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\emph{Implicit Social Network} is a connected social structure among a group of persons, where two of them are linked if they have some common interest. One real\mbox{-}life example of such networks is the implicit social network among the customers of an online commercial house, where there exists an edge between two customers if they like similar items. Such networks are often useful for different commercial applications such as \textit{target advertisement}, \textit{viral marketing}, etc. In this article, we study two fundamental problems in this direction. The first one is that, given the user\mbox{-}item rating data of an E\mbox{-}Commerce house, how we can design implicit social networks among its users and the second one is at the time of designing itself can we obtain the connectivity information among the users. Formally, we call the first problem as the \textsc{Implicit User Network Design} Problem and the second one as \textsc{Implicit User Network Design with Connectivity Checking} Problem. For the first problem, we propose three different algorithms, namely \emph{`Exhaustive Search Approach'}, \emph{`Clique Addition Approach'}, and \textit{`Matrix Multiplication\mbox{-}Based Approach'}. For the second problem, we propose two different approaches. The first one is the sequential approach: designing and then connectivity checking, and the other one is a concurrent approach, which is basically an incremental algorithm that performs designing and connectivity checking simultaneously. Proposed methodologies have experimented with three publicly available rating network datasets such as \emph{Flixter}, \textit{Movielens}, and \textit{Epinions}.

Designing and connectivity checking of implicit social networks from the user-item rating data