Random or preferential? Evolutionary mechanism of user behavior in co-creation community

Abstract

The objective of this paper is to examine the evolutionary mechanism regarding how a co-creation community network evolves as the growth of user interaction, which differs from the existing studies concentrating on the explanation of the forward problems of information management systems (e.g. motivational identification of user participation and examination of users’ outcomes). To achieve this objective, network generation model is formulated as nodes of users, ties of user’s interactions, random process, and preferential attachment. Then, real networks formulated by practice and artificial networks generated by the proposed model are compared by cumulative degree distribution, so as to validate the feasibility of the proposed model and to explain user behavior from the perspective of link formulation. Results indicate that: (i) new users account for main contributions for the development of co-creation community; (ii) new users prefer to interact high-influence all the time, while old users interchangeably choose preferential attachment or random linking in different time periods, (iii) the initial number of users, the probability for choosing preferential attachment or random attachment has a great influence on the properties of a user interactive network.

Appendix 1: Proofs of five possible situations

1.1 Proof of situation 1

Denote $\alpha =1$ and $\beta =1$, the probability that an existing node i with degree ${d}_{i}(t)$ obtains a new connection in period $t+1$ is roughly

$$\frac{\Delta {d}_{i}(t)}{\Delta t}\approx \frac{m+2n}{t}$$

(A.1)

Do integral, the degree of node i is

$$\frac{1}{m+2n}{d}_{i}\left(t\right)=ln t+c$$

(A.2)

According to the initial network, the initial degree of node i is $m ({d}_{i}\left(i\right)=m)$. Put ${d}_{i}\left(i\right)=m$ into Eq. (A.2), and $c$ can be expressed

$$c=\frac{m}{m+2n}-ln i$$

(A.3)

Put the result of c into Eq. (A.2). Besides, based on the method of “mean-field”, a cumulative distribution is calculated by ${F}_{t}\left(d\right)=1-\frac{i(d)}{t}$, if ${d}_{i\left(d\right)}\left(t\right)=d$. Thus, the cumulative distribution of situation 1 is

$${F}_{t}\left(d\right)=1-{e}^{\frac{d-m}{m+2n}}$$

(A.4)

1.2 Proof of situation 2

Denote $\alpha =1$ and $\beta =0$, the probability that an existing node i with degree ${d}_{i}(t)$ obtains a new connection in period t + 1 is roughly

$$\frac{\Delta {d}_{i}(t)}{\Delta t}\approx \frac{m}{t}+\frac{2n{d}_{i}(t)}{2(m+n)t}$$

(A.5)

Do integral, the degree of node i is

$$\frac{1}{{\lambda }_{1}}\mathrm{ln}\left(m+{\lambda }_{1}{d}_{i}\left(t\right)\right)=lnt+c, {\lambda }_{1}=\frac{n}{m+n}$$

(A.6)

According to the initial network, the initial degree of node i is $m ({d}_{i}\left(i\right)=m)$. Put ${d}_{i}\left(i\right)=m$ into Eq. (A.6), and c can be expressed

$$c=\frac{1}{{\lambda }_{1}}\mathrm{ln}\left(m+{\lambda }_{1}m\right)-ln i$$

(A.7)

Put the result of c into Eq. (A.6). Besides, based on the method of “mean-field”, a cumulative distribution is calculated by ${F}_{t}\left(d\right)=1-\frac{i(d)}{t}$, if ${d}_{i\left(d\right)}\left(t\right)=d$. Thus, the cumulative distribution of situation 2 is

$${F}_{t}\left(d\right)=1-{\left(\frac{m+\frac{m}{{\lambda }_{1}}}{d+\frac{m}{{\lambda }_{1}}}\right)}^{\frac{1}{{\lambda }_{1}}}$$

(A.8)

1.3 Proof of situation 3

Denote $\alpha =0$ and $\beta =1$, the probability that an existing node i with degree ${d}_{i}(t)$ obtains a new connection in period t + 1 is roughly

$$\frac{\Delta {d}_{i}(t)}{\Delta t}\approx \frac{m{d}_{i}\left(t\right)}{2\left(m+n\right)t}+\frac{2n}{t}$$

(A.9)

Do integral, the degree of node i is

$$\frac{1}{{\lambda }_{2}}\mathrm{ln}\left(2n+{\lambda }_{2}{d}_{i}\left(t\right)\right)=lnt+c, {\lambda }_{2}=\frac{n}{2(m+n)}$$

(A.10)

According to the initial network, the initial degree of node i is $m ({d}_{i}\left(i\right)=m)$. Put ${d}_{i}\left(i\right)=m$ into Eq. (A.10), and c can be expressed

$$c=\frac{1}{{\lambda }_{2}}\mathrm{ln}\left(2n+{\lambda }_{2}m\right)-lni$$

(A.11)

Put the result of c into Eq. (A.10). Besides, based on the method of “mean-field”, a cumulative distribution is calculated by ${F}_{t}\left(d\right)=1-\frac{i(d)}{t}$, if ${d}_{i\left(d\right)}\left(t\right)=d$. Thus, the cumulative distribution of situation 3 is

$${F}_{t}\left(d\right)=1-{\left(\frac{m+\frac{m}{{\lambda }_{2}}}{d+\frac{m}{{\lambda }_{2}}}\right)}^{\frac{1}{{\lambda }_{2}}}$$

(A.12)

1.4 Proof of situation 4

Denote $\alpha =0$ and $\beta =0$, the probability that an existing node i with degree ${d}_{i}(t)$ obtains a new connection in period t + 1 is roughly

$$\frac{\Delta {d}_{i}(t)}{\Delta t}\approx \frac{m{d}_{i}\left(t\right)+2n{d}_{i}(t)}{2\left(m+n\right)t}$$

(A.13)

Do integral, the degree of node i is

$$\frac{1}{{\lambda }_{3}}\mathrm{ln}\left({\lambda }_{3}{d}_{i}\left(t\right)\right)=ln t+c, {\lambda }_{2}=\frac{m+2n}{2(m+n)}$$

(A.14)

According to the initial network, the initial degree of node i is $m ({d}_{i}\left(i\right)=m)$. Put ${d}_{i}\left(i\right)=m$ into Eq. (A.14), and c can be expressed

$$c=\frac{1}{{\lambda }_{3}}\mathrm{ln}\left({\lambda }_{3}m\right)-ln i$$

(A.15)

Put the result of c into Eq. (A.14). Besides, based on the method of “mean-field”, a cumulative distribution is calculated by ${F}_{t}\left(d\right)=1-\frac{i(d)}{t}$, if ${d}_{i\left(d\right)}\left(t\right)=d$. Thus, the cumulative distribution of situation 4 is

$${F}_{t}\left(d\right)=1-{\left(\frac{d}{m}\right)}^{-\frac{1}{{\lambda }_{3}}}$$

(A.16)

1.5 Proof of situation 5

Denote $0<\alpha <1$ and $0<\beta <1$, the probability that an existing node i with degree ${d}_{i}(t)$ obtains a new connection in period t + 1 is roughly

$$\frac{\Delta {d}_{i}(t)}{\Delta t}\approx \frac{\alpha m+2\beta n}{t}+\frac{(1-\alpha )m{d}_{i}\left(t\right)+2(1-\beta )n{d}_{i}(t)}{2\left(m+n\right)t}$$

(A.17)

Do integral, the degree of node i is

$$\frac{1}{B}\mathrm{ln}\left(A+B{d}_{i}\left(t\right)\right)=ln t+c$$

(A.18)

Specially,

$$A=\alpha m+2\beta n; B=\frac{\left(1-\alpha \right)m+2(1-\beta )n}{2(m+n)}.$$

According to the initial network, the initial degree of node i is $m ({d}_{i}\left(i\right)=m)$. Put ${d}_{i}\left(i\right)=m$ into Eq. (A.18), and c can be expressed

$$c=\frac{1}{B}ln\left(A+Bm\right)-ln i$$

(A.19)

Put the result of c into Eq. (A.18). Besides, based on the method of “mean-field”, a cumulative distribution is calculated by ${F}_{t}\left(d\right)=1-\frac{i(d)}{t}$, if ${d}_{i\left(d\right)}\left(t\right)=d$. Thus, the cumulative distribution of situation 5 is

$${F}_{t}\left(d\right)=1-{\left(\frac{m+A/B}{d+A/B}\right)}^\frac{1}{B}$$

(A.20)

1.6 Appendix 2: Error analysis ($error({G}_{r},{G}_{a})$) of different parameters

Since the small-scale of data in time window 1 shows the anomalous distribution and it may cause a big error for result, the network describing the data from time window 1 is not discussed here. The smallest value with bold underline indicates the best artificial network in different time periods (Tables

Table 6 Time window 2 (N = 1816)

6,

Table 7 Time window 3 (N = 2988)

7,

Table 8 Time window 4 (N = 3710)

8,

Table 9 Time window 5 (N = 4378)

9,

Table 10 Time window 6 (N = 5293)

10,

Table 11 Time window 7 (N = 6238)

11,

Table 12 Time window 8 (N = 7042)

12,

Table 13 Time window 9 (N = 7619)

13, and

Table 14 Time window 10 (N = 8231)

14).