgraphs.
Sankhya A , pages 1–18, 2015.[6] A. D. Barbour and A. R¨ollin. Central limit theorems in the configurationmodel.
Ann. Appl. Probab. , 29(2):1046–1069, 2019.[7] A.D Barbour, M. Kar´nski, and A. Ruci´nski. A central limit theorem fordecomposable random variables with applications to random graphs.
J.Comb. Theory. B , 47:125–145, 1989.[8] S. Bhattacharyya and P. J. Bickel. Subsampling bootstrap of count featuresof networks.
Ann. Statist. , 43:2384–2411, 2015.[9] P. J. Bickel, A. Chen, and E. Levina. The method of moments and degreedistributions for network models.
Ann. Statist. , 39:2280–2301, 2012.[10] J. R. Blum, H. Chernoff, M. Rosenblatt, and H. Teicher. Central limittheorems for interchangeable processes.
Canad. J. Math. , 10:222–229, 1958.[11] B. Bollob´as. Threshold functions for small subgraphs.
Math. Proc. Cam-bridge Philos. Soc. , 90(2):197–206, 1981.3512] B. Bollob´as and O. Riordan. Metrics for sparse graphs. In S. Huczynska,J. D. Mitchell, and C. M. Roney-Dougal, editors,
Surveys in Combinatorics2009 , pages 211–287. Cambridge University Press, Cambridge, UK, 2009.[13] B. Bollob´as, S. Janson, and O. Riordan. The phase transition in inhomoge-neous random graphs.
Random Struct. Algorithms , 31(3):3–122, 2007.[14] B. Bollob´as, S. Janson, and O. Riordan. Sparse random graphs withclustering.
Random Struct. Algorithms , 38:269–323, 2011.[15] C. Bordenave, M. Lelarge, and L. Massouli´e. Non-backtracking spectrum ofrandom graphs: Community detection and non-regular Ramanujan graphs.
Ann. Probab. , 46(1):1–71, 2018.[16] S. Chatterjee. Matrix estimation by Universal Singular Value Thresholding.
Ann. Statist. , 43:177–214, 2015.[17] S. Chatterjee and P. Diaconis. Estimating and understanding exponentialrandom graph models.
Ann. Statist. , 41:2428–2461, 2013.[18] H. Chernoff and H. Teicher. A central limit theorem for sums of interchange-able random variables.
Ann. Math. Statist. , 29:118–130, 1958.[19] M. Coulson, R. E Gaunt, and G. Reinert. Poisson approximation of subgraphcounts in stochastic block models and a graphon model.
ESAIM: Probabilityand Statistics , 20:131–142, 2016.[20] C. M. Cuadras. On the covariance between functions.
J. Multivariate Anal. ,81(1):19–27, 2002.[21] P. Cunningham, M. Harrigan, G. Wu, and D. O’Callaghan. Characterizingego-networks using motifs.
Network Science , 1:170–190, 2013.[22] A. DasGupta.
Asymptotic theory of statistics and probability . Springer,Berlin, 2008.[23] P. Erd¨os and A. R´enyi. On random graphs I.
Publicat. Mathemat. , 6:290–297, 1959.[24] C. Gao, Y. Lu, and H. H. Zhou. Rate-optimal graphon estimation.
Ann.Statist. , 43:2624–2652, 2015.[25] F. Harary. The number of linear, directed, rooted, and connected graphs.
Trans. Amer. Math. Soc. , 78:445–463, 1955.[26] K. M. Harris and R. J. Udry. National longitudinal study of adolescent toadult health (add health) wave i, 1994-1995. 2015. doi: 10.15139/S3/11900.URL https://doi.org/10.15139/S3/11900 .[27] J. Hladk´y, C. Pelekis, and M. ˇSileikis. A limit theorem for small cliques ininhomogeneous random graphs. arXiv:1903.10570, 2019.3628] P. D. Hoff. Modeling homophily and stochastic equivalence in symmetricrelational data. In J. Platt, D. Koller, and S. Roweis, editors,
Advancesin Neural Information Processing Systems 20 , pages 657–664. MIT Press,Cambridge, MA, 2007.[29] P. D. Hoff, A. E. Raftery, and M. S. Handcock. Latent space approaches tosocial network analysis.
J. Amer. Statist. Assoc. , 97(460):1090–1098, 2002.[30] V. Isham, J. Kaczmarska, and M. Nekovee. Spread of information andinfection on finite random networks.
Phys. Rev. E , 83:046128, 2011.[31] S. Janson and A. Ruci´nski. Upper tails for counting objects in randomlyinduced subhypergraphs and rooted random graphs.
Ark. Mat. , 49:79–96,2011.[32] B. Karrer and M. E. J. Newman. Random graphs containing arbitrarydistributions of subgraphs.
Phys. Rev. E , 82:066118, 2010.[33] C. Kleiber and J. Stoyanov. Multivariate distributions and the momentproblem.
J. Multivariate Anal. , 113:7–18, 2013.[34] O. Klopp, A. B. Tsybakov, and N. Verzelen. Oracle inequalities for networkmodels and sparse graphon estimation.
Ann. Statist. , 1:316–354, 2016.[35] J. Lei and A. Rinaldo. Consistency of spectral clustering in stochastic blockmodels.
Ann. Statist. , 43:215–237, 2015.[36] L. Lov´asz. The rank of connection matrices and the dimension of graphalgebras.
Europ. J. Combin. , 27(6):962–970, 2006.[37] L. Lov´asz.
Large Networks and Graph Limits . Am. Mathematical Soc.,Providence, RI, 2012.[38] L. Lov´asz and B. Szegedy. Limits of dense graph sequences.
J. Combin.Theory Ser. B , 96:933–957, 2006.[39] L. Lov´asz and B. Szegedy. The automorphism group of a graphon.
J.Algebra , 421:136–166, 2015.[40] P-A. Maugis, C. Priebe, S. Olhede, and P. Wolfe. Statistical inference fornetwork samples. arXiv:1701.00505, 2017.[41] J. C. Miller. Percolation and epidemics in random clustered networks.
Phys.Rev. E , 80:020901(R), 2009.[42] W. E. Moustafa, A. Deshpande, and L. Getoor. Ego-centric graph patterncensus. In
International Conference on Data Engineering (ICDE) . CurranAssociates, Red Hook, NY, 2012.[43] C. M. Newman and A. L. Wright. An invariance principle for certaindependent sequences.
Ann. Probab. , 9:671–675, 1981.3744] M. E. J. Newman. Random graphs with clustering.
Phys. Rev. Lett. , 103:058701, 2009.[45] C. L. Nickel.
Random dot product graphs a model for social networks.
PhDthesis, Johns Hopkins University, Baltimore, MA, USA, 2006.[46] K Nowicki and J. C. Wierman. Subgraph counts in random graphs usingincomplete u -statistics methods. Discrete Math. , 72(1):299–310, 1988.[47] S. C. Olhede and P. J. Wolfe. Network histograms and universality ofblockmodel approximation.
Proc. Natl. Acad. Sci. USA , 111:14722–14727,2014.[48] P. Orbanz. Subsampling large graphs and invariance in networks.arXiv:1710.04217, 2017.[49] N. Privault and G. Serafin. Normal approximation for sums of weighted u -statistics ? application to kolmogorov bounds in random subgraph counting. Bernoulli , 26(1):587–615, 02 2020.[50] N. Prˇzulj. Biological network comparison using graphlet degree distribution.
Bioinformatics , 23:e177–e183, 2007.[51] A. Rinaldo, S. Petrovi´c, and S. E. Fienberg. Maximum likelihood estimationin the β -model. Ann. Statist. , 41:1085–1110, 2013.[52] K. Rohe, S. Chatterjee, and B Yu. Spectral clustering and the high-dimensional stochastic blockmodel.
Ann. Statist. , 39:1878–1915, 2011.[53] A Ruci´nski. When are small subgraphs of a random graph normally dis-tributed?
Probab. Theory Related Fields , 78(1):1–10, 1988.[54] A. Ruci´nski and A. Vince. Strongly balanced graphs and random graphs.
J. Graph Theory , 10(2):251–264, 1986.[55] P. Sarkar and P. J. Bickel. Role of normalization in spectral clustering forstochastic blockmodels.
Ann. Statist. , 43:962–990, 2015.[56] C. Seshadhri, A. Sharma, A. Stolman, and A. Goel. The impossibility oflow-rank representations for triangle-rich complex networks.
Proc. Natl.Acad. Sci. USA , 117:5631–5637, 2020.[57] M. ˇSileikis and L. Warnke. Counting extensions revisited. arXiv:1911.03012,2019.[58] J. Spencer. Counting extensions.
J. Combin. Theory Ser. A , 55(2):247–255,1990.[59] D. L. Sussman, M. Tang, D. E. Fishkind, and C. E. Priebe. A consistentadjacency spectral embedding for stochastic blockmodel graphs.
J. Amer.Statist. Assoc. , 107:1119–1128, 2012.3860] M. Tang, D. L. Sussman, and C. E. Priebe. Universally consistent vertexclassification for latent positions graphs.
Ann. Statist. , 41:1406–1430, 2013.[61] N. C. Weber. A martingale approach to central limit theorems for exchange-able random variables.
J. Appl. Probab. , 17:662–673, 1980.[62] S. J. Young and E. R. Scheinerman. Random dot product graph modelsfor social networks. In
Algorithms and Models for the Web-Graph , pages138–149. Springer, New York, 2007.[63] Y. Zhao, E. Levina, and J. Zhu. Consistency of community detection innetworks under degree-corrected stochastic block models.