RANDOM IRREGULAR BLOCK-HIERARCHICAL NETWORKS: ALGORITHMS FOR COMPUTATION OF MAIN PROPERTIES

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1 RANDOM IRREGULAR BLOCK-HIERARCHICAL NETWORKS: ALGORITHMS FOR COMPUTATION OF MAIN PROPERTIES Svetlana Avetisyan Mikayel Samvelyan* Matun Kaapetyan Yeevan State Univesity Abstact In this pape, the class of andom iegula block-hieachical s is defined and algoithms fo geneation and calculation of popeties ae descibed. The algoithms pesented fo this class of s ae moe efficient than known algoithms both in computation time and memoy usage and can be used to analyze topological popeties of such s. The algoithms ae implemented in the system ceated by the authos fo the study of topological and statistical popeties of andom s. Keywods: andom s, statistical popeties, algoithms. 1. Intoduction Block-hieachical s have been of paticula inteest in ecent yeas as they tuned out to pesent a convenient model of the spatial stuctue of complex biomolecules, such as DNA and poteins [1]. An active study of vaious complex s gives elevance to the development of a system that can simulate models of andom gaphs of diffeent types and conduct an effective and compehensive analysis of thei topological and statistical popeties [2]. The main popeties of a andom include node degee distibution, node-to-node distance distibution, the distibution of the clusteing coefficients of the nodes, cycle length distibution, length distibution of the connected subs, diamete, etc. In this pape, the class of andom iegula block-hieachical s is defined and algoithms fo geneation and calculation of popeties ae descibed. Blockhieachical s can be epesented as a hieachy of clustes (subnets) connected to each othe. In egula case, which ae discussed in [3,4], the numbe of nested clustes, and hence the numbe of nodes within the clustes of the same level ae identical. In case of an iegula block-hieachical, the numbe of nested clustes may ange fom 1 to p whee p specifies the maximum possible numbe of nested clustes. Algoithms fo the class of iegula block-hieachical s, which coincide with those fo egula s, ae not listed hee (efe to [3]). All the esults ae used in the implementation of Random Netwoks Exploe [4] which enables to cay out a study of statistical popeties of andom s. The esults of the analysis fo the degee distibution of egula and iegula block-hieachical s ae povided as an illustation. 2. An iegula block-hieachical An iegula block-hieachical is a genealized case of a egula blockhieachical [3]. Let s define a the fome fist. A egula block-hieachical G #,% Â #,% is defined by 2 paametes: p - banching index and Г - the numbe of levels in the, p > 1, Г 0. The numbe of nodes {x.,, x 0 }, whee N = p Г and patition into clustes is pecisely defined by these paametes, which detemines the stuctue Poceedings of the IX Annual Scientific Confeence in Russian-Amenian Univesity, 2015, p * Coespondence to: mikayel@samvelyan.com

2 of the nodes in a egula block-hieachical. The is constucted in levels. At each new level γ, 0 γ Γ, new clustes ae fomed though meging p of the clustes that have aleady been constucted at the pevious level. This esults in the fomation of new connections between the nodes of the, as all nodes of a cluste ae connected to evey node of a diffeent cluste, if the two ae connected to each othe on the same level of hieachy. The node-link tee epesents patition into clustes, while the bit sequences that mak all the vetices of the node-link tee detemine connection of the clustes, i.e. the pesence of links in the. M 7 (9) epesents i-th cluste on the level γ, 1 i n 7, whee n 7 is the numbe of clustes on the level γ. M 7 epesents a set of clustes on the level γ, M 7 = {M 7 (.), M 7 (>),..., M 7 ) } V(M (9) 7 ) defines nodes within cluste M (9) 7. All s in the class  #,% have the same numbe of nodes and the same patition into clustes and diffe in the choice of connected pais of nested clustes only. Fo the iegula case, p specifies the maximum numbe of patitions into clustes; the numbe of sub-clustes p fo each cluste. p and Г do not clealy detemine patitioning into clustes and do not set the numbe of nodes, as is the case with egula s. In case of any level of 1 γ C VEM (9) 7 F = VEM (.) Г F = {x.,, x 0 }, N p % Let Count(γ, i) denote the numbe of clustes nested in the cluste M (9) 7, Count(0, i) = 0. Patition into clustes on the level γ > 0 is detemined by the following set: Banch(γ) = PCount(γ, i)q1 i n 7 R. Banch(0) =, and will not be examined futhe. It's obvious that Banch(γ 1) Count(γ, i), 1 γ Γ. Patition of the entie is detemined by the following set: Banch = {Banch(γ) 1 γ Г} Fo this patition of Banch, a class of iegula block-hieachical s is defined G  WXYZ[\. Netwoks of this class have the same stuctue of patition of nodes into clustes and only diffe in connection among nodes. The class of egula blockhieachical s  #,% coincides with that of iegula case  WXYZ[\, if fo each i and γ, 1 i n 7, 1 γ Г, Count(γ, i) = p. In that case, N = p %. The node-link tee is used fo geneating and stoing the. The node-link tee of the G is set in the fom of a (p + 1)-tee that satisfies the following conditions: At each level g, 0 g Γ thee ae n 7 vetices t (9) g, 1 i n 7. The numbe of sub-tees at the vetex of t (9) 7 is Count(γ, i), Count(0, i) = 0. The total numbe of vetices of the node-link tee is equal to the numbe of clustes of the G. Each cluste M (9) 7 of the G is associated with the sub-tee whose oot is the vetex of the node-link tee t (9) (`) g. The nodes in the cluste M γ coespond to the leaves (the end vetices) of the espective sub-tee. The vetex of the node-link tee t (9) g, 1 g Г is maked by a sequence of zeoes and ones bitmap(γ, i), with a length of c (ce.), k = Count(γ, i). The bit > sequence bitmap(γ, i) descibes the connection between the clustes on the level 2

3 of γ 1, nested in the cluste M 7 (9). Let s label bitmap(γ, i) a node-link vecto of the vetex t g (9) of the node-link tee. The node-link tee of an iegula block-hieachical G is defined by the stuctue of patition into clustes: Banch = {Banch(γ) 1 γ Г}, whee Banch(γ) = {Count(γ, i) 1 i n 7 }, and a set of Bitmap - set of node-link vectos: Bitmap = {Bitmap(γ) 1 γ Г}, whee Bitmap(γ) = {bitmap(γ, i) 1 i n 7 }. The block-hieachical G Â WXYZ[\ is detemined pecisely by a set of Banch and Bitmap. The sub-tee coesponding to the cluste will be denoted simila to the cluste as M 7 (9). As an illustation in Figue 1, the G and the coesponding node-link tee ae pesented, wheein Banch = P{3, 4, 2}, {3}R, Bitmap = {{< 011 >, < >, < 1 >}, {< 100>}}, Fig.1. Node-link tee of G Â opq?s, 3. Algoithms fo geneation of iegula block-hieachical s Geneation of G Â WXYZ[\ is ceation of a coesponding node-link tee, i.e. geneation of stuctue of patition of nodes into clustes. Banch = {Banch(γ) 1 γ Г}, whee Banch(γ) = {Count(γ, i) 1 i n 7 }, and geneation of multiple node-link vectos Bitmap = {Bitmap(γ) 1 γ Г}, whee Bitmap(γ) = {bitmap(γ, i) 1 i n 7 }, Geneation of Banch is possible by setting the maximum banching index p and eithe the numbe of elements of the N, o the numbe of levels Г. An algoithm fo both cases is descibed below. To geneate node-link vecto bitmap(γ, i), the pobability of connection ω = k eu, k = VEM 7 (9) F, μ > 0 is detemined. μ specifies the geneated density, while k the numbe of nodes in the cluste M 7 (9) Geneation of the stuctue of patition into clustes when setting maximum banching index p and numbe of nodes N Geneation of the stuctue of patition into clustes Banch is caied out by levels fom the bottom up, i.e. fom the leaves of the tee to the oot. Clustes on the level γ ae constucted, followed by a andom and equipobable combination of some clustes on the level γ with those on the level γ + 1. The numbe of levels Г of an iegula blockhieachical is detemined in the pocess of geneation é N zp ù Г <. 3

4 Input: N, p Output: Г the level of the constucted tee, Banch the stuctue of patition into clustes. 1. Banch, n N, γ 0 2. If n = 1, then Г γ and complete algoithm 3. γ γ + 1, i = 1 4. Geneate a andom intege l fom 1 to p 5. If l > n, then l n 6. Count(γ, i) = l, n: = n l, Banch(γ) = Banch(γ) {l}, i: = i Repeat Steps 4-6 until n 0 8. n = Banch(γ), Banch = Banch { Banch(γ) }, etun to Step Geneation of stuctue of patition into clustes fo a given p and numbe of hieachical levels Г Geneation is caied out by the levels of hieachy fom top to bottom, i.e. the oot is geneated fist, while the leaves of the tee, which coespond to the nodes of the, ae geneated last. The numbe of nodes in an iegula block-hieachical is detemined in the pocess of geneation. Input: p, Г Output: N numbe of nodes, Banch stuctue of patition into clustes. 1. γ = Г, Banch =, n = 1, i = 1, m = 0 2. If γ = 0, then complete the algoithm with the output N = n 3. Geneate a andom intege l fom 1 to p 4. Count(γ, i) = l, Banch(γ) = Banch(γ) {l}, i = i + 1, m = m + l 5. Repeat Steps 3-4 n times 6. n = m, Banch = {Banch(γ)} Banch, γ = γ 1, i = 1, m = 0 7. Retun to Step Geneation of node-link vectos Geneation of node-link vectos Bitmap is caied out on aleady ceated node-link tee Banch. Input: µ > 0, Г, Banch Output: Bitmap multiple node-link vectos. 1. Bitmap =, γ = 0 2. If γ = Г, then complete the algoithm 3. i = 1, γ γ Detemine k = VEM 7 (9) F - numbe of nodes in cluste M 7 (9) 5. Geneate bit sequence bitmap(γ, i) with the length of l (l 1)/2, whee l = Count(γ, i). Pobability of connection ω = k eu is used when geneating 6. Bitmap(γ) Bitmap(γ) {bitmap(γ, i)}, i: = i Repeat Steps 5-6 n times 8. Bitmap = Bitmap {Bitmap(γ)} 9. Retun to Step 2 4

5 Let s estimate the numbe of levels in the geneated. The aveage value of a andom vaiable Count(γ, i) is (p + 1) 2 which means that the aveage numbe of levels is Г q ƒp = log (#.) > N when using geneation algoithm 3.1. This paticula estimate is pacticed to assess the complexity of algoithms fo calculation of the main popeties of an iegula block-hieachical. 4. Algoithms fo calculation of the main popeties of an iegula blockhieachical The node-link tee detemines the stuctue of stoage in a block-hieachical, and geneation of a andom G with nodes N is educed to geneation of a set of Banch and Bitmap. All the algoithms being developed use this stoage stuctue. The calculation of popeties is caied out along the node-link tee which povides high efficiency both in memoy usage and computation time. In this chapte, algoithms fo calculating the degee of a node, the distance between two nodes, the numbe of node-links in a, the numbe of connected subgaphs of a given length, the numbe of cycles with length of 3 and the numbe of cycles with length of 4 ae descibed. The assessment of complexity is pesented fo compaison with the classical algoithms. A definition simila to the one in [3] is intoduced. SEM 7 (9) F, 1 γ Γ denotes a set of clustes on the level γ 1, nested in the cluste M 7 (9). S(M 7 (9) ) = S. (M 7 (9) ), S > (M 7 (9) ),, S c (M 7 (9) )Š, k = Count(γ, i), 1 i n 7, whee S? (M 7 (9) ), 1 n k nested cluste n. Let s define the function ψ 7,9 (n, s) of diect connection of two sub-clustes S? (M 7 (9) ), S (M 7 (9) ) as follows: ψ 7,9 (n, s) = Ž 1, if clustes S Z(M (`) γ ) and S œ (M (`) γ ) ae connected diectly 0, othewise Sub-clustes S? (M (9) 7 ) and S (M (9) 7 ) ae connected diectly if the bit descibing the elation between them is 1 in the node-link vecto bitmap(γ, i) Calculation of the degee of a node in the iegula block-hieachical Suppose G  WXYZ[\ is a block-hieachical, x V(G). Let s conside the set of clustes on the level γ M 7 (.), M 7 (>),, M 7 ) Š. The function ν(x, γ) detemines the numbe of clustes on the level γ, containing the node x. In this case, M γ (ν(x,γ)) is the only cluste on the level γ, containing the node x. Fo convenience, let s denote it M 7 ( ). Claim 1. Let s suppose M γ (x) is a cluste on the level γ, in that case 7 ª«? 9,ν(x,9) degeeex, M γ ( ), F = Eψ 9,ν(x,9) (ν(x, i 1), j) V (S (M 9 (ν(x,9)) )) F G. whee degeeex, M γ ( ) F is the degee of the node x in the cluste M γ (x), 1 i γ. Poof. To calculate the degee of the node x in the cluste M γ (x) it s sufficient to examine the path fom the leaf x to the vetex t 7 (?) on the node-link tee and at each level of i, 5

6 1 i γ and to count the numbe of diect connections in the cluste M (x) 9e. along the nodelink vecto. Claim 1 is poven. Netwok G will have degee(x, G) = degeeex, M % (.) F Assessment of the complexity of the algoithm. To calculate V (S (M 9 (ν(x,9)) )) it is necessay to tavese all the vetices of the node-link tee of the espective cluste M 9e. othe than Level 0. The maximum numbe of vetices of the node-link tee p that ae not leaves is i 1 cg p c. To calculate degee(x, G) it s necessay to cay out the steps of % ((p 1) 9e. cg p c ). The banching index p is a constant and is detemined pio to the calculation. When Г q ƒp = log (#.) > N the time complexity of the algoithm is О(N). ( ) 4.2. Distance between two nodes Claim 2. Let s suppose d(x, y) is the distance between two connected vetices of the. In that case d(x, y) {1,2}, if p = 2 and d(x, y) {1, 2,, p 1}, if p 3. The poof and algoithm fo computation of the distance between two nodes is simila to the egula case [3]. Assessment of the complexity of the algoithm. As with the egula case, it s sufficient to go up the node-link tee fom leaves of the espective nodes to the vetex of the tee that coesponds to the cluste on the lowest level which contains the given nodes. When Г q ƒp = log (#.) > N the time complexity of the algoithm is О(logN) Numbe of edges in the iegula block-hieachical Claim 3. Let s suppose G Â WXYZ[\, E(M 7 (?) ) numbe of edges in the cluste M 7 (?), 1 n n 7, in that case c E(M (?) 7 ) = ²Е S k EM (n) γ Fµ² + Eψ γ,n (i, j) V(S i (M (n) γ ) ) V(S j (M (n) γ ) ) F, k=1 c 1 i=1 c j=i+1 whee = Count(γ, n), E(M (9) ) = Æ, 1 i N. (.) Netwok G will have E( G) = ²Е(M Γ )². Assessment of the complexity of the algoithm. Fist of all, note that, fo the calculation of the numbe of vetices of a given cluste, it s not necessay to tavese all the vetices of the node-link tee of the espective cluste, it s sufficient to sum up the numbe of vetices of the sub-clustes that have aleady been counted by the algoithm V(M (?) 7 ) = V(S c (M (?) 7 ) ). Hence it follows that a maximum of p 2 steps is equied fo the cg. calculation of the addend. Opeating time of the algoithm fo calculation of E(M (?) 7 ) satisfies the following ecuence elation: F(γ) = p F(γ 1) + p>, γ > 0 F(γ) = 0, γ = 0, whee F(γ) denotes the complexity E(M (?) 7 ), 1 n n 7. Solving it, we get F(γ) = p > (p 7 1)/(p 1). Fo level γ = Γ we have O(p > (p Г 1)/(p 1) ). When Г q ƒp = log (#.) > N the time complexity of the algoithm is О(N). 6

7 4.4. Numbe of cycles with length of 3 in the iegula block-hieachical Claim 4. Let s suppose G Â WXYZ[\ and Cyclеs(М 7 (?), 3) is the numbe of cycles with length of 3 between cluste nodes М 7 (?), in that case e> e. Cyclеs(М 7 (?), 3) = Cyclеs ES 9 EM 7 (?) F, 3F + Eψ 7,? (i, j) ²Е S 9 EM 7 (?) Fµ² V(S EM 7 (?) F) F + 9, G. 9½ Eψ 7,? (i, j) ψ 7,? (j, k) ψ 7,? (k, i) V(S 9 EM 7 (?) F) V(S EM 7 (?) F) G9. cg. V(S c EM 7 (?) F) F whee c = Count (γ, n), Cyclеs(М (9), 3) = 0, 1 i N, Netwok G will have Cyclеs(G, 3) = Cyclеs(М (.) À, 3). Assessment of the complexity of the algoithm. Opeating time of the algoithm fo calculation of Cyclеs(М (?) 7, 3) satisfies the following ecuence elation: F(γ) = p F(γ 1) + pá + p >, γ > 0 F(γ) = 0, γ = 0, whee F(γ) denotes the complexity Cyclеs(М 7 (?), 3), 1 n n 7. Solving it, we get F(γ) = p > (p + 1) (p 7 1)/(p 1). When Г q ƒp = log (#.) > N the time complexity of the algoithm is О(N) Numbe of diffeent cycles with length of 3 passing though the same node Let Cyclеs (М ( ) 7, 3) denote the numbe of diffeent cycles with length of 3 passing ( ) though the same node x in the cluste М 7 Claim 5. Let s suppose G Â WXYZ[\ is a block-hieachical, in that case Cyclеs x (М (x) 7, 3) = Cyclеs x (М (x) 7e., 3) + E ψ 7,ν(x,Â) (ν(x, γ 1), j) ²Е(S (M (x) 7 ) )²F + e. Eψ 7,ν(x,Â) (ν(x, γ 1), j) degeeem (x) 7e., xf ²V S EM (x) 7 Fµ²F G. Eψ γ,ν(x,γ) (ν(x, γ 1), i) ψ γ,ν(x,γ) (i, j) ψ 7,ν(x,γ) j, ν(x, γ 1) ²V S 9 EM 7 (x) Fµ² G9. ²V S EM 7 (x) Fµ²F Assessment of the complexity of the algoithm. As in the pevious cases, the numbe of cycles with length of 3, passing though the same node in the cluste, can be calculated by tavesing the node-link tee once. When Г q ƒp = log (#.) > N the time complexity of the algoithm is О(N). + 7

8 4.6. Numbe of cycles with length of 4 in the iegula block-hieachical Simila to Claim 3, an algoithm is developed fo calculating the numbe of cycles with length of 4 in an iegula block-hieachical. When Г = log (#.) > N, the time complexity of the algoithm is О(N). Results of the expeiment. The given algoithms wee tested in the system unde development, Random Netwoks Exploe. The esults of the two expeiments ae as follows: 1. Calculation of cycles with length of 3 and 4 fo egula and iegula blockhieachical s (Table 1). 2. Chats fo the degee distibution of nodes fo egula and iegula block-hieachical s (Fig. 2). The esults wee obtained on an assembly of 100 copies, fo s with banching paamete p = 3 and the numbe of vetices N = 3 É = Cycles with length of 3 Cycles with length of 4 N p μ egula iegula egula iegula Î Î É 5 10 É Ï 6 10.Á Ð Ñ 2 10.Á > Î 10 Ð 4 10 É Table 1. Compaison of cycles with length of 3 and 4 fo egula and iegula block-hieachical s whee = 3, N = on an assembly of 100 copies Fig. 2. Compaison of the degee distibution fo iegula (ound) and egula (squae) s whee N = 19683, p = 3, μ = 0.1 (on the ight), μ = 0.3 (on the left) on an assembly of 100 copies Refeences 1. V. A. Avetisov, A. Kh. Bikulov, A. A. Vasilyev, S. K. Nechaev, A. V. Chetovich. On Some Physical Applications of Random Hieachical Matices. JETP 136(3) 566 (2009). 2. Albet R, Baabási A-L. Statistical mechanics of complex s. Rev. Mod. Phys. 74: (2002). 8

9 3. S. Avetisyan, A. Hautyunyan, D. Aslanyan, M. Kaapetyan, A. Kochayan. Algoithms fo Computation of Statistical Popeties of Regula Block-hieachical Netwoks. Poceedings of the 6 th Annual Scientific Confeence, Russian-Amenian (Slavonic) Univesity, Decembe S. Avetisyan, A. Kochayan. The System of Geneation of Random Netwoks and Computation of Thei Topological Popeties. 9 th Intenational Confeence of Compute Science and Infomation Technologies (CSIT-2013), Yeevan 2013, pp