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MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 2019, VOL. 25, NO. 5, 482–498 https://doi.org/10.1080/13873954.2019.1660997 ARTICLE On the evaluation of the takeoﬀ time and of the peak time for innovation diﬀusion on assortative networks Maria Letizia Bertotti and Giovanni Modanese Faculty of Science and Technology, Free University of Bozen-Bolzano, Bolzano, Italy ABSTRACT ARTICLE HISTORY Received 16 January 2019 This paper deals with a generalization of the Bass model for the Accepted 25 August 2019 description of the diﬀusion of innovations. The generalization keeps into account heterogeneity of the interactions of the consumers KEYWORDS and is expressed by a system of several nonlinear diﬀerential equa- Innovation diﬀusion; Bass tions on complex networks. The following contributions can be model; assortative networks; singled out: ﬁrst, explicit algorithms are provided for the construc- takeoﬀ and peak time tion of various families of assortative scale-free networks; second, a method is provided for the identiﬁcation of the takeoﬀ time and of the peak time, which represent important turning points in the life cycle of an innovation/product; third, the emergence of speciﬁc patterns in connection with networks of the same family is observed, whose tentative interpretation is then given. Also, a comparison with an alternative approach is given, within which adoption times of diﬀerent communities are evaluated of a network describing ﬁrm cooperations in South Tyrol. 1. Introduction The ﬁndings described in this paper are related to the identiﬁcation of the time at which the adoption rate of new products in a community of variously interconnected indivi- duals reaches its maximum and of the takeoﬀ time, another quantity of interest in the marketing perspective. The investigation of this problem is carried out here within a framework which extends on complex networks the classical Bass model. Since its ﬁrst appearance in the Sixties, the Bass model  for the description of the innovation diﬀusion process has been extensively applied (the reviews and provide several references in this connection). Also, during the course of time, several versions of this model have been formulated, which provide generalizations in a variety of directions (see, e.g. [4,5]). The original version is expressed by a single ordinary diﬀerential equation, in fact a Riccati one, which is, albeit non-linear, analytically solvable. Its solution describes the evolution in time of the number of adopters of new products within a population, as a consequence of two basic factors, innovation and imitation. An aggregate perspective is taken, in that only the cumulative fraction of adopters is considered and, for a ﬁxed product, the two parameters appearing in the equation, the ‘innovation’ and the ‘imitation’ coeﬃcient, are the same for the whole population. CONTACT Maria Letizia Bertotti email@example.com Faculty of Science and Technology, Free University of Bozen-Bolzano, Bolzano, Italy © 2019 Informa UK Limited, trading as Taylor & Francis Group MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 483 It is, however, to be expected that a thorough investigation keeps into account hetero- geneity of the individuals. This can be done in diﬀerent ways, among whose a signiﬁcant one is to consider the network of interpersonal connections. Indeed, especially in regard to the imitation aspect of the process, it can make a big diﬀerence whether individuals who have already adopted have a few or several contacts . Driven by this motivation, we have introduced in [7,8] a network structure into the model. Speciﬁcally, we have considered networks whose nodes have at most a ﬁxed number n of links. A ﬁrst statistical attribute of networks is given by the so-called degree distribution PðkÞ, which carries information on the fraction of nodes having k links. Further and central importance in relation to the network topology have the degree correlations PðhjkÞ,with PðhjkÞ expressing the condi- tional probability that an individual with k links is connected to one with h links. In [7,8], following the approach developed for the study of epidemic spreading on complex net- works [9–11], we have reformulated the Bass equation dFðtÞ ¼½1 FðtÞ½ p þ qFðtÞ (1) dt (in which FðtÞ is the cumulative adopter fraction at the time t, and p and q are the innovation and the imitation coeﬃcient, respectively) into a system of n ordinary diﬀerential equations, one for each admissible number of links, of the form "# dG ðtÞ ¼½1 G ðtÞ p þ jq PðhjjÞG ðtÞ j ¼ 1; :::; n : (2) j h dt h¼1 The quantity G ðtÞ in (2) represents for any j ¼ 1; :::n the fraction of potential adopters with j links that at the time t have adopted the innovation. More precisely, denoting by F ðtÞ the fraction of the total population composed by individuals with j links, who at the time t have adopted, and admitting that, in the end, all individuals will adopt (in analogy with the fact that the solution FðtÞ of (1) tends to 1 as t tends to inﬁnity), we set G ðtÞ¼ F ðtÞ=PðjÞ. j j In principle, one could consider at this point diﬀerent kinds of networks. We dealt with scale-free ones, having degree distribution of the form PðkÞ¼ c=k where 2< γ < 3, because it is into this category and with power-law exponent into this range that many real-world networks fall [12,13]. We explored various features of the diﬀerent link classes for both correlated and uncorrelated networks, comparing results of numerical simulations relative to system (2) with results relative to Equation (1). In particular, to perform this task, we devised an algorithm for the construction of both assortative and disassortative networks. In this paper, in view of the foreseen application and of the fact that networks in the social sciences are found to be typically assortative [14,15], we further restrict attention to their category. Yet, we observe en passant that in the modelling of diﬀusion and innovation also networks falling beyond the group of the social ones, notably collabora- tion networks among ﬁrms, may play a part. And in fact, we deal with such a network, disassortative indeed, in Section 5, where an approach alternative to that one based on Equation (2) is discussed. It is now worth recalling that some procedures designed to construct correlated networks, in fact, exist in the literature. For instance , and  suggest diﬀerent 484 M. L. BERTOTTI AND G. MODANESE rewiring-based algorithms which generate assortatively mixed networks and assortative mixing to a desired degree; in  a network model which encompasses addition of both new nodes and new links, meant to mimic a real growing network (a preprint- archive) is introduced; besides , and  propose models for (assortative and disassortative) complex networks with weighted links. However, precise explicit (not only approximate) expressions for the correlations PðhjjÞ, suitable to be employed in calculations, can be hardly found in these or other papers. And the numerical solutions of our diﬀerential Equation (2) require the knowledge of a set of values of PðhjjÞ deﬁned for each h and j and satisfying as well suitable conditions of normalization and network closure, see Section 2. For this reason, we start providing in the next section a few ‘recipes’ towards building correlation matrices of assortative networks. In Section 3 we focus on the identiﬁcation in the present analytical context of two speciﬁc times (takeoﬀ time and peak time) which play a signiﬁcant role in the life cycle of an innovation/ product. Then, we calculate these times for several networks belonging to the families devised in Section 2. A comparison between the degree correlation matrices pertaining to diﬀerent networks of the same family for each of three families reveals the existence of unexpected patterns, which call for explanation. This is the subject of Section 4, where also a possible interpretation is proposed. Elements of an approach alternative to that based on a mean-ﬁeld approximation dealt with in Sections 2–4 are discussed in Section 5. In that section also the calculation is done of adoption times of diﬀerent communities in a network model of ﬁrms in South Tyrol. We want to stress however that the purpose of this paper is to construct an analytical framework for the evaluation of the mentioned times, rather than to treat experimental data. Finally, in Section 6, a summary of the results and our conclusions are given. It may be of interest before continuing to point out how lively and vivid the interest in innovation dynamics and spreading phenomena still is. As the following few but indicative recent references show, research related to these topics keeps growing and branching out into a manifold of novel paths. For example, an agent-based model for the description of the Skype technology large-scale adoption process is proposed in . An analysis and a discussion of the diverse roles of rational strategic approaches and serendipity towards the development of innovations are carried out in . An evolutionary game model with interacting innovators and developers is suggested in  to investigate to which extent a system is able to maintain innovators. In  a model for the emergence of innovations is devised which involves random walks on networks whose nodes represent concepts and ideas. A simple model, inspired by the Lotka-Volterra one regarding competition for common resources occurring among species is employed in to ‘measure the pace’ of collective attention through the inﬂationary information ﬂows of popular topics and cultural items. Further articles on the subject can be found in the collection . 2. Families of assortative networks We start here by suggesting a practical way for the construction of diﬀerent families of assortative networks (namely, such that high degree nodes tend to be linked to other high degree nodes, whereas low degree nodes tend to be linked to other low degree nodes). We consider for any ﬁxed natural number n a scale-free network with degree distribution MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 485 PðkÞ¼ c=k for k n (k 1), with 2< γ< 3. We aim at building the n n matrices, whose elements are the correlation coeﬃcients PðhjkÞ. The reason why this is a delicate task is that the PðkÞ and PðhjkÞ must satisfy, besides the normalizations n n X X PðkÞ¼ 1; and PðhjkÞ¼ 1 ; k¼1 k¼1 and the positivity requirements PðkÞ 0; and PðhjkÞ 0 ; also the Network Closure Condition (NCC) hPðkjhÞPðhÞ¼ kPðhjkÞPðkÞ; "h; k ¼ i ¼ 1; :::; n : (3) We recall here that the assortativity of a network can be established by looking at the average nearest neighbour degree function k ðkÞ¼ hPðhjkÞ : (4) nn h¼1 Indeed, if k ðkÞ is increasing in k, the network is assortative. Alternatively, one can nn calculate the Newman assortativity coeﬃcient r (Pearson correlation coeﬃcient) as deﬁned in  (see also ). We propose next the construction of three diﬀerent matrices (in fact, matrix families), starting from a ﬁrst requirement that their largest elements are on the diagonal, whereas their other elements become smaller and smaller the more apart from the diagonal they are. Notice ﬁrst that the NCC condition (3) provides a constraint on the correlation matrix elements expressed by 1γ PðhjkÞ¼ PðkjhÞ : (5) 1γ ‘Power-like’ family.A ﬁrst natural choice is that of taking the elements on and above the diagonal as jh kj if h< k PðhjkÞ¼ (6) 1if h ¼ k for some λ > 0, and the elements under the diagonal, i.e. with h > k,deﬁned through the formula (5). Since the normalization PðhjkÞ¼ 1 has to hold true, we compute for any k¼1 k ¼ 1; :::; n the sum C ¼ PðhjkÞ and call C the greatest of these sums: k max h¼1 C ¼ max C : max k k¼1;...;n At this point, we re-deﬁne the correlation matrix by setting the elements on the diagonal equal to P ðkjkÞ¼ C C ; k ¼ 1; ... ; n; max k 486 M. L. BERTOTTI AND G. MODANESE and leaving the other elements unchanged. For any k ¼ 1; :::; n the column sum P ðhjkÞ is then equal to h¼1 C ¼ C 1 þ C C ¼ C 1 : k max k max Finally, we normalize the entire matrix by setting 00 0 P ðhjkÞ¼ P ðhjkÞ; h; k ¼ 1; .. . ; n: ðC 1Þ max A number of numerical simulations in correspondence to diﬀerent values of n and λ show that the function k ðkÞ¼ hP ðhjkÞ (7) nn h¼1 is increasing. A graphical representation (for the case in which n ¼ 99, γ ¼ 2:5 and λ ¼ 1) is given in the left panel of Figure 1. We then conclude that these correlation matrices deﬁne a one-parameter family (λ being the parameter) of assortative networks. ‘Linear’ family. Another possibility is to start by deﬁning, for μ 2ð0; 1, the elements on and above the diagonal as PðhjkÞ¼ 1 jh kj if h k (8) and the elements under the diagonal as in the formula (5). To get the normalization PðhjkÞ¼ 1, one can then apply the same procedure k¼1 as for the ﬁrst network family. In this case too, k ðkÞ turns out to be increasing in k nn suggesting assortativity for the μ family of networks with correlations P ðhjkÞ. A graphical representation of the increasing character of k ðkÞ (for the case in which nn n ¼ 99, γ ¼ 2:5 and μ ¼ 1) is given in the central panel of Figure 1. ‘Exponential’ family. A further possibility is to start by deﬁning the elements on and above the diagonal as Figure 1. The three panels in this ﬁgure display the increasing character of the function k ðkÞ for nn networks of the three families introduced in this section. Speciﬁcally, the left panel refers to a network of the power-like family, the central panel refers to a network of the linear family, and the right panel refers to a network of the exponential family. In all cases, the maximum number n of links of the network is equal to 99, the parameter γ (the exponent of the power-law degree distribution) is equal to 2:5 Also, the parameters λ; μ and ν are all equal to 1. MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 487 ðhkÞ PðhjkÞ¼ e if h k (9) and the elements under the diagonal as in the formula (5). Once again, the adjustment procedure necessary to get the required normalization is as for the two previous network families. And again, the increasing character of k ðkÞ, nn displayed as a result of a large number of simulations, supports the assortativity of the networks associated to these correlation matrices. The increasing character of k ðkÞ nn (for the case in which n ¼ 99, γ ¼ 2:5 and ν ¼ 1) is given in the right panel of Figure 1. 2.1. Alternative assortative matrices We brieﬂy outline here an alternative possibility for the construction of assortative networks. It consists in starting by assigning symmetrical correlation matrices of the ‘e ’ type (excess-degree correlations, with j; k ¼ 0; ... ; n 1, see [12,13]) with larger jk correlation for similar degree, and derive from them the excess-degree distribution q as P 1 n1 k q k1 q ¼ e and the degree distribution PðkÞ as PðkÞ¼ n . With suitable k jk j¼0 1 j q j1 j¼1 choices of e one can obtain, in this way, a degree distribution which is well approxi- jk mated by a power law. The disadvantage of this procedure is that the degree distribu- tion is not obtained in general in explicit form, but as the sum of a series. Still, this technique can be useful in order to enlarge the choice of the possible assortative correlations. We recall the relations between PðkÞ, PðhjkÞ and q , e : k jk ~e kh PðhjkÞ¼ "h; k ¼ i ¼ 1; :::; n ; (10) ~e kj j¼1 PðhjkÞkPðkÞ ~e ¼ P ; (11) hk n jPðjÞ j¼1 where ~e ¼ e .Bydeﬁnition, e is the fraction of links in the network joining k;h k1;h1 jk nodes of excess degrees j and k. Therefore, it is always symmetrical. This implies that any set of PðkÞ, PðhjkÞ obtained from a given e automatically satisfy the Network jk Closure Condition. As a simple example consider e ¼ c ; (12) jk δ ð1 þ j þ kÞ where c is a normalization constant depending on δ and determined from the condi- n1 tion e ¼ 1. Taking for instance δ ¼ 2:2, δ ¼ 2:4, δ ¼ 2:6 and δ ¼ 2:8 respec- jk j;k¼0 tively for a network with n ¼ 99, we obtain Newman assortativity coeﬃcients r ¼ 0:51, r ¼ 0:53, r ¼ 0:53 and r ¼ 0:53 and degree distributions PðkÞ behaving as c=k with γ ¼ 2:42, γ ¼ 2:61, γ ¼ 2:79 and γ ¼ 2:98 respectively . Figure 2 shows the log-log plots of these degree distributions. 488 M. L. BERTOTTI AND G. MODANESE Figure 2. Log-log plots of the degree distributions obtained for networks with n ¼ 99 by assigning excess-degree correlations e ¼ c =ð1 þ j þ kÞ , with δ ¼ 2:2, δ ¼ 2:4, δ ¼ 2:6 and δ ¼ 2:8. The jk plots are ordered from the upper one (δ ¼ 2:2) to the lower one (δ ¼ 2:8). The degree distributions PðkÞ behave as c=k with γ ¼ 2:42, γ ¼ 2:61, γ ¼ 2:79 and γ ¼ 2:98 respectively. Figure 3. The two panels show the increasing character of the function k ðkÞ for two networks with nn a maximum number n of links equal to 99 and excess-degree correlations as in (12) with δ ¼ 2:2 (left panel) and δ ¼ 2:8 (right panel). The assortativity of these networks is also evident from the increasing character of their average nearest neighbour degree functions k ðkÞ.In Figure 3 those correspond- nn ing to δ ¼ 2:2 and δ ¼ 2:8 are displayed. 3. Looking for the adoption peak time An issue of considerable interest in connection with the diﬀusion of innovations is the identiﬁcation of the peak time, which is the time corresponding to the peak of sales in marketing applications. To be more concrete, let f ðtÞ¼ F ðtÞ,where F ðtÞ is as in the j j j Introduction and the dot denotes the time derivative, denote the fraction of new adop- tions per unit time in the ‘link class j’ i.e. in the subset of individuals having j links. And MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 489 Figure 4. The left panel displays the evolution in time of the population fractions F ðtÞ in thecaseofan underlying network of the ‘power-like’ family with parameters n ¼ 9, γ ¼ 2:9and λ ¼ 1. The innovation coeﬃcient p is equal to 0:03 and the imitation coeﬃcient q is equal to 0:4. The graphs of the functions F ,as in the centre panel those of the functions f , are ordered from the upper one (j ¼ 1) to the lower one (j ¼ 9). One may observe that the largest fraction of new adopters belongs at all times to a single link class, the link class with j ¼ 1, which reaches its adoption peak later than the others. The fact that more connected individuals adopt earlier is a general (and rather intuitive) phenomenon. The centre panel shows the evolution in time of the nine functions f ðtÞ, whereas the right panel displays the evolution in time of the function fðtÞ together with that of the function FðtÞ relative to the original Bass equation. The values of t in correspondence to which the two graphs have a maximum are the peak time of the system on the network and of the homogeneous Bass system, respectively. let f ðtÞ¼ f ðtÞ be the total number of new adoptions per unit time. The three panels in i¼1 Figure 4 help to illustrate the concept. The left panel displays the evolution in time of the population fractions F ðtÞ for j ¼ 1; :::; 9. To ﬁx ideas, an underlying network of the ﬁrst family with parameters n ¼ 9, γ ¼ 2:9and λ ¼ 1 has been chosen. The innovation and the imitation coeﬃcients p and q are taken to be equal to 0:03 and 0:4, respectively, which are conceivable values employed in the traditional literature on the Bass equation .The centre panel shows the evolution in time of the nine functions f ðtÞ,whereas the right panel displays the evolution in time of the function f ðtÞ¼ f ðtÞ, together with that one i¼1 of the function FðtÞ relative to the original Bass equation. The values of t in correspon- dence to which the two graphs have a maximum are the peak time of the system on the network (occurring a little bit earlier) and the one of the homogeneous Bass system (occurring a little bit later on) respectively. Another time value of interest, especially for industry analysts, for managers and ﬁrms producing the innovations, is the so-called takeoﬀ time, which represents the moment when a transition from the initial phase to a growth phase occurs. This time can be found in the context of the generalized Bass model on networks as the time at which the acceleration of the function F ðtÞ¼ F ðtÞ (which expresses the fraction at net j i¼1 time t of the population which has adopted) is maximal and, hence, the third derivative of F ðtÞ vanishes. Equivalently, at this point, the second derivative of f ðtÞ¼ f ðtÞ net j i¼1 vanishes and the graph of f ðtÞ has an inﬂection point. Hence, this time also marks the point when, even if sales continue to increase, their increase increment begins dimin- ishing. A portion of the graph of the function f ðtÞ appearing in Figure 4 is displayed in 490 M. L. BERTOTTI AND G. MODANESE Figure 5. A portion of the graph of the function fðtÞ of Figure 4 is here displayed with time scaling suitable to make evident the existence of an inﬂection point (the takeoﬀ point) and of a maximum point (the peak point). Figure 5 where times have been scaled so as to make evident the existence of an inﬂection and of a maximum point. We calculated both the takeoﬀ and the peak time for several networks belonging to three families of Section 2. Results relative to a sample of networks for each family are reported in the three Tables 1, 2 and 3. There, a speciﬁc network is identiﬁed by the values of the two parameters γ and λ in the case of the ‘power-like’ family, γ and μ in the case of the ‘linear’ family, γ and ν in the case of the ‘exponential’ family. Notice that in the three tables for each network also the assortativity coeﬃcient r is calculated. Also, to improve the readability of the results contained in the tables, Figure 6 is here inserted. The two panels in this ﬁgure display two surfaces, respectively, obtained through interpolation of the values of the takeoﬀ time and of the peak time for the ‘power-like’ family in correspondence to the values of the parameters γ and λ men- tioned in Table 1. Similar ﬁgures for the other two families are qualitatively similar and hence are not reported here. Table 1. This table provides approximate values for the takeoﬀ time, the peak time and the assortativity coeﬃcient of 20 networks of the ‘power-like’ family in Section 2. ‘Power-like’ family λ ¼ 0:5 λ ¼ 1:0 λ ¼ 1:5 λ ¼ 2:0 takeoﬀ time t off γ ¼ 2:1 2.049 1.934 1.843 1.783 γ ¼ 2:3 1.994 1.879 1.786 1.723 γ ¼ 2:5 1.955 1.838 1.744 1.678 γ ¼ 2:7 1.932 1.814 1.717 1.650 γ ¼ 2:9 1.925 1.807 1.708 1.638 peak time t max γ ¼ 2:1 3.967 3.907 3.891 3.903 γ ¼ 2:3 4.060 4.116 4.244 4.386 γ ¼ 2:5 4.277 4.454 4.655 4.816 γ ¼ 2:7 4.585 4.794 4.975 5.108 γ ¼ 2:9 4.891 5.064 5.205 5.307 assortativity coeﬃcient r γ ¼ 2:1 0.594 0.716 0.803 0.857 γ ¼ 2:3 0.626 0.734 0.811 0.861 γ ¼ 2:5 0.652 0.748 0.817 0.863 γ ¼ 2:7 0.673 0.759 0.821 0.863 γ ¼ 2:9 0.690 0.767 0.823 0.861 MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 491 Table 2. This table provides approximate values for the takeoﬀ time, the peak time and the assortativity coeﬃcient of 20 networks of the ‘linear’ family in Section 2. ‘Linear’ family μ ¼ 0:25 μ ¼ 0:5 μ ¼ 0:75 μ ¼ 1:0 takeoﬀ time t off γ ¼ 2:1 2.136 2.113 2.075 2.017 γ ¼ 2:3 2.080 2.059 2.021 1.967 γ ¼ 2:5 2.040 2.020 1.984 1.931 γ ¼ 2:7 2.013 1.995 1.961 1.910 γ ¼ 2:9 2.002 1.985 1.955 1.905 peak time t max γ ¼ 2:1 4.037 4.020 3.992 3.959 γ ¼ 2:3 4.069 4.069 4.065 4.075 γ ¼ 2:5 4.200 4.219 4.248 4.302 γ ¼ 2:7 4.443 4.476 4.525 4.598 γ ¼ 2:9 4.749 4.781 4.824 4.885 assortativity coeﬃcient r γ ¼ 2:1 0.481 0.513 0.563 0.629 γ ¼ 2:3 0.530 0.555 0.598 0.654 γ ¼ 2:5 0.570 0.590 0.627 0.674 γ ¼ 2:7 0.602 0.620 0.650 0.691 γ ¼ 2:9 0.629 0.644 0.668 0.705 Table 3. This table provides approximate values for the takeoﬀ time, the peak time and the assortativity coeﬃcient of 20 networks of the ‘exponential’ family in Section 2. ‘Exponential’ family ν ¼ 0:5 ν ¼ 1:0 ν ¼ 1:5 ν ¼ 2:0 takeoﬀ time t off γ ¼ 2:1 2.128 2.096 2.067 2.036 γ ¼ 2:3 2.073 2.043 2.014 1.987 γ ¼ 2:5 2.033 2.006 1.976 1.949 γ ¼ 2:7 2.007 1.984 1.955 1.928 γ ¼ 2:9 1.997 1.977 1.949 1.924 peak time t max γ ¼ 2:1 4.032 4.008 3.989 3.970 γ ¼ 2:3 4.070 4.067 4.068 4.073 γ ¼ 2:5 4.208 4.229 4.255 4.284 γ ¼ 2:7 4.454 4.492 4.532 4.572 γ ¼ 2:9 4.758 4.791 4.827 4.861 assortativity coeﬃcient r γ ¼ 2:1 0.492 0.535 0.572 0.608 γ ¼ 2:3 0.539 0.574 0.606 0.635 γ ¼ 2:5 0.578 0.605 0.634 0.658 γ ¼ 2:7 0.609 0.631 0.656 0.677 γ ¼ 2:9 0.634 0.651 0.673 0.692 4. Emerging patterns The patterns announced in the Introduction come out as follows: If one ﬁxes a value of the parameter λ or μ or ν, depending on which of the three tables relative to the three network families she is considering, it is immediate to check that the peak times increase as the parameter γ increases passing from 2:1to2:9. As well, the takeoﬀ times diminish as γ increases. It is then tempting to try and compare the degree correlation matrices pertaining to networks with diﬀerent values of γ. Somehow surprisingly, it turns out that each time one subtracts the degree correlation matrix of a network with a certain γ from the degree correlation matrix of a network with a greater γ (keeping the other parameter ﬁxed) one gets a matrix for which all entries 492 M. L. BERTOTTI AND G. MODANESE Figure 6. The panels display two surfaces, respectively, obtained through interpolation of the values of the takeoﬀ time (left panel) and of the peak time (right panel) for the ‘power-like’ family in correspondence to the values of the parameters γ and λ mentioned in Table 1. Similar ﬁgures for the other two families are qualitatively similar and hence are not reported here. below the main diagonal are negative, whereas all elements above the main diagonal are positive. For example, the diﬀerence P ðhjkÞ P ðhjkÞ of the degree ðμ¼0:5;γ¼2:9Þ ðμ¼0:5;γ¼2:7Þ correlation matrices of the networks of the ‘linear’ family with μ ¼ 0:5 and γ ¼ 2:9 and γ ¼ 2:7, respectively, is approximatively 0 1 0:0201 0:0004 0:0004 0:0004 0:0004 0:0003 0:0003 0:0003 0:0003 0:0059 0:0267 0:0004 0:0004 0:0004 0:0004 0:0003 0:0003 0:0003 B C B C 0:0043 0:0057 0:0266 0:0004 0:0004 0:0004 0:0004 0:0003 0:0003 B C B C 0:0030 0:0056 0:0050 0:0229 0:0004 0:0004 0:0004 0:0004 0:0003 B C B C 0:0022 0:0047 0:0057 0:0042 0:0173 0:0004 0:0004 0:0004 0:0004 : B C B C 0:0017 0:0038 0:0052 0:0054 0:0037 0:0110 0:0004 0:0004 0:0004 B C B C 0:0013 0:0030 0:0045 0:0053 0:0050 0:0032 0:0049 0:0004 0:0004 B C @ A 0:0010 0:0024 0:0039 0:0049 0:0052 0:0047 0:0028 0: 0:0004 0:0008 0:0020 0:0032 0:0043 0:0050 0:0051 0:0043 0:0025 0:0028 This is an unexpected and curious phenomenon, which calls for explanation or interpretation. As a matter of fact, in a comparison of two networks belonging to a same family among the three ones deﬁned in Section 2, the following can be observed: the correlation matrix of the network for which the takeoﬀ time occurs earlier and the peak time occurs later (consistently with a longer period of major sales) has each entry PðhjkÞ with k > h greater [respectively, each entry PðhjkÞ with k< h smaller] than the corresponding entry of the correlation matrix of the other network. It seems then than a greater number of individuals connected to individuals who have less connections than them and equivalently a smaller number of individuals connected to individuals who have more connections than them can have a positive eﬀect on the length of the major sales period. Yet, it has to be stressed that this phenomenon just refers to comparisons between networks within the same family. Notice that a ﬁrst glance at Table 1, 2, 3 could induce to conclude that this eﬀect is just the consequence of a greater assortativity. But, a monotonicity character of the assorta- tivity coeﬃcient r with respect to the parameter γ does not always hold true. This can be observed by looking, for example, at the values of r for networks of the ‘power-like’ family with λ ¼ 2:0. And we found a lack of monotonicity of r also in other simulations. MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 493 5. Community eﬀects in the network Bass model The calculations of the previous sections have been performed in the framework of the mean-ﬁeld approximation, which is known to work well, under wide conditions, for diﬀusion processes in two-state systems, including the Bass model and several others . Still, it is interesting to explore the network Bass model also in certain conditions which lie outside the scope of the mean-ﬁeld approximation. In , we have brieﬂy reported the results of numerical solutions of the network Bass model on Barabasi-Albert networks. These were obtained computing the evolution of each single node, connected to the rest of the network according to the adjacency matrix a . In principle, one has to solve a system of N coupled non-linear equations ij where N is the number of nodes, instead of n equations like in the mean-ﬁeld approximation (n being the maximum degree present in the network). In fact, however, all the nodes with low degree, which are the overwhelming majority in a scale-free network, are coupled with few other nodes. In contrast, in the mean-ﬁeld approxima- tion, all the n variables G are completely coupled to each other in all equations, even if with coeﬃcients PðhjkÞ which are In order to study diﬀusion on Barabasi-Albert networks, we have used in  random realizations of the networks, obtained with a preferential attachment algorithm. We have checked that the average diﬀusion times of the link classes are quite close to the mean-ﬁeld predictions, even though there can be signiﬁcant deviations for the single nodes; for instance, considering two nodes of the same degree, one of which is connected to a hub and the other to a peripherical node with low degree, the ﬁrst node always adopts earlier than the second. The equation system employed has the form "# dX ¼ð1 X Þ p þ q a X : (13) i ij j dt j¼1 Such a form can be derived by a ﬁrst moment closure according to the deﬁnition by [29,30]. Indeed, the variable X can be considered as the average hx i (i ¼ 1 ... N) of the i i non-adoption or adoption state of node i (with x respectively equal to 0 and 1) over many stochastic evolutions of the system. Here we report the results of the application of Equation (13) to a small real network suitable for diﬀusion studies, namely the network of the Top-150 companies of the year 2017 in our province, South Tyrol. The list of these companies is published annually by the Commerce Chamber of Bolzano on its website. Companies are selected according to certain listed ranking criteria and also declare their commercial partners. The structure of the connected part of the network is visible in Figures 7 and 8. It has 126 nodes, a global clustering coeﬃcient C ¼ 3=310 and an assortativity coeﬃcient r ¼0:523. The maximum degree is 35 and the average degree is 15=7. Denoting by N the number of nodes with degree k,the values of N for k ¼ 1; .. . ; 13 are equal to 98; 14; 1; 1; 3; 1; 0; 4; 1; 0; 0; 0; 1, whereas the other non-zero elements are N ¼ 1 and N ¼ 1. An analysis of the network 27 35 communities performed with Mathematica according to the Modularity and Spectral 494 M. L. BERTOTTI AND G. MODANESE Figure 7. The cooperation network of the Top 150 companies in South Tyrol (connected part). criteria allows to spot a few communities, among which the most numerous are those of the two largest hubs, visible on the right. usually very small for high degrees. Like for the Barabasi-Albert networks mentioned above, the equations system are obtained formally from a ﬁrst moment closure, but in practice they give a better description of the real situation, because we can suppose that the continuous variable X , with 0 X 1, represents the adoption level of the innovation inside Company i; i i in other words, whereas in applications of the Bass model to the innovation adoption process of single individuals, x jumps from 0 to 1 in a discontinuous way, here (for companies) X evolves in a continuous way, like the average 0 hx i 1 for individuals j i over many realizations. The detailed solution of the Equations (13) for our real network (depicted in Figure 7) displays some interesting features. (1) Looking at the adoption times of the nodes of degree 1 (compare histogram in Figure 8), one ﬁnds as expected that those belonging to the same cluster adopt in the same time, since they can be regarded as identical. For instance, in the cluster of the largest hub, their adoption time is t ¼ 3:4 (for the hub itself, t ¼ 2:2); in the cluster max max of the second-largest hub, the adoption time is t ¼ 3:7(for the hub: t ¼ 2:4). What max max is the origin of the diﬀerences between clusters? Clearly, if the hub of a cluster is connected to the rest of the network better than the hub of another cluster, then it will adopt earlier and will consequently ‘infect’ earlier its cluster. In the SI epidemic model, which is equivalent to a Bass model with p ¼ 0 (pure contagion, no publicity/innovation term), this would actually be the only possible explanation of the diﬀerence between the two hubs. In the Bass model, however, the publicity term plays an essential role, because each degree-1 node in a cluster has an ‘individual’ adoption probability due to the p-term and independent of the state of its neighbours. After adoption, each degree-1 node tends to infect its hub, and the hub, in turn, re-destributes the inﬂuence on the whole cluster. In the end, therefore, the largest hubs tend to adopt earlier, like parents of large families where each kid keeps bringing home new ideas or gadgets. MATHEMATICAL AND COMPUTER MODELLING OF DYNAMICAL SYSTEMS 495 Figure 8. The top panel above shows the distribution of the peak adoption times of the nodes (companies) of the real network of Figure 7, the cooperation network of the Top 150 companies in South Tyrol, accordingtothe Bass modelinthe ﬁrst moment closure (Equation (13)), with parameters q ¼ 0:35, p ¼ 0:03. The adoptions in the two largest clusters are clearly visible at times 3.4 and 3.7 and those of their hubs at 2.2 and 2.4. The three panels 8a, 8b, 8c correspond to snapshots of the network at time t ¼ 2:5, t ¼ 4:0and t ¼ 6:0. In each of them, the red ones in the online version (correspondent to the dark ones in the printed version) are the nodes which have adopted, whereas the white ones in the online version (light in the printed version) are the nodes which have not adopted. 496 M. L. BERTOTTI AND G. MODANESE (2) Since the network contains, as also evident from Figure 7, two large communities with diﬀerent adoption times, if we plot the curves f ðtÞ of the total adoption rates comm for each of these communities, their peaks are shifted. Therefore, the f ðtÞ curve for the tot entire network will have a multi-modal character; even if this does not show up in separate peaks, the consequence is that the plot of f ðtÞ is deformed in comparison to tot the standard logistic curve of the homogeneous Bass model (and also compared to the curve of the network Bass model in mean-ﬁeld approximation). It is possible to check this deformation by trying, with no success, to ﬁt f ðtÞ with the usual Bass logistic tot 2 1 at at function f ðtÞ¼ a p e ½1 þðq=pÞe , where a ¼ p þ q. All this must be taken Bass into account when the Bass diﬀusion is modelled through an agent-based simulation, like in , where the results are ﬁtted with the Bass logistic function. 6. Conclusion The dynamics of diﬀusion of innovations and information on social networks have attracted considerable attention in the last years. From the mathematical point of view, the most distinctive feature of a social network is its assortativity, deﬁned in terms of the Newman coeﬃcient r and of the average nearest neighbour degree function k ðkÞ. nn In this work, we have developed new techniques for the construction of assortative correlation matrices and discussed some peculiar features of these matrices. We have then employed the correlation matrices for the numerical computation of the diﬀusion curves of the Bass model, which is widely used in marketing analysis. The Bass model oﬀers the advantage, in comparison to other epidemic models, that the so-called diﬀusion peak time and takeoﬀ time are well deﬁned, independently of the initial conditions; they depend only on the empirical model parameters (innovation and imitation coeﬃcient) and on the features of the network (scale-free exponent, assorta- tivity). We have thus been able to study relations between these quantities that can be helpful in the analysis of real diﬀusion data. Notes 1. Of course, the value of γ in general changes if a diﬀerent n is taken. 2. We should point out here that, as explained in , when results relative to systems on scale-free networks with diﬀerent exponents γ are to be compared, the coeﬃcient q has to be normalized. 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Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis
Published: Sep 3, 2019
Keywords: Innovation diffusion; Bass model; assortative networks; takeoff and peak time
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