Quantifying Community Evolution in Developer Social Networks: Proof of Indices’ Properties

Let $\mathcal{I}^{\psi}_{c_{t,i}}$ and $\mathcal{I}^{\eta}_{c_{t,i}}$ denote the community split and shrink indices, respectively. Without loss of generality, we assume $m\geq 1$. The properties of the two indices are as follows.

P-1. $\mathcal{I}^{\psi}_{c_{t,i}}$ and $\mathcal{I}^{\eta}_{c_{t,i}}$ are strictly monotonic increasing functions of $m$, given $0<\eta_{i}<1$, and $\hat{\psi}_{i,j}=\frac{1}{m},j=1,2,\cdots,m$.

P-2. $\mathcal{I}^{\psi}_{c_{t,i}}$ / $\mathcal{I}^{\eta}_{c_{t,i}}$ is a strictly monotonic decreasing / increasing function of $\eta_{i}$, respectively, for $\eta_{i}>0$, given $m>1$, and member migration distribution $\hat{\psi}_{i,j},j=1,2,\cdots,m$ with $\mathcal{H}_{c_{t,i}}>0$.

P-3. Given $m$ and $\eta_{i}$, the maximum split index $\mathcal{I}^{\psi}_{c_{t,i}}=(1-\eta_{i})\mathcal{H}^{*}_{t\rightarrow t+1}$ is obtained when the members of $c_{t,i}$ migrate to the communities detected in the next step with a even distribution, i.e., when we have $\hat{\psi}_{i,j}=\frac{1}{m},j=1,2,\cdots,m$. And the minimum split index $\mathcal{I}^{\psi}_{c_{t,i}}=0$ is obtained when $m=1$ or all the members of $c_{t,i}$ who stay in the project migrate to a single community in the next step, i.e., there exists a $j^{\prime}$-th community in time $t+1$ that $\hat{\psi}_{i,j^{\prime}}=1$ and $\hat{\psi}_{i,j}=0,\forall j\neq j^{\prime}$, resulting in $\mathcal{H}_{c_{t,i}}=0$ and $\mathcal{I}^{\psi}_{c_{t,i}}=0$.

P-4. Given $m>1$ and $\eta_{i}$, the maximum shrink index $\mathcal{I}^{\eta}_{c_{t,i}}=\eta_{i}\mathcal{H}^{*}_{t\rightarrow t+1}$ is obtained when the corresponding split index is minimized, i.e., all stayed members of community $c_{t,i}$ migrate to a single community in the next step. And the minimum shrink index $\mathcal{I}^{\eta}_{c_{t,i}}=\eta_{i}^{2}\mathcal{H}^{*}_{t\rightarrow t+1}$ is obtained when the members of $c_{t,i}$ migrate evenly to communities in time $t+1$. For the special case of $m=1$, the shrink index is only determined by $\eta_{i}$.

Fig. 1 illustrates the curves of the split and shrink indices under different conditions, from which we can find correspondence to the above properties.

The proof of the above four properties are as follows.

When $\hat{\psi}_{i,j}=\frac{1}{m},j=1,2,\cdots,m$, we have $\mathcal{H}_{c_{t,i}}=-\sum_{j=1}^{m}\hat{\psi}_{i,j}\log_{2}(\hat{\psi}_{i,j})=-\log_{2}\frac{1}{m}$. As a result, we have

Substitute Eq. (2) into Eq. (1), we have

We can see from Eq. (3) that $\frac{\partial\mathcal{I}^{\psi}_{c_{t,i}}}{\partial m}>0$, for $m\geq 1$, and $0\leq\eta_{i}<1$.

As a result, the community split index $\mathcal{I}^{\psi}_{c_{t,i}}$ is a strictly monotonic increasing function of $m$, for $m\geq 1$, $0\leq\eta_{i}<1$, and $\hat{\psi}_{i,j}=\frac{1}{m},j=1,2,\cdots,m$.

For the case of $\eta_{i}=1$, we always have $\mathcal{I}^{\psi}_{c_{t,i}}=0,\forall m\geq 1$, and regardless the distribution of $\hat{\psi}_{i,j},j=1,2,\cdots,m$, which is intuitively correct that a community cannot split if all of its members leaves the project.

Next, for community shrink index $\mathcal{I}^{\eta}_{c_{t,i}}$, we also take its partial derivative with respect to $m$, which is

Since $\frac{\partial\mathcal{H}^{*}_{t\rightarrow t+1}}{\partial m}=\frac{\partial-\log_{2}(\frac{1}{m})}{\partial m}=\frac{1}{m\ln(2)}$, and $\frac{\partial\mathcal{I}^{\psi}_{c_{t,i}}}{\partial m}=\frac{1-\eta_{i}}{m\ln(2)}$ following Eq. (3), we have

From Eq. (5), we have $\frac{\partial\mathcal{I}^{\eta}_{c_{t,i}}}{\partial m}>0$ for $m\geq 1$, $0<\eta_{i}\leq 1$, and $\hat{\psi}_{i,j}=\frac{1}{m},j=1,2,\cdots,m$.

For the case of $\eta_{i}=0$, we always have $\mathcal{I}^{\eta}_{c_{t,i}}=0,\forall m\geq 1$, given any member migration distribution $\hat{\psi}_{i,j},j=1,2,\cdots,m$, which is reasonable because a community does not shrink if none of its members leave the project, regardless the number of communities detected in the next step (i.e., $m$).

Combining the above two results, we show that: $\mathcal{I}^{\psi}_{c_{t,i}}$ and $\mathcal{I}^{\eta}_{c_{t,i}}$ are strictly monotonic increasing functions of $m$, given $0<\eta_{i}<1$, and $\hat{\psi}_{i,j}=\frac{1}{m},j=1,2,\cdots,m$.

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As a result, we have $\frac{\partial\mathcal{I}^{\psi}_{c_{t,i}}}{\partial\eta_{i}}<0$, meaning that $\mathcal{I}^{\psi}_{c_{t,i}}$ is a strictly monotonic decreasing function of $\eta_{i}$, as long as $\mathcal{H}_{c_{t,i}}>0$.

We have $\mathcal{H}_{c_{t,i}}=0$ and thus $\mathcal{I}^{\psi}_{c_{t,i}}=0,\forall\eta_{i}\in[0,1]$ when only one of the $\hat{\psi}_{i,j}$’s is one with rest of the $\hat{\psi}_{i,j}$’s equal to zero (including the case that $m=1$). Intuitively, this case means that a community is not regarded as splitting if all of its remaining members migrate to a single community in the next step, regardless the amount of members who leave the project.

Next, we show that community shrink index $\mathcal{I}^{\eta}_{c_{t,i}}$ is a strictly monotonic increasing function of $\eta_{i}$ under the given conditions. Taking the partial derivative of $\mathcal{I}^{\eta}_{c_{t,i}}$ with respect to $\eta_{i}$ gives us

The forth term in Eq. (7) is $\eta_{i}\frac{\partial\mathcal{I}^{\psi}_{c_{t,i}}}{\partial\eta_{i}}=\eta_{i}\frac{\partial(1-\eta_{i})\mathcal{H}_{c_{t,i}}}{\partial\eta_{i}}=-\eta_{i}\mathcal{H}_{c_{t,i}}$. And the last term in Eq. (7) is $\eta_{i}\frac{\partial\sigma_{\eta_{i}}}{\partial\eta_{i}}=0.5\eta_{i}$ when $m=1$, and $\eta_{i}\frac{\partial\sigma_{\eta_{i}}}{\partial\eta_{i}}=0$ when $m>1$ following the definition of $\sigma_{\eta_{i}}$.

For the case of $m>1$, Eq. (7) can be written as

Because $\mathcal{H}^{*}_{t\rightarrow t+1}$ is the maximum entropy, we have $\mathcal{H}^{*}_{t\rightarrow t+1}>0$, $\mathcal{H}^{*}_{t\rightarrow t+1}\geq\mathcal{H}_{c_{t,i}}\geq 0$, and $(1-2\eta_{i})<1$ when $\eta_{i}>0$. We then have $\frac{\partial\mathcal{I}^{\eta}_{c_{t,i}}}{\partial\eta_{i}}>0$ for $\eta_{i}>0$ and $m>1$.

For the case of $m=1$, we have $\sigma_{\eta_{i}}=\eta_{i}\frac{\partial\sigma_{\eta_{i}}}{\partial\eta_{i}}=0.5\eta_{i}$, and $\mathcal{H}^{*}_{t\rightarrow t+1}=\mathcal{H}_{c_{t,i}}=0$. Eq. (7) then becomes

We also have $\frac{\partial\mathcal{I}^{\eta}_{c_{t,i}}}{\partial\eta_{i}}>0$ for $\eta_{i}>0$ and $m=1$. As a result, $\mathcal{I}^{\eta}_{c_{t,i}}$ is a strictly monotonic increasing function of $\eta_{i}$ for $\eta_{i}>0$, and $m\geq 1$.

Summarizing the above, we show that: $\mathcal{I}^{\psi}_{c_{t,i}}$ / $\mathcal{I}^{\eta}_{c_{t,i}}$ is a strictly monotonic decreasing / increasing function of $\eta_{i}$, respectively, for $\eta_{i}>0$, given $m>1$, and the distribution of member migration $\hat{\psi}_{i,j},j=1,2,\cdots,m$ with $\mathcal{H}_{c_{t,i}}>0$.

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In the above analysis we have $\sigma_{\eta_{i}}=0$ because we assume $m>1$. When $m=1$, we have $\sigma_{\eta_{i}}=0.5\eta_{i}$, and $\mathcal{H}^{*}_{t\rightarrow t+1}=\mathcal{H}_{c_{t,i}}=0$. The shrink index is $\mathcal{I}^{\eta}_{c_{t,i}}=0.5\eta_{i}^{2}$, which is only determined by $\eta_{i}$.

Following the above analysis, if we consider the change of $\eta_{i}$, we can find that the maximum and minimum possible value of the shrink index is $\mathcal{I}^{\eta}_{c_{t,i}}=-\log_{2}\frac{1}{m}$ when $\eta_{i}=1$, and $\mathcal{I}^{\eta}_{c_{t,i}}=0$ when $\eta_{i}=0$, respectively.

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From properties P-3 and P-4 we can further see that given $m>1$, the community split and shrink indices vary in the same range given by $\mathcal{I}^{\psi}_{c_{t,i}}\in[0,-\log_{2}\frac{1}{m}]$, and $\mathcal{I}^{\eta}_{c_{t,i}}\in[0,-\log_{2}\frac{1}{m}]$, with different values of $\eta_{i}$ and the distribution of member migration specified by $\hat{\psi}_{i,j},j=1,2,\cdots,m$. As a result, it is feasible for us to draw meaningful results, such as the community shows a stronger trend of splitting / shrinking, by directly comparing the values of community split and shrink indices.