Spectral Properties of the Threshold Network Model

Sp ectral Prop erties of the Thre shold Net w ork Mo del Y usuk e Ide ∗ F aculty of Engine ering, Kanagawa Unive rsity, Y okohama, 221-868 6, Jap an Norio Konno † Dep artment of Applie d Mathematics, Y okohama National U niversity, Y okohama, 240-850 1, Jap an Nobuaki Obata ‡ Gr aduate Scho ol of Information Scienc es, T ohoku U niversity, Sendai, 980-857 9, Jap an Abstract W e study t he spectral distribution of the thresh old net work mo del. T he results con tain an explicit description and its asymptotic b eha viour. 1 In tro du ction The thr eshold network mo del G n ( X , θ ), where X is a random v ariable, n ≥ 2 is an in teger and θ ∈ R is a constan t called a threshold, is a ra ndom graph on the v ertex set V = { 1 , 2 , . . . , n } obtained as f o llo ws: let X 1 , X 2 , . . . , X n b e indep enden t copies of X and draw an edge b etw een t wo distinct v ertices i, j ∈ V if X i + X j > θ . In other w ords, G n ( X , θ ) is sp ecified b y the random adjacency matrix A = ( A ij ) defined b y A ij = ( I ( θ, ∞ ) ( X i + X j ) , if i 6 = j , 0 , otherwise , where I B denotes the indicator function of a set B . As a small v ariant one may allo w self-lo ops, see e.g., [4]. In this case the threshold net w ork mo del is denoted b y ˜ G n ( X , θ ), ∗ E-mail: ide@ k anagawa-u.ac.jp † E-mail: konno@ynu.ac.jp ‡ E-mail: o bata@math.is.to hoku.ac.jp Abbr. title: Spectral P rop erties of Threshold Netw o rk Mo del Key wo r ds and phr ases: sp ectral distribution, threshold netw o rk mo del, random g raphs, complex net works. 1 where t wo vertice s i, j ∈ V (p ossibly i = j ) are connected if X i + X j > θ . T he adjacency matrix ˜ A = ( ˜ A ij ) is giv en b y ˜ A ij = I ( θ, ∞ ) ( X i + X j ) , i, j ∈ V . The threshold net work mo del has b een extensiv ely studied as a reasonable candidate mo del of real w orld complex graphs (net w o rks), whic h are often c haracterized by small diameters, high clustering, and p o w er-law (scale-free) degree distributions [1 , 2, 2 0]. In fact, the threshold net w ork mo del b elongs to the so- called hidden v aria ble mo dels [5, 22] and is know n for being capable of generating scale-free net works. Their mean b eha vior [3, 5, 8, 9, 15, 21, 22] and limit theorems [7, 11 – 13] for t he degree, the clustering co efficien ts, the n um b er of subgraphs, and the av erage distance hav e b een a nalyzed. See also [6, 11– 14, 16, 17] for related w orks. Sp ectral prop erties of the threshold net w ork mo del are also of in terest. As a simple case, the binary threshold mo del app ears in [2 3 ]. The stro ng la w of la rge n um b ers and cen tral limit theorem for the rank of the adjacency matrix of the model with self-lo ops are giv en b y [4]. Eigen v alues and eigen vec tors of the Lapla cian matrix of the mo del ha ve b een studied [1 8, 19]. F or general results of sp ectral analysis of graphs see e.g. Hora– Obata [10]. The main purp ose of this pap er is t o study the sp ectral distribution o f the threshold net work mo del. Our result co v ers the preceding study of the rank of the adjacency matrix. This pap er is o rganized as follows : In Section 2 w e recall the hierarchic al structure of the threshold net w o rk mo del and deriv e the sp ectral distribution of eac h sample graph (threshold graph). In Section 3 we obta in similar results for the threshold net w ork mo del whic h admits self-lo ops. In Section 4 w e deriv e some asymptotic b eha viors of the sp ectral distributions and in Sec tion 5 w e giv e a simple example called the binary t hreshold mo del. 2 Sp ectra of thre shold graphs Eac h sample graph G ∈ G n ( X , θ ) has a hierarc hical structure describ ed b y the so-called creation sequence, in tro duced b y Hagb erg–Sc hult–Sw art [9]. Here w e adopt a v ariant b y Diaconis–Holmes–Janson [6]. Eac h G b eing determined b y the v alues of r andom v ariables X 1 , X 2 , . . . , X n , w e a r range them in increasing order: X (1) ≤ X (2) ≤ · · · ≤ X ( n ) . If X (1) + X ( n ) > θ , w e hav e θ < X (1) + X ( n ) ≤ X (2) + X ( n ) ≤ · · · ≤ X ( n − 1) + X ( n ) , whic h means that t he ve rtex corresp onding to X ( n ) is connected with the n − 1 other v ertices. Otherwise, w e hav e θ ≥ X (1) + X ( n ) ≥ · · · ≥ X (1) + X (3) ≥ X (1) + X (2) , whic h means that the v ertex corresp onding to X (1) is isolatedD W e set s n = 1 or s n = 0 according as the former case or the latter o ccurs. Then, according to the case w e remo v e the ra ndo m v ariable X ( n ) or X (1) , w e con tin ue similar pro cedure to define s n − 1 , . . . , s 2 . Finally , we set s 1 = s 2 and o bt a in a { 0 , 1 } -sequence { s 1 , s 2 , . . . , s n } , whic h is called the cr e ation se quenc e of G and is denoted by S G . 2 Giv en a creation seq uence S G let k i and l i denote the nu m b er of consec utiv e bits of 1’s and 0’s, resp ectiv ely , as follows: S G = { k 1 z }| { 1 , . . . , 1 , l 1 z }| { 0 , . . . , 0 , k 2 z }| { 1 , . . . , 1 , l 2 z }| { 0 , . . . , 0 , . . . , k m z }| { 1 , . . . , 1 , l m z }| { 0 , . . . , 0 } . (1) It may happ en that k 1 = 0 or l m = 0, but w e hav e k 2 , . . . , k m , l 1 , . . . , l m − 1 ≥ 1 and m ≥ 1. Moreo ver, by definition w e ha v e tw o case s: (a) k 1 = 0 (equiv alen tly s 1 = 0) a nd l 1 ≥ 2; (b) k 1 ≥ 2 (equiv alen tly , s 1 = 1). F or example, if S G = { 1 , 1 , 0 , 0 , 1 , 0 , 1 , 0 } t hen k 1 = 2 , l 1 = 2 , k 2 = 1 , l 2 = 1 , k 3 = 1 , l 3 = 1 and Fig . 1 show s the shap e of G . 1 0 Figure 1: A threshold graph G corresp o nding to S G = { 1 , 1 , 0 , 0 , 1 , 0 , 1 , 0 } The creation sequence S G giv es rise to a partition of the v ertex set: V = m [ i =1 V (1) i ∪ m [ i =1 V (0) i | V (1) i | = k i , | V (0) i | = l i . The subgraph induced b y V (1) i is the complete graph of k i v ertices, and that induced b y V (0) i is the null graph of l i v ertices. Moreo ver, ev ery v ertex in V (1) i (resp. V (0) i ) is connected (resp. disconnected) with all v ertices in V (1) 1 ∪ · · · ∪ V (1) i ∪ V (0) 1 ∪ · · · ∪ V (0) i − 1 . In general, a graph p ossessing the ab ov e hierarc hical structure is called a thr eshold gr aph [14]. Theorem 1. L et G b e a ther eshold gr aph with a cr e ation se quenc e S G = { s 1 = s 2 , s 3 , . . . , s n } . Define k i and l i as in (1) and set C n ( − 1) = m X i =1 k i − ( m − 1) − I { 1 } ( s 1 ) , C n (0) = m X i =1 l i − ( m − 1) . (2) Then the s p e ctr al dis tribution of G is given by µ n ( G ) = C n ( − 1) n δ − 1 + C n (0) n δ 0 + 1 n J X j =1 δ λ j , J = 2( m − 1) + I { 1 } ( s 1 ) , (3) 3 wher e { λ j } exhausts the eigenvalues of the matrix:            k m − 1 l m − 1 k m − 1 l m − 2 . . . l 1 k 1 k m 0 0 0 . . . 0 0 k m 0 k m − 1 − 1 l m − 2 . . . l 1 k 1 k m 0 k m − 1 0 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . k m 0 k m − 1 0 . . . 0 0 k m 0 k m − 1 0 . . . 0 k 1 − 1            (4) for s 1 = 1 (e quivalently, k 1 ≥ 2 ), or            k m − 1 l m − 1 k m − 1 l m − 2 . . . k 2 l 1 k m 0 0 0 . . . 0 0 k m 0 k m − 1 − 1 l m − 2 . . . k 2 l 1 k m 0 k m − 1 0 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . k m 0 k m − 1 0 . . . k 2 − 1 l 1 k m 0 k m − 1 0 . . . k 2 0            (5) for s 1 = 0 (e quivalently, k 1 = 0 ). Mor e over, a ny λ j in (3) diff e rs fr om 0 and − 1 . Pr o of. Let 1 i,j denote the i × j matrix consisting of o nly 1, 0 i,j the i × j zero matrix, I i the i × i iden tit y matrix, and ¯ 1 i,i = 1 i,i − I i . By the hierarc hical structure men tioned ab ov e, the adjacency matrix of G is represen ted in the form: A G =              0 l m ,l m 0 l m ,k m 0 l m ,l m − 1 0 l m ,k m − 1 0 l m ,l m − 2 . . . 0 l m ,l 1 0 l m ,k 1 0 k m ,l m ¯ 1 k m ,k m 1 k m ,l m − 1 1 k m ,k m − 1 1 l m ,l m − 2 . . . 1 k m ,l 1 1 k m ,k 1 0 l m − 1 ,l m 1 l m − 1 ,k m 0 l m − 1 ,l m − 1 0 l m − 1 ,k m − 1 0 l m − 1 ,l m − 2 . . . 0 l m − 1 ,l 1 0 l m − 1 ,k 1 0 k m − 1 ,l m 1 k m − 1 ,k m 0 k m − 1 ,l m − 1 ¯ 1 k m − 1 ,k m − 1 1 k m − 1 ,l m − 2 . . . 1 k m − 1 ,l 1 1 k m − 1 ,k 1 0 l m − 2 ,l m 1 l m − 2 ,l m 0 l m − 2 ,l m − 1 1 l m − 2 ,k m − 1 0 l m − 2 ,l m − 2 . . . 0 l m − 2 ,l 1 0 l m − 2 ,k 1 . . . . . . . . . . . . . . . . . . . . . . . . 0 l 1 ,l m 1 l 1 ,k m 0 l 1 ,l m − 1 1 l 1 ,k m − 1 0 l 1 ,l m − 2 . . . 0 l 1 ,l 1 0 l 1 ,k 1 0 k 1 ,l m 1 k 1 ,k m 0 k 1 ,l m − 1 1 k 1 ,k m − 1 0 k 1 ,l m − 2 . . . 0 k 1 ,l 1 ¯ 1 k 1 ,k 1              . The adjacency matrix A acts on C n from the left. W e define subspaces of C n b y V i ( − 1) =      0 u i + l i ξ k i 0 d i   : ξ 1 + ξ 2 + · · · + ξ k i = 0    , 1 ≤ i ≤ m, V i (0) =      0 u i η l i 0 k i + d i   : η 1 + η 2 + · · · + η l i = 0    , 1 ≤ i ≤ m − 1 , V m (0) =  η l m 0 k m + d m  , 4 where ξ k =      ξ 1 ξ 2 . . . ξ k      , η l =      η 1 η 2 . . . η l      , 1 j =      1 1 . . . 1      , 0 j =      0 0 . . . 0      and u i = m X j = i +1 ( l j + k j ) , d i = i − 1 X j =1 ( l j + k j ) . Since A G acts on V i ( − 1) as the scalar op erator with − 1, it p ossesses the eigen v alues − 1 with m ultiplicity at least m X i =1 dim V i ( − 1) = m X i =1 ( k i − 1) = m X i =1 k i − m if k 1 ≥ 2 (i.e., s 1 = 1), and m X i =2 dim V i ( − 1) = m X i =2 ( k i − 1) = m X i =2 k i − ( m − 1) if k 1 = 0 (i.e., s 1 = 0). In any case , the m ultiplicit y is at least C n ( − 1) defined in ( 2). Similarly , acting on V i (0) as a scalar op erator with 0, A G p ossesses the eigen v alues 0 with m ultiplicity at least C n (0). Let W b e the orthogona l complemen t to L m i =1 ( V i ( − 1) ⊕ V i (0)). The matr ix represen- tation of A G on W with resp ect to the basis v i =   0 u i + l i 1 k i 0 d i   , 1 ≤ i ≤ m, and w i =   0 u i 1 l i 0 k i + d i   , 1 ≤ i ≤ m − 1 . is g iv en by ( 4 ) o r by (5 ) a ccording as k 1 ≥ 2 or k 1 = 0. Then, o ne ma y v erify easily the eigen v alues of the matrices (4) and (5) are differen t fro m − 1 nor 0. Remark After simple calculation we see that the eigen v alues λ 1 , . . . , λ J in (3) are ob- tained from the c ha r acteristic equations M ( λ ) = 0 , where M ( λ ) = det            k m − 1 − λ l m − 1 k m − 1 l m − 2 . . . l 1 k 1 k m − λ 0 0 . . . 0 0 k m 0 k m − 1 − 1 − λ l m − 2 . . . l 1 k 1 k m 0 k m − 1 − λ . . . 0 0 . . . . . . . . . . . . . . . . . . . . . k m 0 k m − 1 0 . . . − λ 0 k m 0 k m − 1 0 . . . 0 k 1 − 1 − λ            5 if k 1 ≥ 2 (i.e., s 1 = 1), and M ( λ ) = det            k m − 1 − λ l m − 1 k m − 1 l m − 2 . . . k 2 l 1 k m − λ 0 0 . . . 0 0 k m 0 k m − 1 − 1 − λ l m − 2 . . . k 2 l 1 k m 0 k m − 1 − λ . . . 0 0 . . . . . . . . . . . . . . . . . . . . . k m 0 k m − 1 0 . . . k 2 − 1 − λ l 1 k m 0 k m − 1 0 . . . k 2 − λ            if k 1 = 0 (i.e., s 1 = 0). Simple calculation sho ws that M ( − 1) = ( k 1 · · · k m · l 1 · · · l m − 1 , if k 1 ≥ 2 (i.e., s 1 = 1) , k 2 · · · k m · ( l 1 − 1) · l 2 · · · l m − 1 , otherwise , and M (0) = ( ( k 1 − 1) · k 2 · · · k m · l 1 · · · l m − 1 , if k 1 ≥ 2 (i.e., s 1 = 1) , k 2 · · · k m · l 1 · · · l m − 1 , otherwise , from whic h w e see also that { λ j } do not con ta in − 1 or 0. 3 Sp ectra of thre shold graphs with sel f-lo ops The idea of a creation sequence in Section 2 can b e applied to the threshold netw o rk mo del which allow s self-lo ops. With eac h G ∈ e G n ( X , θ ) w e asso ciate a creation seque nce e S G = { ˜ s 1 , ˜ s 2 , . . . , ˜ s n } as follo ws: if X (1) + X ( n ) > θ , w e hav e θ < X (1) + X ( n ) ≤ X (2) + X ( n ) ≤ · · · ≤ X ( n − 1) + X ( n ) ≤ X ( n ) + X ( n ) , whic h implies that the v ertex corresp onding to X ( n ) is connected with the n − 1 other v ertices and has a self-lo op. Otherwise, θ ≥ X (1) + X ( n ) ≥ · · · ≥ X (1) + X (3) ≥ X (1) + X (2) ≥ X (1) + X (1) , whic h means that the v ertex corresp onding to X (1) is isolated and has no self-lo opsD W e set ˜ s n = 1 or ˜ s n = 0 according as the fo r mer case or the latter o ccurs. Then, according to the case w e remo v e the random v ariable X ( n ) or X (1) , w e contin ue similar pro cedure to define ˜ s n − 1 , . . . , ˜ s 2 . Finally , letting X ( ∗ ) b e the last remained random v ariable, set ˜ s 1 = 1 if X ∗ > θ / 2 and ˜ s 1 = 0 otherwise. In this case G is called a thr e s h old gr aph with self-lo ops asso ciated with a creation sequence ˜ S = { ˜ s 1 , ˜ s 2 , . . . , ˜ s n } . W e note that if ˜ s j = 1 the corresp onding v ertex ha s a self-lo op and otherwise no self-lo op. Giv en a creation sequence ˜ S = { ˜ s 1 , ˜ s 2 , . . . , ˜ s n } we define k j and l j as in (1). It ma y happ en that k 1 = 0 and l m = 0, but k 2 , . . . , k m , l 1 , . . . , l m − 1 ≥ 1 and m ≥ 1. The adjacency 6 matrix of G is of the form: ˜ A G =              0 l m ,l m 0 l m ,k m 0 l m ,l m − 1 0 l m ,k m − 1 0 l m ,l m − 2 . . . 0 l m ,l 1 0 l m ,k 1 0 k m ,l m 1 k m ,k m 1 k m ,l m − 1 1 k m ,k m − 1 1 l m ,l m − 2 . . . 1 k m ,l 1 1 k m ,k 1 0 l m − 1 ,l m 1 l m − 1 ,k m 0 l m − 1 ,l m − 1 0 l m − 1 ,k m − 1 0 l m − 1 ,l m − 2 . . . 0 l m − 1 ,l 1 0 l m − 1 ,k 1 0 k m − 1 ,l m 1 k m − 1 ,k m 0 k m − 1 ,l m − 1 1 k m − 1 ,k m − 1 1 k m − 1 ,l m − 2 . . . 1 k m − 1 ,l 1 1 k m − 1 ,k 1 0 l m − 2 ,l m 1 l m − 2 ,l m 0 l m − 2 ,l m − 1 1 l m − 2 ,k m − 1 0 l m − 2 ,l m − 2 . . . 0 l m − 2 ,l 1 0 l m − 2 ,k 1 . . . . . . . . . . . . . . . . . . . . . . . . 0 l 1 ,l m 1 l 1 ,k m 0 l 1 ,l m − 1 1 l 1 ,k m − 1 0 l 1 ,l m − 2 . . . 0 l 1 ,l 1 0 l 1 ,k 1 0 k 1 ,l m 1 k 1 ,k m 0 k 1 ,l m − 1 1 k 1 ,k m − 1 0 k 1 ,l m − 2 . . . 0 k 1 ,l 1 1 k 1 ,k 1              . (6) Rep eating a similar arg ument as in Theorem 1, w e come to the followin g Theorem 2. L et G b e a thr eshold gr aph w i th self-lo ops asso ciate d with a cr e ation se quenc e ˜ S = { ˜ s 1 , ˜ s 2 , . . . , ˜ s n } and i ts adja c ency m atrix give n as in (6) . Set e C n (0) = n − 2 ( m − 1) − I { 1 } ( ˜ s 1 ) . (7) Then the s p e ctr al dis tribution of G is given by e µ n ( G ) = e C n (0) n δ 0 + 1 n J X j =1 δ λ j , J = 2( m − 1) + I { 1 } ( ˜ s 1 ) (8) wher e { λ j } exhaust the eigenvalues of            k m l m − 1 k m − 1 l m − 2 . . . l 1 k 1 k m 0 0 0 . . . 0 0 k m 0 k m − 1 l m − 2 . . . l 1 k 1 k m 0 k m − 1 0 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . k m 0 k m − 1 0 . . . 0 0 k m 0 k m − 1 0 . . . 0 k 1            for ˜ s 1 = 1 (i.e., k 1 ≥ 1 ), or            k m l m − 1 k m − 1 l m − 2 . . . k 2 l 1 k m 0 0 0 . . . 0 0 k m 0 k m − 1 l m − 2 . . . k 2 l 1 k m 0 k m − 1 0 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . k m 0 k m − 1 0 . . . k 2 l 1 k m 0 k m − 1 0 . . . k 2 0            for ˜ s 1 = 0 (i.e., k 1 = 0 ). Mor e over, a ny λ j in (8) diff e rs fr om 0 . 7 Remark The eigenv alues λ 1 , . . . , λ J in (8) are obtained from the c haracteristic equations: det                  − λ 0 . . . 0 0 0 . . . 0 k m λ − λ . . . 0 0 0 . . . k m − 1 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . − λ 0 k 2 . . . 0 0 0 0 . . . λ k 1 − λ 0 . . . 0 0 0 0 . . . l 1 λ − λ . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 l m − 2 . . . 0 0 0 . . . − λ 0 l m − 1 0 . . . 0 0 0 . . . λ − λ                  = 0 for s 1 = 1 (i.e., k 1 ≥ 1), or det                    − λ 0 . . . 0 0 0 0 . . . 0 k m λ − λ . . . 0 0 0 0 . . . k m − 1 0 0 λ . . . 0 0 0 0 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 0 . . . λ − λ k 2 0 . . . 0 0 0 0 . . . 0 l 1 + λ − λ 0 . . . 0 0 0 0 . . . l 2 0 λ − λ . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 l m − 2 . . . 0 0 0 0 . . . − λ 0 l m − 1 0 . . . 0 0 0 0 . . . λ − λ                    = 0 for s 1 = 0 (i.e., k 1 = 0). 4 Limit theo rems In this section we discuss asym ptotic b eha viors of the sp ectral distributions obtained in the previous sections. W e first consider the case where the distribution of X is discrete and g iv en by P ( X = i ) = p i , i = 0 , 1 , . . . , ∞ X i =0 p i = 1 . Let m ≥ 1 b e a fixed integer. T a k e a particlar threshold θ = 2 m − 1 and assume that p i > 0 for i = 0 , 1 , . . . , 2 m − 1 . It follo ws from the strong law of large num b ers that l i = ♯ { j : X j = m − i } , i = 1 , . . . , m, k i = ♯ { j : X j = m − 1 + i } , i = 1 , . . . , m − 1 , k m = ♯ { j : X j ≥ 2 m − 1 } , l i = k i = 0 , i ≥ m + 1 , 8 for large n almost surely . Moreo ver, denoting b y F the distribution function of X , we ha v e lim n →∞ 1 n m X i =1 l i = F ( m − 1) a.s. lim n →∞ 1 n m X i =1 k i = 1 − F ( m − 1) a.s. With these observ a tion w e easily obtain the f ollo wing Theorem 3. Notations and assumptions b eing a s ab ove, the sp e c tr al distributions of G n ( X , 2 m − 1) verifies lim n →∞ µ n ( G ) = (1 − F ( m − 1)) · δ − 1 + F ( m − 1 ) · δ 0 a.s. Similarly, the s p e ctr al dis tributions of ˜ G n ( X , 2 m − 1) verifies lim n →∞ e µ n ( G ) = δ 0 a.s. Remark Similar results ho ld when t he distribution of X is discrete and F has only finite n umber o f jumps in ( −∞ , θ / 2 ] or ( θ / 2 , ∞ ). But no simple description is kno wn fo r a general case. Next we consider t he case where the distribution of X is contin uous. As is stated b y Bose–Sen [4] implicitly , if the distribution of X is con tinuous and symme tric around 0, then the distribution of zero and one entries in the creation sequence e S of each graph generated by e G n ( X , 0) is the same as the distribution o f a sequenc e of i.i.d. Bernoulli random v ariables { e Y i } i =1 , 2 ,...,n with success probabilit y 1 / 2 , that is, P ( e Y i = 0) = P ( e Y i = 1) = 1 / 2 , for i = 1 , 2 , . . . , n. This means that e S = { ˜ s 1 , ˜ s 2 , . . . , ˜ s n } d = { e Y 1 , e Y 2 , . . . , e Y n } . Recall that the creation sequence S of eac h gra ph generated b y G n ( X , 0) is alw a ys satisfied with s 1 = s 2 . Then w e observ e that S = { s 1 = s 2 , s 3 , . . . , s n } d = { Y 2 , Y 2 , Y 3 , . . . , Y n } , where { Y i } i =2 , 3 ,...,n b e the seq uence of i.i.d. Bernoulli rando m v ariables with succe ss probabilit y 1 / 2, similarly . T aking the ab ov e consideration into accoun t , we obtain asymptotic behaviors of co ef- ficien ts of p oint measures on − 1 and 0 app earing in µ n ( G ) and e µ n ( G ). Theorem 4. Assume that the distribution of X is c ontinuous and symmetric ar ound 0 . Define C n ( − 1) , C n (0) and e C n (0) a s in (2) and (7) . T hen we have (1) lim n →∞ C n ( − 1) /n = lim n →∞ C n (0) /n = 1 / 4 a.s. (2) √ n ( C n ( − 1) /n − 1 / 4) ⇒ N (0 , 1 / 4) and √ n ( C n (0) /n − 1 / 4) ⇒ N (0 , 1 / 4) as n → ∞ . (3) lim n →∞ e C n (0) /n = 1 / 2 a.s. (4) √ n ( e C n (0) /n − 1 / 2) ⇒ N (0 , 1 / 4) a s n → ∞ . 9 Pr o of. Note the following relations: C n ( − 1) = m X i =1 k i − ( m − 1) − I { 1 } ( s 1 ) d = Y 2 + n X i =2 Y i ! − n − 1 X i =2 (1 − Y i ) Y i +1 − Y 2 = Y 2 + n − 1 X i =2 Y i Y i +1 , C n (0) = m X i =1 l i − ( m − 1) d = ( (1 − Y 2 ) + n X i =2 (1 − Y i ) ) − n − 1 X i =2 (1 − Y i ) Y i +1 = 2 − Y 2 − Y n + n − 1 X i =2 (1 − Y i )(1 − Y i +1 ) , e C n (0) = n − 2( m − 1 ) − I { 1 } ( ˜ s 1 ) d = n − 2 n − 1 X i =1 (1 − e Y i ) e Y i +1 − e Y 1 = 1 − e Y n + n − 1 X i =1 (1 − e Y i + e Y i +1 )(1 + e Y i − e Y i +1 ) . W e then easily c hec k that E [ C n ( − 1)] = E [ C n (0)] − 1 2 = n 4 and E [ e C n (0)] = n 2 . Applying a similar argumen t as in [4, Theorem 1], w e ha v e t he a ssertion. When the distribution of X is con tin uo us and symmetric around θ / 2, w e can obta in similar results f or G n ( X , θ ) and e G n ( X , θ ) by straig h tf orw a rd mo dification. Study cov ering a more general situation is now in pro gress. 5 Binary thre shold mo del In t his section w e g ive a simple example. The threshold net w ork mo del defined b y Bernoulli tria ls X 1 , X 2 , . . . , X n with success probabilit y p , i.e., 0 < P ( X i = 1) = p < 1, and a threshold 0 ≤ θ < 1 is called the binary thr eshold mo del and is denoted b y G n ( p ). F or G ∈ G n ( p ) the partition of the ve rtex set V is given b y V = V (1) ∪ V (0) , V (1) = { i ; X i = 1 } , V (0) = { i ; X i = 0 } . Theorem 5. F or G ∈ G n ( p ) we set | V (1) | = k and | V (0) | = l . Th e n the sp e ctr al distribu- tion of G is give n by µ k ,l = k − 1 n δ − 1 + l − 1 n δ 0 + 1 n δ λ + + 1 n δ λ − , 10 wher e λ ± = k − 1 ± p ( k − 1) 2 + 4 k l 2 . (9) Pr o of. W e need only to apply Theorem 1 with l 1 = l , l 2 = k 1 = 0, k 2 = k and m = 2. In this case, (5) b ecomes  k − 1 l k 0  , of whic h the eigen v alues are λ ± in (9). Corollary 1. L et µ n ( G ) b e the sp e c tr al distribution of G ∈ G n ( p ) . Then we have lim n →∞ µ n ( G ) = p · δ − 1 + (1 − p ) · δ 0 a.s. Pr o of. By the strong law of la rge n um b ers, see also Theorem 3. As for the the mean sp ectral distribution w e ha v e Theorem 6. The me an sp e ctr al distribution of the b i n ary thr eshold mo del G n ( p ) is given by µ n =  p − 1 n  δ − 1 +  1 − p − 1 n  δ 0 + 1 n n X k =0  n k  p k (1 − p ) n − k  δ λ − ( k ) + δ λ + ( k )  , (10) wher e λ ± ( k ) = k − 1 ± p ( k − 1) 2 + 4 k ( n − k ) 2 , k = 0 , 1 , . . . , n. Pr o of. Since P ( | V (1) | = k , | V (0) | = l ) =  n k  p k (1 − p ) l , k + l = n, the mean sp ectral distribution is giv en b y µ = n X k =0  n k  p k (1 − p ) l µ k ,l . Then (10) follows from Theorem 5 by direct computation. Corollary 2. L et µ n b e me an sp e ctr al distribution of the binary thr eshold mo del G n ( p ) . Then we have lim n →∞ µ n = p · δ − 1 + (1 − p ) · δ 0 . Ac kno wledgmen t. 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