Motif-based tests for bipartite networks
MMotif-based tests for bipartite networks
Sarah Ouadah , , Pierre Latouche , Stéphane Robin , ( ) UMR MIA-Paris, AgroParisTech, INRAE, Université Paris-Saclay, 75005 Paris, France( ) MAP5, UMR CNRS 8145, Université de Paris, 75006 Paris, France( ) CESCO, UMR 7204, MNHN - CNRS - UPMC, Paris, France Abstract
Bipartite networks are a natural representation of the interactions between entities fromtwo different types. The organization (or topology) of such networks gives insight to un-derstand the systems they describe as a whole. Here, we rely on motifs which provide ameso-scale description of the topology. Moreover, we consider the bipartite expected de-gree distribution (B-EDD) model which accounts for both the density of the network andpossible imbalances between the degrees of the nodes. Under the B-EDD model, we provethe asymptotic normality of the count of any given motif, considering sparsity conditions.We also provide close-form expressions for the mean and the variance of this count. Thisallows to avoid computationally prohibitive resampling procedures. Based on these results,we define a goodness-of-fit test for the B-EDD model and propose a family of tests for net-work comparisons. We assess the asymptotic normality of the test statistics and the powerof the proposed tests on synthetic experiments and illustrate their use on ecological datasets.
Keywords: bipartite networks; network motifs; goodness-of-fit; network comparison;expected degree distribution
Bipartite interaction networks are used to represent a diverse range of interactions in variousfields such as biology, ecology, sociology or economics. For instance, in ecology, bipartitegraphs depict interactions between two groups of species such as plants and pollinators [seee.g. Simmons et al., 2019b, Doré et al., 2020] or host and parasites [see e.g. Vacher et al.,2008, D’Bastiani et al., 2020], in agroethnology, they may involve interactions betweenfarmers and crop species [see Thomas et al., 2015] and in economics, country-product tradesas signals of the 2007-2008 financial crisis [see Saracco et al., 2016]. Formally, a bipartiteinteraction network can be viewed as a bipartite graph, the nodes of which being individualspertaining to two different groups, and an edge between two nodes being present if these twoindividuals interact. In the sequel, the two types of nodes will be referred to as top nodesand bottom nodes, respectively. Characterizing the general organization of such a network,namely its topology, is key to understand the behavior of the system as a whole.The topology of a network can be studied at various scales. Micro-scale analyses typ-ically focus on the degree of each node, the betweenness of each edge or on the closenessbetween each pair of nodes. On the opposite, macro-scale analysis focus on global proper-ties of the network such as its density or its modularity. The reader may refer to Newman a r X i v : . [ m a t h . S T ] J a n Motifs in the bipartite expected degree model
We consider a bipartite graph = ( , ) with 𝑁 nodes. The set of nodes is = ( 𝑡 , 𝑏 ) ,where 𝑡 = (cid:74) , 𝑚 (cid:75) (resp. 𝑏 = (cid:74) , 𝑛 (cid:75) ) stands for the set of top (resp. bottom) nodes, andthe set of edges is ⊂ 𝑡 × 𝑏 , meaning than an edge can only connect a top node witha bottom node. The total number of nodes is therefore 𝑁 = 𝑛 + 𝑚 . We denote by 𝐺 thecorresponding 𝑚 × 𝑛 incidence matrix where the entry 𝐺 𝑖𝑗 of 𝐺 is 1 if ( 𝑖, 𝑗 ) ∈ , and 0otherwise. The bipartite expected degree (B-EDD) model is defined as follows: { 𝑈 𝑖 } ≤ 𝑖 ≤ 𝑚 iid , 𝑈 ∼ [0 , , { 𝑉 𝑗 } ≤ 𝑗 ≤ 𝑛 iid , 𝑉 ∼ [0 , , (1) { 𝐺 𝑖𝑗 } ≤ 𝑖 ≤ 𝑚, ≤ 𝑗 ≤ 𝑛 indep. | { 𝑈 𝑖 } ≤ 𝑖 ≤ 𝑚 , { 𝑉 𝑗 } ≤ 𝑗 ≤ 𝑛 , 𝐺 𝑖𝑗 | 𝑈 𝑖 , 𝑉 𝑗 ∼ ( 𝜌𝑔 ( 𝑈 𝑖 ) ℎ ( 𝑉 𝑗 ) ) , where 𝑔, ℎ ∶ [0 , → ℝ + , such that ∫ 𝑔 ( 𝑢 ) d 𝑢 = ∫ ℎ ( 𝑣 ) d 𝑣 = 1 and ≤ 𝜌 ≤ .The parameter 𝜌 controls the density of the graph ( 𝔼 𝐺 𝑖𝑗 = 𝜌 ) whereas the function 𝑔 (resp. ℎ ) encodes the heterogeneity of the expected degrees of the top (resp. bottom) nodes.More specifically, denoting 𝐾 𝑖 = ∑ ≤ 𝑗 ≤ 𝑛 𝐺 𝑖𝑗 the degree of the top node 𝑖 , we have that 𝔼 ( 𝐾 𝑖 ∣ 𝑈 𝑖 ) = 𝑛𝜌𝑔 ( 𝑈 𝑖 ) . The symmetric property holds for bottom nodes. Remark 1.
Lovász and Szegedy [2006] and Diaconis and Janson [2008] introduced ageneric model for exchangeable random graphs called the 𝑊 -graph, which is based ona graphon function Φ ∶ [0 , → [0 , . The B-EDD model is a natural extension of the 𝑊 -graph for bipartite graphs with a product-form graphon function Φ( 𝑢, 𝑣 ) = 𝜌𝑔 ( 𝑢 ) ℎ ( 𝑣 ) .The B-EDD model is obviously exchangeable is the sens that the distribution of the incidencematrix 𝐺 is preserved under permutation of the top nodes and/or the bottom nodes. Remark 2.
The B-EDD model can also be seen has an exchangeable bipartite version of theexpected degree sequence model studied in Chung and Lu [2002] and of the configurationmodel from Newman [2003]. Under these two models, the degree of each node is fixed whichmakes them non exchangeable.
Bipartite motifs.
We are interested in the distribution of the count of motifs (or sub-graphs) in bipartite graphs arising from the B-EDD model. A bipartite motif 𝑠 is defined byits number of top nodes 𝑝 𝑠 , its number of bottom nodes 𝑞 𝑠 and a 𝑝 𝑠 × 𝑞 𝑠 incidence matrix 𝐴 𝑠 . Figures 7 and 8 display the 44 bipartite motifs involving between two and six nodes,from which we see that 𝐴 = ( ) , 𝐴 = ( ) , 𝐴 = ( ) . An important characteristic of a graph motif 𝑠 is its number of automorphisms 𝑟 𝑠 [Stark,2001], that is the number of non-redundant permutations of its incidence matrix (see, e.g.section 2.4 in Picard et al. [2008]): 𝑟 𝑠 = |||||{ 𝐴 𝑠𝜎 𝑡 ,𝜎 𝑏 = ( 𝐴 𝑠𝜎 𝑡 ( 𝑢 ) ,𝜎 𝑏 ( 𝑣 ) ) ≤ 𝑢 ≤ 𝑝 𝑠 , ≤ 𝑣 ≤ 𝑞 𝑠 ∶ 𝜎 𝑡 ∈ 𝜎 ( (cid:74) , 𝑝 𝑠 (cid:75) ) , 𝜎 𝑏 ∈ 𝜎 ( (cid:74) , 𝑞 𝑠 (cid:75) )}||||| . (2) ote that, because pairs of permutations ( 𝜎 𝑡 , 𝜎 𝑏 ) yielding the same matrix 𝐴 𝑠𝜎 𝑡 ,𝜎 𝑏 are notcounted twice, we obviously have that 𝑟 𝑠 ≤ ( 𝑝 𝑠 !) × ( 𝑞 𝑠 !) . In many cases, 𝑟 𝑠 turns out tobe much smaller: in particular, 𝑟 𝑠 = 1 for star-motifs, which will be defined later. Wefurther denote by 𝑑 𝑠𝑢 the degree of the top node 𝑢 ( ≤ 𝑢 ≤ 𝑝 𝑠 ) within motif 𝑠 , thatis 𝑑 𝑠𝑢 = ∑ ≤ 𝑣 ≤ 𝑞 𝑠 𝐴 𝑠𝑢,𝑣 . The degree of the bottom node 𝑣 within 𝑠 is defined similarly as 𝑒 𝑠𝑣 = ∑ ≤ 𝑢 ≤ 𝑝 𝑠 𝐴 𝑠𝑢,𝑣 . Motif occurrence.
Counting the occurrences of motif 𝑠 in simply consists in consid-ering all possible of 𝑝 𝑠 (resp. 𝑞 𝑠 ) top (resp. bottom) nodes among the 𝑚 (resp. 𝑛 ) and checkfor each possible automorphism of 𝑠 if an occurrence is observed. More formally, let usdefine the set 𝑠 of possible positions for motif 𝑠 as the Cartesian product of the set of the ( 𝑚𝑝 𝑠 )( 𝑛𝑞 𝑠 ) possible locations with the set of the 𝑟 𝑠 (top, bottom) permutations giving rise toeach of the automorphisms of 𝑠 . So, a position results from the combination of a location with a permutation . Because the graph is bipartite, any position 𝛼 from 𝑠 decomposes as 𝛼 = ( 𝛼 𝑡 , 𝛼 𝑏 ) where 𝛼 𝑡 stands for an ordered list of top nodes and 𝛼 𝑏 for an ordered list ofbottom nodes. The number of positions for motif 𝑠 in is precisely 𝑐 𝑠 ∶= | 𝑠 | = 𝑟 𝑠 ( 𝑚𝑝 𝑠 )( 𝑛𝑞 𝑠 ) . (3)Now, for a given position 𝛼 = ( 𝛼 𝑡 , 𝛼 𝑏 ) ∈ 𝑠 , we define 𝑌 𝑠 ( 𝛼 ) as the indicator for motif 𝑠 tooccur in position 𝛼 : 𝑌 𝑠 ( 𝛼 ) = ∏ 𝑖 ∈ 𝛼 𝑡 ,𝑗 ∈ 𝛼 𝑏 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 . (4) Remark 3.
Note that the occurrence defined by Equation (4) corresponds to an induced occurrence, which means that we consider that a motif 𝑠 is observed at position 𝛼 as soonas all the present edges that are specified by its incidence matrix 𝐴 𝑠 are observed, even ifadditional edges are also observed. In other words, we do not check for the absent edgesspecified by 𝐴 𝑠 . Remark 4.
As opposed to an induced occurrence, an exact occurrence is observed whenboth the presence and the absence of edges are satisfied. The indicator variable corre-sponding to an exact occurrence writes ∏ 𝑖 ∈ 𝛼 𝑡 ,𝑗 ∈ 𝛼 𝑏 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 (1 − 𝐺 𝑖𝑗 ) 𝐴 𝑠𝑖𝑗 . Counting inducedand exact occurrences in a graph is actually equivalent, as these counts are related in adeterministic manner. For example, each exact occurrence of motif 6 corresponds to twoinduced occurrences of motif 5. Motif probability.
The B-EDD model is an exchangeable bipartite graph model in thesense that, for any pair of permutations ( 𝜎 𝑡 ∈ 𝜎 ( (cid:74) , 𝑚 (cid:75) ) , 𝜎 𝑏 ∈ 𝜎 ( (cid:74) , 𝑛 (cid:75) ) ) , we have that ℙ { 𝐺 = { 𝑔 𝑖𝑗 } ≤ 𝑖 ≤ 𝑚, ≤ 𝑗 ≤ 𝑛 } = ℙ { 𝐺 = { 𝑔 𝜎 𝑡 ( 𝑖 ) 𝜎 𝑏 ( 𝑗 ) } ≤ 𝜎 𝑡 ( 𝑖 ) ≤ 𝑚, ≤ 𝜎 𝑏 ( 𝑗 ) ≤ 𝑛 } [see e.g. Lovász andSzegedy, 2006, Diaconis and Janson, 2008, for simple graphs]. For any exchangeable graphmodel, we may define 𝜙 𝑠 as the probability for motif 𝑠 to occur in position 𝛼 = ( 𝛼 𝑡 , 𝛼 𝑏 ) : 𝜙 𝑠 ∶= ℙ ( 𝑌 𝑠 ( 𝛼 ) = 1 ) . Importantly, because the model is exchangeable, this probability does not depend on 𝛼 . tar motifs. We define a star as a bipartite motif 𝑠 for which either 𝑞 𝑠 = 1 or 𝑝 𝑠 = 1 (or both). More specifically, we name top stars (resp bottom stars) motifs for which 𝑝 𝑠 = 1 (resp. 𝑞 𝑠 = 1 ). The top stars in Figures 7 and 8 are motifs 1, 2, 7, 17 and 44, and the bottomstars are motifs 1, 3, 4, 8 and 18. Observe that 𝑟 𝑠 = 1 for all star motifs, that 𝑑 𝑠𝑣 = 1 for all 𝑣 in all top star motifs, and that 𝑒 𝑠𝑢 = 1 for all 𝑢 in all bottom star motifs.Because they will play a central role in the sequel, we adopt a specific notation for theprobability of star motifs, denoting 𝛾 𝑑 the occurrence probability of the top star with degree 𝑑 and 𝜆 𝑑 for the occurrence probability of the bottom star with degree 𝑑 . As a consequence,we have that 𝛾 = 𝜙 , 𝛾 = 𝜙 , 𝛾 = 𝜙 𝛾 = 𝜙 , 𝛾 = 𝜙 , (5) 𝜆 = 𝜙 , 𝜆 = 𝜙 , 𝜆 = 𝜙 𝜆 = 𝜙 , 𝜆 = 𝜙 . Expected count.
Let us now denote by 𝑁 𝑠 the count, that is the number of occurrencesof a motif 𝑠 in a graph . We simply have that 𝑁 𝑠 = ∑ 𝛼 ∈ 𝑠 𝑌 𝑠 ( 𝛼 ) As a consequence, the expected count of 𝑠 in is 𝔼 ( 𝑁 𝑠 ) = 𝑐 𝑠 𝜙 𝑠 . We also define the nor-malized frequency of motif 𝑠 as 𝐹 𝑠 = 𝑁 𝑠 ∕ 𝑐 𝑠 , which is an unbiased estimate of 𝜙 𝑠 . Illustration.
As an illustration, we consider two of the networks studied by Simmonset al. [2019a], which include both plant-pollinator and seed dispersal networks extractedfrom the Web of Life database ( ). More specifically, we considerthe two largest networks of each type, which were first published by Robertson [1929] andSilva [2002], respectively. The plant-pollinator network involves 546 plant species and 1044insects and the seed dispersal network 207 plant species and 110 seed dispersers (birds orinsects). Table 1 gives the counts and the frequency of the star motifs with up to four branch.For the sake of clarity, we will limit ourselves to motifs up to five nodes in the illustrations.Observe that both the counts 𝑁 𝑠 and the number of possible positions 𝑐 𝑠 range over hugeorder of magnitudes. Main property of motif probabilities under B-EDD.
The tests we propose rely onthe comparison between the observed count (or normalized frequency) of a motif, with itstheoretical counterpart under a B-EDD model. More specifically, the motif probabilitieshave a close form expression under the B-EDD model.
Proposition 1.
Under the B-EDD model (1) , we have that 𝜙 𝑠 = 𝑝 𝑠 ∏ 𝑢 =1 𝛾 𝑑 𝑠𝑢 𝑞 𝑠 ∏ 𝑣 =1 𝜆 𝑒 𝑠𝑣 / ( 𝜙 ) 𝑑 𝑠 + . (6) where 𝑑 𝑠 + ∶= ∑ 𝑢 𝑑 𝑠𝑢 = ∑ 𝑣 𝑒 𝑠𝑣 stands for the total number of edges in 𝑠 . 𝑚 = 546 , 𝑛 = 1044 [Robertson, 1929]edge top stars bottom stars 𝑠 𝑐 𝑠 𝑁 𝑠 𝐹 𝑠 −2 −3 −5 −6 −3 −4 −5 seed dispersal: 𝑚 = 207 , 𝑛 = 110 [Silva, 2002]edge top stars bottom stars 𝑠 𝑐 𝑠 𝑁 𝑠 𝐹 𝑠 −2 −3 −4 −4 −3 −4 −4 Table 1: Coefficients 𝑐 𝑠 , counts 𝑁 𝑠 and frequency 𝐹 𝑠 of all star motifs. Top: plant-pollinatornetwork, bottom: seed dispersal network. The motif number 𝑠 refers to Figure 7. Proof.
This follows from the fact that, under B-EDD, the edges are independent condition-ally on the latent coordinates 𝑈 𝑖 and 𝑉 𝑗 defined in (1), which are all independent with respectto one other. Consider an arbitrary position 𝛼 = ( 𝛼 𝑡 , 𝛼 𝑏 ) ; for the sake of clarity, we identifythe elements of 𝛼 𝑡 with (cid:74) , 𝑝 𝑠 (cid:75) and the elements of 𝛼 𝑏 with (cid:74) , 𝑞 𝑠 (cid:75) . We have 𝜙 𝑠 = 𝔼 ( 𝑈 𝑖 ) ≤ 𝑖 ≤ 𝑝𝑠 , ( 𝑉 𝑗 ) ≤ 𝑗 ≤ 𝑞𝑠 ( ℙ { ∏ ≤ 𝑖 ≤ 𝑝 𝑠 , ≤ 𝑣 ≤ 𝑞 𝑠 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 = 1 |||||| ( 𝑈 𝑖 ) ≤ 𝑖 ≤ 𝑝 𝑠 , ( 𝑉 𝑗 ) ≤ 𝑗 ≤ 𝑞 𝑠 }) = 𝔼 ( 𝑈 𝑖 ) ≤ 𝑖 ≤ 𝑝𝑠 , ( 𝑉 𝑗 ) ≤ 𝑗 ≤ 𝑞𝑠 ⎛⎜⎜⎝ ∏ ≤ 𝑖 ≤ 𝑝 𝑠 , ≤ 𝑗 ≤ 𝑞 𝑠 ∶ 𝐴 𝑠𝑖𝑗 =1 𝜌𝑔 ( 𝑈 𝑖 ) ℎ ( 𝑉 𝑗 ) ⎞⎟⎟⎠ = 𝔼 ( 𝑈 𝑖 ) ≤ 𝑖 ≤ 𝑝𝑠 , ( 𝑉 𝑗 ) ≤ 𝑗 ≤ 𝑞𝑠 ( 𝜌 𝑑 𝑠 + ∏ ≤ 𝑖 ≤ 𝑝 𝑠 𝑔 ( 𝑈 𝑖 ) 𝑑 𝑠𝑖 ∏ ≤ 𝑗 ≤ 𝑞 𝑠 ℎ ( 𝑉 𝑖 ) 𝑒 𝑠𝑗 ) = 𝜌 𝑑 𝑠 + ∏ ≤ 𝑖 ≤ 𝑝 𝑠 ( ∫ 𝑔 ( 𝑢 ) 𝑑 𝑠𝑖 d 𝑢 ) ∏ ≤ 𝑗 ≤ 𝑞 𝑠 ( ∫ ℎ ( 𝑣 ) 𝑒 𝑠𝑗 d 𝑣 ) . The result then results from the fact that 𝛾 𝑑 = 𝜌 𝑑 ∫ 𝑔 ( 𝑢 ) 𝑑 d 𝑢, 𝜆 𝑑 = 𝜌 𝑑 ∫ ℎ ( 𝑣 ) 𝑑 d 𝑣, 𝜌 = 𝜙 . (7) ■ An important consequence of Proposition 1 is that, under B-EDD, the motif probabilityof any motif can be expressed in terms of probabilities of star motifs. Figure 1 provides anintuition of this: a motif can be decomposed in terms of top and bottom stars arising fromeach of its nodes.In the sequel, to distinguish the motif probability 𝜙 𝑠 under an arbitrary exchangeablemodel from the probability under the B-EDD model, we will denote by 𝜙 𝑠 the probabilityof motif 𝑠 under B-EDD. Figure 7 provides the list of all 𝜙 𝑠 expressions.
15 2 7 3 3 1 ⚪ ⚪◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻
Figure 1: Decomposition of motif 15 as an overlap of 2 top stars (motifs 2 and 7) and 3bottom stars (motifs 3, 3 and 1). Because each edge is accounted for twice, we get 𝜙 = 𝜙 𝜙 𝜙 𝜙 𝜙 ∕ 𝜙 = 𝜙 𝜙 𝜙 ∕ 𝜙 . Probability estimate under B-EDD.
Proposition 1 suggests a natural plug-in estima-tor for the B-EDD motif probability 𝜙 𝑠 : 𝐹 𝑠 = ∏ 𝑝 𝑠 𝑢 =1 Γ 𝑑 𝑠𝑢 ∏ 𝑞 𝑠 𝑣 =1 Λ 𝑒 𝑠𝑣 𝐹 𝑑 𝑠 + , (8)where Γ 𝑑 (resp Λ 𝑑 ) denotes the normalized frequency of the top (resp. bottom) star motifwith degree 𝑑 . Obviously, Γ 𝑑 (resp Λ 𝑑 ) is an unbiased estimated of 𝛾 𝑑 (resp. 𝜆 𝑑 ). Variance of the count.
We now consider the variance of the count, that is 𝕍 ( 𝑁 𝑠 ) = 𝔼 ( 𝑁 𝑠 ) − 𝔼 ( 𝑁 𝑠 ) , where 𝑁 𝑠 = ∑ 𝛼,𝛽 ∈ 𝑠 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) (9) = ∑ 𝛼 ∈ 𝑠 𝑌 𝑠 ( 𝛼 ) + ∑ 𝛼,𝛽 ∈ 𝑠 ∶ | 𝛼 ∩ 𝛽 | =0 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) + ∑ 𝛼,𝛽 ∈ 𝑠 ∶ 𝛼 ≠ 𝛽, | 𝛼 ∩ 𝛽 | > 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) . When positions 𝛼 and 𝛽 are equal, the product 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) is simply given by 𝑌 𝑠 ( 𝛼 ) , theindicator of the presence of 𝑠 at position 𝛼 . Then, when positions 𝛼 and 𝛽 do not overlap( | 𝛼 ∩ 𝛽 | = 0 ), the product 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) simply indicates that two occurrences of motif 𝑠 occurin position 𝛼 and 𝛽 , which are independent under the B-EDD model. When positions 𝛼 and 𝛽 are different and do overlap ( | 𝛼 ∩ 𝛽 | > ), the product 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) becomes the indicatorof a super-motif, that is a motif made of two overlapping automorphisms of 𝑠 . We denoteby ( 𝑠 ) the set of super-motifs generated by the overlaps of two occurrences of the motif 𝑠 ; Figure 2 provides some examples of super-motifs.An expression similar to (9) can be derived for the covariance between two counts: ℂ ov ( 𝑁 𝑠 , 𝑁 𝑡 ) = 𝔼 ( 𝑁 𝑠 𝑁 𝑡 ) − 𝔼 ( 𝑁 𝑠 ) 𝔼 ( 𝑁 𝑡 ) , where 𝑁 𝑠 𝑁 𝑡 = ∑ 𝛼 ∈ 𝑠 ,𝛽 ∈ 𝑡 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑡 ( 𝛽 ) (10) = ∑ 𝛼 ∈ 𝑠 ,𝛽 ∈ 𝑡 ∶ | 𝛼 ∩ 𝛽 | =0 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑡 ( 𝛽 ) + ∑ 𝛼 ∈ 𝑠 ,𝛽 ∈ 𝑡 ∶ 𝛼 ≠ 𝛽, | 𝛼 ∩ 𝛽 | > 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑡 ( 𝛽 ) . Again, the last term corresponds to occurrences of super-motifs resulting from an overlapbetween an occurrence of motif 𝑠 and an occurrence of motif 𝑡 . We denote by 𝑆 ( 𝑠, 𝑡 ) theset of these super-motifs. We use the strategy described in Picard et al. [2008] to determinethe sets of super-motifs ( 𝑠 ) and ( 𝑠, 𝑠 ′ ) . Observe that these sets do not depend on the bserved networks, so, to alleviate the computational burden, they can be determined andstored once for all.Eq. (9) shows that 𝔼 ( 𝑁 𝑠 ) only depends on 𝔼 ( 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 )) , which is 𝜙 𝑠 when positions 𝛼 and 𝛽 do not overlap and the probability of the corresponding super-motif when theyoverlap. As a consequence, we have that 𝔼 ( 𝑁 𝑠 ) = 𝜅 𝑚,𝑛,𝑠 𝜙 𝑠 + 𝜅 ′ 𝑚,𝑛,𝑠 𝜙 𝑠 + ∑ 𝑆 ∈ ( 𝑠 ) 𝜅 ′′ 𝑚,𝑛,𝑠,𝑆 𝜙 𝑆 , (11)where the 𝜅 𝑚,𝑛,𝑠 , 𝜅 ′ 𝑚,𝑛,𝑠 , 𝜅 ′′ 𝑚,𝑛,𝑠,𝑆 are constants, which depend on the dimensions of the graph,on the motif 𝑠 and on the super-motif 𝑆 . The order of magnitude of 𝜅 𝑚,𝑛,𝑠,𝑆 for large 𝑚 and 𝑛 will be studied in Section 5.1.2.Because super-motifs are actually motifs, their respective occurrence probability 𝜙 𝑆 under B-EDD are given by Proposition 1 as well, so the expectation and the variance of 𝑁 𝑠 under B-EDD can be expressed as functions of the 𝜙 𝑠 and { 𝜙 𝑆 } 𝑆 ∈ ( 𝑠 ) . An estimate 𝐹 𝑆 ofeach 𝜙 𝑆 can be obtained using Eq. (8) in the same way. ⚪ ⚪ ⚪◻ ◻ ⚪ ⚪ ⚪ ⚫ ⚫ ⚫◻ ⬔ ◼ ⚪ ⚪ ⚪ ⚫ ⚫ ⚫ ⬔ ⬔ ⚪ ⚪ ◒ ⚫ ⚫◻ ◻ ◼ ◼ ⚪ ⚪ ◒ ⚫ ⚫◻ ⬔ ◼ | 𝛼 𝑡 ∩ 𝛽 𝑡 | | 𝛼 𝑏 ∩ 𝛽 𝑏 | Figure 2: Some super-motifs from ( 𝑠 ) for motif 𝑠 = 9 (top left) with 𝑝 𝑠 = 3 top nodes and 𝑞 𝑠 = 2 bottom nodes. | 𝛼 𝑡 ∩ 𝛽 𝑡 | (resp. | 𝛼 𝑏 ∩ 𝛽 𝑏 | ): number of top (resp. bottom) nodes sharedby the overlapping positions 𝛼 and 𝛽 . Black: nodes from 𝛼 , white: nodes from 𝛽 , black/white:nodes from 𝛼 ∩ 𝛽 . There are actually | (9) | = 396 such super-motifs of motif 9. Remark 5.
The estimate defined in (8) is only based on empirical quantities (the counts ofstars motifs) and does not depend on any parameter estimation. Especially, the functions 𝑔 and ℎ do not need to be estimated as the frequency of star motifs provides all necessaryinformation about the degree distributions. As a consequence, we may define plug-in esti-mates of the occurrence probability, the expected count and the variance of the count of anymotif under B-EDD. Illustration.
Table 2 compares the empirical frequencies 𝐹 𝑠 of a selection of motifs withtheir respective estimated probability 𝐹 𝑠 . The probability estimates are computed accordingto Equation 8, using the star motifs frequencies Γ 𝑑 and Λ 𝑒 given in Table 1. Observe thatthe difference between the observed frequency 𝐹 𝑠 and their estimated expectation under theB-EDD model 𝐹 𝑠 are of the same order of magnitude, if not smaller, than their estimatedstandard deviations. Asymptotic framework.
We consider a sequence of B-EDD random graphs defined asfollows. { 𝑁 } 𝑁 ≥ is a sequence of independent graphs, where 𝑁 is a B-EDD random graphwith 𝑚 = ⌊ 𝜆𝑁 ⌋ top nodes with 𝜆 ∈ (0 , , 𝑛 = 𝑁 − 𝑚 bottom nodes and parameters 𝜌 𝑁 , ℎ 𝑠 𝐹 𝑠 −5 −5 −6 −7 −8 𝐹 𝑠 −5 −6 −6 −7 −8 √ ̂ 𝕍 ( 𝐹 𝑠 ) −5 −6 −6 −8 −9 seed dispersal 𝑠 𝐹 𝑠 −4 −4 −5 −5 −6 𝐹 𝑠 −4 −4 −5 −5 −6 √ ̂ 𝕍 ( 𝐹 𝑠 ) −4 −5 −5 −5 −6 Table 2: Empirical frequency 𝐹 𝑠 , estimated probability 𝐹 𝑠 and estimated standard-deviation ofthe frequency according to the B-EDD model for a selection of motifs. All estimates are derivedfrom the star motifs frequencies given in Table 1. and 𝑔 , where the sequence { 𝜌 𝑁 } 𝑁 ≥ satisfies 𝜌 𝑁 = Θ( 𝑚 − 𝑎 𝑛 − 𝑏 ) with 𝑎, 𝑏 > . All quantitiescomputed on 𝑁 should be indexed by 𝑁 as well but for the sake of clarity, we will dropthat index in the rest of the paper. This section is devoted to the asymptotic normality of motif frequencies under the B-EDDmodel. More precisely, our first main result states the asymptotic normality of the followingstatistic 𝑊 𝑠 relying on 𝐹 𝑠 the empirical frequency of a given motif 𝑠 in : 𝑊 𝑠 = 𝐹 𝑠 − 𝐹 𝑠 √ ̂ 𝕍 ( 𝐹 𝑠 ) , (12)where 𝐹 𝑠 denotes the estimator of 𝜙 𝑠 defined in (8) and ̂ 𝕍 ( 𝐹 𝑠 ) the one of 𝕍 ( 𝐹 𝑠 ) obtainedby the plug-in of 𝐹 𝑆 ( 𝑆 being any super-motif generated by two occurrences of 𝑠 ) in theexpressions of 𝕍 ( 𝑁 𝑠 ) given in (9)-(11). Theorem 1. If 𝑎 + 𝑏 < 𝑑 𝑠 + , then for all non-star motif 𝑠 and under the B-EDD model, thestatistic 𝑊 𝑠 is asymptotically normal as 𝑚 ∼ 𝑛 → ∞ : 𝑊 𝑠 𝐷 ←→ (0 , . The proof is based on three results given hereafter in Proposition 2, Lemma 1 andLemma 2.
Sketch of proof.
Let first consider the following decomposition of the numerator of 𝑊 𝑠 : 𝐹 𝑠 − 𝐹 𝑠 ∶= 𝐿 𝑠 + 𝐶 𝑠 where 𝐿 𝑠 = 𝐹 𝑠 − 𝜙 𝑠 and 𝐶 𝑠 = 𝜙 𝑠 − 𝐹 𝑠 . Under the null B-EDD model, we show that, ( 𝑖 ) 𝐿 𝑠 ∕ √ 𝕍 ( 𝐹 𝑠 ) is asymptotically normal inProposition 2, it is the leader term, ( 𝑖𝑖 ) 𝐶 𝑠 ∕ √ 𝕍 ( 𝐹 𝑠 ) is negligible in Lemma 1, it is the eminder term. Then, we conclude using Slutsky Theorem Lemma 2 which states that ̂ 𝕍 ( 𝐹 𝑠 )∕ 𝕍 ( 𝐹 𝑠 ) → in probability. ■ Remark 6.
Like 𝐹 𝑠 , 𝑊 𝑠 is only based on empirical quantities, that is 𝑖 ) the empirical fre-quency of motif 𝑠 and 𝑖𝑖 ) the empirical frequencies of the stars motifs forming 𝑠 . The ex-pected frequencies of the supermotifs of 𝑠 involved in ̂ 𝕍 ( 𝐹 𝑠 ) also depend only on empiricalstar frequencies. Remark 7.
Gao and Lafferty [2017] proved a similar result as Theorem 1 in the EDDmodel, for a test statistic which is a linear combination of edges, vees and triangles empiricalfrequencies in the case of simple graphs, and under a specific condition on the graph density.Though their result is not comparable to ours since triangles can not occur in bipartitegraphs and we do not account for stars motifs. Although they seem similar, a fair comparisonbetween Theorem 1 and the result from Gao and Lafferty [2017] is not easy ( 𝑖 ) because themodel is not the same (we consider bipartite graphs whereas they consider simple graphs)and ( 𝑖𝑖 ) because they only consider vees (which are star-motifs) and triangles (which do notoccur in bipartite graphs). In the following proposition, the asymptotic normality of the statistic ruling the law of 𝑊 𝑠 is stated under the null. This statistic involves the empirical frequency of a given nonstar motif 𝑠 and its theoretical expectation and variance. The proof of its asymptotic nor-mality mostly relies on tools of martingale theory. We show that we can exhibit conditionalmartingale difference sequences relative to a specific filtration. This filtration is generatedby the sequence of graphs 𝑁 (see a proper definition of the filtration in Section 5.1.1). So,we could apply the central limit theorem of Hall and Heyde [2014]. Proposition 2. If 𝑎 + 𝑏 < 𝑑 𝑠 + , then for all star motif 𝑠 and under the B-EDD model, wehave, as 𝑚 ∼ 𝑛 → ∞ , 𝐹 𝑠 − 𝜙 𝑠 √ 𝕍 ( 𝐹 𝑠 ) 𝐷 ←→ (0 , . The complete proof is given in Section 5.2, it relies especially on Lemma 6 and Lemma7.
Sketch of proof.
We first consider the decomposition 𝐿 𝑠 = 𝐹 𝑠 − 𝜙 𝑠 = 𝑀 𝑠 + 𝑅 𝑠 with 𝑀 𝑠 being the difference between 𝐹 𝑠 and its expectation conditionally to the considered filtra-tion and 𝑈 , 𝑉 , and 𝑅 𝑠 the difference between the latter conditional expectation and 𝜙 𝑠 ; theproper definitions are given in Section 5.2.1. Lemma 6 shows that, under the null B-EDDmodel, the reminder term 𝑅 𝑠 ∕ √ 𝕍 ( 𝐹 𝑠 ) | 𝑈 , 𝑉 → a.s. as 𝑚 ∼ 𝑛 → ∞ . Lemma 7 showsthat, under the B-EDD model, the leader term 𝑀 𝑠 ∕ √ 𝕍 ( 𝐹 𝑠 ) | 𝑈 , 𝑉 is asymptotically normalwith variance 𝕍 ( 𝑁 𝑠 | 𝑈 , 𝑉 )∕ 𝕍 ( 𝑁 𝑠 ) . Slutsky theorem implies the asymptotic normality of 𝐿 𝑠 ∕ √ 𝕍 ( 𝐹 𝑠 ) conditional on ( 𝑈 , 𝑉 ) . Then, Lemma 4 shows that 𝕍 ( 𝑁 𝑠 | 𝑈 , 𝑉 )∕ 𝕍 ( 𝑁 𝑠 ) tendsto 1 in probability for all ( 𝑈 , 𝑉 ) , which allows deconditionning. ■ The two following lemmas combined with Proposition 2 permit to conclude to Theorem1. Their proofs are given in sections 5.3 and 5.4 respectively.
Lemma 1. If 𝑎 + 𝑏 < 𝑑 𝑠 + , then for all non-star motif 𝑠 and under the B-EDD model, wehave, as 𝑚 ∼ 𝑛 → ∞ , 𝐹 𝑠 − 𝜙 𝑠 √ 𝕍 ( 𝐹 𝑠 ) → a.s. emma 2. If 𝑎 + 𝑏 < 𝑑 𝑠 + , then for all star motif 𝑠 and under the B-EDD model, we have,as 𝑚 ∼ 𝑛 → ∞ , ̂ 𝕍 ( 𝐹 𝑠 )∕ 𝕍 ( 𝐹 𝑠 ) → a.s. We consider a bipartite network and we want to test if it arises from the B-EDD model: { 𝐻 ∶ follows a B-EDD model ,𝐻 ∶ does not follow a B-EDD model . To this aim, we consider the test statistic 𝑊 𝑠 = ( 𝐹 𝑠 − 𝐹 𝑠 )∕ √ ̂ 𝕍 ( 𝐹 𝑠 ) defined in (12). Theidea is thus to compare the frequency of a motif observed in the network with its expectedvalue under the B-EDD model. Remark 8.
We can consider more specific hypothesis. Suppose we want to test the topnode heterogeneity under B-EDD, more specifically 𝐻 ∶ follows a B-EDD model and 𝑔 is constant. Then, according to (7) , we have that 𝛾 𝑑 = 𝜌 𝑑 under 𝐻 , so a similar statistic to 𝑊 𝑠 can be designed by considering 𝐹 𝑠 = ∏ 𝑝 𝑠 𝑢 =1 𝐹 𝑑 𝑠𝑢 ∏ 𝑞 𝑠 𝑣 =1 Λ 𝑒 𝑠𝑣 ∕ 𝐹 𝑑 𝑠 + . In the same manner, astatistic can be designed to test the bottom node heterogeneity. Illustration.
Table 3 gives the test statistics 𝑊 𝑠 for goodness of fit to the B-EDD modelfor the same motifs as in Table 2. According to Theorem 1, these statistics should be com-pared with the quantiles of standard normal distribution (0 , . Almost no motif frequencydisplays a significant deviation from its expectation under the B-EDD model. Only motif16 in the plant-pollinator network displays a higher frequency than expected under B-EDD(with 𝑝 -value 7.5 −3 ). plant-pollinator 𝑠 𝑊 𝑠 -6.45 −2 −1 -6.63 −2 −1 𝑠 𝑊 𝑠 -2.14 −1 -2.14 −1 -2.93 −1 -2.95 −1 -3.56 −1 Table 3: Test statistics 𝑊 𝑠 for the goodness-of-fit of B-EDD for the same motifs as in Table 2. This section is devoted to network comparison test. More specifically, considering twonetworks assumed to arise from two B-EDD models, we want to test if they arise from thesame B-EDD model, or for, instance, from two different B-EDD model with same function 𝑔 . The rational behind the tests we propose is to compare the frequency of a motif observedin one network with its expected value according to the parameters of the other network. Tothis aim, we need to introduce specific notations. otations. The B-EDD model is parametrized with the ( 𝑚, 𝑛, 𝜌, 𝑔, ℎ ) but all momentsdepend on ( 𝑚, 𝑛, 𝜌, 𝛾, 𝜆 ) , where 𝛾 (resp. 𝜆 ) stands for the sequence of occurrence probabil-ity of all the top (resp. bottom) star motifs. In the sequel we denote by 𝐸 𝑠 the expectedfrequency of motif 𝑠 : 𝐸 𝑠 ( 𝑚, 𝑛, 𝜌, 𝛾, 𝜆 ) ∶= 𝜙 𝑠 , so its plug-in estimate is 𝐸 𝑠 ( 𝑚, 𝑛, 𝐹 , Γ , Λ) = 𝐹 𝑠 . Similarly, we denote the variance ofthe frequency by 𝑉 𝑠 ( 𝑚, 𝑛, 𝜌, 𝛾, 𝜆 ) ∶= 𝕍 ( 𝐹 𝑠 ) and its plug-in estimate 𝑉 𝑠 ( 𝑚, 𝑛, 𝐹 , Γ , Λ) ∶= ̂ 𝕍 𝑠 ( 𝐹 𝑠 ) . A global test.
We consider two bipartite networks 𝐴 and 𝐵 supposed to arise fromB-EDD models with respective dimensions and parameters ( 𝑚 𝐴 , 𝑛 𝐴 , 𝜌 𝐴 , 𝛾 𝐴 , 𝜆 𝐴 ) and ( 𝑚 𝐵 , 𝑛 𝐵 , 𝜌 𝐵 , 𝛾 𝐵 , 𝜆 𝐵 ) .We want to test { 𝐻 ∶ { ( 𝜌 𝐴 , 𝑔 𝐴 , ℎ 𝐴 ) = ( 𝜌 𝐵 , 𝑔 𝐵 , ℎ 𝐵 ) } ,𝐻 ∶ { 𝜌 𝐴 ≠ 𝜌 𝐵 or 𝑔 𝐴 ≠ 𝑔 𝐵 or ℎ 𝐴 ≠ ℎ 𝐵 } . This is to test that, although the two networks may have different dimensions ( 𝑚, 𝑛 ), theyhave the same density ( 𝜌 ), the same top node heterogeneity ( 𝑔 ) and the same bottom nodeheterogeneity ( ℎ ). Test statistics.
The test statistic is based on 𝐹 𝐴𝑠 and 𝐹 𝐵𝑠 the empirical frequencies ofmotif 𝑠 in 𝐴 and 𝐵 respectively. The superscript 𝐴 (resp. 𝐵 ) is added to all quantitiesobserved in 𝐴 (resp. 𝐵 ). 𝑊 𝑠 = ( 𝐹 𝐴𝑠 − 𝐸 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝐹 𝐵 , Γ 𝐵 , Λ 𝐵 ) ) − ( 𝐹 𝐵𝑠 − 𝐸 𝑠 ( 𝑚 𝐵 , 𝑛 𝐵 , 𝐹 𝐴 , Γ 𝐴 , Λ 𝐴 ) )√ 𝑉 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝐹 𝐵 , Γ 𝐵 , Λ 𝐵 ) + 𝑉 𝑠 ( 𝑚 𝐵 , 𝑛 𝐵 , 𝐹 𝐴 , Γ 𝐴 , Λ 𝐴 ) . (13) Theorem 2.
If both 𝑚 𝐴 ∕ 𝑚 𝐵 and 𝑛 𝐴 ∕ 𝑛 𝐵 tends to constants, if 𝑎 + 𝑏 < 𝑑 𝑠 + , then for allnon-star motif 𝑠 and under 𝐻 , the statistic 𝑊 𝑠 is asymptotically normal as 𝑚 𝐴 ∼ 𝑛 𝐴 ∼ 𝑚 𝐵 ∼ 𝑛 𝐵 → ∞ : 𝑊 𝑠 𝐷 ←→ (0 , . Proof.
We decompose 𝐹 𝐴𝑠 − 𝐸 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝐹 𝐵 , Γ 𝐵 , Λ 𝐵 ) = 𝐿 𝐴𝑠 + 𝐶 𝐴𝑠 where 𝐿 𝐴𝑠 = 𝐹 𝐴𝑠 − 𝐸 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝜙 𝐵 , 𝛾 𝐵 , 𝜆 𝐵 ) and 𝐶 𝐴𝑠 = 𝐸 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝜙 𝐵 , 𝛾 𝐵 , 𝜆 𝐵 ) − 𝐸 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝐹 𝐵 , Γ 𝐵 , Λ 𝐵 ) . Because ( 𝑚 𝐴 , 𝑛 𝐴 ) go to infinity at the same speed as ( 𝑚 𝐵 , 𝑛 𝐵 ) , under 𝐻 , 𝐿 𝐴𝑠 ∕ 𝑉 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝜙 𝐵 , 𝛾 𝐵 , 𝜆 𝐵 ) is asymptotically normal according to Proposition 2, whereas 𝐶 𝐴𝑠 ∕ 𝑉 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝜙 𝐵 , 𝛾 𝐵 , 𝜆 𝐵 ) tends to zero according to Lemma 1. Using the same arguments for the symmetric term, weget that and the negligible one ( 𝐶 𝐴𝑠 ∕ 𝑉 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝜙 𝐵 , 𝛾 𝐵 , 𝜆 𝐵 ) , 𝐶 𝐵𝑠 ∕ 𝑉 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝜙 𝐴 , 𝛾 𝐴 , 𝜆 𝐴 ) ) ,replacing 𝑉 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝜙 𝐵 , 𝛾 𝐵 , 𝜆 𝐵 ) and 𝑉 𝑠 ( 𝑚 𝐵 , 𝑛 𝐵 , 𝜙 𝐴 , 𝛾 𝐴 , 𝜆 𝐴 ) with their plug-in estimate 𝑉 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝐹 𝐵 , Γ 𝐵 , Λ 𝐵 ) and 𝑉 𝑠 ( 𝑚 𝐵 , 𝑛 𝐵 , 𝐹 𝐴 , Γ 𝐴 , Λ 𝐴 ) . We conclude using Lemma 2 and Slutsky Theorem. ■ esting equal top nodes heterogeneity. Suppose we want to test that, although thetwo networks may have different dimensions, different densities, and different bottom nodeheterogeneity, they have the same top node heterogeneity, that is { 𝐻 ∶ { 𝑔 𝐴 = 𝑔 𝐵 } ,𝐻 ∶ { 𝑔 𝐴 ≠ 𝑔 𝐵 } . Since we allow the two networks to have different densities, one might normalize theprobabilities of star motifs given in (5) as follows: ̃𝛾 = 1 , ̃𝛾 = 𝜙 ∕ 𝜙 ̃𝛾 = 𝜙 ∕ 𝜙 ̃𝛾 = 𝜙 ∕ 𝜙 , ̃𝛾 = 𝜙 ∕ 𝜙 ,̃𝜆 = 1 , ̃𝜆 = 𝜙 ∕ 𝜙 , ̃𝜆 = 𝜙 ∕ 𝜙 ̃𝜆 = 𝜙 ∕ 𝜙 , ̃𝜆 = 𝜙 ∕ 𝜙 . This allows to see that we can rewrite 𝐸 𝑠 ( 𝑚, 𝑛, 𝜌, 𝛾, 𝜆 ) = 𝜙 𝑠 as an expression of 𝑔 on whichrelies the test we consider. According to (6) and to the definition of 𝜙 𝑠 under the B-EDDmodel, we get: 𝐸 𝑠 ( 𝑚, 𝑛, 𝜌, 𝑔, ℎ ) = 𝜌 𝑑 𝑠 + 𝑝 𝑠 ∏ 𝑢 =1 ̃𝛾 𝑑 𝑠𝑢 𝑞 𝑠 ∏ 𝑣 =1 ̃𝜆 𝑒 𝑠𝑣 = 𝜌 𝑑 𝑠 + 𝑝 𝑠 ∏ 𝑢 =1 𝑞 𝑠 ∏ 𝑣 =1 𝑔 𝑑 𝑠𝑢 ℎ 𝑒 𝑠𝑣 , where 𝑔 𝑑 = ∫ 𝑔 ( 𝑢 ) 𝑑 d 𝑢 and ℎ 𝑒 = ∫ ℎ ( 𝑣 ) 𝑒 d 𝑣 . We may consider the following test statistic: 𝑊 𝑔𝑠 = ( 𝐹 𝐴𝑠 − 𝐸 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝐹 𝐴 , ̃ Γ 𝐵 , ̃ Λ 𝐴 ) ) − ( 𝐹 𝐵𝑠 − 𝐸 𝑠 ( 𝑚 𝐵 , 𝑛 𝐵 , 𝐹 𝐵 , ̃ Γ 𝐴 , ̃ Λ 𝐵 ) )√ 𝑉 𝑠 ( 𝑚 𝐴 , 𝑛 𝐴 , 𝐹 𝐴 , ̃ Γ 𝐵 , ̃ Λ 𝐴 ) + 𝑉 𝑠 ( 𝑚 𝐵 , 𝑛 𝐵 , 𝐹 𝐵 , ̃ Γ 𝐴 , ̃ Λ 𝐵 ) , where ̃ Γ and ̃ Λ are the plug-in estimates of ̃𝛾 and ̃𝜆 respectively. Similar statistics can bedesigned to test 𝜌 𝐴 = 𝜌 𝐵 , ℎ 𝐴 = ℎ 𝐵 or any combination. Illustration.
Both the plant-pollinator and the seed dispersal networks involve plantsspecies. Although these species are not the same, one may be interested in comparing ifthe level of heterogeneity across plants (encoded in the function 𝑔 ) is the same in bothnetworks. From an ecological point of view, this amounts to test if there is the same thedegree of imbalance between specialists and generalists among plants regarding pollinationand seed dispersion, that are two of the main reproduction means.Table 4 provides the results of the network comparison test presented above. No significantdifference is observed, suggesting that, although generalist and specialist plants may existfor both types of interactions, the degree of imbalance between them is comparable. We designed a simulation study to illustrate Theorem 1 and to assess the performance ofthe goodness-of-fit test and the comparison test described in Section 3.2 and Section 3.3 re-spectively. More specifically, our purpose is to illustrate the asymptotic normality of the teststatistics and evaluate the power of the tests for various graph sizes, densities and sparsityregimes. 𝐹 𝐴𝑠 −5 −5 −6 −7 −8 ̂ 𝔼 𝐹 𝐴𝑠 −4 −5 −5 −6 −6 𝐹 𝐵𝑠 −4 −4 −5 −5 −6 ̂ 𝔼 𝐹 𝐵𝑠 −4 −5 −5 −6 −7 𝐹 𝐵𝑠 − 𝐹 𝐴𝑠 -4.21 −4 -1.05 −4 -4.26 −5 -1.76 −5 -5.91 −6 ̂ 𝔼 ( 𝐹 𝐵𝑠 − 𝐹 𝐴𝑠 ) -6.96 −5 −6 -1.11 −5 −6 −6 √ ̂ 𝕍 ( 𝐹 𝐴𝑠 ) + ̂ 𝕍 ( 𝐹 𝐵𝑠 ) −4 −5 −5 −5 −6 𝑊 𝑠 -1.56 -1.56 -0.97 -1.28 -0.96Table 4: Network comparison test for 𝐻 = { 𝑔 𝐴 = 𝑔 𝐵 } as defined in Section 3.3 for the samemotifs as in Table 2. Networks: 𝐴 = plant-pollinator, 𝐵 = seed dispersal. ̂ 𝔼 ( ⋅ ) is a shorthandfor the notation 𝐸 𝑠 ( ⋯ ) (idem for ̂ 𝕍 ( ⋅ ) and 𝑉 𝑠 ( ⋯ ) ). Simulation design.
We simulated series of networks with parameters ( 𝑚, 𝑛, 𝜌, 𝜇 𝑔 , 𝜇 ℎ )varying according to the following design: Network dimension:
We simulated networks with equal dimensions 𝑚 = 𝑛 , with valuesin {50 , , , , , ; Sparsity regime:
We considered equal parameters 𝑎 = 𝑏 in {1∕3 , , , ; Network density:
The resulting density is 𝜌 = 𝜌 𝑚 − 𝑎 𝑛 − 𝑏 , 𝜌 being fixed so that 𝜌 = . when 𝑚 = 𝑛 = 100 ; Degree imbalance:
We considered the functions 𝑔 ( 𝑢 ) = 𝜇 𝑔 𝑢 𝜇 𝑔 −1 and ℎ ( 𝑣 ) = 𝜇 ℎ 𝑣 𝜇 ℎ −1 ;observe that 𝜇 𝑔 = 1 means that 𝑔 is constant so no imbalance does exist top nodes(resp. for 𝜇 ℎ , ℎ and bottom nodes). We set 𝜇 𝑔 = 2 , 𝜇 ℎ = 3 .For each configuration, 𝑆 = 100 networks were sampled and the test applied. Results.
The results are displayed in Figure 3 and Figure 4. In Figure 3, the QQ-plots ofthe 𝑊 𝑠 statistic (black dots) defined in (12) and the ̃𝑊 𝑠 statistic (blue dots) defined in (14)hereafter, are given for four motifs in a network with dimension 𝑚 = 𝑛 = 1000 and sparsityregime 𝑎 = 𝑏 = 1∕3 . Remember that the larger the power 𝑎 , the sparser the graph. Weobserve that normality of 𝑊 𝑠 holds for motifs 6 and 15, but not for motifs 5 and 10.Actually, the latter case is due to the fluctuations of 𝐹 𝑠 . More specifically, for non-starmotifs, 𝐹 𝑠 is not an unbiased estimate of 𝜙 𝑠 and it is not independent from 𝐹 𝑠 . As a con-sequence, for finite dimensions 𝑚 and 𝑛 , we both have that 𝔼 ( 𝐹 𝑠 ) ≠ 𝜙 𝑠 = 𝔼 ( 𝐹 𝑠 ) and 𝕍 ( 𝐹 𝑠 − 𝐹 𝑠 ) ≠ 𝕍 ( 𝐹 𝑠 ) . Both the bias of 𝐹 : 𝔹 ( 𝐹 𝑠 ) = 𝔼 ( 𝐹 ) − 𝜙 𝑠 and the variance of thenumerator of 𝑊 𝑠 : 𝕍 ( 𝐹 𝑠 − 𝐹 𝑠 ) can be estimated using the delta method, which requires thecovariance given in Equation (10). This enables us to define a corrected version ̃𝑊 𝑠 of thetest statistic 𝑊 𝑠 : ̃𝑊 𝑠 ∶= ̂ 𝕍 ( 𝐹 𝑠 − 𝐹 𝑠 ) −1∕2 ( 𝐹 𝑠 − 𝐹 𝑠 + ̂ 𝔹 ( 𝐹 𝑠 ) ) , (14)where the bias ̂ 𝔹 ( 𝐹 𝑠 ) and ̂ 𝕍 ( 𝐹 𝑠 − 𝐹 𝑠 ) are both plug-in estimates. llustration. We provide in Table 5 the values of corrected corrected statistics ̃𝑊 𝑠 for theplant-pollinator and the seed dispersal networks, to be compared with Table 3. Observe thatthe correction does not yield in different conclusions, in terms of fit to the B-EDD modelfor both networks. 𝑠 ̃𝑊 𝑠 for the goodness-of-fit of B-EDD for the same motifs as inTable 2. motif 6 motif 15 motif 5 motif 10 −3 −2 −1 0 1 2 3 − − − l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l −3 −2 −1 0 1 2 3 − − − l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l −3 −2 −1 0 1 2 3 − − − l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l ll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l −3 −2 −1 0 1 2 3 − − − l lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l ll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll l l Figure 3: Qq-plots of the test statistics 𝑊 𝑠 for 4 motifs in a network with dimension 𝑚 = 𝑛 =2000 and sparsity regime 𝑎 = 1∕3 (black dots). Blue dots: qq-plot for the corrected statistic ̃𝑊 𝑠 defined in Equation (14). Red line: 95% confidence interval for a qq-plot with sample size 𝑆 = 100 . Figure 4 displays the QQ-plots of the corrected test statistics ̃𝑊 𝑠 gathered accordingto the order of magnitude of the expected motif frequencies. All network sizes, sparsityregimes and non-star motifs are thus considered here together. As expected, the normalitybecomes more accurate when the motifs frequency increases. Simulation design.
In order to illustrate the power of the goodness-of-fit test, we simu-lated a series of networks from a mixture of a B-EDD model and a latent block model (LBM)[Govaert and Nadif, 2008], characterizing the presence of clusters of rows and columns inincidence matrices. Thus, a mixing weight 𝛼 varying from to was considered so that 𝛼 = 0 corresponds to a B-EDD that is 𝐻 . In details, the following simulation setup wasinvestigated: Network dimension and density:
We considered dimensions similar to the pollination andseed dispersal binary networks studied in Simmons et al. [2019b], that is 𝑚 = 𝑛 ∈{10 , … , } . To mimic the sparsity of the same networks, we fitted the density viaa linear regression and obtained log ( 𝜌 ) = 0 . . ( 𝑚𝑛 ) ; B-EDD model:
We used the same functions 𝑔 and ℎ as in Section 4.1, with 𝜇 𝑔 = 2 , 𝜇 ℎ = 3 ; LBM model:
We considered groups in rows and groups in columns, all groups withproportion and all connection probabilities 𝛾 𝑘 𝓁 = 𝐶𝛾 min for all ≤ 𝑘, 𝓁 ≤ , ( 𝑁 𝑠 ) < ≤ 𝔼 ( 𝑁 𝑠 ) <
10 10 ≤ 𝔼 ( 𝑁 𝑠 ) < 𝔼 ( 𝑁 𝑠 ) ≥ −3 −2 −1 0 1 2 3 − − − −3 −2 −1 0 1 2 3 − − − −3 −2 −1 0 1 2 3 − − − −3 −2 −1 0 1 2 3 − − − Figure 4: Qq-plots of the corrected test statistics ̃𝑊 𝑠 . The plot displays the results of the sim-ulation design (i.e for all network size 𝑛 , sparsity regime 𝑎 and non-star motifs 𝑠 . The qq-plotsare gathered accorded to the order of magnitude of the expected count 𝔼 ( 𝑁 𝑠 ) , from the smallest(top left) to the largest (bottom right). Red line: same legend as Figure 3. except 𝛾 = 𝐶𝛾 max , with 𝐶 set such that 𝐶 ( 𝛾 max + 3 𝛾 min )∕4 = 1 . Two regimes wereconsidered: 𝛾 max = 0 . (scenario I: easy) and 𝛾 max = 0 . (scenario II: hard); Connection probability:
We sampled the { 𝑈 𝑖 } ≤ 𝑖 ≤ 𝑚 and { 𝑉 𝑗 } ≤ 𝑗 ≤ 𝑛 all independently anduniformly over [0 , , and set the { 𝑍 𝑖 } ≤ 𝑖 ≤ 𝑚 and { 𝑊 𝑗 } ≤ 𝑗 ≤ 𝑛 as 𝑍 𝑖 = 𝕀 { 𝑈 𝑖 > .
5} + 1 and 𝑊 𝑗 = 𝕀 { 𝑉 𝑗 > .
5} + 1 . Finally, the edges were sampled with probability ℙ { 𝐺 𝑖𝑗 = 1 ∣ 𝑈 𝑖 , 𝑉 𝑗 } = 𝜌 ( (1 − 𝛼 ) 𝑔 ( 𝑈 𝑖 ) ℎ ( 𝑉 𝑗 ) + 𝛼𝛾 𝑍 𝑖 𝑊 𝑗 ) . For each configuration, 𝑆 = 500 networks were sampled and the test applied. Again thetest corrected statistic ̃𝑊 𝑠 was used. Results.
The results are given in Figure 5. For illustration purposes, we only present theresults we obtained for 𝑚 = 𝑛 ranging from to . Moreover, for the sake of clarity, weonly consider motifs 5, 6, 10, and 15 which constitute a representative panel of the set ofmotifs with size and .As the network dimensions increase, we can clearly observe that the tests become morepowerful. For small networks with 𝑚 = 𝑛 = 50 and 𝑚 = 𝑛 = 100 , the LBM regimewith 𝛾 max = 0 . is easier and leads to tests associated with motifs and with higherpower. These differences vanish for larger values of 𝑛 and 𝑚 . Overall, we found that motifs and lead to more powerful tests. These results illustrate that the methodology proposedis relevant and that the goodness-of-fit tests for different motifs can be used to detect thedeparture from a B-EDD model. Simulation design.
We also studied the power of the test for network comparison in-troduced in Section 3.3. To this aim, we simulated series of networks 𝐴 with parameters( 𝑚 𝐴 , 𝑛 𝐴 , 𝜌 𝐴 , 𝜇 𝐴𝑔 , 𝜇 𝐴ℎ ) varying according to the same design as in Section 4.1, where 𝜇 𝐴𝑔 wasset to .We focused on the test of 𝐻 = { 𝑔 𝐴 = 𝑔 𝐵 } so, for each network 𝐴 , we simulated a sequenceof networks 𝐵 with same dimensions ( 𝑚 𝐵 = 𝑚 𝐴 , 𝑛 𝐵 = 𝑛 𝐴 ), but a with a different parameter 𝜇 𝐵𝑔 . More specifically, setting 𝜇 ∗ 𝑔 = 1 (absence of degree imbalance between top nodes), wesampled networks 𝐵 with 𝜇 𝐵𝑔 = (1 − 𝛼 ) 𝜇 𝐴 + 𝛼𝜇 ∗ 𝑔 , with 𝛼 = 0 , . , . , … 1 , so that 𝛼 = 0 = 𝑛 = 50 𝑚 = 𝑛 = 100 𝑚 = 𝑛 = 200 𝑚 = 𝑛 = 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure 5: Empirical power of the goodness-of-fit tests, averaged over 𝑆 = 500 simulations.Top: scenario I (easy: 𝛾 max = 0 . ); bottom: scenario II (hard: 𝛾 max = 0 . ). From left to right: 𝑚 = 𝑛 = 50 , , , . Color = motif: black=5, red=6, green=10, blue=15. corresponds to 𝐻 .Regarding the two remaining parameters 𝜌 𝐵 and 𝜇 𝐵ℎ , we considered two scenarios: I (easy): 𝜌 𝐵 = 𝜌 𝐵 , 𝜇 𝐵ℎ = 𝜇 𝐴ℎ , so that the two networks only differ with respect to 𝜇 𝑔 ; II (hard): 𝜌 𝐵 = 𝜌 𝐴 ∕2 , 𝜇 𝐵ℎ = 2 , so that the two network differ in all parameters, but onlythe difference in 𝜇 𝑔 is tested.The ’hard’ scenario is designed to assess the ability of the proposed test statistic to accom-modate to differences in density and bottom node imbalance between the two networks,when testing the equality of their top node imbalance. For each configuration, 𝑆 = 500 pairs of networks ( 𝐴 , 𝐵 ) were sampled and compared.Following the simulation results presented in Section 4.1, we used the delta-method to de-rive a corrected version ̃𝑊 𝑠 of the test statistic 𝑊 𝑠 defined in Equation (13). Similarly toSection 4.1, the performances of the uncorrected test statistic 𝑊 𝑠 become similar to theseof the corrected version ̃𝑊 𝑠 for large networks (results not shown). Illustration.
Again, to illustrate the effect of the proposed correction, we provide in Ta-ble 6 the values of corrected statistics ̃𝑊 𝑠 testing 𝐻 = { 𝑔 𝐴 = 𝑔 𝐵 } , network 𝐴 beingplant-pollinator and network 𝐵 being seed dispersal. These results can be compared withTable 4: The correction yields in (moderately) higher absolute values, suggesting a gain ofpower. Results.
The results are displayed in Figure 6. We only present the results for 𝑚 𝐴 = 𝑛 𝐴 = 𝑚 𝐵 = 𝑛 𝐵 ranging for 50 to 500. Moreover, as in the previous section, we only consider mo-tifs 5, 6, 10 and 15.As expected, the test becomes more powerful when the networks dimensions increase. Moreinterestingly, for small networks, the smaller motifs (5 and 6, with size 4) turn out to yield ̃𝑊 𝑠 -2.71 -1.90 -1.76 -1.34 -0.96Table 6: Corrected test statistics ̃𝑊 𝑠 for 𝐻 = { 𝑔 𝐴 = 𝑔 𝐵 } for the same motifs as in Table 2 andsame networks as in Table 4. a higher power. The difference vanishes when the dimensions increase.These conclusions hold under the two scenarios, which shows that the proposed test statis-tic does accommodate for departures that may exist between two networks, not being thedeparture under study (scenario II ’hard’). Still, the power is always better under scenario I:obviously, the test performs better when focusing on the only difference that actually exists(scenario I ’easy’). 𝑚 = 𝑛 = 50 𝑚 = 𝑛 = 100 𝑚 = 𝑛 = 200 𝑚 = 𝑛 = 500 . . . . . . l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l . . . . . . l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l . . . . . . l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l . . . . . . l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l . . . . . . l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l . . . . . . l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l . . . . . . l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l . . . . . . l l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l ll l l l l l l l l l l Figure 6: Empirical power of the network comparison test for 𝐻 = { 𝑔 𝐴 = 𝑔 𝐵 } , averaged over 𝑆 = 500 simulations. Top: scenario I (easy); bottom: scenario II (hard). From left to right: 𝑚 = 𝑛 = 50 , , , . Color = motif: same legend as Figure 5. In this section, we introduce notations and useful technical lemmas for establishing proofsof Proposition 2 in Section 5.2, Lemma 1 in Section 5.3 and Lemma 2 in Section 5.4.
Let remind that we consider a bipartite graph = ( , ) with 𝑁 nodes. The set of nodesis = ( 𝑡 , 𝑏 ) , where 𝑡 = (cid:74) , 𝑚 (cid:75) (resp. 𝑏 (cid:74) , 𝑛 (cid:75) ) stands for the set of top (resp. bottom) odes, and the set of edges is ⊂ 𝑡 × 𝑏 , meaning than an edge can only connect a topnode with a bottom node. The total number of nodes is therefore 𝑁 = 𝑛 + 𝑚 . We denote by 𝐺 the corresponding 𝑚 × 𝑛 incidence matrix where the entry 𝐺 𝑖𝑗 of 𝐺 is 1 if ( 𝑖, 𝑗 ) ∈ , and0 otherwise.Let consider now a collection of bipartite graphs ( 𝓁 ) 𝓁 ∈ (cid:74) ,𝑁 (cid:75) = ( 𝓁 , 𝓁 ) with 𝓁 nodes.In the following, we introduce notations for subsets of interest and a filtration we will useto construct differences of martingales involving motif counts. Subsets definitions.
Let introduce the following subsets definitions:• 𝓁 = {( 𝑘 , … , 𝑘 𝓁 ) ⊂ 𝑡 ∪ 𝑏 with at least one top node and one bottom node } , 𝓁 ∈ (cid:74) , 𝑁 (cid:75) , it is the set of nodes of 𝓁 meaning the 𝓁 selected nodes among , and 𝑘 𝓁 denotes the 𝓁 -th and last selected one; we will use 𝑘 𝓁 several times hereafter;• 𝑉 𝑡 𝓁 = 𝓁 ∩ 𝑡 and 𝑉 𝑏 𝓁 = 𝓁 ∩ 𝑏 , these are the sets of top and bottom nodes in 𝓁 ;• 𝑠, 𝓁 = { ( 𝑖 , … , 𝑖 𝑝 𝑠 ) ⊂ 𝑉 𝑡 𝓁 } × { ( 𝑗 , … , 𝑗 𝑞 𝑠 ) ⊂ 𝑉 𝑏 𝓁 } , 𝓁 ∈ (cid:74) 𝑝 𝑠 + 𝑞 𝑠 , 𝑁 (cid:75) , it is the posi-tions set of motif 𝑠 in 𝓁 ;• 𝑇 𝓁 = { 𝑘 𝓁 ∈ 𝑡 } is an event;• 𝑠, 𝓁 = { { 𝑠, 𝓁 −1 ⧵ 𝑖 𝑝 𝑠 } ∪ { 𝑖 𝑝 𝑠 = 𝑘 𝓁 } if 𝑇 𝓁 , { 𝑠, 𝓁 −1 ⧵ 𝑗 𝑞 𝑠 } ∪ { 𝑗 𝑞 𝑠 = 𝑘 𝓁 } otherwise , it is the positions set of motif 𝑠 in 𝓁 with the particularity that 𝑘 𝓁 the last node addedto 𝓁 is part of motif 𝑠 . Filtration.
The filtration ( 𝓁 ) 𝓁 ∈ (cid:74) ,𝑁 (cid:75) is defined by the 𝜎 -algebra 𝓁 = 𝜎 ( 𝓁 ) . We present here three lemmas which are key arguments in the proofs of Proposition 2,Lemma 1 and Lemma 2.The following lemma gives the order of magnitude of the variance of a count. Before,its statement let give the order of magnitude of the expected count of a motif 𝑠 with 𝑝 𝑠 topnodes and 𝑞 𝑠 bottom nodes. It writes 𝔼 ( 𝑁 𝑠 ) = 𝑐 𝑠 𝜙 𝑠 , with 𝑐 𝑠 = Θ( 𝑚 𝑝 𝑠 𝑛 𝑞 𝑠 ) (normalizing coefficient specific to 𝑠 ) (15) 𝜌 = Θ( 𝑚 − 𝑎 𝑛 − 𝑏 ) , with 𝑎, 𝑏 > (graph density) (16) 𝜙 𝑠 = Θ( 𝜌 𝑑 𝑠 + ) = Θ( 𝑚 − 𝑎𝑑 𝑠 + 𝑛 − 𝑏𝑑 𝑠 + ) (expected frequency of 𝑠 ) , (17)where 𝑑 𝑠 + stands for the total number of edges in 𝑠 and 𝑐 𝑠 being defined in (3). Lemma 3.
We have, 𝕍 ( 𝑁 𝑠 ) = Θ ( max( 𝑚 𝑝 𝑠 −2 𝑎𝑑 𝑠 + −1 𝑛 𝑞 𝑠 −2 𝑏𝑑 𝑠 + , 𝑚 𝑝 𝑠 −2 𝑎𝑑 𝑠 + 𝑛 𝑞 𝑠 −2 𝑏𝑑 𝑠 + −1 , 𝑚 𝑝 𝑠 − 𝑎𝑑 𝑠 + −1 𝑛 𝑞 𝑠 − 𝑏𝑑 𝑠 + −1 ) ) . Proof.
Let observe that, for 𝛼, 𝛽 ∈ 𝑠,𝑁 , 𝑁 𝑠 = ∑ 𝛼 𝑌 𝑠 ( 𝛼 ) + ∑ 𝛼 ∩ 𝛽 ≠ ∅ 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) + ∑ 𝛼 ∩ 𝛽 =∅ 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) . Thus, a general form for the variance is the following: 𝕍 ( 𝑁 𝑠 ) = 𝔼 ( 𝑁 𝑠 ) + ∑ 𝑡 ∈ ( 𝑠 ) 𝔼 ( 𝑁 𝑡 ) + (| 𝑠 | − 𝑐 𝑠 ) 𝜙 𝑠 , (18) here 𝑠 = { 𝛼, 𝛽 ∈ 𝑠,𝑁 ∶ 𝛼 ∩ 𝛽 = ∅ } and ( 𝑠 ) denotes the set of supermotifs of 𝑠 whichare formed by two overlapping occurrences of 𝑠 .Let evaluate the orders of the three added terms of assertion (18). Considering that 𝜌 = Θ( 𝑚 − 𝑎 𝑛 − 𝑏 ) , the first term of (18) is Θ( 𝑚 𝑝 𝑠 − 𝑎𝑑 𝑠 + 𝑛 𝑞 𝑠 − 𝑏𝑑 𝑠 + ) . Then denoting ( 𝑎 ) 𝑏 = 𝑎 ( 𝑎 −1) … ( 𝑎 − 𝑏 ) , we see that | 𝑠 | − 𝑐 𝑠 = ( 𝑚 𝑝 𝑠 −1 )( 𝑝 𝑠 !) ( 𝑛 ) 𝑞 𝑠 −1 ( 𝑞 𝑠 !) − ( 𝑚 𝑝 𝑠 −1 ) ( 𝑝 𝑠 !) ( 𝑛 ) 𝑞 𝑠 −1 ( 𝑞 𝑠 !) (19) = Θ ( (−1) 𝑝 𝑠 −1 𝑝 𝑠 𝑚 𝑝 𝑠 −1 𝑛 𝑞 𝑠 + (−1) 𝑞 𝑠 −1 𝑞 𝑠 𝑚 𝑝 𝑠 𝑛 𝑞 𝑠 −1 ( 𝑝 𝑠 !) ( 𝑞 𝑠 !) ) = Θ ( max( 𝑚 𝑝 𝑠 −1 𝑛 𝑞 𝑠 , 𝑚 𝑝 𝑠 𝑛 𝑞 𝑠 −1 ) ) . Thus the third term is Θ ( max( 𝑚 𝑝 𝑠 −2 𝑎𝑑 𝑠 + −1 𝑛 𝑞 𝑠 −2 𝑏𝑑 𝑠 + , 𝑚 𝑝 𝑠 −2 𝑎𝑑 𝑠 + 𝑛 𝑞 𝑠 −2 𝑏𝑑 𝑠 + −1 ) ) . Let focus now on the second term. When 𝑡 ∈ 𝑘 ( 𝑠 ) , it can result of an overlap of ( 𝑖 )only top nodes, ( 𝑖𝑖 ) only bottom nodes, or ( 𝑖𝑖𝑖 ) both. For each case we have( 𝑖 ) 𝑝 𝑡 < 𝑝 𝑠 , 𝑞 𝑡 = 2 𝑞 𝑠 , 𝑑 𝑡 + = 2 𝑑 𝑠 + so 𝔼 𝑁 𝑡 = 𝑂 ( 𝑚 𝑝 𝑠 −1 𝑛 𝑞 𝑠 𝜌 𝑑 𝑠 + ) = 𝑂 ( 𝑚 𝑝 𝑠 −2 𝑎𝑑 𝑠 + −1 𝑛 𝑞 𝑠 −2 𝑏𝑑 𝑠 + ) ;( 𝑖𝑖 ) 𝑝 𝑡 = 2 𝑝 𝑠 , 𝑞 𝑡 < 𝑞 𝑠 , 𝑑 𝑡 + = 2 𝑑 𝑠 + so 𝔼 𝑁 𝑡 = 𝑂 ( 𝑚 𝑝 𝑠 𝑛 𝑞 𝑠 −1 𝜌 𝑑 𝑠 + ) = 𝑂 ( 𝑚 𝑝 𝑠 −2 𝑎𝑑 𝑠 + 𝑛 𝑞 𝑠 −2 𝑏𝑑 𝑠 + −1 ) ;( 𝑖𝑖𝑖 ) 𝑝 𝑡 < 𝑝 𝑠 , 𝑞 𝑡 < 𝑞 𝑠 , 𝑑 𝑠 + < 𝑑 𝑡 + < 𝑑 𝑠 + so 𝔼 𝑁 𝑡 = 𝑂 ( 𝑚 𝑝 𝑠 − 𝑎𝑑 𝑠 + −1 𝑛 𝑞 𝑠 − 𝑏𝑑 𝑠 + −1 ) .Combining the orders of the three terms of assertion (18), we get that the order of mag-nitude of the variance of a count is 𝕍 ( 𝑁 𝑠 ) = Θ ( max( 𝑚 𝑝 𝑠 −2 𝑎𝑑 𝑠 + −1 𝑛 𝑞 𝑠 −2 𝑏𝑑 𝑠 + , 𝑚 𝑝 𝑠 −2 𝑎𝑑 𝑠 + 𝑛 𝑞 𝑠 −2 𝑏𝑑 𝑠 + −1 , 𝑚 𝑝 𝑠 − 𝑎𝑑 𝑠 + −1 𝑛 𝑞 𝑠 − 𝑏𝑑 𝑠 + −1 ) ) . ■ The last argument of proof of Proposition 2, Lemma 7 and Lemma 1 relies on the fol-lowing result.
Lemma 4.
We have, as 𝑚 ∼ 𝑛 → ∞ , 𝕍 ( 𝑁 𝑠 | 𝑈 , 𝑉 )∕ 𝕍 ( 𝑁 𝑠 ) → in probability . Proof.
First let us write that 𝔼 ( 𝑁 𝑠 | 𝑈 , 𝑉 ) = ∑ 𝛼 ∈ 𝑠 ℙ ( 𝑌 𝑠 ( 𝛼 ) = 1 | 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 ) 𝔼 ( 𝑁 𝑠 | 𝑈 , 𝑉 ) = ∑ 𝛼,𝛽 ∈ 𝑠 ℙ ( 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) = 1 | 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 , 𝑈 𝛽 𝑡 , 𝑉 𝛽 𝑏 ) . The proof relies on showing the convergence in probability of the two above expectationstowards ∑ 𝛼 ∈ 𝑠 ℙ ( 𝑌 𝑠 ( 𝛼 ) = 1 ) and ∑ 𝛼,𝛽 ∈ 𝑠 ℙ ( 𝑌 𝑠 ( 𝛼 ) 𝑌 𝑠 ( 𝛽 ) = 1 ) , respectively. Let us nowintroduce the equivalence relation R 𝑠 and the set 𝑅 𝑠 defined as follows: R 𝑠 ∶ ( 𝜎 𝑡 , 𝜎 𝑏 ) ∼ ( ̃𝜎 𝑡 , ̃𝜎 𝑏 ) ⇔ 𝐴 𝑠𝜎 𝑡 ,𝜎 𝑏 = 𝐴 𝑠̃𝜎 𝑡 , ̃𝜎 𝑏 and 𝑅 𝑠 = ( 𝜎 ( (cid:74) , 𝑝 𝑠 (cid:75) ) ⊗ 𝜎 ( (cid:74) , 𝑞 𝑠 (cid:75) )) ∕ R 𝑠 . Then, we can exhibit the two following quantities which are two-samples U-Statistics (seeSection 12.2, p.165 in Van der Vaart [2000]): 𝑟 𝑠 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠 𝑘 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 ) and 𝑟 𝑠 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠 𝑘 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 , 𝑈 𝛽 𝑡 , 𝑉 𝛽 𝑏 ) , ith 𝑟 𝑆 and 𝑐 𝑠 being defined in (2), (3), respectively, 𝑠 denoting the location relative to agiven position for motif 𝑠 and where 𝑘 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 ) = ∑ 𝜎 ∈ 𝑅 𝑠 ℙ ( 𝑌 𝑠 ( 𝜎 ( 𝛼 )) = 1 | 𝑈 𝜎 𝑡 ( 𝛼 𝑡 ) , 𝑉 𝜎 𝑏 ( 𝛼 𝑏 ) ) 𝑘 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 , 𝑈 𝛽 𝑡 , 𝑉 𝛽 𝑏 ) = ∑ ( 𝜎 𝛼 ,𝜎 𝛽 )∈ 𝑅 𝑠 ℙ ( 𝑌 𝑠 ( 𝜎 𝛼 ( 𝛼 )) 𝑌 𝑠 ( 𝜎 𝛽 ( 𝛽 )) = 1 | 𝑈 𝜎 𝑡𝛼 ( 𝛼 𝑡 ) , 𝑉 𝜎 𝑏𝛼 ( 𝛼 𝑏 ) , 𝑈 𝜎 𝑡𝛽 ( 𝛽 𝑡 ) , 𝑉 𝜎 𝑏𝛽 ( 𝛽 𝑏 ) ) , with 𝑘 (·) and 𝑘 (·) being permutation symmetric kernels in ( 𝑈 𝑖 ) 𝑖 and ( 𝑉 𝑗 ) 𝑗 separetely. Weconclude by applying the central limit theorem for two-sample U-Statistics (see Theorem12.6 in Van der Vaart [2000]) which holds under the assumption that the kernel of the U-statistic has a finite moment of order two. Here, as it concerns probabilities this assumptionis obviously fulfilled. ■ In proofs of Lemma 2, Lemma 6 and Lemma 7, we need to know the cardinal order ofthe sets ⊗𝑘𝑠, 𝓁 ⧵ ( 𝑘 ) 𝑠, 𝓁 , 𝑘 = 2 , which contains only dependent 𝑘 -uplets of positions of motif 𝑠 on the event 𝑇 𝓁 for which the last node added to 𝓁 is a top node. Recall that 𝑠, 𝓁 is thepositions set of motif 𝑠 in the subgraph of with nodes in 𝓁 and the particularity that 𝑘 𝓁 the last node added to 𝓁 is part of motif 𝑠 . The definition of the other set of interest is thefollowing: ( 𝑘 ) 𝑠, 𝓁 = { 𝛼 , … 𝛼 𝑘 ∈ 𝑠, 𝓁 ∶ ( 𝛼 𝑡 ⧵ 𝑘 𝓁 ) × 𝛼 𝑏 ∩ … ∩ ( 𝛼 𝑏𝑘 ⧵ 𝑘 𝓁 ) × 𝛼 𝑏𝑘 = ∅ } . Lemma 5.
We have, on 𝑇 𝓁 , | 𝑠, 𝓁 | 𝑘 − | ( 𝑘 ) 𝑠, 𝓁 | = Θ ( 𝓁 𝑘 ( 𝑝 𝑠 −1)−1 𝑡 𝓁 𝑘𝑞 𝑠 𝑏 ) , with 𝓁 𝑡 and 𝓁 𝑏 denoting respectively top and bottom nodes in 𝓁 .Proof. Let observe that | 𝑠, 𝓁 | = ( 𝓁 𝑡 − 1 𝑝 𝑠 − 1 )( 𝓁 𝑏 𝑞 𝑠 ) , | ( 𝑘 ) 𝑠, 𝓁 | = ( 𝓁 𝑡 − 1 𝑝 𝑠 − 1 ) 𝑘 ( 𝓁 𝑏 𝑞 𝑠 … 𝑞 𝑠 𝓁 𝑏 − 𝑘𝑞 𝑠 ) + ( 𝓁 𝑡 𝑝 𝑠 … 𝑝 𝑠 𝓁 𝑡 − 𝑘𝑝 𝑠 )( 𝓁 𝑏 𝑞 𝑠 ) 𝑘 + ( 𝓁 𝑡 𝑝 𝑠 … 𝑝 𝑠 𝓁 𝑡 − 𝑘𝑝 𝑠 )( 𝓁 𝑏 𝑞 𝑠 … 𝑞 𝑠 𝓁 𝑏 − 𝑘𝑞 𝑠 ) . The leader term of order Θ ( 𝓁 𝑘 ( 𝑝 𝑠 −1) 𝑡 𝓁 𝑘𝑞 𝑠 𝑏 ) obviously vanishes and imply the lost of oneorder (the calculation omitted here are simply based on the same arguments as in (19)). ■ For establishing the proof of Proposition 2, we first consider a decomposition of 𝐿 𝑠 = 𝐹 𝑠 − 𝜙 𝑠 in Section 5.2.1 , then we focus on the reminder term of this decomposition in Lemma 6and finally show the asymptotic normality of the leading term in Lemma 7. .2.1 Decomposition of 𝐿 𝑠 Let use the sets introduced in Section 5.1.1 to express 𝐿 𝑠 as follows: 𝐿 𝑠 ( 𝑈 , 𝑉 ) = 𝐹 𝑠 − 𝜙 𝑠 ( 𝑈 , 𝑉 ) = 1 𝑐 𝑠 ∑ 𝛼 =( 𝛼 𝑡 ,𝛼 𝑏 )∈ 𝑠,𝑁 { 𝑌 𝑠 ( 𝛼 ) − 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 )}= 1 𝑐 𝑠 𝑁 ∑ 𝓁 =1 ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝑌 𝑠 ( 𝛼 ) − 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 )} , with the random variables 𝑈 , 𝑉 of the B-EDD model (1). Then let decompose 𝐿 𝑠 as the sumof two expressions, the first one corresponding to a martingale difference sequence relativeto the filtration ( 𝓁 ) 𝓁 ∈ (cid:74) ,𝑁 (cid:75) , the second one being a term of rest: 𝐿 𝑠 ( 𝑈 , 𝑉 ) ∶= 𝑀 𝑠 ( 𝑈 , 𝑉 ) + 𝑅 𝑠 ( 𝑈 , 𝑉 ) , where 𝑀 𝑠 ( 𝑈 , 𝑉 ) = 1 𝑐 𝑠 𝑁 ∑ 𝓁 =1 ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝑌 𝑠 ( 𝛼 ) − 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 )} 𝑅 𝑠 ( 𝑈 , 𝑉 ) = 1 𝑐 𝑠 𝑁 ∑ 𝓁 =1 ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) − 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 )} . Observe that by construction, 𝑀 𝑠, 𝓁 = ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝑌 𝑠 ( 𝛼 ) − 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 )} is a condi-tional martingale difference with respect to ( 𝓁 ) 𝓁 ∈ (cid:74) ,𝑁 (cid:75) : 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) = 0 . 𝑅 𝑠 Lemma 6.
Under the B-EDD model and condition 𝑎 + 𝑏 < 𝑑 𝑠 + , 𝑅 𝑠 ( 𝑈 , 𝑉 )∕ √ 𝕍 ( 𝐹 𝑠 ) | 𝑈 , 𝑉 → a.s. as 𝑚 ∼ 𝑛 → ∞ , where 𝑅 𝑠 ( 𝑈 , 𝑉 ) = 𝑐 𝑠 ∑ 𝑁 𝓁 =1 ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) − 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 )} .Proof. The proof consists in showing the two following assertions:(A1) 𝔼 ( 𝑅 𝑠 ( 𝑈 , 𝑉 )∕ √ 𝕍 ( 𝐹 𝑠 ) | 𝑈 , 𝑉 ) = 0 ;(A2) 𝕍 ( 𝑐 𝑠 𝑅 𝑠 ( 𝑈 , 𝑉 )∕ √ 𝕍 ( 𝑁 𝑠 ) | 𝑈 , 𝑉 ) → almost surely as 𝑛 tends to infinity under con-dition 𝑎 + 𝑏 < 𝑑 𝑠 + .Let show assertion (A1): 𝔼 ( 𝑅 𝑠 ( 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ) = 𝔼 ⎛⎜⎜⎝ 𝑐 𝑠 𝑁 ∑ 𝓁 =1 ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) − 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 )} | 𝑈 , 𝑉 ⎞⎟⎟⎠ = 𝔼 ⎛⎜⎜⎝ 𝑐 𝑠 𝑁 ∑ 𝓁 =1 ∑ 𝛼 ∈ 𝑠, 𝓁 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) ⎞⎟⎟⎠ − 1 𝑐 𝑠 𝑁 ∑ 𝓁 =1 ∑ 𝛼 ∈ 𝑠, 𝓁 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 )= 1 𝑐 𝑠 ∑ 𝛼 =( 𝛼 𝑡 ,𝛼 𝑏 )∈ 𝑠,𝑁 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝑈 , 𝑉 ) − 1 𝑐 𝑠 ∑ 𝛼 =( 𝛼 𝑡 ,𝛼 𝑏 )∈ 𝑠,𝑁 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 ) = 0 . et focus now on assertion (A2). Let first observe that, 𝕍 ( 𝑅 𝑠 ( 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ) = 𝕍 ⎛⎜⎜⎝ 𝑐 𝑠 𝑁 ∑ 𝓁 =1 ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) − 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 )} | 𝑈 , 𝑉 ⎞⎟⎟⎠ = 𝑁 ∑ 𝓁 =1 𝕍 ⎛⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠, 𝓁 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ⎞⎟⎟⎠ , by independance of successive choices of 𝓁 . Using definition (4) of the indicator motif,we see that 𝑁 ∑ 𝓁 =1 𝕍 ⎛⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠, 𝓁 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ⎞⎟⎟⎠ = 𝑁 ∑ 𝓁 =1 𝕍 ⎛⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠, 𝓁 𝔼 ( ∏ 𝑖 ∈ 𝛼 𝑡 ,𝑗 ∈ 𝛼 𝑏 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 | 𝓁 −1 ; 𝑈 , 𝑉 ) |
𝑈 , 𝑉 ⎞⎟⎟⎠ . Then according to measurability with respect to 𝓁 −1 and the position (top or bottom) of 𝑘 𝓁 the last selected node, we get 𝑁 ∑ 𝓁 =1 𝕍 ⎛⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠, 𝓁 𝔼 ( ∏ 𝑖 ∈ 𝛼 𝑡 ,𝑗 ∈ 𝛼 𝑏 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 | 𝓁 −1 ; 𝑈 , 𝑉 ) |
𝑈 , 𝑉 ⎞⎟⎟⎠ = ℙ ( 𝑇 𝓁 ) 𝑁 ∑ 𝓁 =1 𝕍 ⎛⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠, 𝓁 ( ∏ 𝑗 ∈ 𝛼 𝑏 𝔼 ( 𝐺 𝑘 𝓁 𝑗 | 𝑈 , 𝑉 ) 𝐴 𝑠 ( 𝑘 𝓁 𝑗 ) ) ⎛⎜⎜⎝ ∏ 𝑖 ∈ 𝛼 𝑡 ⧵ 𝑘 𝓁 ,𝑗 ∈ 𝛼 𝑏 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 ⎞⎟⎟⎠ | 𝑈 , 𝑉 ⎞⎟⎟⎠ + (1 − ℙ ( 𝑇 𝓁 )) 𝑁 ∑ 𝓁 =1 𝕍 ⎛⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠, 𝓁 (∏ 𝑖 ∈ 𝛼 𝑡 𝔼 ( 𝐺 𝑖𝑘 𝓁 | 𝑈 , 𝑉 ) 𝐴 𝑠 ( 𝑖𝑘 𝓁 ) ) ⎛⎜⎜⎝ ∏ 𝑖 ∈ 𝛼 𝑡 ,𝑗 ∈ 𝛼 𝑏 ⧵ 𝑘 𝓁 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 ⎞⎟⎟⎠ | 𝑈 , 𝑉 ⎞⎟⎟⎠ , and using the usual notation of the conditional expectation of 𝐺 𝑖𝑗 ’s, we have 𝕍 ( 𝑅 𝑠 ( 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ) = ℙ ( 𝑇 𝓁 ) 𝑁 ∑ 𝓁 =1 𝕍 ⎛⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠, 𝓁 ( ∏ 𝑗 ∈ 𝛼 𝑏 𝜙 ( 𝑈 𝑘 𝓁 , 𝑉 𝑗 ) 𝐴 𝑠 ( 𝑘 𝓁 𝑗 ) ) ⎛⎜⎜⎝ ∏ 𝑖 ∈ 𝛼 𝑡 ⧵ 𝑘 𝓁 ,𝑗 ∈ 𝛼 𝑏 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 ⎞⎟⎟⎠ | 𝑈 , 𝑉 ⎞⎟⎟⎠ + (1 − ℙ ( 𝑇 𝓁 )) 𝑁 ∑ 𝓁 =1 𝕍 ⎛⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠, 𝓁 (∏ 𝑖 ∈ 𝛼 𝑡 𝜙 ( 𝑈 𝑖 , 𝑉 𝑘 𝓁 ) 𝐴 𝑠 ( 𝑖𝑘 𝓁 ) ) ⎛⎜⎜⎝ ∏ 𝑖 ∈ 𝛼 𝑡 ,𝑗 ∈ 𝛼 𝑏 ⧵ 𝑘 𝓁 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 ⎞⎟⎟⎠ | 𝑈 , 𝑉 ⎞⎟⎟⎠ . Then, considering the fact that 𝕍 ( ∑ 𝑖 𝑎 𝑖 𝑋 𝑖 ) ≤ (∑ 𝑖 𝑎 𝑖 √ 𝕍 ( 𝑋 𝑖 ) ) , we get 𝕍 ( 𝑅 𝑠 ( 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ) ≤ 𝑁 ∑ 𝓁 =1 ⎛⎜⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ 𝑠, 𝓁 ( ∏ 𝑗 ∈ 𝛼 𝑏 𝜙 ( 𝑈 𝑘 𝓁 , 𝑉 𝑗 ) 𝐴 𝑠 ( 𝑘 𝓁 𝑗 ) ) √√√√√ 𝕍 ⎛⎜⎜⎝ ∏ 𝑖 ∈ 𝛼 𝑡 ⧵ 𝑘 𝓁 ,𝑗 ∈ 𝛼 𝑏 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 | 𝑈 , 𝑉 ⎞⎟⎟⎠⎞⎟⎟⎟⎠ . rom now, we will work on the set ⊗ 𝑠, 𝓁 ⧵ (2) 𝑠, 𝓁 which contains only dependent pairs ofpositions. It follows from the Bernoulli conditional distribution of 𝐺 𝑖𝑗 combined with thefact that 𝑎 √ 𝑏 < √ 𝑎𝑏 when 𝑎 < , that 𝕍 ( 𝑅 𝑠 ( 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ) ≤ 𝑁 ∑ 𝓁 =1 ⎛⎜⎜⎜⎝ 𝑐 𝑠 ∑ 𝛼 ∈ ⊗ 𝑠, 𝓁 ⧵ (2) 𝑠, 𝓁 √ ∏ 𝑗 ∈ 𝛼 𝑏 𝜙 ( 𝑈 𝑘 𝓁 , 𝑉 𝑗 ) 𝐴 𝑠 ( 𝑘 𝓁 𝑗 ) ∏ 𝑖 ∈ 𝛼 𝑡 ⧵ 𝑘 𝓁 ,𝑗 ∈ 𝛼 𝑏 𝜙 ( 𝑈 𝑖 , 𝑉 𝑗 ) 𝐴 𝑠 ( 𝑖𝑗 ) ⎞⎟⎟⎟⎠ ≤ 𝑁 ∑ 𝓁 =1 𝑐 𝑠 ⎛⎜⎜⎜⎝ ∑ 𝛼 ∈ ⊗ 𝑠, 𝓁 ⧵ (2) 𝑠, 𝓁 √ 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 ) ⎞⎟⎟⎟⎠ ≤ 𝑐 𝑠 𝑁 ∑ 𝓁 =1 (| 𝑠, 𝓁 | − | (2) 𝑠, 𝓁 |) max 𝛼 ∈ ⊗ 𝑠, 𝓁 ⧵ (2) 𝑠, 𝓁 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 ) . In order to evaluate the right-hand side term of the above inequality, recall that 𝑐 𝑠 =Θ( 𝑚 𝑝 𝑠 𝑛 𝑞 𝑠 ) by (15), 𝜙 𝑠 = Θ ( 𝑚 − 𝑎𝑑 𝑠 + 𝑛 − 𝑏𝑑 𝑠 + ) by (17) and | 𝑠, 𝓁 | − | (2) 𝑠, 𝓁 | = Θ ( 𝓁 𝑝 𝑠 −3 𝑡 𝓁 𝑞 𝑠 𝑏 ) byLemma 5, 𝓁 𝑡 and 𝓁 𝑏 denoting respectively top and bottom nodes in 𝓁 . Thus, we get 𝑐 𝑠 𝑁 ∑ 𝓁 =1 (| 𝑠, 𝓁 | − | 𝑠, 𝓁 |) max 𝛼 ∈ ⊗ 𝑠, 𝓁 ⧵ (2) 𝑠, 𝓁 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 )= Θ ( 𝑚 −2 𝑝 𝑠 𝑛 −2 𝑞 𝑠 ) 𝑁 ∑ 𝓁 = 𝓁 𝑡 + 𝓁 𝑏 =1 Θ ( 𝓁 𝑝 𝑠 −3 𝑡 𝓁 𝑞 𝑠 𝑏 𝓁 − 𝑎𝑑 𝑠 + 𝑡 𝓁 − 𝑏𝑑 𝑠 + 𝑏 ) = Θ ( 𝑚 −2 𝑝 𝑠 𝑛 −2 𝑞 𝑠 ) 𝑁 ∑ 𝓁 = 𝓁 𝑡 + 𝓁 𝑏 =1 Θ ( 𝓁 𝑝 𝑠 +2 𝑞 𝑠 − 𝑎𝑑 𝑠 + − 𝑏𝑑 𝑠 + −3 ) = Θ ( 𝑁 − 𝑎𝑑 𝑠 + − 𝑏𝑑 𝑠 + −3 ) . By taking the normalization √ 𝕍 ( 𝐹 𝑠 ) = √ 𝕍 ( 𝑁 𝑠 )∕ 𝑐 𝑠 which order is Θ ( max ( 𝑁 −2 𝑎𝑑 𝑠 + −2 𝑏𝑑 𝑠 + −1 , 𝑁 − 𝑎𝑑 𝑠 + − 𝑏𝑑 𝑠 + −2 )) by Lemma 2 and (15), we conclude to 𝕍 ( 𝑅 𝑠 ( 𝑈,𝑉 ) √ 𝕍 ( 𝐹 𝑠 ) | 𝑈 , 𝑉 ) → almost surely as 𝑛 tends toinfinity under condition 𝑎 + 𝑏 < 𝑑 𝑠 + . ■ 𝑀 𝑠 Lemma 7.
Under the B-EDD model and condition 𝑎 + 𝑏 < 𝑑 𝑠 + , 𝑀 𝑠 ( 𝑈 , 𝑉 )∕ √ 𝕍 ( 𝐹 𝑠 ) | 𝑈 , 𝑉 𝐷 ←→ ( , 𝕍 ( 𝑁 𝑠 | 𝑈 , 𝑉 ) 𝕍 ( 𝑁 𝑠 ) ) ) , as 𝑚 ∼ 𝑛 → ∞ , where 𝑀 𝑠 ( 𝑈 , 𝑉 ) = 𝑐 𝑠 ∑ 𝑁 𝓁 =1 ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝑌 𝑠 ( 𝛼 ) − 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 )} . roof. We will apply the following martingale central limit theorem to the conditional mar-tingale difference sequence 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) = ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝑌 𝑠 ( 𝛼 ) − 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 )} with re-spect to ( 𝓁 ) 𝓁 ∈ (cid:74) ,𝑁 (cid:75) . Theorem 3 ([Hall and Heyde, 2014]) . Suppose that for every 𝑛 ∈ ℕ and 𝑘 𝑛 → ∞ the ran-dom variables 𝑋 𝑛, , … , 𝑋 𝑛,𝑘 𝑛 are a martingale difference sequence relative to an arbitraryfiltration 𝑛, ⊂ 𝑛, ⊂ … ⊂ 𝑛,𝑘 𝑛 . If1. ∑ 𝑘 𝑛 𝑖 =1 𝔼 ( 𝑋 𝑛,𝑖 | 𝑛,𝑖 −1 ) → in probability,2. ∑ 𝑘 𝑛 𝑖 =1 𝔼 ( 𝑋 𝑛,𝑖 𝕀 { | 𝑋 𝑛,𝑖 | > 𝜖 } | 𝑛,𝑖 −1 ) → in probability for every 𝜖 > ,then ∑ 𝑘 𝑛 𝑖 =1 𝑋 𝑛,𝑖 ←→ (0 , . Here 𝑋 𝑛,𝑖 and 𝑛,𝑖 would be 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 )∕( 𝑐 𝑠 √ 𝕍 ( 𝐹 𝑠 )) and 𝓁 respectively, and we haveto verify the two following conditions:(C1) 𝕍 ( 𝑁 𝑠 ) ∑ 𝑁 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) → 𝕍 ( 𝑁 𝑠 | 𝑈,𝑉 ) 𝕍 ( 𝑁 𝑠 ) in probability,(C2) 𝕍 ( 𝑁 𝑠 ) ∑ 𝑁 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) 𝕀 { | 𝑀 𝑠, 𝓁 ( 𝑈,𝑉 ) |√ 𝕍 ( 𝑁 𝑠 ) > 𝜖 } | 𝓁 −1 ; 𝑈 , 𝑉 ) → in probability forevery 𝜖 > .Let verify condition (C1). First observe that it follows from properties of martingale dif-ferences, meaning variance decomposition, null conditional expectation and conditional or-thogonality of differences, that 𝕍 ( 𝑁 ∑ 𝓁 =1 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ) = 𝔼 [ 𝕍 ( 𝑁 ∑ 𝓁 =1 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 )] + 𝕍 [ 𝔼 ( 𝑁 ∑ 𝓁 =1 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 )] = 𝔼 [ 𝕍 ( 𝑁 ∑ 𝓁 =1 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 )] = 𝔼 ⎡⎢⎢⎣ 𝔼 ⎛⎜⎜⎝( 𝑁 ∑ 𝓁 =1 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) ) | 𝓁 −1 ; 𝑈 , 𝑉 ⎞⎟⎟⎠⎤⎥⎥⎦ = 𝔼 ( 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 )) , and further notice that 𝕍 (∑ 𝑁 𝓁 =1 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ) = 𝕍 ( 𝑐 𝑠 𝑀 𝑠 ( 𝑈 , 𝑉 ) | 𝑈 , 𝑉 ) . Since 𝑀 𝑠 = 𝐿 𝑠 − 𝑅 𝑠 (see Section 5.2.1), 𝔼 ( 𝕍 ( 𝑁 𝑠 ) 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) ) = 𝕍 ( 𝑐 𝑠 𝐿 𝑠 ( 𝑈 , 𝑉 ) − 𝑅 𝑠 ( 𝑈 , 𝑉 ) √ 𝕍 ( 𝑁 𝑠 ) | 𝑈 , 𝑉 ) → 𝕍 ( 𝑁 𝑠 | 𝑈 , 𝑉 )∕ 𝕍 ( 𝑁 𝑠 ) , as 𝑛 → ∞ , in probability and under condition 𝑎 + 𝑏 < 𝑑 𝑠 + , because 𝕍 ( 𝑐 𝑠 𝐿 𝑠 ( 𝑈,𝑉 ) √ 𝕍 ( 𝑁 𝑠 ) | 𝑈 , 𝑉 ) = 𝕍 ( 𝑁 𝑠 | 𝑈,𝑉 ) 𝕍 ( 𝑁 𝑠 ) and 𝕍 ( 𝑐 𝑠 𝑅 𝑠 ( 𝑈,𝑉 ) √ 𝕍 ( 𝑁 𝑠 ) | 𝑈 , 𝑉 ) → a.s. under condition 𝑎 + 𝑏 < 𝑑 𝑠 + by Lemma 6. ow, let verify condition (C2). First, by applying the Cauchy-Schwartz inequality, weget 𝕍 ( 𝑁 𝑠 ) 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) 𝕀 { | 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) |√ 𝕍 ( 𝑁 𝑠 ) > 𝜖 } | 𝓁 −1 ; 𝑈 , 𝑉 ) ≤ 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) 𝕍 ( 𝑁 𝑠 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) × 𝑁 ∑ 𝓁 =1 ℙ ( | 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) |√ 𝕍 ( 𝑁 𝑠 ) > 𝜖 | 𝓁 −1 ; 𝑈 , 𝑉 ) , then applying Bienaymé-Tchebychev inequality implies that 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) 𝕍 ( 𝑁 𝑠 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) × 𝑁 ∑ 𝓁 =1 ℙ ( | 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) |√ 𝕍 ( 𝑁 𝑠 ) > 𝜖 | 𝓁 −1 ; 𝑈 , 𝑉 ) ≤ 𝕍 ( 𝑁 𝑠 ) 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) × 1 𝜖 𝕍 ( 𝑁 𝑠 ) 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) , and by condition (C1), we get 𝕍 ( 𝑁 𝑠 ) 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) 𝕀 { | 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) |√ 𝕍 ( 𝑁 𝑠 ) > 𝜖 } | 𝓁 −1 ; 𝑈 , 𝑉 ) ≤ 𝜖 𝕍 ( 𝑁 𝑠 ) 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) × 𝕍 ( 𝑁 𝑠 | 𝑈 , 𝑉 ) 𝕍 ( 𝑁 𝑠 ) . Then, we use the following notation for expressing 𝑀 𝑠, 𝓁 : 𝑀 𝑠, 𝓁 = ∑ 𝛼 ∈ 𝑠, 𝓁 { 𝑌 𝑠 ( 𝛼 ) − 𝔼 ( 𝑌 𝑠 ( 𝛼 ) | 𝓁 −1 ; 𝑈 , 𝑉 )} = 𝑁 𝑠, 𝓁 − 𝔼 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) . By the binomial formula we thus have 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) = 𝔼 (( 𝑁 𝑠, 𝓁 − 𝔼 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) ) | 𝓁 −1 ; 𝑈 , 𝑉 ) = 𝔼 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) − 4 𝔼 ( 𝑁 𝑠, 𝓁 𝔼 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) +6 𝔼 ( 𝑁 𝑠, 𝓁 𝔼 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) −4 𝔼 ( 𝑁 𝑠, 𝓁 𝔼 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) + 𝔼 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) . Using the same arguments as in the proof of Lemma 6, observe that 𝐸 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) ≤ ∑ 𝛼 ∈ 𝑠, 𝓁 ( ∏ 𝑗 ∈ 𝛼 𝑏 𝜙 ( 𝑈 𝑘 𝓁 , 𝑉 𝑗 ) 𝐴 𝑠 ( 𝑘 𝓁 𝑗 ) ) 𝔼 ⎛⎜⎜⎝⎛⎜⎜⎝ ∏ 𝑖 ∈ 𝛼 𝑡 ,𝑗 ∈ 𝛼 𝑏 ⧵ 𝑘 𝓁 𝐺 𝐴 𝑠𝑖𝑗 𝑖𝑗 ⎞⎟⎟⎠ | 𝑈 , 𝑉 ⎞⎟⎟⎠ ≤ ∑ 𝛼 ∈ 𝑠, 𝓁 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 ) , and we have, 𝔼 ( 𝑁 𝑘𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) = 𝔼 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) + ∑ 𝑡 ∈ 𝑘 ( 𝑠 ) 𝔼 ( 𝑁 𝑡, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) + 𝔼 ( 𝑁 𝑠, 𝓁 | 𝓁 −1 ; 𝑈 , 𝑉 ) 𝑘 , here 𝑘 ( 𝑠 ) denotes here the set of supermotifs of 𝑠 which are here formed by 𝑘 overlappingoccurrences of 𝑠 . Finally, we get 𝕍 ( 𝑁 𝑠 ) 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) 𝕀 { | 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) |√ 𝕍 ( 𝑁 𝑠 ) > 𝜖 } | 𝓁 −1 ; 𝑈 , 𝑉 ) ≤ 𝜖 𝕍 ( 𝑁 𝑠 ) 𝑁 ∑ 𝓁 =1 𝔼 ( 𝑀 𝑠, 𝓁 ( 𝑈 , 𝑉 ) | 𝓁 −1 ; 𝑈 , 𝑉 ) × 𝕍 ( 𝑁 𝑠 | 𝑈 , 𝑉 ) 𝕍 ( 𝑁 𝑠 ) ≤ 𝜖 𝕍 ( 𝑁 𝑠 ) 𝑁 ∑ 𝓁 =1 | 𝑠, 𝓁 | × ( max 𝛼 ∈ 𝑠, 𝓁 𝜙 𝑠 ( 𝑈 𝛼 𝑡 , 𝑉 𝛼 𝑏 ) ) × 𝕍 ( 𝑁 𝑠 | 𝑈 , 𝑉 ) 𝕍 ( 𝑁 𝑠 ) . Condition (C2) holds since 𝕍 ( 𝑁 𝑠 ) = Θ ( max ( 𝑁 𝑝 𝑠 +4 𝑞 𝑠 −4 𝑎𝑑 𝑠 + −4 𝑏𝑑 𝑠 + −2 , 𝑁 𝑝 𝑠 +4 𝑞 𝑠 −2 𝑎𝑑 𝑠 + −2 𝑏𝑑 𝑠 + −4 )) by Lemma 3, | 𝑠, 𝓁 | = Θ ( 𝓁 𝑝 𝑠 −4 𝑡 𝓁 𝑞 𝑠 𝑏 ) (see the proof of Lemma (5)), 𝜙 𝑠 = Θ ( 𝑁 −4 𝑎𝑑 𝑠 + −4 𝑏𝑑 𝑠 + ) by (17) and 𝕍 ( 𝑁 𝑠 | 𝑈 , 𝑉 )∕ 𝕍 ( 𝑁 𝑠 ) = Θ(1) by Lemma 4. ■ Proof.
Let show that ( 𝐹 𝑠 − 𝜙 𝑠 )∕ √ 𝕍 ( 𝐹 𝑠 ) → a.s. as 𝑛 → ∞ under the B-EDD model andcondition 𝑎 + 𝑏 < 𝑑 𝑠 + ruling the graph density. Recall (8) the definition of 𝐹 𝑠 : 𝐹 𝑠 = ∏ 𝑝 𝑠 𝑢 =1 Γ 𝑑 𝑠𝑢 ∏ 𝑞 𝑠 𝑣 =1 Λ 𝑒 𝑠𝑣 𝐹 𝑑 𝑠 + , where Γ 𝑑 (resp Λ 𝑑 ) denote the normalized empirical frequencies of the top (resp bottom)star motif with degree 𝑑 and 𝐹 the one of the edge.Let begin with a Taylor expansion of order 1 of 𝐹 𝑠 in parameters ( 𝛾, 𝜆, 𝜙 ) denoting thetop star motif, bottom star motif and edge probabilities respectively: 𝐹 𝑠 (Γ , Λ , 𝐹 ) = 𝐹 𝑠 ( 𝛾, 𝜆, 𝜙 ) + 𝜕𝐹 𝑠 ( 𝛾, 𝜆, 𝜙 ) ( (Γ , Λ , 𝐹 ) − ( 𝛾, 𝜆, 𝜙 ) ) + 𝑜 ( (Γ , Λ , 𝐹 ) − ( 𝛾, 𝜆, 𝜙 ) ) = 𝜙 𝑠 + 𝜙 𝑠 𝜕 log( 𝐹 𝑠 ( 𝛾, 𝜆, 𝜙 )) ( (Γ , Λ , 𝐹 ) − ( 𝛾, 𝜆, 𝜙 ) ) + 𝑜 ( (Γ , Λ , 𝐹 ) − ( 𝛾, 𝜆, 𝜙 ) ) = 𝜙 𝑠 + 𝜙 𝑠 { 𝑝 𝑠 ∑ 𝑢 =1 𝛾 𝑑 𝑠𝑢 (Γ 𝑑 𝑠𝑢 − 𝛾 𝑑 𝑠𝑢 ) + 𝑞 𝑠 ∑ 𝑣 =1 𝜆 𝑒 𝑠𝑣 (Λ 𝑒 𝑠𝑣 − 𝜆 𝑒 𝑠𝑣 ) − 𝑑 + 𝜙 ( 𝐹 − 𝜙 ) } + 𝑜 ( Γ − 𝛾, Λ − 𝜆, 𝐹 − 𝜙 ) ) . Given the two following observations: i) the asymptotic normality of ( 𝐹 𝑠 − 𝜙 𝑠 )∕ √ 𝕍 ( 𝐹 𝑠 ) holds for any motif 𝑠 , including star motifs, under the B-EDD model and condition 𝑎 + 𝑏 < 𝑑 𝑠 + by Proposition 2, ii) the empirical frequencies of motifs converge to the expected onesby the law of large numbers, we get 𝐹 𝑠 − 𝜙 𝑠 √ 𝕍 ( 𝐹 𝑠 )= 𝑝 𝑠 ∑ 𝑢 =1 Θ ⎛⎜⎜⎝ 𝜙 𝑠 𝛾 𝑑 𝑠𝑢 √ 𝕍 (Γ 𝑑 𝑠𝑢 ) 𝕍 ( 𝐹 𝑠 ) ⎞⎟⎟⎠ + 𝑞 𝑠 ∑ 𝑣 =1 Θ ⎛⎜⎜⎝ 𝜙 𝑠 𝜆 𝑒 𝑠𝑣 √ 𝕍 (Λ 𝑒 𝑠𝑣 ) 𝕍 ( 𝐹 𝑠 ) ⎞⎟⎟⎠ + Θ ⎛⎜⎜⎝ 𝜙 𝑠 𝜙 √ 𝕍 ( 𝐹 ) 𝕍 ( 𝐹 𝑠 ) ⎞⎟⎟⎠ + 𝑜 (1)= 𝑝 𝑠 ∑ 𝑢 =1 Θ ⎛⎜⎜⎝ 𝜙 𝑠 𝑐 𝑠 𝛾 𝑑 𝑠𝑢 𝑐 𝛾 √ 𝕍 ( 𝑁 Γ 𝑑𝑠𝑢 ) 𝕍 ( 𝑁 𝑠 ) ⎞⎟⎟⎠ + 𝑞 𝑠 ∑ 𝑣 =1 Θ ⎛⎜⎜⎝ 𝜙 𝑠 𝑐 𝑠 𝜆 𝑒 𝑠𝑣 𝑐 𝜆 √ 𝕍 ( 𝑁 Λ 𝑒𝑠𝑣 ) 𝕍 ( 𝑁 𝑠 ) ⎞⎟⎟⎠ + Θ ⎛⎜⎜⎝ 𝜙 𝑠 𝑐 𝑠 𝜙 𝑐 √ 𝕍 ( 𝑁 ) 𝕍 ( 𝑁 𝑠 ) ⎞⎟⎟⎠ + 𝑜 (1) . ere and only here, 𝑁 Γ 𝑑 (resp. 𝑁 Λ 𝑑 ) and 𝑐 𝛾 (resp. 𝑐 𝜆 ) denote, by abuse of notation, thecount of top stars (resp. bottom stars) of degree 𝑑 and their number of positions in the graph.Considering only non-star motifs 𝑠 , according to the orders of magnitude of 𝑐 𝑠 , 𝜙 𝑠 and 𝕍 ( 𝑁 𝑠 ) given in (15), (17) and Lemma 3 respectively, we conclude to ( 𝐹 𝑠 − 𝜙 𝑠 )∕ √ 𝕍 ( 𝐹 𝑠 ) → a.s.as 𝑛 → ∞ because −2 𝑑 ( 𝑎 + 𝑏 ) < , with 𝑑 = 𝑑 𝑠𝑢 , 𝑒 𝑠𝑢 or 1. ■ Proof.
Let show that ̂ 𝕍 ( 𝐹 𝑠 )∕ 𝕍 ( 𝐹 𝑠 ) → a.s., as 𝑛 → ∞ . First, observe that according to(18), we can write: 𝕍 ( 𝑁 𝑠 ) = ∑ 𝑡 ∈{ 𝑠 }∪ ( 𝑠 ) 𝔼 ( 𝑁 𝑡 ) − 𝔼 ( 𝑁 𝑠 ) = 𝑐 𝑠 𝜙 𝑠 + ∑ 𝑡 ∈ ( 𝑠 ) 𝑐 𝑡 𝜙 𝑡 − 𝑐 𝑠 𝜙 𝑠 , where ( 𝑠 ) denotes here the set of super-motifs of 𝑠 which are formed by two overlappingoccurrences of 𝑠 . Then considering ̂ 𝕍 ( 𝑁 𝑠 ) its plug-in version, meaning 𝐹 𝑠 replaces 𝜙 𝑠 , weget ̂ 𝕍 ( 𝑁 𝑠 ) − 𝕍 ( 𝑁 𝑠 ) = 𝑐 𝑠 ( 𝐹 𝑠 − 𝜙 𝑠 ) + ∑ 𝑡 ∈ ( 𝑠 ) 𝑐 𝑡 ( 𝐹 𝑡 − 𝜙 𝑡 ) − 𝑐 𝑠 ( 𝐹 𝑠 − 𝜙 𝑠 ) . Now we use Lemma 1 stating that, under the B-EDD model and condition 𝑎 + 𝑏 < 𝑑 𝑠 + , 𝐹 𝑠 − 𝜙 𝑠 = 𝑜 ( √ 𝕍 ( 𝐹 𝑠 )) for all motif 𝑠 and the continuous mapping theorem, to obtain that ̂ 𝕍 ( 𝑁 𝑠 ) − 𝕍 ( 𝑁 𝑠 ) = 𝑐 𝑠 𝑜 ( √ 𝕍 ( 𝐹 𝑠 )) + ∑ 𝑡 ( 𝑠 ) 𝑐 𝑡 𝑜 ( √ 𝕍 ( 𝐹 𝑡 )) − 𝑐 𝑠 𝑜 ( 𝕍 ( 𝐹 𝑠 ))= 𝑜 ( √ 𝕍 ( 𝑁 𝑠 )) + ∑ 𝑡 ∈ ( 𝑠 ) 𝑜 ( √ 𝕍 ( 𝑁 𝑡 )) − 𝑜 ( 𝕍 ( 𝑁 𝑠 )) . (20)Let discuss now the order of ( ̂ 𝕍 ( 𝑁 𝑠 ) − 𝕍 ( 𝑁 𝑠 ))∕ 𝕍 ( 𝑁 𝑠 ) . The first and last terms of (20)divided by 𝕍 ( 𝑁 𝑠 ) obviously vanish. When 𝑡 ∈ ( 𝑠 ) , we refer to the order of magnitude of 𝕍 ( 𝑁 𝑠 ) given in Lemma 3 and its proof (see ( 𝑖 )-( 𝑖𝑖 )-( 𝑖𝑖𝑖 )) to get that √ 𝕍 ( 𝑁 𝑡 )∕ 𝕍 ( 𝑁 𝑠 ) vanishesunder condition 𝑎 + 𝑏 < ( 𝑝 𝑠 + 𝑞 𝑠 )∕ 𝑑 𝑠 + . We can finally conclude to ̂ 𝕍 ( 𝐹 𝑠 )∕ 𝕍 ( 𝐹 𝑠 ) → a.s., as 𝑛 → ∞ under condition of Theorem 1. ■ References
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Asymptotic statistics , volume 3. Cambridge university press, 2000. ⚪◻ 𝑐 𝑠 𝑚𝑛𝜙 𝑠 𝜙 𝑠 ⚪◻ ◻ ⚪ ⚪◻ 𝑐 𝑠 𝑚 ( 𝑛 ) 𝑛 ( 𝑚 ) 𝜙 𝑠 𝛾 𝜆 𝑠 ⚪ ⚪ ⚪◻ ⚪ ⚪◻ ◻ ⚪ ⚪◻ ◻ ⚪◻ ◻ ◻ 𝑐 𝑠 𝑛 ( 𝑚 ) ( 𝑚 )( 𝑛 ) ( 𝑚 )( 𝑛 ) 𝑚 ( 𝑛 ) 𝜙 𝑠 𝜆 𝛾 𝜆 ∕ 𝜙 𝛾 𝜆 ∕ 𝜙 𝛾 𝑠 ⚪ ⚪ ⚪ ⚪◻ ⚪ ⚪ ⚪◻ ◻ ⚪ ⚪ ⚪◻ ◻ ⚪ ⚪ ⚪◻ ◻ ⚪ ⚪ ⚪◻ ◻ 𝑐 𝑠 𝑛 ( 𝑚 ) ( 𝑚 )( 𝑛 ) ( 𝑚 )( 𝑛 ) ( 𝑚 )( 𝑛 ) ( 𝑚 )( 𝑛 ) 𝜙 𝑠 𝜆 𝛾 𝜆 ∕ 𝜙 𝛾 𝜆 ∕ 𝜙 𝛾 𝜆 𝜆 ∕ 𝜙 𝛾 𝜆 ∕ 𝜙 𝑠
13 14 15 16 17 ⚪ ⚪◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ⚪◻ ◻ ◻ ◻ 𝑐 𝑠 ( 𝑚 )( 𝑛 ) ( 𝑚 )( 𝑛 ) ( 𝑚 )( 𝑛 ) ( 𝑚 )( 𝑛 ) 𝑚 ( 𝑛 ) 𝜙 𝑠 𝛾 𝜆 ∕ 𝜙 𝛾 𝜆 ∕ 𝜙 𝛾 𝛾 𝜆 ∕ 𝜙 𝛾 𝜆 ∕ 𝜙 𝛾 Figure 7: Bipartite motifs of size 2, 3, 4 and 5 as given in Simmons et al. [2019b].30
18 19 20 21 22 23 ⚪ ⚪ ⚪ ⚪ ⚪◻ ⚪ ⚪ ⚪ ⚪◻ ◻ ⚪ ⚪ ⚪ ⚪◻ ◻ ⚪ ⚪ ⚪ ⚪◻ ◻ ⚪ ⚪ ⚪ ⚪◻ ◻ ⚪ ⚪ ⚪ ⚪◻ ◻ 𝑠
24 25 26 27 28 29 ⚪ ⚪ ⚪ ⚪◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ 𝑠
30 31 32 33 34 35 ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ 𝑠
36 37 38 39 40 41 ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪ ⚪◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ◻ 𝑠
42 43 44 ⚪ ⚪◻ ◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ◻ ⚪◻ ◻ ◻ ◻⚪ ⚪◻ ◻ ◻ ◻ ⚪ ⚪◻ ◻ ◻ ◻ ⚪◻ ◻ ◻ ◻