Discuss the balance theory and status theory of signed network.

 Collective Properties of Positive and Negative Network

As discussed in the previous subsection, distinct properties are observed for positive and negative links. When we consider positive and negative links together, they present collective properties, which can be explained by two important social theories in signed networks, i.e., balance theory and status theory. Here, we present these collective properties by introducing these two social theories, which have been proven to be very helpful in mining signed social networks. For example, the signed clustering coefficient and the relative signed clustering coefficient are defined, based on the intuition "the enemy of my enemy is my friend" implied by balance theory. Note that balance theory is developed for undirected signed social networks, whereas status theory is developed for directed signed social networks.

Balance Theory

Balance theory is originally introduced by Heider at the individual level and generalized by Cartwright and Harary in the graph-theoretical formation at the group level. When the signed network is not restricted to be complete, the network is balanced if all its cycles have an even number of negative links. Using this definition, it is proven that "a signed graph is balanced if and only if nodes can be separated into two mutually exclusive subsets such that each positive link joins two nodes of the same subset and each negative link joins nodes from different subsets." It is difficult to represent real-world signed networks by balanced structure. Therefore, Davis introduced the notion of a customizable graph - a signed graph is customizable if there exists a partition of the nodes such that nodes with positive links are in the same subset and nodes with negative links are between different subsets.

For a triad, four possible sign combinations exist as demonstrated in figure 8.3. Among these four combinations, A and C are balanced. The way to measure the balance of signed networks in social media is to examine all these triads and then to compute the ratio of A and Cover A, B, C, and D Existing work reported that triads in signed networks in social media are highly balanced. For example, it is found that the ratios of balanced triads of signed networks in Epinions, Slashdot, and Wikipedia are 0.941, 0.912, and 0.909, respectively, and more than 90% of triads are balanced in other social media datasets, Furthermore, the ratio of balanced triads increases while that of unbalanced triads decreases over time.

Status Theory

While balance theory is naturally defined for undirected networks, the status theory is relevant for directed networks. Social status can be represented in a variety of ways, such as the rankings of nodes in social networks, and it represents the prestige of nodes. In its most basic form, status theory suggests that u has a higher status than u, if there is a positive link from u to u or a negative link from u to

As shown in figure 8.4, there are two types of triads in directed networks, which correspond to acyclic and cyclic triads. Note that flipping the directions of all the links has no impact on the type of the cyclic triad. Since there are four possible sign combinations, there are four types of cyclic signed triads for T: as shown in figure 8.5. Each link in an acyclic triad can be positive or negative and the signs of links in an acyclic triad are not exchangeable; hence, there are eight types of acyclic signed triads as depicted in figure 8.6. Overall, there are 12 types of triads in directed signed networks.

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