Overlaying Social Networks of Different Perspectives for Inter-network Community Evolution

Abstract

In many real-life social networks, a group of individuals may be involved in multiple kinds of activities such as professional, leisure and friendship ones. Even though individuals may belong to a social network with a very precise type of links such as professional ties in LinkedIn, the interactions that may happen in other social networks such as Facebook are not reflected in the original network. We believe that overlaying networks with various types of links helps discover interesting patterns. The objective of this paper is then to overlay two or many social networks with different kinds of social activities in order to unveil homogeneous groups that could not appear in a unique social network. To that end, we propose a community detection approach based on possibility theory, which identifies time-based perspective communities for each kind of social activities that occur within a sequence of time windows. Furthermore, different perspectives are layered to detect communities that may belong to several networks in a given time period. Discovered communities in a given network for a time period can be perceived as views or perspectives in one or many networks.

Appendices

Appendix A: Inferring Possibility Distribution from Probability Distribution

A consistency principle between probability and possibility can be stated in a non-formal way [29]: “what is probable should be possible”. This requirement can then be translated via the inequality:

$$\begin{aligned} P(A)\le \varPi (A) \quad \forall A \subseteq \varOmega \end{aligned}$$

(7)

where $P$ and $\varPi$ are, respectively, a probability and a possibility measure on the domain $\varOmega$. In this case, $\varPi$ is considered as dominating $P$.

Transforming a probability measure into a possibilistic one then amounts to choosing a possibility measure in the set of possibility measures dominating $P$. This should be done, by adding a strong order preservation constraint, which ensures the preservation of the shape of the distribution:

$$\begin{aligned} p_i < p_j \Leftrightarrow \pi _i<\pi _j \quad \forall i,j \in \lbrace 1,\ldots ,q\rbrace , \end{aligned}$$

(8)

where $p_i=P(\lbrace E_{\omega _i} \rbrace )$ and $\pi _i=\varPi (\lbrace E_{\omega _i} \rbrace )$, $\forall i \in \lbrace 1,\ldots ,q\rbrace$. It is possible to search for the most specific possibility distribution verifying (7) and (8). The solution of this problem exists, is unique and can be described as follows. One can define a strict partial order $\mathsf P$ on $\varOmega$ represented by a set of compatible linear extensions $\varLambda (\mathsf P )=\lbrace l_u, u=1,L\rbrace$. To each possible linear order $l_u$, one can associate a permutation $\upsigma _u$ of the set $\lbrace 1,\ldots ,q \rbrace$ such that:

$$\begin{aligned} {\upsigma _{u}}(i)< \upsigma _{u}(j) \Leftrightarrow (\omega _{\upsigma _{u}(i)},\omega _{\upsigma _{u}(j)}) \in l_u, \end{aligned}$$

(9)

The most specific possibility distribution, compatible with the probability distribution $p = (p_1, p_2,\ldots , p_q)$ can then be obtained by taking the maximum over all possible permutations:

$$\begin{aligned} \pi _i=\displaystyle {\max _{u=1,L}}\displaystyle {\sum _{\lbrace j |\upsigma _{u}^{-1}(j)\le \upsigma _{u}^{-1}(i)\rbrace }}p_j \end{aligned}$$

(10)

The permutation $\upsigma$ is a bijection and the reverse transformation $\upsigma ^{-1}$ gives the rank of each $p_i$ in the list of the probabilities sorted in the ascending order. The number $L$ of permutations depends on the duplicated $p_i$ in $p$. It is equal to $1$ if there is no duplicate $p_i, \forall i$ and for this case $\mathsf P$ is a strict linear order on $\varOmega$.

Appendix B: Inferring Possibility Distribution for Classes of Activities

Let $\textit{n}_\textit{k}$ denote the number of observations (activities) of class $k$ in a sample of size $N$. Then, the random vector $n = (n_1, \ldots , \textit{n}_\textit{K})$ can be considered as a multinomial distribution with parameter $p = (p_{1}, p_{2},\ldots , p_{K})$. A confidence region for $p$ at level $1-\alpha$ can be computed using simultaneous confidence intervals as described in [19]. Such a confidence region can be considered as a set of probability distributions.

It is proposed to characterize the probabilities $p = (p_{1}, p_{2},\ldots , p_{K})$ of generating the different classes by simultaneous confidence intervals with a given confidence level $1 - \alpha$. Here, $p_{k}$ represents the probability of generating the class of events $A_{\omega _k}$. From this imprecise specification, a procedure for constructing a possibility distribution is described, insuring that the resulting possibility distribution will dominate the true probability distribution in at least $100(1 - \alpha )$ of the cases.

Since the probabilities $p$ of generating classes are unknown, we can build confidence intervals for each one of them. In interval estimation, a scalar population parameter is typically estimated as a range of possible values, namely a confidence interval, with a given confidence level $1 - \alpha$.

To build confidence intervals for multinomial proportions, it is possible to find simultaneous confidence intervals with a joint confidence level $1 - \alpha$. The method attempts to find a confidence region $\fancyscript{C}_n$ in the parameter space $p = (p_1,\ldots , \textit{p}_\textit{K}) \in [0; 1]^{K}| \displaystyle {\sum \nolimits _{i=1}^{K}} p_i=1$ as the Cartesian product of $K$ intervals $[p_1^{-} , p_1^{+}] \ldots [\textit{p}_\textit{K}^{-}, \textit{p}_\textit{K}^{+}]$ such that we can estimate the coverage probability with:

$$\begin{aligned} \mathbb {P}(p \in \fancyscript{C}_n)\ge 1-\alpha \end{aligned}$$

(11)

We can use the Goodman [9] formulation in a series of derivations to solve the problem of building the simultaneous confidence intervals.

$$\begin{aligned} A=\chi ^{2}(1-\alpha /K,1)+N \end{aligned}$$

(12)

where $\chi ^{2}(1-\alpha /K,1)$ denotes the quantile of order $1-\alpha /K$ of the chi-square distribution with one degree of freedom, and $N= \displaystyle {\sum \nolimits _{i=1}^{K}} n_i$, denotes the size of the sample. We have also the following quantities:

$$\begin{aligned} B_i=\chi ^{2}(1-\alpha /K,1)+2n_i, \end{aligned}$$

(13)

$$\begin{aligned} C_i=\frac{n_i^{2}}{N}, \end{aligned}$$

(14)

$$\begin{aligned} \varDelta _i=B_i^{2}-4AC_i, \end{aligned}$$

(15)

Finally, for each class of activities $A_{\omega _K}$ the bounds of the confidence intervals are defined as follows:

$$\begin{aligned}{}[p_i^{-}, p_i^{+}]=\left[ \frac{B_i-\varDelta _i^{\frac{1}{2}}}{2A},\frac{B_i+\varDelta _i^{\frac{1}{2}}}{2A}\right] \end{aligned}$$

(16)

It is now possible, based on these above interval-valued probabilities, to compute the most possibility distributions of a class dominating any particular probability measure. Let $\mathsf P$ denote the partial order induced by the intervals $[p_i]=[p_i^{-},p_i^{+}]$:

$$\begin{aligned} (\omega _i,\omega _j) \in \mathsf P \Leftrightarrow p_i^{+} <p_j^{-} \end{aligned}$$

(17)

This partial order may be represented by the set of its compatible linear extensions $\varLambda (\mathsf P )=\lbrace l_u, u=1,L\rbrace$, or equivalently, by the set of the corresponding permutations $\lbrace \upsigma _u, u=1,L\rbrace$. Then, for each possible permutation $\upsigma _u$ associated with each linear order in $\varLambda (\mathsf P )$, and each class $A_{\omega _i}$, we can solve the following linear program:

$$\begin{aligned} \pi _i^{\upsigma _u}=\displaystyle {\max _{p_1,\ldots , p_K}}\displaystyle {\sum _{\lbrace j |\upsigma _{u}^{-1}(j)\le \upsigma _{u}^{-1}(i)\rbrace }}p_j \end{aligned}$$

(18)

under the constraints:

$$\begin{aligned} \left\{ \begin{array}{lll} \displaystyle {\sum _{i=1}^{K}} p_i=1 \\ p_k^{-}\le p_k \le p_k^{+} \quad \forall k \in \lbrace 1,\ldots ,K \rbrace \\ p_{\upsigma _u(1)}\le p_{\upsigma _u(2)}\le \cdots \le p_{\upsigma _u(K)} \end{array} \right. \end{aligned}$$

(19)

Finally, we can take the distribution of the class $A_{\omega _k}$ dominating all the distributions $\pi ^{\upsigma _u}$:

$$\begin{aligned} \pi _i=\displaystyle {\max _{u=1, L}}\pi _i^{\upsigma _u} \quad \forall i \in \lbrace 1,\ldots , K \rbrace \end{aligned}$$

(20)

1.1 Complexity

The complexity of our computational procedure is related to the discover of the possibility degrees of the $K$ classes. To solve this problem, the conceptually simplest approach is to generate all the linear extensions compatible with the partial order induced by the probability intervals, and then to solve the associated linear programs (i.e. Eq. (10)). However, this approach is unfortunately limited to small values of $K$ (e.g., $K < 10$) due to the complexity of the algorithms generating linear extensions of $O(L)$, where $L$ is the number of linear extensions. Even for moderate values of $K$, $L$ can be very large ($K!$ in the worst case) and generating all the linear extensions and solving the linear programs soon becomes intractable. A new formulation of the solution can be derived to reduce considerably the computations. This formulation is based on several steps. First, all the linear programs to be solved will be grouped in different subsets; then, an analytic expression for the best solution in each subset will be given; and lastly, it will be shown that it is not necessary to evaluate the solution for every subset. A simple computational algorithm will be derived (see [19] for more details). The actual complexity might actually be close to $O(|P_i|)$ where $P_i$ denotes the set of indices of the classes with a rank possibly, but not necessarily smaller than $\omega _i$.