Define signed network (graph). Explain the properties of negative network.

 Definition of Signed Network

Given a set of nodes N= (n1,n2, nm) and a set of edges E = (e1,e2,,,en), where each edge is a pair of nodes, e_k = (ni, nj). A signed network/graph is a triple G₂ = <N, E, S> consisting of a set of nodes N, a set of edges E, and a mapping S which is a function S: E(+,-), ie, the mapping S associates with every edge aE either a positive valence, typically denoted by (+), or a negative valence, denoted by (-). The positive valence of an edge usually denotes the fact that the relationship modeled by the edge has some positive quality, such as kindness, friendship, or trust. Likewise, the negative valence represents. antagonizing feelings between nodes, such as enmity, dislike, or distrust. Edges can be lacking directional information; in such cases, the relationship is considered symmetrical. If edges are directional, such a network/graph is called a signed digraph. Some formulations also allow for the existence of multi edges, as well as half-edges (which are edges with only one endpoint) and loose edges (which are edges without any endpoints), but half-edges and lose edges are not signed. A complete signed network is a signed network in which each pair of nodes belongs to the set of edges.

Properties of Negative Network

There are some well-known properties of positive links such as power-law degree distributions, high clustering coefficient, high reciprocity, transitivity, and strong correlation with similarity. However, we cannot easily extend these properties of positive links to negative links. Here, we will review important properties of negative links in social media, which are analogous to those of positive links.

Power-law distributions: It is well known that the distributions of incoming or outgoing positive links for users usually follow power-law distributions - a few users te with large degrees while most users have small degrees. In incoming or outgoing negative links for each user are calculated and there are two important findings - (a) in a signed network, positive links are denser than negative links and there are many users without any incoming and outgoing negative links; and (b) for users with negative links, the degree distributions also follow power-law distributions - a few users have a large number of negative links, while most users have few negative links.

Clustering coefficient: Nodes in networks with positive links are often easy to cluster. This property is often reflected by their high clustering coefficients (CC). High values of CC are expected because of the inherently cohesive nature of positive links. However, the values of clustering coefficients for negative links are significantly lower than those for positive links. This suggests that many useful properties such as triadic closure cannot be applied to negative links. winya d blane hostin& Cas

Reciprocity: Positive links show high reciprocity. Networks with positive links are strongly reciprocal, which indicates that pairs of nodes tend to form bi-directional b connections, whereas networks with negative links show significantly lower He reciprocity. Asymmetry in negative links is confirmed in the correlations between the in- and out-degrees of nodes. In- and out-degrees of positive links are almost balanced, while negative links show an obvious suppression of such reciprocity.

Transitivity: Positive links show strong transitivity, which can be explained as "friends' friends are friends". The authors examined the transitivity of negative links on two social media signed networks Epinions and Slashdot and found that negative links maybe not be transitive since they observed both "enemies' enemies are friends" and "enemies' enemies are enemies".

Correlation with a similarity: Positive links have strong correlations with similarity, which can be explained by two important social theories, ie., homophily and social influence. Homophily suggests that users are likely to connect to other similar users, while social influence indicates that users' behaviors are likely to be influenced by their friends, Via analyzing two real-world signed social networks Epinions and Slashdot, it is found that users are likely to be more similar to users with negative links than those without any links, while users with positive links are likely to be more similar than those with negative links, These observations suggest that negative links in signed social networks may denote neither similarity nor dissimilarity. 

In addition, recent work conducted a comprehensive signed link analysis and found: (1) users with positive (negative) emotions are likely to establish positive (negative) links; (2) users are likely to like their friends' friends and dislike their friends' foes, and (3) users with higher optimism (pessimism) are more likely to create positive (negative) links.