User online consumption behaviour based on fractional differential equation

When potential consumers in the dynamic consumer social network are internally influenced by the adoption probability a>γ(c+μ)〈k〉c〈k2〉 a > {{\gamma (c + \mu)\left\langle k \right\rangle} \over {c\left\langle {{k^2}} \right\rangle}} of buying products, there is always a positive balance point in the system of formula (1). Prove that the right side of Eq. (1) is equal to 0, then we have (6) {c(1−xk(t)−yk(t))−kaxk(t)∑k'=1Mp(k'|k)yk'(t)−μxk(t)=0kaxk(t)∑k'=1Mp(k'|k)yk'(t)−γyk(t)=0 \left\{{\matrix{{c(1 - {x_k}(t) - {y_k}(t)) - ka{x_k}(t)\sum\limits_{{k^{'}} = 1}^M p({k^{'}}|k){y_{{k^{'}}}}(t) - \mu {x_k}(t) = 0} \cr {ka{x_k}(t)\sum\limits_{{k^{'}} = 1}^M p({k^{'}}|k){y_{{k^{'}}}}(t) - \gamma {y_k}(t) = 0} \cr}} \right.

Two equilibrium solutions Eo=(cc+μ0) {E_o} = \left({{c \over {\mathop 0\limits^{c + \mu}}}} \right) , Ek*=(xk(t)yk(t))=cc+μ+c+γγkaθ(kaθ1γ) E_k^* = \left({\matrix{{{x_k}(t)} \cr {{y_k}(t)} \cr}} \right) = {c \over {c + \mu + {{c + \gamma} \over \gamma}ka\theta}}\left({{{\mathop {ka\theta}\limits^1} \over \gamma}} \right) of the equation system (6) can be obtained. By substituting y_k(t) in Ek* E_k^* into Eq. (2), we can bring the self-consistent equation θ=cγ〈k〉∑k'=1Mcc+μ+c+γγk'aθ(1kaθγ) \theta = {c \over {\gamma \left\langle k \right\rangle}}\sum\limits_{{k^{'}} = 1}^M {c \over {c + \mu + {{c + \gamma} \over \gamma}{k^{'}}a\theta}}\left({{{\matrix{1 \cr {ka\theta} \cr}} \over \gamma}} \right) of θ. We make Fθ=caγ〈k〉∑k'=1M(k')2θp(k')c+μ+c+γγk'aθ−θ F\theta = {{ca} \over {\gamma \left\langle k \right\rangle}}\sum\limits_{{k^{'}} = 1}^M {{{{({k^{'}})}^2}\theta p({k^{'}})} \over {c + \mu + {{c + \gamma} \over \gamma}{k^{'}}a\theta}} - \theta obviously θ = 0, F(0) = 0, and d2F(θ)dθ2=−2〈k〉∑k'=1Mca2γ(c+μ)(c+γ)(k')3p(k')[γ(c+μ)+(c+γ)k'aθ]3 {{{d^2}F(\theta)} \over {d{\theta ^2}}} = {{- 2} \over {\left\langle k \right\rangle}}\sum\limits_{{k^{'}} = 1}^M {{c{a^2}\gamma (c + \mu)(c + \gamma)({k^{'}}{)^3}p({k^{'}})} \over {{{[\gamma (c + \mu) + (c + \gamma){k^{'}}a\theta ]}^3}}} . Among them, θ changes in [0, 1], and the value ranges of a, c, μ and γ are both (0, 1), so d2F(θ)dθ2<0 {{{d^2}F(\theta)} \over {d{\theta ^2}}} < 0 , Fθ is a concave function on 0 ≤ θ ≤ 1. At the same time (7) F(1)=1〈k〉∑k'=1Mk'p(k')ck'aγ(c+μ)+(c+γ)k'a−1<0 F(1) = {1 \over {\left\langle k \right\rangle}}\sum\limits_{{k^{'}} = 1}^M {{{k^{'}}p({k^{'}})c{k^{'}}a} \over {\gamma (c + \mu) + (c + \gamma){k^{'}}a}} - 1 < 0

Therefore, the necessary and sufficient condition for the existence of a unique positive solution of formula (1) is dF(θ)dθ|θ=0>0 {{dF(\theta)} \over {d\theta}}{|_{\theta = 0}} > 0 , that is, dF(θ)dθ|θ=0=ca〈k2〉γ(c+μ)〈k〉−1>0 {{dF(\theta)} \over {d\theta}}{|_{\theta = 0}} = {{ca\left\langle {{k^2}} \right\rangle} \over {\gamma (c + \mu)\left\langle k \right\rangle}} - 1 > 0 . Where 〈k2〉=∑k=1N〈k2〉p(k) \left\langle {{k^2}} \right\rangle = \sum\limits_{k = 1}^N \left\langle {{k^2}} \right\rangle p(k) represents the second moment of the node degree of the consumer’s social network. Therefore, when a>γ(c+μ)〈k〉c〈k2〉 a > {{\gamma (c + \mu)\left\langle k \right\rangle} \over {c\left\langle {{k^2}} \right\rangle}} , there is always a positive equilibrium point in the system (1).

When R0=ca〈k2〉γ(c+μ)〈k〉>1 {R_0} = {{ca\left\langle {{k^2}} \right\rangle} \over {\gamma (c + \mu)\left\langle k \right\rangle}} > 1 , there is only a positive equilibrium point in the social system of formula (1), which makes the products in the system continue to spread for a long time. It is almost impossible for consumers to communicate with all purchasers in the network to obtain information [9]. Therefore, suppose that the maximum number of contacts of potential consumers is Z. When k > Z, p(k) = 0. For the subspace Ω = {(x₁,y₁,x₂,y₂,… ,x_Z,y_Z)^T ∈ R^2Z|x_k ≥ 0,y_k ≥ 0,x_k + y_k ≤ 1,1 ≤ k ≤ Z} of system (1). It is easy to verify that the space Ω is a positively invariant set and a compact convex set. (8) (λ+c+μ)Z(λ+γ)Z−1(λ+γ−∑j=1Zbj)=(λ+c+μ)Z(λ+γ)Z−1(λ−γ(ca〈k2〉γ(c+μ)〈k〉−1)) {(\lambda + c + \mu)^Z}{(\lambda + \gamma)^{Z - 1}}(\lambda + \gamma - \sum\limits_{j = 1}^Z {b_j}) = (\lambda + c + \mu {)^Z}{(\lambda + \gamma)^{Z - 1}}(\lambda - \gamma ({{ca\left\langle {{k^2}} \right\rangle} \over {\gamma (c + \mu)\left\langle k \right\rangle}} - 1))

Therefore, when R0=ca〈k2〉γ(c+μ)〈k〉>1 {R_0} = {{ca\left\langle {{k^2}} \right\rangle} \over {\gamma (c + \mu)\left\langle k \right\rangle}} > 1 , the Jacobian matrix of the system (1) at the equilibrium point (cc+μ,0)T {({c \over {c + \mu}},0)^T} has a unique positive eigenvalue. According to the Perron-Frobenius principle, the most significant fundamental part of all the eigenvalues of the Jacobian matrix when R₀ > 1 is a positive number. Therefore, from Lemma 2, we know that Proposition 2 is established.

Proposition 1 and Proposition 2 show that when potential consumers’ decision to purchase products is only affected by the purchasers. Therefore, the probability of potential consumers’ purchases of products affected by the internal influence of the consumer’s social network must be at least higher than a critical value to ensure the monopoly firm’s Products continue to increase in dynamic consumer social networks [10]. On the other hand, when the adoption probability is lower than the critical value, the monopolistic firm’s products cannot continue to spread, and the spread fails. This critical value is related to dynamic consumer social networks (node degree distribution and network scale). Therefore, the dynamics of consumer social networks (the proportion of new potential consumers, the probability of potential consumers losing their desire to buy, and the non-purchasing probability of publishing information or publishing invalid input) are related.

When companies implement advertising and penetration pricing strategies, there are no constraints on the continuous diffusion of products in dynamic consumer social networks. The proportion of new purchases per unit time is more significant than when only advertising strategies are implemented [11]. Prove that the right side of Eq. (4) is equal to 0 to obtain the positive equilibrium solution of the equations (9) (xk(t)yk(t))=c(c+μ)+(c+γ)gL(akθ+b)γ(gLakθ+b1)γ \left({\matrix{{{x_k}(t)} \cr {{y_k}(t)} \cr}} \right) = {c \over {(c + \mu) + (c + \gamma){{{g_L}(ak\theta + b)} \over \gamma}}}{{\left({\mathop {{g_L}ak\theta + b}\limits^1} \right)} \over \gamma}

Assuming ag_L = a′, bg_L = b′, the positive equilibrium solutions of Eqs. (4) and (3) are similar. Therefore, when we consider both advertising and penetration pricing strategies, our conclusions are identical to the findings when we only consider advertising strategies. That is, there are no restrictions on the continuous spread of products in dynamic consumer social networks. In system (3), in which the enterprise adopts the advertising strategy, the proportion y_k(t) of buyers with node degree k gradually increases with time and finally reaches a stable state. Because of g_L > 1, ag_L = a′ > a,ag_L = b′ > b, the balance of newly added purchasers affected by advertising and internal influence in the system (4) per unit time is more significant than that in the case of only implementing the advertising strategy, and the proportion coefficient that affects the exit of purchasers from the system has not changed. Therefore, the proportion of newly added purchasers in the system (4) per unit time is more significant than that in the system (3).

Proposition 3s shows that when companies implement advertising and penetration pricing strategies simultaneously, products continue to spread in dynamic consumer social networks. However, penetration pricing amplifies the effect of communication between advertising and consumers and makes product diffusion faster.

In a dynamic consumer social network where the degree distribution of the network obeys a power-law distribution, when companies implement advertising and skimming pricing strategies at the same time, γc≥14 {\gamma \over c} \ge {1 \over 4} or γc<14 {\gamma \over c} < {1 \over 4} and ω_H < ω₁ make the product continue to spread [12]. Prove that the right side of Eq. (5) is equal to 0 to obtain the positive equilibrium solution of the system of equations (10) (xk(t)yk(t))=cgH(cgH+μ)gH+(cg2H+γ)gHγ(akθ+b)(gHγ(akθ+b)1) \left({\matrix{{{x_k}(t)} \cr {{y_k}(t)} \cr}} \right) = {{c{g_H}} \over {(c{g_H} + \mu){g_H} + (c{g^2}_H + \gamma){{{g_H}} \over \gamma}(ak\theta + b)}}\left({\mathop {{{{g_H}} \over \gamma}(ak\theta + b)}\limits^1} \right)

We substitute y_k(t) in Eq. (10) into Eq. (2): (11) θ=1〈k〉∑k'=1Mk'p(k')cgH(ak'θ+b)γ(cgH+μ)+(cg2H+γ)(b+ak'θ) \theta = {1 \over {\left\langle k \right\rangle}}\sum\limits_{{k^{'}} = 1}^M {k^{'}}p({k^{'}}){{c{g_H}(a{k^{'}}\theta + b)} \over {\gamma (c{g_H} + \mu) + (c{g^2}_H + \gamma)(b + a{k^{'}}\theta)}}

F(θ)=1〈k〉∑k'=1Mk'p(k')cgH(ak'θ+b)γ(cgH+μ)+(cg2H+γ)(b+ak'θ)−θ F(\theta) = {1 \over {\left\langle k \right\rangle}}\sum\limits_{{k^{'}} = 1}^M {k^{'}}p({k^{'}}){{c{g_H}(a{k^{'}}\theta + b)} \over {\gamma (c{g_H} + \mu) + (c{g^2}_H + \gamma)(b + a{k^{'}}\theta)}} - \theta . Since F(0)=cbgHγ(cgH+μ)+(cg2H+γ)>0,d2F(θ)dθ2<0 F(0) = {{cb{g_H}} \over {\gamma (c{g_H} + \mu) + (c{g^2}_H + \gamma)}} > 0,{{{d^2}F(\theta)} \over {d{\theta ^2}}} < 0 , where the value ranges of a, b, c, μ and γ are both (0, 1), g_H > 0, the necessary and sufficient conditions for F(0) = 0 to have a unique positive solution in θ ∈ [0, 1] are: (12) F(1)=1〈k〉∑k'=1Mk'p(k')cgH(ak'+b)γ(cgH+μ)+(cg2H+γ)(ak'θ+b)−1≤0 F(1) = {1 \over {\left\langle k \right\rangle}}\sum\limits_{{k^{'}} = 1}^M {{{k^{'}}p({k^{'}})c{g_H}(a{k^{'}} + b)} \over {\gamma (c{g_H} + \mu) + (c{g^2}_H + \gamma)(a{k^{'}}\theta + b)}} - 1 \le 0

The degree distribution of the network obeys the power-law distribution of dynamic consumer social networks, which can be obtained from Eq. (12) (13) g2H−gH+γc≥0 {g^2}_H - {g_H} + {\gamma \over c} \ge 0

When γc≥14 {\gamma \over c} \ge {1 \over 4} formula (13) always holds. At this time, the price change range of the company’s implementation of the skimming pricing strategy only needs to ensure that the proportion of new purchasers is not too small.

Proposition 5 shows that when companies implement advertising and skimming pricing strategies simultaneously, they must ensure that the company’s products continue to spread in the dynamic consumer social network where the network’s degree distribution obeys the power-law distribution. There are constraints on product prices [13]. This price constraint has nothing to do with the network characteristics of the dynamic consumer social network. Still, it is only related to the proportion of new potential consumers in the dynamic consumer social network and the probability that the consumers who have purchased the product will not publish information or publish invalid information.