Limit Laws for the Distance to Fréchet Means of Random Graphs

Limit La ws for the Distance to F r ´ ec het Means of Random Graphs Qunqiang F eng, Zixin T ang, and Zhish ui Hu ∗ Departmen t of Statistics and Finance, Sc ho ol of Managemen t Univ ersity of Science and T echnology of China Hefei 230026, China Abstract This pap er inv estigates the F r´ ec het mean of the Erd˝ os-R ´ en yi random graph G n,p with resp ect to the F rob enius distance on graph Laplacians, a metric that captures global structural information beyond lo cal edge flips. W e first c haracterize the F r ´ echet mean set as consisting of quasi-regular graphs (i.e., graphs where all vertex degrees differ by at most one). W e then analyze the asymptotic b ehavior of the F rob enius distance F n = d F ( G n,p , R ) as n → ∞ , where R is an y F r´ echet mean. Closed-form expressions for the mean and v ariance of F 2 n are derived, whic h are in v ariant to the choice of R . Leveraging these results, w e establish sev eral w eak con vergence la ws for the F rob enius distance ov er all regimes of p ∈ (0 , 1) as n → ∞ . Finally , under the scaling condition n 2 p (1 − p ) → ∞ we pro ve the asymptotic normality of this distance, whic h exhibits a phase transition gov erned by the growth rate of np (1 − p ). Our results rev eal ho w metric selection fundamen tally shap es F r´ echet mean geometry in random graphs. Keywor ds: Erd˝ os-R ´ enyi random graph; F rob enius metric; regular graph; dep endency graph; Stein’s metho d AMS 2020 Subje ct Classific ation : Primary 05C80, 60C05; secondary 60F05 1 In tro duction The F r´ echet me an , in tro duced b y Maurice F r´ echet [ 21 ], generalizes the concept of exp ectation from Euclidean space to general metric spaces ( X , d ). It is defined as an y minimizer of the F r ´ echet function f µ ( x ) = Z X d 2 ( x, y )d µ ( y ) , x ∈ X , (1) where µ is a probability measure on X . The existence and uniqueness of the F r´ echet mean dep end on the top ological prop erties of the underlying metric space [ 25 , 23 , 12 ]. When uniqueness fails, all the minimizers usually form a closed set [ 18 ], termed the F r ´ echet me an set . Owing to its applicability to non-Euclidean data, the F r´ echet mean has gained significan t traction in probabilit y theory [ 4 , 32 , 18 ], statistics [ 5 , 15 , 31 , 28 , 29 , 1 , 34 ], and machine learning [ 38 , 10 ]. It has found particular applications in netw ork data analysis [ 22 , 26 , 27 ], where graph-v alued data inherently resides in non-Euclidean spaces. Mey er [ 30 ] recently derived the F r´ echet mean for inhomogeneous Erd˝ os-R ´ en yi random graphs [ 8 ] using the Hamming distance (defined via adjacency matrices; see the next section). F or the homogeneous Erd˝ os-R´ en yi random graph G n,p on n v ertices with edge probabilit y p , an immediate consequence of Mey er [ 30 ] is that the F r´ echet mean of G n,p is the empty graph when p ≤ 1 / 2 and the complete graph otherwise. Ho wev er, these extreme graphs lie on the “b oundary” of the graph space and fail to capture t ypical top ological features of G n,p for general p ∈ (0 , 1). While the Hamming distance measures lo cal edge flips and is sensitive to sparsity [ 16 ], the F rob enius distance defined through graph Laplacians incorporates global structural information ∗ Email: huzs@ustc.edu.cn 1 b ey ond local connectivity [ 14 ]. As this distance is increasingly used in netw ork analysis [ 38 , 11 ], in this pap er w e prop ose studying the F r ´ ec het mean of Erd˝ os-R´ enyi random graphs under the F rob enius metric. In contrast to the results in Meyer [ 30 ], w e establish that under the F rob enius distance, the F r´ echet mean of G n,p is the empt y graph only if np < 1, and the complete graph only if n (1 − p ) < 1. F urthermore, for any positive integer n and p ∈ (0 , 1), the F r ´ ec het mean set of G n,p comprises quasi-regular graphs (see remarks following Theorem 1 ). Although uniqueness do es not alw ays hold, w e additionally c haracterize the asymptotic b eha vior of the F rob enius distance b et ween G n,p and elemen ts of its F r ´ ec het mean set, as the graph size n tends to infinit y . T o this end, w e first deriv e an explicit expression for the F r´ echet function of G n,p in terms of vertex degrees, whic h reduces the problem to minimizing a separable quadratic form. By analyzing the resulting degree sequences, we obtain a complete description of the F r´ echet mean set. The c haracterization reveals a ric h structure: when np is an integer, an y ( np − 1)-regular or np -regular graph can b e a F r´ ec het mean; otherwise, the F r ´ ec het mean set consists of nearly regular graphs whose degrees differ by at most one. The main technical con tribution of the pap er is a detailed analysis of the asymptotic distribution of the F rob enius distance F n b et w een G n,p and any F r´ echet mean graph of it. The limiting b ehavior exhibits a phase transition go verned b y the growth rate of np (1 − p ). When np (1 − p ) → 0 but n 2 p (1 − p ) → ∞ (v ery sparse regime) or np (1 − p ) → c > 0 (sparse regime), the squared distance F 2 n con verges to a normal distribution after appropriate centering and scaling; we prov e this using a dependency graph approac h. In the dense regime where np (1 − p ) → ∞ , the dep endency graph b ecomes to o dense for a direct application, and we instead employ a refined normal approximation method based on discrete-difference Stein b ounds recen tly dev elop ed by Shao and Zhang [ 33 ]. This allows us to establish asymptotic normalit y for F 2 n with an explicit v ariance form ula. Applying the delta metho d then yields the corresp onding limit la ws for F n itself. Our results sho w that the F rob enius distance behav es differen tly from the Hamming distance, capturing the in trinsic global structure of Erd˝ os-R´ enyi random graphs. Throughout this pap er, we shall use the follo wing notation. F or any real n umber x , let ⌊ x ⌋ denote its flo or function (i.e., the greatest integer less than or equal to x ), and let { x } = x − ⌊ x ⌋ represen t its fractional part. F or t wo sequences of p ositiv e n umbers a n and b n , w e write a n ∼ b n if a n /b n → 1 as n → ∞ . W e write a n = Θ( b n ) if there exist constants 0 < c 1 ≤ c 2 < ∞ such that c 1 ≤ lim inf n →∞ a n b n ≤ lim sup n →∞ a n b n ≤ c 2 . W e denote by diag( x 1 , x 2 , . . . , x n ) the diagonal matrix whose i -th diagonal en try is x i for eac h i = 1 , 2 , . . . , n . F or an n × n matrix A = ( a ij ), the F rob enius norm of A is defined as ∥ A ∥ F = ( P n i =1 P n j =1 a 2 ij ) 1 / 2 . F or probabilistic con vergence, let D − → and P − → denote conv ergence in distribution and con vergence in probability , resp ectiv ely . The rest of the pap er is organized as follo ws. Section 2 reformulates the F r´ echet function for G n,p via vertex degrees and characterizes its F r´ ec het mean set as consisting of quasi-regular graphs. Section 3 deriv es the mean and v ariance of the squared F rob enius distance b et ween G n,p and its F r´ echet means, and establishes w eak conv ergence results with a phase transition driv en b y n 2 p (1 − p ). Section 4 prov es the asymptotic normality of this distance under the condition n 2 p (1 − p ) → ∞ : a dep endency graph metho d is used for the very sparse and sparse regimes, while a refined Stein b ounds-based normal approximation is adopted for the dense regime; the delta metho d then yields the corresp onding asymptotic normality for the F rob enius distance itself. 2 F r ´ ec het mean set for Erd˝ os-R ´ en yi random graphs W e b egin b y defining a metric space on the set G n of all simple graphs with vertex set [ n ] := { 1 , 2 , . . . , n } , where n ≥ 2 is a fixed in teger. Clearly , the space G n has cardinality |G n | = 2 ( n 2 ) . Here we consider t wo fundamen tal matrices for graphs: the adjacency matrix and the Laplacian matrix [ 14 ]. F or a graph G ∈ G n , its adjac ency matrix A = ( a ij ) is the n × n matrix defined b y a ij =  1 , if there is an edge b et w een vertices i and j in G ; 0 , otherwise , 2 with a ii = 0 for all i ∈ [ n ]. The L aplacian matrix L = ( l ij ) of G is given b y L = D − A, where D = diag( D 1 , D 2 , . . . , D n ) is the degree matrix with D i = P n j =1 a ij recording the degree of vertex i . F rom the basic knowledge of graph theory , there exist bijective corresp ondences b et ween G , A , and L . Consequen tly , w e can equip G n with matrix-induced distances . The F rob enius distance is arguably the simplest of these metrics, and it is a common c hoice for the metrics on the space of graph Laplacians (see, e.g., [ 38 ]). The F r ob enius distanc e b et ween graphs G, G ′ ∈ G n can b e defined using adjacency matrices, d A F ( G, G ′ ) = ∥ A − A ′ ∥ F =   X i,j ∈ [ n ]  a ij − a ′ ij  2   1 2 , (2) or Laplacian matrices, d L F ( G, G ′ ) = ∥ L − L ′ ∥ F =   X i,j ∈ [ n ]  l ij − l ′ ij  2   1 2 , (3) where A ′ and L ′ corresp ond to G ′ . Since all entries in the adjacency matrix are equal to 0 or 1, we hav e | a ij − a ′ ij | = ( a ij − a ′ ij ) 2 for every pair ( i, j ). Hence, the Hamming distance betw een G and G ′ satisfies d H ( G, G ′ ) = 1 2 X i,j ∈ [ n ] | a ij − a ′ ij | = 1 2  d A F ( G, G ′ )  2 , b y ( 2 ). It thus follows that d A F and d H are equiv alent metrics on G n . As describ ed in the introduction, we will work exclusively with the Laplacian-based distance ( 3 ), and use the notation d F for brevity (omitting the sup erscript L ). F or any G ∈ G n , let G ∈ G n denote its complement graph, i.e., an y tw o distinct vertices are adjacen t in G if and only if they are not adjacent in G . Since the F rob enius norm is inv ariant under sign change, one can obtain the symmetry d F ( G, G ′ ) = d F ( G, G ′ ) . (4) Define the space of adjacency matrices as A n =  A ∈ { 0 , 1 } n × n : a ii = 0 , a ij = a j i for all i, j ∈ [ n ]  . Let G n,p b e the Erd˝ os-R ´ en yi random graph on [ n ] with edge probability p ∈ (0 , 1). Then, for any fixed graph G ∈ G n with adjacency matrix A = ( a ij ) ∈ A n , the probability mass function of G n,p is P ( G n,p = G ) = Y 1 ≤ i 1 2 ; ⌊ np ⌋ + 1 or ⌊ np ⌋ − 1 , if { np } = 1 2 ; ⌊ np ⌋ − 1 , if { np } < 1 2 . Remark 1. The pro of follo ws directly from the discussion ab o ve: the ob jective forces each degree to b e as close as p ossible to np − 1 2 , and the graphicality constrain ts reduce to the parity conditions enco ded in the three cases. The v erification that these degree sequences are realizable follows directly from the Erd˝ os-Gallai theorem (see, e.g., [ 35 ]), which gives a necessary and sufficien t condition for a non-negative integer sequence to b e graphical; the details are straightforw ard and thus omitted for brevity . Theorem 1 rev eals that every F r´ ec het mean of G n,p is a quasi-r e gular graph (i.e., all vertex degrees in it differ by at most one). This reflects the well-kno wn homogeneit y of the Erd˝ os-R´ enyi random graph. In particular, case (ii) asserts that whenever n ⌊ np ⌋ is even, ev ery ⌊ np ⌋ -regular graph (see [ 9 , 7 ]) is a F r ´ ec het mean of G n,p . F or enumeration results on regular graphs, we refer to [ 6 , 36 ]. Tw o extreme regimes are worth highligh ting: • If np < 1, then ⌊ np ⌋ = 0 and n ⌊ np ⌋ = 0 is ev en, so case (ii) applies and the only p ossible degree sequence is all zeros. Hence the F r´ echet mean set consists solely of the empt y graph. • If n (1 − p ) < 1 (i.e., p > 1 − 1 /n ), then np > n − 1. Consequen tly ⌊ np ⌋ = n − 1 and n ⌊ np ⌋ = n ( n − 1) is alw ays even, so again case (ii) applies and every vertex must hav e degree n − 1. Thus the only F r´ echet mean is the complete graph K n . 5 Apart from these t wo b oundary situations, the F r´ echet mean is never unique provided n is sufficiently large (t ypically n ≥ 4). This non-uniqueness follo ws from the fact that the degree sequences considered in the theorem can corresp ond to multiple non-isomorphic graphs. 3 First t w o momen ts of the squared F rob enius distance The purpose of this pap er is to in vestigate the asymptotic b eha vior of the F rob enius distance b et ween an Erd˝ os-R´ enyi random graph G n,p and an arbitrary elemen t of its F r´ ec het mean set as n → ∞ . In what follo ws, w e may only fo cus on the case where np is not an in teger and n ⌊ np ⌋ is even, which satisfies Theorem 1 (ii). T ypically , we let p = p ( n ) ∈ (0 , 1) dep end on n . F or ease of notation, we set m = ⌊ np ⌋ hereafter. Let R ∈ G n b e an arbitrary but fixed m -regular graph with adjacency matrix M = ( m ij ) ∈ A n . F rom Theorem 1 , the graph R is a F r´ echet mean of a random graph G n,p . Define the F rob enius distance to this mean as F n = d F ( G n,p , R ) . Although the distribution of F n ma y dep end on R , w e will show that the asymptotic distribution of F 2 n , as w ell as its mean and v ariance, do es not dep end on the particular choice of the m -regular graph R . Recall that A n,p = ( I ij ) and D n,p = diag( X 1 , X 2 , . . . , X n ) denote the adjacency and degree matrices of G n,p , resp ectively , where I ij is the edge indicator and X i = P j  = i I ij . Since each v ertex degree D i in R is equal to m , it follows b y ( 10 ) that the squared distance F 2 n = n X i =1  m 2 + m − 2( m − 1) X i − 2 X j  = i m ij I ij + X j  = i X k  = i,j I ij I ik  = m ( m + 1) n − 2 U n + W n , (11) where U n = n X i =1 X j  = i ( m − 1 + m ij ) I ij and W n = n X i =1 X j  = i X k  = i,j I ij I ik . Note that the sum in U n coun ts each undirected edge twice, so U n / 2 is the total weigh t of edges in G n,p with w eights m − 1 + m ij . Similarly , W n coun ts ordered triples ( i, j, k ) with j  = k , whic h corresponds to twice the num b er of wedges (a w edge is an unordered triple ( i, j, k ) such that both ij and ik are edges). F rom ( 11 ), one can see that it is more conv enient to study the squared distance F 2 n but not F n directly . F or the first t wo momen ts of F 2 n , we hav e the following. Prop osition 2. F or the squar e d F r ob enius distanc e F 2 n , E [ F 2 n ] = ( np − m )( np − m − 1) n + (3 n − 2) np (1 − p ) , (12) V ar( F 2 n ) = 2 np (1 − p )  4( np − m + 1 − 3 p ) 2 n + ( n 2 + n + 18) p (1 − p ) + (2 np − 2 m − 1) 2 − 5  . (13) Pr o of. W e derive the mean and v ariance of F 2 n via the first tw o moments of random v ector ( U n , W n ). Since P j  = i m ij = m for each i ∈ [ n ], E [ U n ] = n X i =1 X j  = i ( m − 1 + m ij ) p = n X i =1 [( n − 1)( m − 1) p + mp ] = ( mn − n + 1) np. (14) 6 Noting that there are 3  n 3  p ossible w edges in G n,p and each of them o ccurs with probabilit y p 2 , we hav e E [ W n ] = 2 · 3  n 3  p 2 = n ( n − 1)( n − 2) p 2 . (15) T aking exp ectations on b oth sides of ( 11 ) gives E [ F 2 n ] = m ( m + 1) n − 2 E [ U n ] + E [ W n ] . By ( 14 ) and ( 15 ), after straightforw ard calculations we ha ve E [ F 2 n ] = m ( m + 1) n − 2( mn − n + 1) np + n ( n − 1)( n − 2) p 2 =  ( np − m ) 2 − ( np − m ) + (3 n − 2) p (1 − p )  n, whic h prov es ( 12 ). Since each edge indicator I ij app ears t wice in U n , for the v ariance of U n w e hav e V ar( U n ) = 2 n X i =1 X j  = i ( m − 1 + m ij ) 2 p (1 − p ) = 2 p (1 − p ) n X i =1 X j  = i [( m − 1) 2 + (2 m − 1) m ij ] (b y m 2 ij = m ij ) = 2 np (1 − p )[( m − 1) 2 ( n − 1) + (2 m − 1) m ] = 2 np (1 − p )[ n ( m − 1) 2 + m 2 + m − 1] . (16) F or V ar( W n ), note that W n / 2 is the n umber of w edges, and any tw o wedge indicators are indep enden t if they do not share common edges. With a basic combinatorial argument, one can see that in a complete graph on n vertices, there are 3  n 3  (2 n − 5) pairs of w edges sharing exactly one edge. By symmetry and indep endence, w e thus ha ve V ar  W n 2  = 3  n 3  · V ar( I 12 I 13 ) + 2 · 3  n 3  (2 n − 5) · Co v ( I 12 I 13 , I 12 I 14 ) = 1 2 n ( n − 1)( n − 2) p 2 (1 − p 2 ) + n ( n − 1)( n − 2)(2 n − 5) p 3 (1 − p ) , whic h implies that V ar( W n ) = 2 n ( n − 1)( n − 2) p 2 (1 − p )[1 + (4 n − 9) p ] . (17) Similarly to ( 17 ), for the co v ariance we ha ve Co v ( U n , W n ) = 2 n X i =1 X j  = i X k  = i,j  Co v  ( m − 1 + m ij ) I ij , I ij I ik  + Co v  ( m − 1 + m ik ) I ik , I ij I ik  = 4( m − 1) · 6  n 3  Co v ( I 12 , I 12 I 13 ) + 2 n X i =1 X j  = i X k  = i,j ( m ij + m ik )Co v ( I 12 , I 12 I 13 ) = 4( m − 1) n ( n − 1)( n − 2) p 2 (1 − p ) + 2 p 2 (1 − p ) n X i =1 X j  = i X k  = i,j ( m ij + m ik ) = 4( m − 1) n ( n − 1)( n − 2) p 2 (1 − p ) + 4 p 2 (1 − p ) mn ( n − 2) = 4 n ( n − 2) p 2 (1 − p )( mn − n + 1) , (18) 7 where in the  p en ultimate equality w e used the following identit y: for each i ∈ [ n ], X j  = i X k  = i,j ( m ij + m ik ) = 2 m ( n − 2) , whic h, in fact, is due to that P j  = i m ij = m , and eac h term m ij ( j  = i ) app ears 2( n − 2) times in the double sum. Also by ( 11 ), we hav e V ar( F 2 n ) = 4V ar( U n ) + V ar( W n ) − 4Co v ( U n , W n ) . Substituting ( 16 )–( 18 ) into this yields V ar( F 2 n ) = 2 np (1 − p )  4( m − 1) 2 n + 4( m 2 + m − 1) − (8 mn − 9 n + 9)( n − 2) p + ( n − 1)( n − 2)(4 n − 9) p 2  , whic h, together with some basic algebra, implies ( 13 ). The first tw o moments of F 2 n giv en in ( 12 ) and ( 13 ) lead to the follo wing weak conv ergence results. Prop osition 3. L et F n b e the F r ob enius distanc e b etwe en G n,p and an arbitr ary F r´ echet me an gr aph. (i) If n 2 p (1 − p ) → 0 , then F n P − → 0 . (ii) If n 2 p (1 − p ) → λ for some c onstant λ > 0 , then F n D − → 2 r P oi  λ 2  , wher e Poi( λ 2 ) denotes a Poisson r andom variable with me an λ 2 . (iii) If n 2 p (1 − p ) → ∞ , then F n p E [ F 2 n ] P − → 1 . (19) In p articular, if we further have that np (1 − p ) c onver ges to a non-ne gative limit or diver ges to infinity, F n p n 2 p (1 − p ) P − → √ a, (20) wher e the c onstant a =          2 , if np (1 − p ) → 0; 2 +  c − ⌊ c ⌋  2 + ⌊ c ⌋ c , if np (1 − p ) → c for some c onstant c > 0; 3 , if np (1 − p ) → ∞ . (21) Pr o of. F or any F r´ echet mean R of G n,p , it follows b y ( 4 ) that F n = d F ( G n,p , R ) , where R is the complement graph of R . This implies that R is also a F r´ echet mean of G n,p . Consequently , without loss of generality we can alwa ys assume 0 < p ≤ 1 2 . Esp ecially , we may let p → 0 in (i) and (ii), since if n 2 p (1 − p ) → c for some constant c ≥ 0, the edge probabilit y p m ust tend to 0 or 1. It should also b e noted that, under the condition n 2 p → c < ∞ , the F r´ echet mean R is unique to b eing empty for sufficiently large n (see the comments b elo w Theorem 1 ). 8 When n 2 p → 0, we hav e mn → 0 and np → 0. It thus follows b y ( 12 ) that E [ F 2 n ] tends to 0. Therefore, the squared distance F 2 n con verges in probabilit y to 0, which prov es (i). Consider (ii). Since R is empty for sufficien tly large n (i.e., m = m ij = 0), we can simplify ( 11 ) to F 2 n = 4 E n + W n , where E n = P 1 ≤ i 0 such that V ar( F 2 n ) ( E [ F 2 n ]) 2 ≤ C np (1 − p )[ n + n 2 p (1 − p )] [ n 2 p (1 − p )] 2 = C n 2 p (1 − p ) + C n → 0 , whic h prov es ( 22 ) by Cheb yshev’s inequalit y . F urthermore, the conv ergence result ( 20 ) follows immediately from ( 19 ) and Slutsky’s Theorem. This completes the pro of of Prop osition 3 . Remark 2. F rom ( 25 ), one can conduct a refined order-of-magnitude analysis of the v ariance V ar( F 2 n ) under the condition that np (1 − p ) → c ≥ 0 (for a constan t c ) or np (1 − p ) → ∞ . More precisely , the asymptotic equiv alence of the v ariance is giv en by V ar( F 2 n ) ∼      8 n 2 p (1 − p ) , if np (1 − p ) → 0; 2 c  4( c − ⌊ c ⌋ + 1) 2 + c  n, if np (1 − p ) → c for some non-integer constant c > 0; 2 n 3 p 2 (1 − p ) 2 , if np (1 − p ) → ∞ . (26) It is critical to note that when c is a positive integer, the ratio V ar( F 2 n ) /n does not p ossess a unique limit, whic h we rigorously verify via the following asymptotic argument. First, supp ose that np is monotonically 9 decreasing and conv erges to a p ositive integer c . Since np is non-in teger for all n ≥ 1 by assumption, w e ha ve np > c for every n ≥ 1, and th us m = c for all sufficiently large n . Substituting m = c into ( 25 ) and taking the asymptotic limit, we obtain V ar( F 2 n ) n → 2 c (4 + c ) , as n → ∞ . Conv ersely , if np is monotonically increasing and conv erges to the p ositiv e integer c , w e ha ve m = ⌊ np ⌋ = c − 1 for all sufficiently large n , which yields a distinct limit V ar( F 2 n ) n → 2 c (16 + c ) . Accordingly , w e conclude that V ar( F 2 n ) = Θ( n ) when c is a p ositive integer, but there exists no universal constan t a suc h that V ar( F 2 n ) ∼ an as n → ∞ . 4 Asymptotic normalit y In this section, w e establish the asymptotic normalit y for F 2 n under the condition n 2 p (1 − p ) → ∞ . Prop osition 3 implies that when n 2 p (1 − p ) fails to diverge to infinity , F n cannot b e asymptotically normal under any normalization. W e first state the main results of this pap er in the followin g. Theorem 2. F or the distanc e F n = d F ( G n,p , R ) with an arbitr ary F r´ echet me an gr aph R , we have that as n 2 p (1 − p ) → ∞ , F 2 n − E [ F 2 n ] p V ar( F 2 n ) D − → N (0 , 1) . (27) In p articular, the fol lowing assertions hold. (i) If np (1 − p ) → 0 and n 2 p (1 − p ) → ∞ , then F 2 n − E [ F 2 n ] p n 2 p (1 − p ) D − → N (0 , 8) . (ii) If np (1 − p ) → c for some non-inte ger c onstant c > 0 , then F 2 n − E [ F 2 n ] √ n D − → N  0 , 2 c  4( c − ⌊ c ⌋ + 1) 2 + c  . (iii) If np (1 − p ) → ∞ , then F 2 n − E [ F 2 n ] p n 3 p 2 (1 − p ) 2 D − → N (0 , 2) . Theorem 2 reveals a phase transition in the asymptotic behavior of F 2 n , characterized b y the scaling of np (1 − p ), which determines the appropriate normalization and limiting v ariance. By the symmetry ( 4 ), without loss of generality w e can assume p ≤ 1 2 in the pro of of the theorem. Then, these three sp ecial regimes corresp ond, respectively , to v ery sparse regime ( np → 0 but n 2 p → ∞ ), sparse regime ( np → c > 0), and dense regime ( np → ∞ ) for Erd˝ os-R´ enyi random graphs. When c is a p ositive integer, the asymptotic normalit y of the abov e form requires additional constraints, as indicated in Remark 2; for instance, np should con verge to c monotonically from ab o ve or below. F ollowing the existing results, the precise formulation of the corresp onding conclusions can b e readily deriv ed, and th us we omit the details here. Applying the delta method (see, e.g., [ 37 , Chapter 3]) to the results of Theorem 2 , we obtain the follo wing asymptotic distributions for the F rob enius distance itself. 10 Theorem 3. Under the same c onditions of The or em 2 , we have the fol lowing. (i) If np (1 − p ) → 0 and n 2 p (1 − p ) → ∞ , then F n − p E [ F 2 n ] D − → N (0 , 1) . (ii) If np (1 − p ) → c for some non-inte ger c onstant c > 0 , then F n − p E [ F 2 n ] D − → N 0 , c  4( c − ⌊ c ⌋ + 1) 2 + c  2  ( c − ⌊ c ⌋ ) 2 + ⌊ c ⌋ + 2 c  ! . (iii) If np (1 − p ) → ∞ , then F n − p E [ F 2 n ] p np (1 − p ) D − → N  0 , 1 6  . 4.1 Pro of of Theorem 2 (i) and (ii) This subsection and the one that follows are dev oted to the proof of Theorem 2 . Throughout the entire pro of, we assume without loss of generality that 0 < p ≤ 1 2 . By a standard subsequence argument, it suffices to verify that the conv ergence ( 27 ) holds under each of the conditions (i), (ii), and (iii) to complete the pro of of Theorem 2 . W e adopt differen t techniques to handle the three regimes in Theorem 2 : • When np (1 − p ) → c for some constant c ≥ 0, the random v ariables contributing to F 2 n exhibit a sparse dep endency structure. In these cases, we construct a suitable dep endency graph [ 2 , 3 ] for the summands and verify the conditions of Lemma 1 to establish asymptotic normality . • When np (1 − p ) → ∞ , the dep endencies b ecome to o in tricate for a simple dep endency graph approach. Instead, w e emplo y recen t Berry–Esseen b ounds for functionals of independent random v ariables de- v elop ed by Shao and Zhang [ 33 ], which pro vide sharp er control on the con vergence rate. The proof for (i) and (ii) using dep endency graphs is presented first; the pro of for (iii) using Berry–Esseen b ounds is giv en in the next subsection. Let us first recall some fundamen tal concepts of the dependency graph for random v ariables. Let { X i } i ∈I b e a family of random v ariables defined on a common probabilit y space. A dep endency gr aph for these random v ariables is any graph L with vertex set I suc h that if A and B are tw o disjoint subsets of I with no edges b et w een A and B , then the families { X i } i ∈ A and { X j } j ∈ B are mutually independent. F or any p ositiv e in teger r and i 1 , i 2 , . . . , i r ∈ I , we also denote by N L ( i 1 , i 2 , . . . , i r ) = r [ k =1 { j ∈ I : j = i k or j is adjacent to i k in L } the closed neighborho od of { i 1 , i 2 , . . . , i r } in L . The following auxiliary lemma, stated as Theorem 6.33 in [ 24 ], plays an imp ortan t role in our pro of of the asymptotic normality of F 2 n . Lemma 1. Supp ose that { S n } ∞ n =1 is a se quenc e of r andom variables such that S n = P α ∈A n X nα , wher e for e ach n , { X nα , α ∈ A n } is a family of r andom variables with dep endency gr aph L n . Supp ose further that ther e exist numb ers Q 1 n and Q 2 n such that P α ∈A n E [ | X nα | ] ≤ Q 1 n and, for every α 1 , α 2 ∈ A n , X α ∈ N L n ( α 1 ,α 2 ) E [ | X nα |   X nα 1 , X nα 2 ] ≤ Q 2 n . 11 As n → ∞ , if Q 1 n Q 2 2 n / (V ar( S n )) 3 / 2 → 0 , then S n − E [ S n ] p V ar( S n ) D − → N (0 , 1) . Pr o of of (i) and (ii) in The or em 2 . Recall from ( 11 ) that F 2 n can b e expressed as a linear combination of edge and w edge indicators. Let us denote b y { T α } α ∈B n the set of all p ossible edges and w edges in G n,p , where B n stands for an index set with the cardinality |B n | =  n 2  + 3  n 3  = 1 2 n ( n − 1) 2 . F or each α ∈ B n , define Y nα =    − 4( m − 1 + m ij ) I ij , if T α is the edge b et ween v ertices i and j ; 2 I ij I ik , if T α is the wedge cen tered at i with neigh b ors j  = k . Then, by ( 11 ), e F 2 n := F 2 n − m ( m + 1) n = X α ∈B n Y nα . (28) Note that V ar( e F 2 n ) = V ar( F 2 n ). T o pro ve the desired result ( 27 ) in Theorem 2 , it is no w sufficient to show that e F 2 n − E [ e F 2 n ] q V ar( e F 2 n ) D − → N (0 , 1) . (29) W e will apply Lemma 1 to the collection { Y nα } α ∈B n . First, construct a dependency graph L n with v ertex set B n b y connecting t wo vertices α, β with an edge if and only if the corresp onding subgraphs T α and T β share at least one edge. (If they share no edge, the corresponding random v ariables are independent b ecause they inv olve disjoin t sets of edge indicators.) No w we v erify the conditions of Lemma 1 . Without loss of generalit y , as in the pro of of Prop osition 3 , w e may assume np → c for some constan t c ≥ 0, so that p → 0 and m = ⌊ np ⌋ satisfies | m − 1 + m ij | ≤ c + 1 for large n . Since | Y nα | ≤ 4( c + 1) for an edge and | Y nα | ≤ 2 for a wedge, X α ∈B n E [ | Y nα | ] ≤ 4( c + 1)  n 2  p + 2 · 3  n 3  p 2 ≤ 3( c + 1) n 2 p := Q 1 n , whic h is of order n 2 p . Fix an y t wo vertices α 1 , α 2 ∈ B n and let N L n ( α 1 , α 2 ) be their closed neighborho o d in L n . Note that the union T α 1 ∪ T α 2 con tains at most six vertices. It follows that the n umber of edges in this union is at most  6 2  , and that the n um b er of edges w edges that intersect this union is at most 3  6 3  + 2  6 2  ( n − 6). These simple facts implies X α ∈ N L n ( α 1 ,α 2 ) E [ | Y nα | | Y nα 1 , Y nα 2 ] ≤  6 2  + 3  6 3  + 2  6 2  ( n − 6) p. Th us, we ma y take Q 2 n = O (1), since np is b ounded in Cases (i) and (ii). By ( 26 ) and ( 28 ), we ha ve V ar( e F 2 n ) = V ar( F 2 n ) of order n 2 p in Case (i) and of order n in Case (ii). Consequen tly , Q 1 n Q 2 2 n (V ar( e F 2 n )) 3 / 2 =          O  n 2 p ( n 2 p ) 3 / 2  = O 1 p n 2 p ! → 0 , in Case (i) , O  n 2 p n 3 / 2  = O  1 √ n  → 0 , in Case (ii) . Th us the condition of Lemma 1 is satisfied, and ( 29 ) holds for np → c for some constant c ≥ 0. If np tends to 0 or a non-integer constant c > 0, then (i) and (ii) of Theorem 2 follows b y ( 27 ) and Slutsky’s theorem. 12 W e next consider (iii), and b egin with some basic analysis. Note that we no w hav e 0 < p ≤ 1 2 and np → ∞ , which is the defining regime of (iii). Recall that E n denotes the n umber of edges in G n,p , and W n is t wice the num b er of w edges. By ( 11 ) and ( 28 ), we can rewrite e F 2 n as e F 2 n = − 2 e U n − 4( n − 2) pE n + W n (30) where e U n = n X i =1 X j  = i ( m − np − 1 + m ij + 2 p ) I ij . Here we take 4( n − 2) p as the co efficien t of E n instead of 4 np , since this choice will mak e the subsequen t computations more concise. Note that these co efficien ts satisfy | m − np − 1 + m ij + 2 p | ≤ 2 , since 0 < np − m < 1 , m ij ∈ { 0 , 1 } , and 0 < p ≤ 1 2 . No w V ar( e F 2 n ) = V ar( F 2 n ) by ( 28 ), and from ( 26 ) we ha ve, as np → ∞ , V ar( e F 2 n ) ∼ 2 n 3 p 2 (1 − p ) 2 . (31) A direct computation gives V ar( e U n ) ≤ 16 V ar( E n ) = 16  n 2  p (1 − p ) = O ( n 2 p ) = o  V ar( e F 2 n )  , (32) since np → ∞ . Consequently , 2( e U n − E [ e U n ]) q V ar( e F 2 n ) P − → 0 b y Cheb yshev’s inequality . Moreo ver, from ( 30 ) and Slutsky’s theorem, to prov e ( 29 ) it suffices to sho w that ( W n − 4( n − 2) pE n ) − E [ W n − 4( n − 2) pE n ] p 2 n 3 p 2 (1 − p ) 2 D − → N (0 , 1) . (33) W e now examine wh y the dep endency graph metho d used for (i) and (ii) cannot b e directly applied to establish ( 33 ). Under the assumptions p ≤ 1 / 2 and np → ∞ , it is kno wn that E n and W n satisfy a joint cen tral limit theorem: r 2 p 1 − p     E n − E [ E n ] np W n − E [ W n ] 4 n 2 p 2     D − →  N N  , where N denotes a standard normal random v ariable (see [ 19 , Theorem 3(iii)] and [ 20 , Theorem 1(ii)]; note that b oth results are obtained via dep endency graph argumen ts). Since the limit in distribution is a degenerate normal, after straigh tforward calculations it follo ws that the linear combination W n − 4( n − 2) pE n satisfies W n − 4( n − 2) pE n − E [ W n − 4( n − 2) pE n ] p n 4 p 3 P − → 0 , whic h indicates that the joint conv ergence is degenerate for this particular linear functional; the asymptotic normalit y of W n − 4( n − 2) pE n requires a finer analysis that captures the next-order fluctuations. Con- sequen tly , the dep endency graph approach do es not provide sufficient precision to prov e ( 33 ), and a more sophisticated method—such as the Berry–Esseen b ounds for functionals of indep enden t v ariables dev elop ed in [ 33 ]—is needed. This justifies our separate treatment of Theorem 2 (iii). 13 4.2 Pro of of Theorem 2 (iii) Giv en the limitations of the dep endency graph approach, we now turn to a more refined normal appro ximation tec hnique. Recen tly , Shao and Zhang [ 33 ] dev elop ed a pow erful metho d for establishing Berry–Esseen b ounds for functionals of indep enden t random v ariables. Their approach is based on a discrete-difference version of Stein’s metho d, inspired by the generalized p erturbativ e framework of Chatterjee [ 13 ]. This metho d yields a b ound on the Kolmogorov distance b etw een the distribution of a standardized statistic and the standard normal distribution. T o prepare for the pro of of (iii), we first recall the k ey result from [ 33 ] that will b e instrumen tal in our analysis. Let N ≥ 2 b e an in teger, X = ( X 1 , . . . , X N ) a v ector of indep enden t random v ariables taking v alues in a measurable space X , and X ′ = ( X ′ 1 , . . . , X ′ N ) an indep enden t copy . F or any subset A ⊂ [ N ], define X A b y replacing X s with X ′ s for s ∈ A and leaving others unchanged. F or a measurable function h : X N → R , set ∆ s h ( X A ) = h ( X A ) − h ( X A ∪{ s } ) , s ∈ [ N ] . (34) In particular, ∆ s h ( X ) = ∆ s h ( X ∅ ). Consider the standardized random v ariable H = h ( X ) − E [ h ( X )] σ , where σ 2 = V ar( h ( X )). The following lemma, stated as Theorem 2.1 of [ 33 ], provides a b ound on the Kolmogoro v distance b et ween H and the standard normal random v ariable N . Lemma 2. L et A s = { 1 , 2 , . . . , s − 1 } for s ≥ 2 and A 1 = ∅ . Define r andom variables V = 1 2 σ 2 N X s =1 ∆ s h ( X )∆ s h ( X A s ) and V ∗ = 1 σ 2 N X s =1 ∆ s h ( X )   ∆ s h ( X A s )   . Then the Kolmo gor ov distanc e b etwe en H and N satisfies d K ( H , N ) ≤ E | 1 − E [ V | H ] | + 2 E | E [ V ∗ | H ] | . T o apply Lemma 2 , we encode the random graph mo del as a random vector of edge indicators. Let N =  n 2  b e the total num b er of p ossible edges in G n,p . W e fix an ordering of the edges by sorting pairs ( i, j ) lexicographically: first by increasing i , then by increasing j . More precisely , for s ∈ [ N ] define i ( s ) = min  1 ≤ t ≤ n − 1 : s ≤ (2 n − t − 1) t 2  , j ( s ) = s + i ( s ) − [2 n − i ( s )][ i ( s ) − 1] 2 , whic h giv es a bijection betw een [ N ] and the set of all p ossible edges in G n,p . Consequen tly , the random v ector Z = ( Z 1 , . . . , Z N ) with Z s = I i ( s ) j ( s ) represen ts the edge indicators of G n,p . Both E n and W n can b e view ed as functions of Z ; we will occasionally write E n ( Z ) and W n ( Z ) to emphasize this dep endence. Define h ( Z ) = 1 2 W n ( Z ) − 2( n − 2) pE n ( Z ) = 1 2 W n − 2( n − 2) pE n , (35) i.e., the num b er of w edges minus 2( n − 2) p times the num b er of edges. Note that 2 h ( Z ) = W n − 4( n − 2) pE n , so proving asymptotic normalit y of h ( Z ) is equiv alent to establishing ( 33 ). Let σ 2 n = V ar[ h ( Z )]. F rom ( 30 )-( 32 ), we obtain σ 2 n = 1 2 n 3 p 2 (1 − p ) 2 (1 + o (1)) . (36) W e then in tro duce the standardized statistic H n = h ( Z ) − E [ h ( Z )] σ n . 14 Our goal is to prov e that under the condition that p ≤ 1 2 and np → ∞ , H n D − → N (0 , 1) . (37) T o define the discrete difference op erator required in Lemma 2 , we need an indep enden t copy of Z . Let { I ′ ij : 1 ≤ i < j ≤ n } b e an indep enden t copy of { I ij : 1 ≤ i < j ≤ n } , and set Z ′ = ( Z ′ 1 , . . . , Z ′ N ) with Z ′ s = I ′ i ( s ) j ( s ) . Then Z ′ is an indep enden t copy of Z . F or any subset A ⊂ [ N ], denote b y Z A the vector obtained from Z b y replacing Z s with Z ′ s for s ∈ A and leaving the other coordinates unchanged. F ollowing [ 33 ], we define the discrete difference ∆ s h ( Z A ) = h ( Z A ) − h ( Z A ∪{ s } ) , s ∈ [ N ] . In particular, ∆ s h ( Z ) = h ( Z ) − h ( Z { s } ). No w set A s = { 1 , . . . , s − 1 } for s ≥ 2 and A 1 = ∅ . Then, in accordance with Lemma 2 , we define V n = 1 2 σ 2 n N X s =1 ∆ s h ( Z ) ∆ s h ( Z A s ) , (38) and V ∗ n = 1 σ 2 n N X s =1 ∆ s h ( Z )   ∆ s h ( Z A s )   . (39) Lemma 2 then yields the following b ound on the Kolmogoro v distance betw een H n and a standard normal v ariable N : d K ( H n , N ) ≤ E   1 − E [ V n | H n ]   + 2 E   E [ V ∗ n | H n ]   . (40) In the next step, w e will estimate the right-hand side and show that it tends to zero as n → ∞ , thereb y establishing ( 37 ). W e now do further analysis on V n and V ∗ n , esp ecially using their explicit expressions. F or any fixed v ertex pair 1 ≤ i < j ≤ n , we define a set of edge v ariables { I ( i,j ) kℓ : 1 ≤ k  = ℓ ≤ n } , where I ( i,j ) kℓ = ( I ′ kℓ , if min { k , ℓ } < i or min { k , ℓ } = i < max { k , ℓ } < j, I kℓ , otherwise . (41) In other w ords, I ( i,j ) kℓ equals the indep enden t copy if the edge ( k , ℓ ) app ears b efore ( i, j ) in the ordering, and equals the original indicator otherwise. This precisely captures the effect of the replacement set A s . Lemma 3. F or any fixe d s ∈ [ N ] with the c orr esp onding vertex p air ( i, j ) = ( i ( s ) , j ( s )) , we have ∆ s h ( Z ) = X k  = i,j M ij k , (42) and for A s = { 1 , 2 , . . . , s − 1 } with A 1 = ∅ , ∆ s h ( Z A s ) = X ℓ  = i,j f M ij ℓ , (43) wher e M ij k =  I ij − I ′ ij  ( I ik + I j k − 2 p ) , f M ij ℓ =  I ij − I ′ ij  I ( i,j ) iℓ + I ( i,j ) j ℓ − 2 p  . (44) Pr o of. F or any subset A ⊂ [ N ], b y ( 34 ) and ( 35 ) w e hav e ∆ s h ( Z A ) = 1 2 W n ( Z A ) − 2( n − 2) pE n ( Z A ) − 1 2 W n ( Z A ∪{ s } ) + 2( n − 2) pE n ( Z A ∪{ s } ) 15 = 1 2 ∆ s W n ( Z A ) − 2( n − 2) p ∆ s E n ( Z A ) . (45) If A = ∅ , it follo ws by definition that ∆ s W n ( Z ) = 2 X k  = i,j ( I ij − I ′ ij )( I ik + I j k ) , and ∆ s E n ( Z ) = I ij − I ′ ij . Th us, combining ( 45 ) yields ∆ s h ( Z ) = X k  = i,j  I ij − I ′ ij  ( I ik + I j k ) − 2( n − 2) p  I ij − I ′ ij  = X k  = i,j  I ij − I ′ ij  ( I ik + I j k − 2 p ) , whic h prov es ( 42 ). F or A s = { 1 , . . . , s − 1 } , also by definition w e hav e ∆ s W n ( Z A s ) = 2 X ℓ  = i,j  I ij − I ′ ij  ( I ( i,j ) iℓ + I ( i,j ) j ℓ ) , and ∆ s E n ( Z A s ) = I ij − I ′ ij . By ( 45 ), similarly w e hav e that for an y s ∈ [ N ], ∆ s h ( Z A s ) = X ℓ  = i,j  I ij − I ′ ij  I ( i,j ) iℓ + I ( i,j ) j ℓ  − 2( n − 2) p  I ij − I ′ ij  = X ℓ  = i,j  I ij − I ′ ij  I ( i,j ) iℓ + I ( i,j ) j ℓ − 2 p  , and thus ( 43 ) holds. The pro of of Lemma 3 is complete. By ( 38 ) and ( 39 ), Lemma 3 provides the explicit expressions for V n and V ∗ n as follows: V n = 1 2 σ 2 n X 1 ≤ ii j 1  = j 2 Co v ( Λ ij 1 , Λ ij 2 ) = O  n 3 · n 2 p 4 n 6 p 4  = O  1 n  . As for Case 3, w e now consider the cov ariance Cov( Λ i 1 j 1 , Λ i 2 j 2 ), where { i 1 , j 1 } ∩ { i 2 , j 2 } = ∅ . Since in this case, J 2 i 1 j 1 and J 2 i 2 j 2 are indep enden t and each is indep enden t of B i 1 j 1 and B i 2 j 2 , we hav e Co v ( Λ i 1 j 1 , Λ i 2 j 2 ) = E  J 2 i 1 j 1  E  J 2 i 2 j 2  Co v ( B i 1 j 1 , B i 2 j 2 ) = [2 p (1 − p )] 2 X k 1 ,ℓ 1 / ∈{ i 1 ,j 1 } X k 2 ,ℓ 2 / ∈{ i 2 ,j 2 } Co v  J (1) i 1 j 1 k 1 J (2) i 1 j 1 ℓ 1 , J (1) i 2 j 2 k 2 J (2) i 2 j 2 ℓ 2  . (54) 19 Consider the co v ariance inside the sum. If k 1 or ℓ 1 lies outside { i 2 , j 2 } , then J (1) i 1 j 1 k 1 or J (2) i 1 j 1 ℓ 1 is inde- p enden t of J (1) i 2 j 2 k 2 J (2) i 2 j 2 ℓ 2 , and the corresp onding cov ariance is equal to zero. Similarly for k 2 , ℓ 2 . A necessary condition for a non-zero contribution is therefore { k 1 , ℓ 1 } ⊆ { i 2 , j 2 } and { k 2 , ℓ 2 } ⊆ { i 1 , j 1 } . (55) Under ( 55 ), each of k 1 , ℓ 1 can only b e i 2 or j 2 , and each of k 2 , ℓ 2 can only b e i 1 or j 1 , giving at most 16 index quadruples. A direct calculation (e.g., for the pattern k 1 = ℓ 1 = i 2 , k 2 = ℓ 2 = i 1 ) sho ws that each suc h cov ariance is of order p . Hence, X k 1 ,ℓ 1 / ∈{ i 1 ,j 1 } X k 2 ,ℓ 2 / ∈{ i 2 ,j 2 } Co v  J (1) i 1 j 1 k 1 J (2) i 1 j 1 ℓ 1 , J (1) i 2 j 2 k 2 J (2) i 2 j 2 ℓ 2  = O ( p ) . By ( 54 ), this implies that Co v ( Λ i 1 j 1 , Λ i 2 j 2 ) = O ( p 3 ) , and thus the total contribu tion from Case 3 is 1 4 σ 4 n X i 1 0. Using the expressions ( 12 ) and ( 24 ) for E [ F 2 n ], a straigh tforward calculation yields E [ F 2 n ] −  ( c − ⌊ c ⌋ ) 2 + ⌊ c ⌋ + 2 c  n = 2 c  c − ⌊ c ⌋ + 1  δ √ n (1 + o (1)) . This discrepancy is of order √ n , the same as the normalizing constant in Theorem 2 (ii), confirming that cen tering by the exact exp ectation is necessary . Remark 4. It is also w orth noting that the metho d used for case (iii) of Theorem 2 , which relies on the Berry–Esseen b ounds dev elop ed by Shao and Zhang [ 33 ], is not directly transferable to cases (i) and (ii). In these regimes, V ar( e U n ) is not of smaller order than V ar( F 2 n ); hence e U n cannot b e neglected, and the represen tation F 2 n = − 2 e U n − 4( n − 2) pE n + W n do es not reduce to a simple functional form as tractable as in the sparse cases. Moreov er, e U n explicitly in volv es the F r´ echet mean graph through the coefficients m ij . Since the F r ´ ec het mean set in cases (i) and (ii) ma y contain multiple non-isomorphic graphs, these coefficients are not uniquely determined. Consequently , constructing the quantities V n and V ∗ n in Lemma 2 would dep end on the particular choice of the F r ´ ec het mean, making a uniform v ariance estimate prohibitiv ely complicated. In contrast, the dep endency graph approac h adopted for cases (i) and (ii) circumv ents these difficulties by w orking directly with F 2 n and exploiting the sparse dep endence structure, which b ecomes increasingly dense in case (iii) and thus necessitates the more refined normal appro ximation of [ 33 ]. 4.3 Pro of of Theorem 3 Based on Theorem 2 , we no w give a formal pro of for Theorem 3 using the delta metho d. Pr o of of The or em 3 . It follows b y ( 21 ) and ( 24 ) that B E [ F 2 n ] ∼      2 n 2 p (1 − p ) , if np (1 − p ) → 0;  ( c − ⌊ c ⌋ ) 2 + ⌊ c ⌋ + 2 c  n, if np (1 − p ) → c for some constant c > 0; 3 n 2 p (1 − p ) , if np (1 − p ) → ∞ . (57) Note that E [ F 2 n ] > 0 in all three regimes, and the function g ( x ) = √ x is differentiable at E [ F 2 n ] with g ′ ( x ) = 1 / (2 √ x ) > 0. Therefore, applying the delta metho d to ( 27 ) yields the asymptotic normality of F n cen tered at p E [ F 2 n ], with asymptotic v ariance  g ′ ( E [ F 2 n ])  2 · V ar( F 2 n ) = V ar( F 2 n ) 4 E [ F 2 n ] . Using the asymptotic expressions for V ar( F 2 n ) from ( 26 ) and for E [ F 2 n ] from ( 57 ), a straigh tforward calculation giv es V ar( F 2 n ) 4 E [ F 2 n ] ∼          1 , if np (1 − p ) → 0; c  4( c − ⌊ c ⌋ + 1) 2 + c  2  ( c − ⌊ c ⌋ ) 2 + ⌊ c ⌋ + 2 c  , if np (1 − p ) → c for some non-integer constant c > 0; 1 6 np (1 − p ) , if np (1 − p ) → ∞ . Com bining this and applying Slutsky’s theorem completes the pro of of Theorem 3 . References [1] A. Av eni. Uniform L ar ge Sample The ory of Gener alize d F r´ echet Me ans . PhD thesis, Duke Universit y , 2025. 21 [2] P . Baldi and Y. Rinott. Asymptotic normalit y of some graph-related statistics. Journal of Applie d Pr ob ability , 26(1):171–175, 1989. [3] P . Baldi, Y. Rinott, and C. Stein. A normal approximation for the num b er of lo cal maxima of a random function on a graph. In T.W. Anderson, K. B. Athrey a, and D. L. Iglehart, editors, Pr ob ability, Statistics, and Mathematics , pages 59–81. Academic Press, 1989. [4] D. Barden, H. Le, and M. Ow en. Central limit theorems for Fr´ echet means in the space of ph ylogenetic trees. Ele ctr onic Journal of Pr ob ability , 18:1–25, 2013. [5] A. Bhattac harya and R. Bhattachary a. Nonp ar ametric Infer enc e on Manifolds: With Applic ations to Shap e Sp ac es . Cambridge Univ ersity Press, Cam bridge, 2012. [6] B. Bollob´ as. A probabilistic pro of of an asymptotic formula for the num b er of lab elled regular graphs. Eur op e an Journal of Combinatorics , 1(4):311–316, 1980. [7] B. Bollob´ as. R andom Gr aphs, 2nd Edition . Cambridge Universit y Press, Cam bridge, 2001. [8] B. Bollob´ as, S. Janson, and O. Riordan. The phase transition in inhomogeneous random graphs. R andom Structur es and Algorithms , 31(1):3–122, 2007. [9] R. C. Bose. Strongly regular graphs, partial geometries and partially balanced designs. Pacific Journal of Mathematics , 13(2):389–419, 1963. [10] F. Bouc hard, A. Mian, M. Tiomok o, G. Ginolhac, and F. P ascal. Random matrix theory impro ved Fr ´ ec het mean of symmetric p ositiv e definite matrices. In F orty-first International Confer enc e on Ma- chine L e arning , ICML’24, pages 4403–415, 2024. [11] A. E. Brouw er and W. H. Haemers. Sp e ctr a of Gr aphs . Springer, New Y ork, 2012. [12] Y. Cao and A. Monod. A geometric condition for uniqueness of Fr´ echet means of p ersistence diagrams. Computational Ge ometry , 128:102162, 2025. [13] S. Chatterjee. A new metho d of normal approximation. The Annals of Pr ob ability , 36(4):1584–1610, 2008. [14] F. R. K. Chung. Sp e ctr al Gr aph The ory . American Mathematical So ciet y , 1997. [15] X. Dai and H.-G. M¨ uller. Principal comp onen t analysis for functional data on Riemannian manifolds and spheres. The Annals of Statistics , 46(6B):3334–3361, 2018. [16] C. Donnat and S. Holmes. T rac king net work dynamics: A survey using graph distances. The Annals of Applie d Statistics , 12(2):971–1012, 2018. [17] R. Durrett. Pr ob ability: The ory and Examples, 5th Edition . Cambridge Univ ersity Press, Cambridge, 2019. [18] S. N. Ev ans and A. Q. Jaffe. Limit theorems for F r´ echet mean sets. Bernoul li , 30(1):419–447, 2024. [19] Q. F eng, Z. Hu, and C. Su. The Zagreb indicies of random graphs. Pr ob ability in the Engine ering and Informational Scienc es , 27(2):247–260, 2013. [20] Q. F eng, H. Ren, and Y. Tian. Limit laws for the generalized Zagreb indices of random graphs. A dvanc es in Applie d Pr ob ability , 57:1539–1558, 2025. [21] M. F r´ echet. Les ´ el´ emen ts al´ eatoires de nature quelconque dans un espace distanci´ e. Annales de l’institut Henri Poinc ar´ e , 10(4):215–310, 1948. 22 [22] C. E. Ginestet, J. Li, P . Balachandran, S. Rosen b erg, and E. D. Kolaczyk. Hyp othesis testing for net work data in functional neuroimaging. The Annals of Applie d Statistics , 11(2):725–750, 2017. [23] T. Hotz and S. Huck emann. Intrinsic means on the circle: uniqueness, lo cus and asymptotics. Annals of the Institute of Statistic al Mathematics , 67(1):177–193, 2015. [24] S. Janson, T. Luczak, and A. Ruci ´ nski. R andom Gr aphs . John Wiley , New Y ork, 2000. [25] H. Karcher. Riemannian center of mass and mollifier smo othing. Communic ations on Pur e and Applie d Mathematics , 30(5):509–541, 1977. [26] E. D. Kolaczyk, L. Lin, S. Rosenberg, J. W alters, and J. Xu. Averages of unlab eled netw orks: Geometric c haracterization and asymptotic b eha vior. The Annals of Statistics , 48(1):514–538, 2020. [27] S. Lunag´ omez, S. C. Olhede, and P . J. W olfe. Mo deling netw ork p opulations via graph distances. Journal of the A meric an Statistic al Asso ciation , 116(536):2023–2040, 2021. [28] A. McCormack and P . Hoff. The Stein effect for Fr´ echet means. The Annals of Statistics , 50(6):3647– 3676, 2022. [29] A. McCormac k and P . D. Hoff. Equiv ariant estimation of Fr ´ ec het means. Biometrika , 110(4):1055–1076, 2023. [30] F. G. Mey er. The Fr ´ ec het mean of inhomogeneous random graphs. In International Confer enc e on Complex Networks and Their Applic ations , pages 207–219. Springer, 2021. [31] A. P etersen and H.-G. M ¨ uller. F r´ echet regression for random ob jects with Euclidean predictors. The A nnals of Statistics , 47(2):691–719, 2019. [32] C. Sc h¨ otz. Strong la ws of large n umbers for generalizations of Fr´ ec het mean sets. Statistics , 56(1):34–52, 2022. [33] Q.-M. Shao and Z.-S Zhang. Berry-Esseen b ounds for functionals of indep enden t random v ariables. Sto chastic Pr o c esses and their Applic ations , 183:104574, 2025. [34] A. T orres-Signes, M. P . F r ´ ıas, and M. D. Ruiz-Medina. Global Fr´ echet regression from time correlated biv ariate curve data in manifolds. Statistic al Pap ers , 66:73, 2025. [35] A. T ripathi and S. Vija y . A note on a theorem of Erd˝ os & Gallai. Discr ete Mathematics , 265(1):417–420, 2003. [36] R. v an der Hofstad. R andom Gr aphs and Complex Networks . Cam bridge Univ ersity Press, Cambridge, 2016. [37] A. W. V an der V aart. Asymptotic Statistics . Cambridge Univ ersity Press, Cam bridge, 1998. [38] Y. Zhou and H.-G. M ¨ uller. Netw ork regression with graph Laplacians. Journal of Machine L e arning R ese ar ch , 23(320):1–41, 2022. 23