Artif Intell Rev DOI 10.1007/s10462-016-9462-1
Influence of social network on method musical composition Roman Anselmo Mora-Gutiérrez1 · Eric Alfredo Rincón-García1 · Antonin Ponsich1 · Javier Ramírez-Rodríguez1 · Iris Iddaly Méndez-Gurrola1
© Springer Science+Business Media Dordrecht 2016
Abstract The method of musical composition (MMC) is a metaheuristic based on sociocultural creativity systems. Within the MMC, models of social influence and social learning are used and integrated in a social network, which is composed of a set of individuals with links between them and involves a set of interaction rules. In this paper, a comparative study on the performance of the MMC with different network structures is proposed. Sixteen benchmark nonlinear optimization problems are solved, taking into account nine social topologies, which are: (a) linear, (b) tree, (c) star, (d) ring, (e) platoons, (f) von Neumann, (g) full connection, (h) random and (i) small world. In addition, the update of each topology structure was tested according to four different strategies: one static, two dynamic and one self-adaptive states. An exhaustive statistical analysis of the obtained numerical results indicates that the social dynamics has no significant impact on the MMC’s behavior. However, the topology structures can be classified into groups that consistently influence the performance level of the MMC. More precisely, a structure characterized by a low value of its mean number of neighbors and a rather fast information transfer process (star topology) performs in a radically opposite way as structures where each agent has many neighbors (random and complete topologies). These observations allow to provide some guidelines for the selection of a network topology used within a social algorithm.
B
Roman Anselmo Mora-Gutiérrez
[email protected];
[email protected] Eric Alfredo Rincón-García
[email protected] Antonin Ponsich
[email protected] Javier Ramírez-Rodríguez
[email protected] Iris Iddaly Méndez-Gurrola
[email protected]
1
Departamento de Sistemas, Universidad Autónoma Metropolitana, C.P. 02200 Mexico, DF, Mexico
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Keywords
Network topologies · Multi-agent search systems · Non-linear optimization
1 Introduction In real life, everybody organizes and reorganizes his/her social networks, in order to reach some goals, since links (kinship, friendship, political affiliation etc.) between members of a social system allow out communication, exchange of both ideas and objects and knowledge diffusion and sharing (Thomas 1996; Christakis and Fowler 2010). So, the behaviour of the society members depends on the relationship configuration, direct links and indirect linkages (Scott 2009). In other words, the social network topology influences social behaviors, since it provides a framework for individuals to interact routinely (Sen and Sen 2010). “Social norms” emerge when individuals repeatedly interact with their neighbors , which can be seen as patterns of actions or behaviors. Those patterns spread through the social network until one behavior is adopted as the default one for all members of the society (Sen and Sen 2010; Mendes 2004). Social and cultural algorithms classically mimic the operating mode of real social systems such as bee colonies, ant colonies, bird flocks, human societies, etc., and are constructed on an artificial society composed of artificial agents. Generally, social algorithms combine principles of interaction, communication, socio-cognition between agents, population structure, dynamic environment with ideas drawn from evolutionary computation. These algorithms have been used to solve several optimization problems with satisfactory results, see for instance Landa Becerra and Coello Coello (2005), Ray and Liew (2003), Tang and Li (2008), among others. The method of musical composition (MMC) is a social algorithm that mimics the social creativity system involved in the process of musical composition. Within the MMC, models of social influence and social learning are used and integrated in a social network, which is composed of a set of individuals with links between them (Christakis and Fowler 2010; Scott 2009) and involves a set of interaction rules. This metaheuristic was proposed in Mora-Gutiérrez et al. (2012a) and it has been used to solve some kinds of optimization problems, obtaining promising results (see Mora-Gutiérrez et al. 2012b, 2013b; Mora-Gutiérrez 2013; Silva López et al. 2013). The MMC’s operating mode is based on an artificial society where interactions between agents, called composers, involve changes of the features of each composer’s artwork. These changes occur at several levels: (a) personal, (b) local and (c) social (Mora-Gutiérrez et al. 2013a; Mora-Gutiérrez 2013). The social network topology used within the MMC is defined through of a neigborhood structure for each composer in the artificial society. The present work proposes a comparative studies on the effects of several social networks topologies. The remainder of this paper is organized as follows: in the next section, previous related works are outlined. In Sect. 3, the experimental design used to analyze the effects of the social network topology in MMC is described. Numerical results are discussed in Sect. 4 while conclusions and perspectives are provided in Sect. 5.
2 Related work This section first describes the MMC, then an overview of studies dealing with social network topologies within metaheuristics is proposed.
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2.1 Description of the MMC As already mentioned in this paper, the MMC is an algorithm that mimics a creativity system associated with musical composition. The MMC is based on a multi-agent model, with a specific social network. This latter is composed of a set of N c vertices or agents (composers) and a set E of edges or links (which represent relationships between composers). In this model, a solution is called a “tune” and it is represented by an n-dimensional vector represented as tune = [x1 , x2 , . . . , xn ], where xi is value of the ith decision variable. Each composer has a personal knowledge, i.e. a set of tunes. Besides, a set of mechanisms and policies governs the possible interactions and information exchange between composers. A brief description of the MMC is as follows. Initially, the MMC builds a social network with N c composers and a set of randomly generated links between composers. For each composer i, an artwork is created randomly (an artwork is set of solutions used as the previous knowledge of composers) and stored in a score matrix (P,,i ). Then, while a stopping criterion is not met, the following steps are performed: (a) the social network’s links are updated, (b) each composer i (i = 1, . . . , N c) establishes contact with composer j from i’s neighborhood (∀ j ∈ Ni ) and examine his/her personal artwork (P,, j ), (c) each composer i (i = 1, . . . N c) greedily decides whether selecting information from composer j (∀ j ∈ Ni ): if the worst solution in P,, j (tune j,wor st ) is better than the worst solution in P,,i (tunei,wor st ), then composer i randomly selects one tune from j’s artwork. (d) each composer i (i = 1, . . . N c) builds his/her acquired knowledge, composed of the tunes selected from other composers, and stores it the matrix (I SC,,i ), (e) each composer i (i = 1, . . . N c) builds his/her knowledge background, stored in matrix K M,,i , as the concatenation of P,,i and I SC,,i , (f) each composer i (i = 1, . . . , N c) creates a new tune (xi,new ) based on his/her KM,,i matrix and finally, (g) each composer i (i = 1, . . . , N c) greedily replaces, or not, his/her worst tune tunei,wor st by xi,new in P,,i , according to the respective quality of both tunes. Generally, the stopping criterion is the maximum number of arrangements (maxarrangement ), i.e. the total number of composed tunes (equivalent to a maximum number of computed solutions). The basic structure of MMC is presented in Algorithm 1. The parameters used within the MMC algorithm are: the maximum number of arrangements (max _arrangement), the number of composers (Nc) and number of tunes each composer’s artwork (Ns) has. Furthermore, the factor of genius over innovation (ifg) represents the probability that a composer generates a new tune without considering foreknowledge while cfg is the probability of generating a single decision variable without considering previous knowledge. Finally, parameter factor of exchange between agents (fcla) is the probability of creating/eliminating an edge in the social network. For more detailed information about the MMC algorithm, we refer the reader to Mora-Gutiérrez et al. (2012a, b, 2013a), MoraGutiérrez (2013).
2.2 Overview of studies of social network topologies in metaheuristics In the last decade, several studies addressing the effects of neighborhood topologies on the performance of social algorithms have developed. Specially swarm-based algorithm have represented the main topic of these works (Kennedy and Mendes 2006; Toscano Pulido et al. 2011). Generally, the topologies used in particle swarm optimization (PSO) neighborhoods are based on global or local structures, called gbest and lbest, respectively (Eberhart et al. 1996; Kennedy 1999). According to the gbest neighborhood, every individual is attracted towards the best solution found by any member of the entire population. On the other hand, the
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Algorithm 1: The basic MMC algorithm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Input: MMC parameters and data about the instance to solve. Output: The best tunes generated by the composers society. Create an artificial society with specific interaction rules between agents. for every agent-composer i in the society do repeat Randomly generate tune q for composer i (tunei,q ). Evaluate tunei,q . until N s tunes are created; Store the N s created tunes in the score matrix (P,,i ) of composer i end repeat Update the artificial society. Exchange information between agents. for each agent-composer i in the society do Update the knowledge matrix K M,,i . Evaluate the fitness of every tune in the knowledge matrix. Set tunei,wor st ← tune in the knowledge matrix with the worst objective value. Generate a new tune (tunei,new ). Evaluate the new tune. if tunei,new is better than tunei,wor st (in terms of the objective function) then Replace tunei,wor st by tunei,new in P,,i . end end Build the new set of solutions. until The termination criterion is satisfied;
lbest structure makes that individuals are attracted towards the best solution achieved by some agent from a restricted neighborhood (Shi 2004; Kennedy 1999). The gbest structure is equivalent to a fully connected social network, where every individual is able to compare its performance with that of every other member of the population, imitating only the very best. The effects of several classical social structures, initially proposed in Bavelas (1950), Watts and Strogatz (1998), on the behavior of a PSO algorithm were compared in Kennedy (1999) on four standard test functions. The authors consider four topologies, which are: circle (also called ring), wheel (also called star), fully connected (also called complete or gbest) and random. The numerical results showed that the sociometrics of the particle swarm has a significant effect on its ability to find optimal solutions. Similar experiments carried out in Suganthan (1999) suggested that the lbest topology seems to achieve a better exploration of the search space while the gbest structure obtained faster convergence times. Besides, particle interactions controlled by a feedforward neural networks were proposed in Mendes et al. (2002), in conjunction with the study of five topologies: gbest, ring, square, pyramid and four clusters. In Mendes et al. (2003), five population topologies (gbest, ring, pyramid, von Neumann and four clusters) were employed to solve five standard test functions. The numerical results of these studies coincide on the importance of the used topology for the PSO performance. Particularly, as empirically demonstrated in Mendes (2004), the robustness of the algorithm and its ability to avoid premature convergence can be affected by the social topology used. Similar comparative and statistical studies on the effects of neighborhood topologies on the convergence rate of a PSO-based approach were performed in Toscano Pulido et al. (2011), Reyes Medina et al. (2009), Hu et al. (2004). In Günther and Nissen (2009), an application
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of several neighborhood topologies within a PSO algorithm applied to the staff scheduling problem was developed. An overview on PSO (Poli et al. 2007), specifically in Sects. 2 and 3, presents an outline over population dynamics and topologies. A method for choosing a specific topological neighborhood when dealing with multimodal functions, called NichePSO, was introduced in Brits et al. (2002, 2007). The NichePSO was compared with classical PSO implementations, using the gbest and lbest architectures and two niched genetic algorithms, in Brits et al. (2007); the numerical results obtained on multimodal functions showed that NichePSO successfully located and maintained multiple optimal solutions. A PSO niching algorithm using the ring topology was proposed by Li (2010). The ring topology allowed that this algorithm does not require any niching parameters. Experimental results showed that PSO algorithms using the ring topology was able to provide superior and more consistent performance over some existing PSO niching algorithms that require niching parameters. An hybrid PSO-based algorithm, based on a multi-neighborhood, is introduced in Hamdan (2008). The specificity of the algorithm is that three population topologies (star, ring and von Neumann) are simultaneously used in the hybrid algorithm, each particle using the topology that produces the best results. Besides, an investigation on the effect of eight PSO topologies on the performance level of the PSO-ELM (hybridization between PSO with extreme learning machine) was presented in Figueiredo and Ludermir (2013). Regarding the study of different topologies within other population-based techniques, their effect on the central force optimization (CFO), initially introduced in Formato (2007), was analyzed in Green (xxxx). Similarly, the behavior of the original MMC with five different population topologies, in a static state, has been studied in Silva López et al. (2013).
3 Experimental methodology This section presents the design of the computational experiments that will subsequently performed (see Sect. 3.1) to evaluate the impact of the neighborhood topologies used within the MMC. This includes the description of the considered topology structures as well as the rules that govern interactions between agents, concluding with the used test functions, the experimental methodology and parameter settings. Note that all the computational experiments are carried out for unconstrained nonlinear optimization problems (UNOPs), whose general formulation is provided in Eq. 1. min x∈ f (x) such that: n f L U : R → R xi ∈ xi , xi ⊂ R ∀i = 1, 2, . . . , n
(1)
3.1 Population topologies and social network dynamics As above-mentioned, the behavior of the MMC with nine different social topologies is studied and analyzed in this paper. Furthermore, four population dynamics, which determine how the social network is updated, are considered. These different features are described in the Sects. 3.1.1 and 3.1.2.
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3.1.1 Population topologies As shown in the works mentioned in Sect. 2.2, the structure of the population topology used within a search method is an essential feature for a good performance level in social metaheuristics. The nine population topologies tested in this paper (see Fig. 1) are described straightforward: (SN1) Linear Composers are placed on a line, where most composers are connected to two other composers while the first and last composers have only one neighboring agent. (SN2) Tree This topology is based on a hierarchy of nodes. Here, composers are placed on a binary tree structure, each composer having at most two children. (SN3) Star Each composer is connected to a central composer, who can be seen as a signal regulator. (SN4) Ring This topology is built in a circular fashion: each composer is connected to exactly two neighbors (called predecessor and successor, respectively). This topology can be seen as a linear structure whose ends are connected.
SN1
SN2
SN3
SN4
SN5
SN6
SN7
SN8
SN9
Fig. 1 Neighborhood topologies used in this study
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(SN5) Platoons In this topology, k clusters of composers are designed, each composer being connected with agents from the same cluster. k is a random number generated according to a discrete uniform distribution between 2 and N2c . There are not links between clusters. (SN6) von Neumann Each composer is connected to its leftmost, rightmost, top and bottom neighbors on a two dimensional lattice. (SN7) Full connection (also called complete, standard or gbest). Each composer is connected with all the other composers in the society. (SN8) Random The edges of the social network are randomly generated according to a randomy drawn edge density. (SN9) Small world This structure, based on the Watts and Strogatz’s model (Watts and Strogatz 1998), establishes the existence of links between composers according to a power law distribution function. Note that these topologies may be classified, according to their structure, into local, global and mixed types. In a local type topology, each individual is influenced by some small portion of the population (for instance: linear, tree, ring, platoons, von Neumann, random and small world topologies). On the other hand, with a global type topology, each individual is linked with other members of the population without any restriction. This is the case of the full connection topology. Finally, a mixed type topology involves that some members of the society are influenced by some small portion of the population, while other members are connected with all the society. The star topology is representative of this last structure.
3.1.2 Social network dynamics In addition to the topological structure of the social network, another important aspect is the evolution of this structure during the search process. Indeed, a structure might be set at the beginning of an execution and maintained unchanged during the run. On the other hand, this initial structure might also be updated with a preset frequency. In this work, four different updating strategies of the social network are considered, in combination with the nine structures described in Sect. 3.1.1. These states are briefly described in what follows: – Static (S) This state involves that the structure of the social network does not change in time, so that each composer cannot modify his/her links with other composers. This search strategy is obtained by setting the fcla parameter to zero. – Dynamic 1 (D1) The structure of the social network changes in time, in such a way that each composer can have his/her links with other composers modified. This process of edge elimination or creation is performed according to a probability fcla. Note that the value of fclai is set by the user at the beginning of the run and maintained constant afterwards. It is the same for all composers. – Dynamic 2 (D2) The structure of the social network also changes in time, but each composer i has his/her own parameter fclai , ∀i ∈ {1, . . . , Nc}. At the beginning of a run, the Nc values for the fclai parameters are randomly generated from a uniform distribution between 0 and 1. These values remain constant during the whole run. – Self-adaptive (SA) In this last case, the value of the fclai parameters might vary among composers and in time. Initially, Nc fclai values are randomly generated from a uniform distribution between 0 and 1. Subsequently, each composer i updates, in each iteration, its own fclai value according to his/her results and his/her previous knowledge. Algorithm 2 describes in detail the mechanism for fclai update.
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Algorithm 2: Self-adaptation of f clai of composer i
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
Input: fclaiv−1 , Nc, Pi,, Output: Update fclaiv v ← iteration number. if v ≥ 2 then for i = 1 : N c do v−1 P,,i ← score matrix of composer i in iteration v − 1.
v−2 P,,i ← score matrix of composer i in iteration v − 2.
v−1 v−1 tunei,wor st ← worst solution in P,,i .
v−2 v−2 tunei,wor st ← worst solution in P,,i . v−1 v−1 tunei,best ← best solution in P,,i .
v−2 v−2 tunei,best ← best solution in P,,i . v−2 v−2 a1 = f tunei,wor st − f tunei,best +1 v−1 v−1 a2 = f tunei,wor st − f tunei,best v−2 v−1 a3 = f tunei,best − f tunei,best a2 −a3 1 τi = a + N c 1 v−1 v−2 if f tunei,wor st = f tunei,wor st then Draw βi as a random number from normal distribution with mean 0 and standard deviation 1. α = f claiv−1 + τi ∗ βi else α = f claiv−1 end while α > 1 do α = α−1 2 end while α < 0 do α = α+1 2 end end end f claiv = α
3.2 Test problems In this paper, 16 instances of UNOP were considered for the numerical experimentation. These test functions were proposed in Liang et al. (2013b) and, according to their properties, they are divided into two groups: unimodal functions and multimodal functions. All of them can defined as minimization problems: min f (x) x = [x1 , x2 , . . . , x D ]T xi ∈ [−100, 100] ∀i = 1, . . . , D
(2)
where D is the dimension of the search space. In some problems, several mathematical objects are to be specified, such as:
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– o is a shifting vector of the global optimum, which is defined as o = [o1 , o2 , . . . , o D ]T , oi is uniformly distributed in [−80, 80]. The vector o used in this work was proposed by Liang et al. (2013a, b). – M j is an orthogonal matrix, which is generated from standard normally distributed entries by Gram–Schmidt orthonormalization. Matrices M j used in this work were proposed by Liang et al. (2013a, b). – Λα is a diagonal matrix with dimension D. Diagonal elements of Λα are computed as i−1
2∗(D−1) for all i = 1, 2, . . . , D. λii = α √
1+β
i−1
xi
xi = xi D−1 if xi > 0 , f or all i = 1, . . . , D xi = xi other wise – Tosz (xi ) = sing(xi )⎧expxˆi +0.049(sin(c1 xˆi )+sin(c2 xˆi ) for all i = 1, . . . , D if xi < 0 ⎨ −1 log(|xi |) if xi = 0 0 if xi = 0 , xˆ = where: sing(xi ) = 0 other wise ⎩ 1 other wise 10 if xi > 0 7.9 if xi > 0 , c2 = c1 = 5.5 other wise 3.1 other wise β
– Tany =
The formulation of the considered instances are presented below: – Unimodal functions (a) Sphere function min f (x) :
D
i=1
z i2 − 1400
subject to z = x −o (b) Rotated Bent Cigar function min f (x) : z 12 + 106
D
i=2
z i2 − 1200
subject to 0.5 (M (x − o)) z = M2 Tany 1 (c) Different powers function min f (x) :
D
i=1 |z i |
i−1 2+4 D−1
− 1000
subject to z = x −o – Multimodal functions (d) Rotated Rosenbrock’s function min f (x) :
D−1
i=1
2 100 z i2 − z i+1 + (z i − 1)2 − 900
z = M1
subject to 2.048(x−o) 100
+1
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(e) Rotated Schaffers F7 function min f (x) : zi =
1 D−1
D−1
√
zi +
√
i=1
z i sin2 (50z i0.2 )
2 − 800
subject to
2 yi2 + yi+1 for all i = 1, . . . , D − 1 10 0.5 (M (x − o)) y = Λ M2 Tany 1
(f) Rotated Ackley’s function min f (x) : −20 exp
D 2 −0.2 D1 i=1 z i
1 D
D
cos(2π z i )
− exp subject to 0.5 (M (x − o)) z = Λ10 M2 Tany 1 i=1
+20 + exp −700
(g) Rotated Weierstrass function kmax D
k 0.5 cos 2π 3k (z i + 0.5) min f (x) : k
i=1 k=0 k −D kmax k=0 0.5 cos 2π 3 (0.5) − 600 subject to kmax = 20 0.5 M 0.5(x−o) z = Λ10 M2 Tany 1 100
(h) Rotated Griewank’s function min f (x) :
D
i=1
z i2 4000
z=
−
D
cos
i=1
subject to
Λ10 M
1
zi √ i
600(x−o) 100
+ 1 − 500
(i) Rastrigin’s function min f (x) :
D
i=1
z i2 − 10 cos(2π z i ) + 10 − 400
subject to 5.12(x−o) 0.2 T z = Λ10 Tany osz 100 (j) Rotated Rastrigin’s function min f (x) :
D
i=1
z = M1
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Λ10 M
z i2 − 10 cos(2π z i ) + 10 − 300 subject to Tosz M1 5.12(x−o) 100
0.2 2 Tany
Influence of social network on method musical composition
(k) Non-continuous rotated Rastrigin’s Function D
min f (x) :
i=1
z i2 − 10 cos(2π z i ) + 10 − 200
subject to 0.2 (T (y)) z =M1 Λ10 M2 Tany osz xˆi if |xˆi | ≤ 0.5 yi = r ound (2xˆi ) other wise 2
xˆ =
5.12(x−o) 100
(l) Rotated Katsuura function min f (x) :
10 D2
D i=1
1+i
32
j=1
j 2 z i −r ound 2 j z i 2j
10 D 1.2
−
10 D2
+ 200
subjectto z = M2 Λ100 5(x−o) 100 (m) Lunacek Bi_Rastrigin function D D 2 2
xˆi − μ0 , d D + s xˆi − μ1 min f (x) : min i=1 i=1 D
+ 10 D − cos(2π z i ) + 300 i=1
subject to μ0 =2.5
μ1 = − μ s−d 1 s = 1 − 2√ D+20−8.2 d=1 xˆi = 2sing (xi ∗) yi + μ0 for all i = 1, . . . , D y = 10(x−o) 100 z = Λ100 xˆ − μ0 2
(n) Rotated Lunacek Bi_Rastrigin function D D 2 2
min f (x) : min xˆi − μ0 , d D + s xˆi − μ1 i=1 i=1 D
+ 10 D − cos (2π z i ) + 400 i=1
subject to μ0 =2.5
μ1 = − μ s−d 1 s = 1 − 2√ D+20−8.2 d=1 xˆi = 2sing (yi ∗) yi + μ0 for all i = 1, . . . , D y = 10(x−o) 100 z = M2 Λ100 M1 xˆ − μ0 2
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(o) Expanded Griewank’s plus Rosenbrock’s function min f (x) : g1 (g2 (z 1 , z 2 )) + g1 (g2 (z 2 , z 3 )) + · · · + g1 (g2 (z D−1 , z D ) + g1 (g2 (z D , z 1 )) + 500 subject to D x2 D
xi i √ g1 (x) = +1 − cos 4000 g2 (x) =
i=1 D−1
i=1
i=1
i
2
100 xi2 − xi+1 + (xi − 1)2 z = M1 5(x−o) 100
(p) Expanded Scaffer’s F6 function min f (x) : g (z 1 , g2 ) + g (z 1 , z 2 ) + g (z 2 , z 3 ) + · · · + g (z D−1 , z D ) +g (z D , g1 ) + 600 subject to √ 2 g(x, y) = 0.5 + z=
sin
x 2 −y 2 −0.5
(1−0.001(x 2 −y 2 )) 0.5 (M (x − o)) M2 Tany 1
2
3.3 Design of the experimental tests In light of the above-mentioned experimental conditions, thirty six social configurations were tested, resulting from the combination of the nine networks topologies and the four social dynamics. For each configuration and each UNOP instance, 20 independent replications were carried out. At the end of each run, both the computational execution time and the best found objective function value were registered. Subsequently, classical statistics (best and worst solutions found, as well as average and standard deviation values) were computed over the 20 replications. Furthermore, the obtained results were further analyzed through statistical tests. In different cases (but not all of them, as will be explained in the next section), a non-parametric Wilcoxon rank sum test was applied to each social configuration. The null hypothesis, tested with a 95 % significance level, is that the data from two social configurations for the ith instance are independent. Besides, a two-samples Kolmogorov–Smirnov (KS) test was also applied to the same cases and with the same significance level, in order to contrast the results provided by the Wilcoxon test. Note that both of these tests do not require any normality assumption (which is actually not true for the obtained experimental data). Regarding the parameter settings, the number of variables was chosen D = 10 for all experiments. The number of objective evaluations for the MMC was set to 10,000 × D as proposed in Liang et al. (2013b). The other MMC operating parameters were tuned through a brute-force approach (Birattari 2009) and set constant for all configurations and instances. Table 1 sums up the used value for these parameters.
4 Experimental results and discussions The MMC algorithm was implemented in Matlab R2011a and executed with a 1.8 GHz Intel Core i7 processor. Due to the huge number of computational experiments (four social dynamics, nine network topologies, 16 instances and 20 replications for each instance sum
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Influence of social network on method musical composition Table 1 Parameter settings for the MMC algorithm
Parameter
Value
maxa
5000
ifg
0.1
cfg
0.1
fcla
0.5
Nc
20
Ns
6
up to 11,520 runs), all the results are not exhaustively presented. However, a synthesis of these results is available in “Appendix”, presenting the best, worst and mean values, as well as the associated standard deviation computed over the 20 runs performed for each instance and each configuration {social dynamics, topology}. Therefore, a presentation and interpretation of the relevant observations is provided in what follows. Note that the interpretation will not include discussion about the computing times. Actually, the runtimes range, according to the instances and social dynamics, from 110 to 220 s, the lower CPU times being for a static strategy while the higher times are observed for self-adaptive social dynamics. As predictable, the experiments do not highlight any influence of the used topology on execution times. First, considering unimodal problems [instances (a), b) and (c)], as shown in Tables 3, 4 and 5 of “Appendix”, all the configurations found optimum values for all executions. As a result, it can be deduced that the MMC algorithm is robust enough to deal with this kind of problems, regardless the used social dynamics or network structure. The following analysis will thus focuse on multimodal problems [instances (d)–(p)], on which the different configurations involved within the MMC show more contrasted behaviors. Tables 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 and 18 in “Appendix” present a synthesis of all the obtained results. However, it is impossible to identify the best configuration {social dynamics, topology}, that would outperform all the other ones over all the tackled instances. Therefore, two analysis are performed, independently focusing on the effects of the social dynamics on the one hand, and on the network topologies on the other hand. Note that, when studying the social dynamics, all results for each strategy (i.e. including all the tackled topologies) are considered, and reversely. Besides, concerning the statistical tests, it is relevant to note that normality of the solution distributions cannot be assumed: three statistical tests (Lilliefors, one-sample Kolmogorox– Smirnov and Jarque–Bera) are performed for some randomly chosen samples (for different instances and topologies/dynamics). In all cases, the null hypothesis (standing for the normality assumption) is rejected with a 95 % significance level. Therefore, a classical two-ways ANOVA (considering the social dynamics and the network structure as the two factors) cannot be rigorously taken into account. However, despite the lack of accuracy of its conclusions, such test has been carried out, in order to estimate the possible effects of interaction between both factors: for all instances, the interaction appears to be insignificant (the p values are always much higher than 5 %), justifying that the subsequent analysis considers dynamics and topologies as independent parameters. Regarding the influence of the social dynamics used, Wilcoxon tests and two-samples KS tests are performed with a 95 % significance level. These tests implement pairwise comparisons of the distribution of samples obtained by different strategies. They are first carried out
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Fig. 2 Two-samples KS tests comparing topologies for all instances
considering the results obtained for all instances and, subsequently, for each problem independently. The results are not exhaustively presented here, since the tests failed, in all cases, in rejecting the null hypothesis, which states that the solutions obtained by two dynamics arouse from the same population. As a consequence, the first conclusion of these experiments is that the social dynamics does not have any influence on the efficiency of the MMC algorithm. However, since, as above-mentioned, the computational times are lower when using a static strategy, this latter should be generally used. Concerning the comparative study on topologies, just as for the dynamics, the possible effects of this factor on the MMC performance are analyzed through Wilcoxon and twosamples KS tests. Pairwise comparisons are performed for each topology against all the remaining ones. The null hypothesis, stating that both samples are drawn from the same population, is tested with a 95 % significance level. Additionally, mean values over the 80 (=four social dynamics × 20 replications for each configuration) runs carried out for each test instance and each topology, as well as boxplots, are used for further interpretation. However, for space reasons, only results relevant for the discussion are presented in this paper. First, the Wilcoxon and two-samples KS tests are applied considering considering the results obtained for all the tested instances. This test fails in rejecting the null hypothesis for all pairs of structures, concluding that there are no significant differences between topologies when considering all problems. However, the KS test identifies significant differences between some topologies, with the same significance level. The results of the KS tests, available in Fig. 2 (where black squares mean that there is a significant difference between both topologies), enable to establish some groups among the nine topologies. Topologies SN1, SN3 and SN4 seem to be significantly different from topologies SN7 and SN8. Besides, topologies SN2, SN5, SN6 and SN9 show an intermediate behavior between the two extreme
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Influence of social network on method musical composition
Fig. 3 Wilcoxon tests (leftmost), two-samples KS tests (center) and boxplot (rightmost) comparing topologies for instance (l)
groups (particularly, SN2 and SN5 do not have significantly different performances with all other topologies). In order to provide more detailed insights on the effects of topologies, similar statistical tests are also performed for each multimodal instance, separately. They highlight that different trends can be identified according to each instance. For instances (f), (l), (n) and (p), no significant differences are identified, except for some marginal and isolated cases that do not seem to follow any relevant pattern. As an illustration of these cases, Fig. 3 show the results of the Wilcoxon tests, the KS tests and of the boxplot of the obtained data and the mean values computed for each topology, for instance (l) (the mean values are indicated by the solid line within the boxplot). Similar behaviors are observed for instances (f), (n) and (p). On the other hand, for instances (d), (e), (g), (h)–(k), (m) and (o), interesting trends confirm the preliminary observations conducted over all the tackled instances. These test problems are individually analyzed in what follows: Instance (d) Figure 4 shows the boxplot of the achieved results, highlighting that the topologies globally differ only regarding the extension of their first quartile, much
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Fig. 4 KS tests (leftmost) and boxplot (rightmost) comparing topologies for instance (d)
Instance (e)
Instance (g)
Instance (h)
Instance (i)
123
bigger for topologies SN2 and particularly SN3. The Wilcoxon test, based on median value, cannot identify differences which are visually clear thanks to the boxplot. For instance, the test cannot reject the null hypothesis when comparing SN3 with SN7 or SN8, which are characterized by radically different mean values: −892.11 for SN3, −890.74 for SN7 and −890.62 for SN8. The KS test (see Fig. 4), identifies correctly the differences between SN3 and the other topologies, except with SN2 which also shows a distribution characterized by a large left tail. Besides, topologies SN1, SN4 and SN9 show globally similar behaviors. Thus, this instance seems to propose to groups of topologies according to their performance level: SN2 and SN3 perform significantly better than all the other ones. Here, all the analysis (see Wilcoxon and KS tests, as well as the boxplot and the mean values), presented in Fig. 5, coincide identifying two groups. In this case, contrary to instance (d), topologies SN1, SN2 and SN3 perform significantly worse than the other ones. The boxplot, however, proves that this difference is mainly due to a large right-hand-side tail of the three first topologies, while their minimum and median values are not so far away from those of the other network structures. As illustrated by Fig. 6, the Wilcoxon identifies four groups. First, the best performing topologies are SN1–SN4 and SN9. The second group is composed of SN6 and SN5, while SN8 and finally SN7 are significantly different (and worse) from all the other topologies. Similar conclusions are drawn from the KS tests and are reflected by the boxplot and mean values also available in Fig. 6, although they also highlight some similarities between topologies SN5 and SN8 on the one hand, and between the group {SN1, SN2, SN4} and SN6. The Wilcoxon and KS tests results have some slight differences, but they coincide on the fact that SN3 is only similar to SN2 (failure on null hypothesis rejection), being the two worst alternatives. In addition, the best performing group is composed by SN6, SN7 and SN8. These trends are confirmed by the boxplot and mean values, available in Fig. 7. For this instance, the information provided by the Wilcoxon and KS tests, the boxplot and the mean values(see Fig. 8) highlight that almost all topologies
Influence of social network on method musical composition
Fig. 5 Wilcoxon tests (leftmost), two-samples KS tests (center) and boxplot (rightmost) comparing topologies for instance (e)
Instance (j)
Instance (k)
Instance (m)
Instance (o)
show similar performances, except SN3 and SN7, which achieve the best and worst results respectively, confirming their opposite behavior. The Wilcoxon and KS tests results are here completely identical (only the first one is shown in Fig. 9) identifying three groups (in decreasing order of performance): {SN3, SN4, SN1, SN2, SN9}, {SN6, SN5} and {SN8, SN7}. This conclusion is confirmed by the boxplot and mean values, also available in Fig. 9. Similarly with instance (j), the different analysis tools (see Fig. 10) establish the four following groups (in decreasing order of performance): {SN3, SN4}, {SN9, SN1, SN6, SN5, SN2}, {SN8} and {SN7}. According to Fig. 11, the topologies can clearly be divided into four groups (in decreasing order of performance): {SN7, SN8}, {SN6, SN5}, {SN2, SN9, SN1, SN4} and {SN3}. For this last instance, it results more complicated to define clear and nonoverlapping groups. However, from the boxplot and mean values, available in Fig. 12, topologies SN2 and SN3 are the worst performing ones, while SN8 and, marginally, SN5 and SN6, achieve the best solution distributions.
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Fig. 6 Wilcoxon tests (leftmost), two-samples KS tests (center) and boxplot (rightmost) comparing topologies for instance (g)
Thus, the individual analysis of each one of the “relevant” instances enable to refine the preliminary observations, obtained considering the whole set of multimodal instances. For some problems, the network structure has no significant impact on the behavior of the MMC [instances (f), (l), (n) and (p)]. However, for those problems where significant differences between topologies are observed, some structures definitely show opposite behaviors. As illustrated in Table 2, when SN3 obtains the best results, SN7 and/or SN8 achieve the worst performance levels; and reversely. Moreover, topology SN2 and, in a more marginal way, topologies SN1 and SN4 show behaviors similar to that of SN3. On the other hand, structures SN5, SN6 and SN9 usually achieve intermediate solutions. These observations might be explained by the involved topological structures. Let us recall that SN3 is a star topology, SN1, SN2 and SN4 are linear, tree and ring topologies respectively, while SN7 and SN8 have fully connected and random structures respectively. These structures first differ one from another according the number of neighbors each agent has. The mean number of neighbors for each of the above-mentioned relevant structures is detailed straightforward: – SN3: considering Nc agents, Nc − 1 agents have one neighbor while one agent has Nc − 1 neighbors. The average number of neighbors is therefore 2Nc−2 Nc .
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Influence of social network on method musical composition
Fig. 7 Wilcoxon tests (leftmost), two-samples KS tests (center) and boxplot (rightmost) comparing topologies for instance (h)
– SN2: since this study considers binary trees, the root node has two neighbors, while there are 2log2 (Nc)−1 − 2 intermediate nodes, each one having 3 neighbors and Nc − 2log2 (Nc)−1 −1 leave nodes, each one having one neighbor. The resulting average number of neighbors is therefore greater than 2Nc−3 Nc . – SN1: the two extreme agents have one neighbor while the Nc − 2 intermediate agents have two neighbors, resulting in an average number of neighbors equal to 2Nc2 Nc . – SN4: all agents have two neighbors. – SN7: all agents have Nc − 1 neighbors. – SN8: considering an edge density d < 1, agents have a mean number of d(Nc − 1) neighbors. However, since for each topology, a great part of the analyzed results include those ones obtained for a population structure dynamically updated, it can be considered that, all agents are likely to have some links with any other agent in the society. Thus, it seems reasonable to state that the mean number of neighbors is close to Nc − 1. These observations highlight the fact that, for the first group of topologies (SN3, SN2, SN1 and SN4), agents have about two neighbors, while the second group (SN7 and SN8) have much more neighbors (about Nc − 1). Therefore, it can be concluded that the average number of neighbors has an influence on the performances of the MMC: for some problems, it seems that structures with few neighbors work better than structures with many neighbors, and
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Fig. 8 Wilcoxon tests (leftmost), two-samples KS tests (center) and boxplot (rightmost) comparing topologies for instance (i)
Fig. 9 Wilcoxon tests (leftmost) and boxplot (rightmost) comparing topologies for instance (j)
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Influence of social network on method musical composition
Fig. 10 Wilcoxon tests (leftmost), two-samples KS tests (center) and boxplot (rightmost) comparing topologies for instance (k)
reversely for some other problems. On the other hand, structures having an intermediate average number of neighbors (SN5, SN6, SN9) usually also show intermediate performance levels. Note that these conclusions can be compared with the topology classification proposed in Sect. 3.1.1. Topologies with global (random an full connection) and mixed (star) structures show opposite behaviors. Topologies with a local structure have intermediate performance levels. The obtained results might be further analyzed regarding the way information propagates within the population. The main issue is how a good solution produced by a random agent is conveyed to another random agent, so that the latter learns from the former. It can be intuitively expected that the more intermediate agents lie on the path between these two random agents, slower the information transfer. In addition, the transfer process is carried out through the creation of a new solution by intermediate agents, involving modifications made to the “conveyed good solution”, at each step of the transfer. Thus, the more intermediate agents, the more important the magnitude of these modifications. Observing again the relevant topologies, SN3 involves, in most cases, only one intermediate agent, e.g. the center of the star. Similarly, for topologies SN7 and SN8, there will be no intermediate agent. However, it has already been empirically proved that SN3 and the group {SN7, SN8} show opposite behaviors. Therefore, it seems that the information propagation
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Fig. 11 Wilcoxon tests (leftmost), two-samples KS tests (center) and boxplot (rightmost) comparing topologies for instance (m)
does not have as much impact as the number of neighbors. But, if one considers topologies with similar neighbor numbers (SN1, SN2, SN3 and SN4), the group {SN2, SN1, SN4} is clearly characterized by a number of intermediate agents (between two random nodes of the network) that is much greater than that of SN3. This observation is reflected by the behaviors of these topologies, since SN3 always reaches “more extreme” performance levels than SN2, SN1 or SN4. So, the information transfer might have a small impact, making light differences between topologies characterized by similar numbers of neighbors. To conclude this section, the observations emerging from the exhaustive statistical analysis of the experimental results might be summarized in general guidelines concerning the choice of the network topology used within a social intelligence based optimization technique. As above mentioned, the social dynamics have no impact on the efficacy of the search process, but only on the computational time. A static strategy should be selected since it does not involve topology updates and therefore performs faster. Regarding the topology structure, those obtaining the best (or the worst) results are consistently the star topology and random or full connection topologies. Thus, these three structures should be the first ones to be tested, since the other ones are likely to produce intermediate quality solutions only.
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Influence of social network on method musical composition
Fig. 12 Wilcoxon tests (leftmost), two-samples KS tests (center) and boxplot (rightmost) comparing topologies for instance (o)
Table 2 Extreme performing groups for some relevant multimodal instances
Instance
Best structures
Worst structures
(d)
SN3, SN2
SN7, SN8
(e)
SN4–SN9
SN1, SN2, SN3
(g)
SN1, SN2, SN3 SN4, SN9
SN8, SN7
(h)
SN6, SN7, SN8
SN2, SN3
(i)
SN3
SN7
(j)
SN3, SN4, SN1, SN2, SN9
SN8, SN7
(k)
SN3, SN4
SN8, SN7
(m)
SN7, SN8
SN3
(o)
SN8
SN2, SN3
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R. A. Mora-Gutiérrez et al.
5 Conclusions During the two last decades, many social intelligence based optimization techniques have been introduced and a particular interest has been devoted to the way agents interact within artificial societies. The reported works typically focus on how the topology of social networks might influence the efficacy of the search process. The optimization technique considered in this paper is a recently developed social algorithm, namely the method of musical composition. Two features were investigated: the structure of the network topology and its respective dynamics (comparing static topology vs. strategies that dynamically update the edges of the underlying graph). Computational experiments were carried out over a challenging benchmark of Unconstrained Nonlinear Optimization Problems. Exhaustive statistical tests first proved that the social network dynamics have no significant influence on the performances of the MMC. On the other hand, when comparing the topology structures, differences were observed for some part of the addressed multimodal instances. More precisely, a structure characterized by a low value of its mean number of neighbors and a rather fast information transfer process (star topology) performs in a radically opposite way as structures where each agent has many neighbors (random and complete topologies). Note that, by opposite behaviors, we mean that when one obtains the best results, the others achieves the worst performances, and reversely. Therefore, these observations allow to provide some guidelines when using social intelligence based methods: star and random or complete structures should be the first ones to be tested, since the remaining ones usually achieve intermediate performance levels. The perspectives of future work may be formulated according to two research lines. First, the influence of information transfer within an artificial society might be studied in depth, in order to provide more detailed insights concerning, for instance, convergence speeds. Besides, it would be interesting to examine if the conclusions of the present work are still valid when addressing different classes of optimization problems, such as restricted nonlinear or discrete optimization problems.
Appendix See Tables 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18.
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Influence of social network on method musical composition Table 3 Results of instance (a) Social dynamic S
D1
D2
SA
Network
Worst
Best
Average
s2
s
SN1
−1400
−1400
−1400
0
0
SN2
−1400
−1400
−1400
0
0
SN3
−1400
−1400
−1400
0
0
SN4
−1400
−1400
−1400
0
0
SN5
−1400
−1400
−1400
0
0
SN6
−1400
−1400
−1400
0
0
SN7
−1400
−1400
−1400
0
0
SN8
−1400
−1400
−1400
0
0
SN9
−1400
−1400
−1400
0
0
SN1
−1400
−1400
−1400
0
0
SN2
−1400
−1400
−1400
0
0
SN3
−1400
−1400
−1400
0
0
SN4
−1400
−1400
−1400
0
0
SN5
−1400
−1400
−1400
0
0
SN6
−1400
−1400
−1400
0
0
SN7
−1400
−1400
−1400
0
0
SN8
−1400
−1400
−1400
0
0
SN9
−1400
−1400
−1400
0
0
SN1
−1400
−1400
−1400
0
0
SN2
−1400
−1400
−1400
0
0
SN3
−1400
−1400
−1400
0
0
SN4
−1400
−1400
−1400
0
0
SN5
−1400
−1400
−1400
0
0
SN6
−1400
−1400
−1400
0
0
SN7
−1400
−1400
−1400
0
0
SN8
−1400
−1400
−1400
0
0
SN9
−1400
−1400
−1400
0
0
SN1
−1400
−1400
−1400
0
0
SN2
−1400
−1400
−1400
0
0
SN3
−1400
−1400
−1400
0
0
SN4
−1400
−1400
−1400
0
0
SN5
−1400
−1400
−1400
0
0
SN6
−1400
−1400
−1400
0
0
SN7
−1400
−1400
−1400
0
0
SN8
−1400
−1400
−1400
0
0
SN9
−1400
−1400
−1400
0
0
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R. A. Mora-Gutiérrez et al. Table 4 Results of instance (b) Social dynamic S
D1
D2
SA
123
Network
Worst
Best
Average
s2
s
SN1
−1200
−1200
−1200
0
0
SN2
−1200
−1200
−1200
0
0
SN3
−1200
−1200
−1200
0
0
SN4
−1200
−1200
−1200
0
0
SN5
−1200
−1200
−1200
0
0
SN6
−1200
−1200
−1200
0
0
SN7
−1200
−1200
−1200
0
0
SN8
−1200
−1200
−1200
0
0
SN9
−1200
−1200
−1200
0
0
SN1
−1200
−1200
−1200
0
0
SN2
−1200
−1200
−1200
0
0
SN3
−1200
−1200
−1200
0
0
SN4
−1200
−1200
−1200
0
0
SN5
−1200
−1200
−1200
0
0
SN6
−1200
−1200
−1200
0
0
SN7
−1200
−1200
−1200
0
0
SN8
−1200
−1200
−1200
0
0
SN9
−1200
−1200
−1200
0
0
SN1
−1200
−1200
−1200
0
0
SN2
−1200
−1200
−1200
0
0
SN3
−1200
−1200
−1200
0
0
SN4
−1200
−1200
−1200
0
0
SN5
−1200
−1200
−1200
0
0
SN6
−1200
−1200
−1200
0
0
SN7
−1200
−1200
−1200
0
0
SN8
−1200
−1200
−1200
0
0
SN9
−1200
−1200
−1200
0
0
SN1
−1200
−1200
−1200
0
0
SN2
−1200
−1200
−1200
0
0
SN3
−1200
−1200
−1200
0
0
SN4
−1200
−1200
−1200
0
0
SN5
−1200
−1200
−1200
0
0
SN6
−1200
−1200
−1200
0
0
SN7
−1200
−1200
−1200
0
0
SN8
−1200
−1200
−1200
0
0
SN9
−1200
−1200
−1200
0
0
Influence of social network on method musical composition Table 5 Results of instance (c) Best
Average
s2
s
−999.99
−1000
−1000
0
0
−999.99
−1000
−1000
0
0
SN3
−998.83
−1000
−999.94
0.07
0.26
SN4
−999.48
−1000
−999.97
0.01
0.1
SN5
−999.99
−1000
−1000
0
0
SN6
−1000
−1000
−1000
0
0
SN7
−1000
−1000
−1000
0
0
SN8
−1000
−1000
−1000
0
0
SN9
−975.95
−1000
−998.56
29.47
5.43
SN1
−1000
−1000
−1000
0
0
SN2
−1000
−1000
−1000
0
0
SN3
−1000
−1000
−1000
0
0
SN4
−1000
−1000
−1000
0
0
SN5
−1000
−1000
−1000
0
0
SN6
−1000
−1000
−1000
0
0
SN7
−1000
−1000
−1000
0
0
SN8
−1000
−1000
−1000
0
0
SN9
−1000
−1000
−1000
0
0
SN1
−1000
−1000
−1000
0
0
SN2
−999.99
−1000
−1000
0
0
SN3
−1000
−1000
−1000
0
0
SN4
−1000
−1000
−1000
0
0
SN5
−1000
−1000
−1000
0
0
SN6
−1000
−1000
−1000
0
0
SN7
−1000
−1000
−1000
0
0
SN8
−1000
−1000
−1000
0
0
SN9
−1000
−1000
−1000
0
0
SN1
−1000
−1000
−1000
0
0
SN2
−934.48
−1000
−996.72
214.63
14.65
SN3
−1000
−1000
−1000
0
0
SN4
−1000
−1000
−1000
0
0
SN5
−1000
−1000
−1000
0
0
SN6
−1000
−1000
−1000
0
0
SN7
−1000
−1000
−1000
0
0
SN8
−1000
−1000
−1000
0
0
SN9
−1000
−1000
−1000
0
0
Social dynamic
Network
Worst
S
SN1 SN2
D1
D2
SA
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R. A. Mora-Gutiérrez et al. Table 6 Results of instance (d) Social dynamic S
D1
D2
SA
123
Network
Worst
Best
Average
s2
s
SN1
−890.5
−897.11
−891.11
2.84
1.69
SN2
−890.51
−895.64
−891.64
3.1
1.76
SN3
−890.5
−896.09
−892.33
4.46
2.11
SN4
−890.51
−895.68
−891.47
3.22
1.79
SN5
−890.49
−897.02
−891.62
4.76
2.18
SN6
−890.53
−893
−890.87
0.57
0.75 1.28
SN7
−890.57
−896.32
−890.88
1.64
SN8
−890.54
−891.4
−890.62
0.03
0.17
SN9
−890.51
−895.58
−891.83
2.95
1.72
SN1
−890.51
−895.46
−891.01
1.47
1.21
SN2
−890.51
−898
−892.24
5.9
2.43
SN3
−890.51
−895.26
−892.55
3.78
1.94
SN4
−890.51
−894.01
−890.71
0.6
0.77
SN5
−890.51
−894.87
−891.21
1.68
1.3
SN6
−890.54
−894.61
−890.93
1.33
1.15
SN7
−890.57
−890.67
−890.6
0
0
SN8
−890.55
−893.39
−890.71
0.4
0.63
SN9
−890.52
−895.81
−891.3
2.66
1.63
SN1
−890.51
−894
−890.86
0.78
0.88
SN2
−890.49
−897.21
−891.75
3.79
1.95
SN3
−890.51
−895.96
−891.8
4.02
2
SN4
−890.52
−896.77
−891.4
3.49
1.87 0.77
SN5
−890.51
−893.99
−890.73
0.59
SN6
−890.53
−894.69
−891.16
1.91
1.38
SN7
−890.59
−890.64
−890.6
0
0 0
SN8
−890.54
−890.61
−890.58
0
SN9
−890.51
−894.71
−891.08
1.52
1.23
SN1
−890.52
−897.11
−891.18
2.5
1.58
SN2
−890.49
−894.25
−890.74
0.69
0.83
SN3
−890.51
−895.47
−891.75
3.25
1.8
SN4
−890.52
−897
−891.62
4.55
2.13
SN5
−890.51
−890.85
−890.6
0.01
0.1
SN6
−890.54
−891.73
−890.63
0.07
0.26
SN7
−890.57
−896.12
−890.88
1.52
1.23
SN8
−890.56
−890.66
−890.59
0
0
SN9
−890.51
−894.86
−890.77
0.93
0.96
Influence of social network on method musical composition Table 7 Results of instance (e) Social dynamic S
D1
D2
SA
Network
Worst
Best
Average
s2 14.8
SN1
−788.88
−799.97
−795.63
SN2
−792.32
−799.93
−797.83
SN3
−786.5
−799.98
−795
SN4
−790.94
−799.86
−797.42
3.67
s 3.85 1.92
17.02
4.13
7.81
2.79
SN5
−786.62
−799.99
−795.5
SN6
−790.56
−800
−798.33
SN7
−788.24
−800
−797.57
10.96
3.31
SN8
−784.15
−799.97
−796.79
20.36
4.51
SN9
−783.48
−800
−797.2
16.48
4.06
SN1
−785.52
−799.98
−796.24
21.24
4.61
15.8 6.29
3.97 2.51
SN2
−783.67
−799.99
−795.1
22.19
4.71
SN3
−788.17
−799.86
−795.58
15.16
3.89
SN4
−793.66
−800
−798.17
3.55
1.88
SN5
−790.23
−799.91
−798.08
6.23
2.5
SN6
−791.65
−799.94
−797.81
5.18
2.28
SN7
−784.73
−800
−797.5
20.23
4.5
SN8
−791.02
−799.98
−798.19
5.73
2.39
SN9
−789.72
−799.9
−796.86
11.74
3.43
SN1
−783.83
−799.94
−795.61
15.4
3.92
SN2
−781.23
−799.97
−794.22
42.16
6.49
SN3
−787.78
−799.95
−796.18
12.08
3.48
SN4
−789.47
−799.84
−797.13
8.07
2.84
SN5
−787.87
−799.89
−797.65
9.34
3.06
SN6
−794.43
−799.99
−798.43
3.92
1.98
SN7
−789.81
−799.97
−797.55
8.65
2.94
SN8
−791.5
−799.94
−797.32
6.43
2.54 2.23
SN9
−793.75
−799.95
−797.5
4.98
SN1
−789.17
−799.98
−796.26
14.44
3.8
SN2
−784.45
−799.99
−794.36
25.56
5.06 2.92
SN3
−788.61
−799.75
−796.44
8.51
SN4
−791.51
−799.98
−798.05
4.54
2.13
SN5
−790.66
−799.99
−797.67
7.36
2.71 2.92
SN6
−789.14
−799.92
−797.18
8.5
SN7
−782.42
−800
−796.17
20.08
4.48
SN8
−790.25
−800
−797.75
8.23
2.87
SN9
−789.51
−800
−797.42
9.87
3.14
123
R. A. Mora-Gutiérrez et al. Table 8 Results of instance (f) s2
s
−679.64
0
0
−679.63
0.01
0.1
−679.75
−679.65
0
0
−679.53
−679.72
−679.62
0
0
−679.56
−679.75
−679.64
0
0
SN6
−679.5
−679.83
−679.64
0.01
0.1
SN7
−679.53
−679.78
−679.63
0
0
SN8
−679.55
−679.82
−679.65
0
0
SN9
−679.51
−679.84
−679.63
0.01
0.1
SN1
−679.53
−679.78
−679.63
0.01
0.1
SN2
−679.54
−679.73
−679.63
0
0
SN3
−679.48
−679.81
−679.66
0.01
0.1
SN4
−679.56
−679.78
−679.66
0
0
SN5
−679.52
−679.84
−679.66
0.01
0.1
SN6
−679.55
−679.81
−679.64
0.01
0.1
SN7
−679.5
−679.79
−679.65
0
0
SN8
−679.57
−679.79
−679.67
0
0
SN9
−679.52
−679.76
−679.65
0
0
SN1
−679.56
−679.76
−679.65
0
0
SN2
−679.56
−679.83
−679.67
0
0
SN3
−679.52
−679.78
−679.64
0.01
0.1
SN4
−679.55
−679.81
−679.67
0.01
0.1
SN5
−679.53
−679.75
−679.66
0
0
SN6
−679.46
−679.8
−679.66
0.01
0.1
SN7
−679.57
−679.8
−679.65
0
0
SN8
−679.54
−679.77
−679.67
0
0
SN9
−679.54
−679.76
−679.62
0
0
SN1
−679.57
−679.77
−679.65
0
0
SN2
−679.47
−679.81
−679.66
0.01
0.1
SN3
−679.53
−679.85
−679.65
0.01
0.1
SN4
−679.56
−679.8
−679.67
0
0
SN5
−679.56
−679.8
−679.67
0
0
SN6
−679.57
−679.79
−679.67
0
0
SN7
−679.53
−679.7
−679.63
0
0
SN8
−679.53
−679.7
−679.63
0
0
SN9
−679.53
−679.7
−679.63
0
0
Social dynamic
Network
Worst
Best
Average
S
SN1
−679.54
−679.76
SN2
−679.5
−679.78
SN3
−679.53
SN4 SN5
D1
D2
SA
123
Influence of social network on method musical composition Table 9 Results of instance (g) Social dynamic S
D1
D2
SA
Network
Worst
Best
Average
s2
s
SN1
−595.71
−598.52
−597.01
0.59
0.77
SN2
−595.91
−597.91
−596.95
0.27
0.52
SN3
−595.73
−597.96
−596.92
0.4
0.63
SN4
−596
−598.77
−597.07
0.55
0.74
SN5
−593.35
−598.18
−596.56
1.66
1.29
SN6
−594.53
−597.86
−596.69
0.82
0.91
SN7
−590.68
−599.24
−594.73
7.02
2.65
SN8
−590.78
−598.21
−595.9
3.42
1.85
SN9
−595.79
−599
−596.95
0.94
0.97
SN1
−596.24
−599
−597.14
0.61
0.78
SN2
−595.5
−598.72
−596.81
0.77
0.88
SN3
−595.69
−598.13
−596.76
0.49
0.7
SN4
−595.56
−598.81
−596.96
0.83
0.91
SN5
−590.94
−597.8
−596.4
2.27
1.51
SN6
−595.58
−598.91
−596.76
0.51
0.71 2.12
SN7
−591.01
−597.86
−593.68
4.51
SN8
−594.14
−598.86
−596.31
1.31
1.14
SN9
−595.49
−598.36
−597.12
0.42
0.65
SN1
−595.87
−598.27
−596.81
0.41
0.64
SN2
−595.48
−598.46
−596.92
0.6
0.77
SN3
−596.08
−598.51
−597.22
0.4
0.63
SN4
−596.14
−598.72
−597.03
0.57
0.75
SN5
−594.41
−598.09
−596.46
0.62
0.79
SN6
−594.07
−598.06
−596.75
1.06
1.03
SN7
−590.67
−597.48
−593.64
5.18
2.28
SN8
−592.43
−597.63
−595.9
1.35
1.16
SN9
−595.84
−598.83
−597.11
0.36
0.6
SN1
−595.87
−598.15
−596.86
0.41
0.64
SN2
−595.85
−598.71
−597.1
0.48
0.69
SN3
−595.64
−598.28
−597.02
0.48
0.69
SN4
−595.71
−597.85
−596.84
0.3
0.55
SN5
−591.56
−598.14
−595.96
3.2
1.79
SN6
−593.99
−598
−596.24
1.22
1.1
SN7
−591.02
−598.87
−594.26
7.12
2.67
SN8
−592.15
−598.83
−595.9
3.25
1.8
SN9
−593.86
−598.44
−596.57
0.69
0.83
123
R. A. Mora-Gutiérrez et al. Table 10 Results of instance (h) Social dynamic S
D1
D2
SA
123
Network
Worst
Best
Average
s2
s 0.37
SN1
−498.55
−499.79
−499.33
0.14
SN2
−498.57
−499.87
−499.28
0.1
0.32
SN3
−498.31
−499.8
−499.14
0.22
0.47
SN4
−498.47
−499.76
−499.36
0.1
0.32
SN5
−498.76
−499.76
−499.37
0.09
0.3
SN6
−499.01
−499.92
−499.44
0.08
0.28
SN7
−499.16
−499.99
−499.59
0.04
0.2
SN8
−498.85
−499.8
−499.46
0.06
0.24
SN9
−498.31
−499.7
−499.29
0.13
0.36
SN1
−499.13
−499.77
−499.41
0.05
0.22
SN2
−498.56
−499.89
−499.32
0.13
0.36
SN3
−498.92
−499.91
−499.33
0.07
0.26
SN4
−498.85
−499.86
−499.38
0.08
0.28
SN5
−499.15
−499.81
−499.52
0.04
0.2
SN6
−499.14
−499.83
−499.54
0.03
0.17
SN7
−499.08
−499.79
−499.53
0.03
0.17 0.22
SN8
−498.95
−499.91
−499.58
0.05
SN9
−498.98
−499.84
−499.48
0.06
0.24
SN1
−498.84
−499.85
−499.42
0.06
0.24
SN2
−498.19
−499.8
−499.16
0.24
0.49
SN3
−498.66
−499.76
−499.23
0.11
0.33
SN4
−499.05
−499.86
−499.44
0.05
0.22
SN5
−498.88
−499.64
−499.38
0.05
0.22
SN6
−499.08
−499.97
−499.58
0.07
0.26
SN7
−499.17
−499.89
−499.58
0.03
0.17
SN8
−499.22
−499.96
−499.62
0.03
0.17
SN9
−498.9
−499.8
−499.35
0.06
0.24
SN1
−498.57
−499.82
−499.35
0.08
0.28
SN2
−498.38
−499.95
−499.23
0.17
0.41
SN3
−498.7
−499.88
−499.31
0.09
0.3
SN4
−498.7
−499.88
−499.31
0.09
0.3
SN5
−498.78
−499.93
−499.42
0.09
0.3
SN6
−499.22
−499.86
−499.57
0.04
0.2
SN7
−499.27
−499.73
−499.55
0.02
0.14
SN8
−499.25
−499.98
−499.61
0.02
0.14
SN9
−498.93
−499.81
−499.32
0.06
0.24
Influence of social network on method musical composition Table 11 Results of instance (i) Social dynamic S
D1
D2
SA
Network
Worst
Best
Average
s2
s
SN1
−396.83
−400
−398.35
0.71
0.84
SN2
−397.29
−399.91
−398.47
0.54
0.73
SN3
−397.21
−400
−398.52
0.61
0.78
SN4
−396.57
−399.92
−398.34
0.88
0.94
SN5
−396.65
−400
−398.59
0.86
0.93
SN6
−396.6
−400
−398.52
0.74
0.86
SN7
−387.89
−400
−398.06
6.86
2.62
SN8
−396.45
−399.97
−398.65
1.25
1.12
SN9
−396.96
−399.94
−398.57
0.69
0.83
SN1
−397.76
−400
−398.95
0.46
0.68
SN2
−396.74
−400
−398.81
0.76
0.87
SN3
−398
−400
−399.21
0.33
0.57
SN4
−397.01
−400
−398.69
0.63
0.79
SN5
−396.64
−400
−398.56
0.86
0.93
SN6
−397.48
−400
−398.94
0.67
0.82
SN7
−395.63
−400
−398.43
1.99
1.41
SN8
−395.05
−400
−398.43
1.46
1.21
SN9
−397.44
−400
−398.99
0.72
0.85
SN1
−396.99
−399.98
−398.91
0.65
0.81
SN2
−397.57
−400
−398.82
0.55
0.74
SN3
−397.96
−400
−399.46
0.38
0.62
SN4
−397.58
−399.96
−398.74
0.46
0.68
SN5
−395.96
−399.92
−398.3
1.26
1.12
SN6
−396.96
−399.82
−398.62
0.44
0.66 2.34
SN7
−391.28
−400
−397.96
5.48
SN8
−396.83
−400
−398.53
1.14
1.07
SN9
−397.93
−400
−399.12
0.42
0.65 0.72
SN1
−397.47
−400
−399.06
0.52
SN2
−397.67
−400
−398.67
0.66
0.81
SN3
−397.9
−400
−399.31
0.52
0.72 0.53
SN4
−397.84
−399.91
−398.87
0.28
SN5
−396.9
−400
−399.14
0.76
0.87
SN6
−397.52
−400
−398.97
0.58
0.76
SN7
−390.94
−400
−397.59
4.73
2.17
SN8
−395.57
−400
−398.77
1.36
1.17
SN9
−397.49
−400
−399.01
0.58
0.76
123
R. A. Mora-Gutiérrez et al. Table 12 Results of instance (j) Network
S
SN1
−278.59
−296.43
−290.27
15.82
3.98
SN2
−280.08
−295.21
−288.75
19.53
4.42
D1
D2
SA
123
Worst
Best
Average
s2
Social dynamic
s
SN3
−288.75
−296.23
−291.06
5.38
2.32
SN4
−280.59
−295.72
−290.52
12.61
3.55
SN5
−274.1
−296.66
−288.53
37.17
6.1
SN6
−272.56
−296.78
−287.77
36.06
6
SN7
−269.27
−295.68
−277.78
58.44
7.64
SN8
−270.15
−292.71
−279.74
54.55
7.39
SN9
−277.74
−296.85
−289.09
34.27
5.85
SN1
−277.73
−294.67
−290.52
16.46
4.06
SN2
−280.35
−296.99
−290.44
19.32
4.4
SN3
−282.51
−297.43
−291.47
11.36
3.37
SN4
−284.34
−295.66
−290.86
5.71
2.39
SN5
−271.2
−295.36
−287.63
39.83
6.31
SN6
−272.05
−297.48
−287.57
44.31
6.66
SN7
−269.12
−296.44
−279.46
58.61
7.66
SN8
−271.57
−293.43
−281.78
56.88
7.54
SN9
−278.8
−294.92
−288.1
24.34
4.93
SN1
−283.5
−296.67
−289.85
15.66
3.96
SN2
−282.03
−296.12
−289.74
14.08
3.75
SN3
−276.17
−295.9
−290.42
22.06
4.7
SN4
−281.7
−296.78
−290.1
19.13
4.37
SN5
−270.21
−298
−286
52.45
7.24
SN6
−276.09
−296.82
−286.14
46.52
6.82
SN7
−269.25
−295.38
−279.01
54.16
7.36
SN8
−271.66
−294.63
−280.16
53.67
7.33
SN9
−278.35
−295.11
−290.07
15.12
3.89
SN1
−281.48
−295.76
−289.56
13.09
3.62
SN2
−282.61
−295.03
−290.01
13.21
3.63
SN3
−284.25
−294.3
−290.53
7.5
2.74
SN4
−281.11
−297.62
−290.15
13.49
3.67
SN5
−272.66
−295.21
−284.4
49.59
7.04
SN6
−272.89
−294.47
−285.48
53.98
7.35
SN7
−274.14
−293.47
−280.33
30.65
5.54
SN8
−272.57
−295.26
−281.82
54.89
7.41
SN9
−287.16
−296.56
−291.74
5.19
2.28
Influence of social network on method musical composition Table 13 Results of instance (k) Social dynamic S
D1
D2
SA
Network
Worst
Best
Average
s2
s 4.75
SN1
−177.68
−194.81
−185.59
22.54
SN2
−172.68
−194.73
−186.2
41.9
6.47
SN3
−177.93
−194.08
−188.3
14.49
3.81 4.37
SN4
−178.81
−194.76
−188.12
19.08
SN5
−170.09
−194.93
−185.35
50.33
7.09
SN6
−173.55
−196.14
−186.55
37.43
6.12 7.22
SN7
−170.11
−197.15
−181.34
52.08
SN8
−170.88
−196.74
−184.76
57.39
7.58
SN9
−173.64
−194.85
−186.54
25.86
5.09 4.92
SN1
−178.43
−195.16
−186.85
24.21
SN2
−178.16
−196.2
−185.51
31.09
5.58
SN3
−179.63
−195.46
−188.03
23.68
4.87 4.92
SN4
−172.07
−194.4
−189.97
24.24
SN5
−174.75
−194.63
−183.66
44.71
6.69
SN6
−173.69
−194.44
−185.74
37.33
6.11
SN7
−171.77
−190.27
−177.98
25.18
5.02
SN8
−171
−193.13
−179.69
38.55
6.21
SN9
−178.24
−196.03
−188.96
22.43
4.74 5.6
SN1
−173.04
−195.44
−188.24
31.35
SN2
−177.02
−194.91
−184.91
31.28
5.59
SN3
−182.12
−195.92
−189.03
14.71
3.84
SN4
−179.92
−193.73
−188.82
17.03
4.13
SN5
−171.43
−197.02
−186.3
58.8
7.67
SN6
−176.17
−193.53
−186.33
33.97
5.83
SN7
−172.42
−193
−180.15
42.59
6.53
SN8
−171.67
−195.41
−180.88
40.48
6.36
SN9
−180.43
−192.22
−187.43
11.47
3.39 5.67
SN1
−175.68
−196.02
−186.81
32.15
SN2
−170.49
−194.39
−184.45
46.13
6.79
SN3
−181.22
−195.1
−187.77
17.68
4.2
SN4
−181.3
−194.98
−188.8
16.1
4.01
SN5
−172.95
−195.7
−187.32
34.16
5.84
SN6
−174.89
−194.3
−184.54
35.29
5.94
SN7
−168.8
−196.29
−179.29
62.43
7.9
SN8
−174.08
−194.41
−182.59
49.43
7.03
SN9
−178.69
−195.9
−187.37
16.61
4.08
123
R. A. Mora-Gutiérrez et al. Table 14 Results of instance (l) Social dynamic S
D1
D2
SA
123
Network
Worst
Best
Average
s2
s
SN1
201.44
200.62
201.12
0.06
0.24
SN2
201.54
200.73
201.15
0.05
0.22
SN3
201.5
200.81
201.21
0.04
0.2
SN4
201.44
200.75
201.07
0.04
0.2
SN5
201.47
200.87
201.17
0.02
0.14
SN6
201.42
200.81
201.09
0.03
0.17
SN7
201.38
200.8
201.14
0.04
0.2
SN8
201.48
200.62
201.07
0.06
0.24
SN9
201.47
200.67
201.12
0.05
0.22
SN1
201.6
200.86
201.19
0.03
0.17
SN2
201.5
200.8
201.16
0.03
0.17
SN3
201.52
200.61
201.15
0.04
0.2
SN4
201.51
200.64
201.15
0.05
0.22
SN5
201.46
200.52
201.05
0.07
0.26
SN6
201.42
200.66
201.1
0.05
0.22
SN7
201.52
200.72
201.04
0.04
0.2
SN8
201.55
200.5
201.08
0.05
0.22
SN9
201.47
200.84
201.15
0.03
0.17
SN1
201.61
200.83
201.11
0.04
0.2
SN2
201.57
200.85
201.23
0.04
0.2
SN3
201.45
200.68
201.08
0.06
0.24
SN4
201.48
200.67
201.15
0.04
0.2
SN5
201.39
200.49
200.97
0.06
0.24
SN6
201.5
200.45
201.11
0.07
0.26
SN7
201.7
200.44
201.12
0.07
0.26
SN8
201.52
200.64
201.1
0.05
0.22 0.24
SN9
201.56
200.43
201.25
0.06
SN1
201.49
200.85
201.17
0.03
0.17
SN2
201.42
200.78
201.1
0.03
0.17
SN3
201.31
200.89
201.1
0.02
0.14
SN4
201.55
200.73
201.12
0.05
0.22
SN5
201.47
200.87
201.16
0.02
0.14
SN6
201.58
200.8
201.17
0.05
0.22
SN7
201.26
200.6
201.05
0.03
0.17
SN8
201.56
200.76
201.21
0.03
0.17
SN9
201.47
200.71
201.17
0.05
0.22
Influence of social network on method musical composition Table 15 Results of instance (m) Social dynamic S
D1
D2
SA
Network
Worst
Best
Average
s2
s
SN1
303.6
300.67
301.94
0.68
0.82
SN2
303.96
300.58
302.18
0.83
0.91
SN3
305.22
301.16
303.08
1.43
1.2
SN4
304.65
300.63
302.15
0.98
0.99
SN5
303.66
300.08
301.73
1.26
1.12
SN6
302.08
300.08
300.91
0.38
0.62
SN7
301.17
300.11
300.46
0.08
0.28
SN8
301.53
300.09
300.57
0.15
0.39
SN9
305.19
300.64
302.38
1.47
1.21
SN1
304.01
300.87
302.24
0.82
0.91
SN2
303.58
300.22
301.66
0.86
0.93
SN3
304.43
301.68
303.28
0.64
0.8
SN4
303.94
300.88
302.49
0.82
0.91
SN5
302.8
300.15
301.14
0.78
0.88
SN6
302.45
300.03
300.96
0.39
0.62
SN7
300.91
300.04
300.48
0.07
0.26
SN8
301.84
300.12
300.54
0.21
0.46
SN9
303.18
300.72
301.92
0.41
0.64
SN1
304.05
301.42
302.41
0.71
0.84
SN2
304.99
300.28
301.66
1.17
1.08
SN3
305.7
300.53
303.31
1.88
1.37 0.82
SN4
303.71
300.12
302.13
0.68
SN5
304.01
300.18
301.37
1.28
1.13
SN6
303.52
300.18
301.43
1.14
1.07
SN7
303.9
300.05
300.7
0.65
0.81
SN8
301.59
300.28
300.62
0.11
0.33
SN9
303.58
300.35
301.67
1.03
1.01
SN1
303.61
300.75
302.05
0.57
0.75 0.99
SN2
304.97
300.6
301.74
0.98
SN3
305.74
301.98
303.59
1.12
1.06
SN4
305.83
300.5
302.48
1.56
1.25
SN5
302.69
300.06
301.23
0.55
0.74
SN6
301.83
300.2
300.89
0.22
0.47
SN7
301.23
300.14
300.52
0.1
0.32
SN8
301.34
300.17
300.61
0.14
0.37
SN9
303.58
300.65
301.97
0.69
0.83
123
R. A. Mora-Gutiérrez et al. Table 16 Results of instance (n) Social dynamic S
D1
D2
SA
123
Network
Worst
Best
Average
s2
s 4.02
SN1
427.88
412.51
421.17
16.19
SN2
429.02
407.46
419.64
32.97
5.74
SN3
429.07
409.63
421.01
31.33
5.6
SN4
430.01
411.47
419.68
29.07
5.39
SN5
428.95
404.74
421.33
37.54
6.13
SN6
430.35
407.8
420
37.51
6.12
SN7
429.27
409.76
421.72
35.61
5.97
SN8
433.44
405.57
417.9
61.75
7.86
SN9
430.79
414.19
422.74
24.32
4.93
SN1
429.68
408.24
420.2
21.54
4.64
SN2
433.95
405.26
419.72
61.93
7.87
SN3
428.58
416.4
422.41
7.83
2.8 4.77
SN4
429.24
411.8
420.87
22.75
SN5
427.22
406.62
417.29
43.44
6.59
SN6
428.4
411.51
421.11
33.9
5.82
SN7
429.42
405.04
419.29
54.53
7.38
SN8
428.03
406.08
420.93
45.88
6.77
SN9
430.13
413.08
420.52
21.92
4.68
SN1
430.24
412.52
420.81
25.46
5.05
SN2
427.73
403.43
419.85
44.29
6.66 4.31
SN3
425.93
408.29
419.85
18.61
SN4
428.28
402.16
421.81
39.89
6.32
SN5
430.66
403.38
416.95
59.1
7.69
SN6
427.31
404.82
417.28
51.92
7.21
SN7
431.44
403.33
419.72
65.14
8.07 4.87
SN8
426.51
407.53
419.62
23.73
SN9
428.16
404.75
419.92
36.06
6
SN1
428.14
407.28
419.61
25.7
5.07
SN2
429.38
408.59
422.35
25.97
5.1
SN3
431.15
412.8
420.88
19.74
4.44 4.08
SN4
428.88
411.87
420.76
16.61
SN5
428.6
409.57
420.77
28.13
5.3
SN6
427.44
407.05
417.85
46.28
6.8 4.88
SN7
428.82
409.63
421.98
23.79
SN8
430.01
404.07
417.83
50.62
7.11
SN9
427.89
402.38
419.17
45.78
6.77
Influence of social network on method musical composition Table 17 Results of instance (o) Social dynamic S
D1
D2
SA
Network
Worst
Best
Average
s2
s
SN1
501.77
500.58
501.19
0.12
0.35
SN2
501.89
500.76
501.14
0.12
0.35
SN3
501.81
500.89
501.2
0.06
0.24
SN4
502.02
500.66
501.27
0.15
0.39
SN5
502.03
500.53
501.08
0.14
0.37
SN6
501.65
500.59
501.03
0.07
0.26
SN7
502.14
500.63
501.14
0.15
0.39
SN8
501.89
500.62
501.11
0.14
0.37
SN9
501.89
500.67
501.12
0.09
0.3
SN1
501.76
500.51
501.15
0.12
0.35 0.41
SN2
501.97
500.45
501.21
0.17
SN3
502.03
500.8
501.3
0.14
0.37
SN4
501.47
500.56
501.05
0.07
0.26
SN5
501.96
500.39
500.98
0.17
0.41
SN6
501.62
500.35
500.93
0.07
0.26
SN7
502
500.49
500.98
0.12
0.35
SN8
501.88
500.5
501.04
0.19
0.44
SN9
501.75
500.52
501.08
0.09
0.3
SN1
501.77
500.65
501.23
0.13
0.36
SN2
501.53
500.73
501.07
0.04
0.2
SN3
501.65
500.72
501.18
0.06
0.24
SN4
501.81
500.64
501.29
0.12
0.35
SN5
501.57
500.39
501.01
0.08
0.28
SN6
501.73
500.6
501.1
0.1
0.32
SN7
501.97
500.36
501.29
0.26
0.51
SN8
501.98
500.65
500.97
0.12
0.35
SN9
501.67
500.57
501.18
0.08
0.28
SN1
501.89
500.77
501.2
0.07
0.26
SN2
502.05
500.41
501.1
0.13
0.36
SN3
501.99
500.99
501.38
0.09
0.3
SN4
501.83
500.86
501.28
0.07
0.26
SN5
501.77
500.4
501.11
0.12
0.35 0.35
SN6
502.09
500.66
501.1
0.12
SN7
501.64
500.59
501.14
0.09
0.3
SN8
501.21
500.51
500.87
0.06
0.24
SN9
501.65
500.58
501.06
0.09
0.3
123
R. A. Mora-Gutiérrez et al. Table 18 Results of instance (p) Social dynamic S
D1
D2
SA
123
Network
Worst
Best
Average
s2
s
SN1
602.77
601.82
602.4
0.05
0.22
SN2
602.79
601.66
602.28
0.12
0.35
SN3
603.34
601.5
602.36
0.17
0.41
SN4
602.82
601.68
602.27
0.06
0.24
SN5
603.02
601.68
602.32
0.09
0.3
SN6
602.7
601.41
602.2
0.13
0.36
SN7
602.65
601.53
602.25
0.1
0.32
SN8
602.64
601.74
602.32
0.06
0.24 0.32
SN9
602.85
601.72
602.35
0.1
SN1
602.82
602.16
602.43
0.03
0.17
SN2
602.95
601.87
602.42
0.07
0.26
SN3
602.75
601.86
602.33
0.06
0.24
SN4
602.67
601.69
602.25
0.07
0.26
SN5
602.89
601.79
602.33
0.06
0.24
SN6
602.86
601.62
602.21
0.11
0.33
SN7
602.69
601.6
602.15
0.11
0.33
SN8
602.8
601.45
602.27
0.11
0.33 0.2
SN9
602.77
601.91
602.42
0.04
SN1
602.63
601.89
602.22
0.04
0.2
SN2
602.8
601.96
602.43
0.06
0.24
SN3
602.92
601.68
602.21
0.09
0.3
SN4
602.63
601.5
602.29
0.07
0.26
SN5
602.78
601.68
602.33
0.08
0.28
SN6
602.62
601.68
602.2
0.06
0.24
SN7
603.23
601.35
602.28
0.14
0.37
SN8
603.1
601.84
602.36
0.08
0.28 0.35
SN9
602.68
601.44
602.14
0.12
SN1
602.72
601.83
602.31
0.05
0.22
SN2
602.97
601.89
602.44
0.09
0.3
SN3
602.93
601.73
602.33
0.1
0.32
SN4
602.79
601.9
602.29
0.07
0.26
SN5
602.79
601.85
602.36
0.04
0.2
SN6
602.68
601.61
602.26
0.13
0.36
SN7
603.08
601.31
602.33
0.16
0.4
SN8
602.79
601.15
602.26
0.15
0.39
SN9
602.71
601.64
602.28
0.1
0.32
Influence of social network on method musical composition
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