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Link Streams as Evolving Social Networks
In social network analysis (SNA), a link stream is a special type of evolving social network that is defined over a discrete set of times $$T$$, which is a subset of the set of real numbers $$\mathbb{R}$$. Link streams are characterized by the fact that their codomain is always a simple, single directed or undirected edge, denoted by $$G = E $$.

Link streams can be thought of as the basis of all evolving social networks, since they can be accumulated to create more complex graphs. They are also the simplest type of evolving social network, since they do not require the measurable and bounded function conditions that are present in more general evolving social networks.

Definition
Formally, a link stream $$\gamma_L$$ is defined as a function $$\gamma_L:T\rightarrow E$$, where $$T$$ is a discrete subset of the set of real numbers $$\mathbb{R}$$ and $$E $$ represents the set of edges in the link stream. The function $$\gamma_L$$ maps each element in the set of times $$T$$ to a unique edge in the set $$E $$.

Example
One example of a link stream is the evolving social network of the first scene of Shakespeare's play "Macbeth".

In this scene, three witches meet on a heath in Scotland during a thunderstorm. They discuss their plans to meet again at a later time, when they will encounter Macbeth and Banquo, two generals who are returning from a battle. The witches prophesize that Macbeth will become the Thane of Cawdor and eventually the King of Scotland, and that Banquo will be the father of future kings, though he will not himself be king. The witches then depart, leaving the audience to anticipate the events of the rest of the play.

In this case, the graph is defined over a set of vertices V={All, 1st Witch, 2nd Witch, 3rd Witch}. Therefore, the link stream is a function $$N:V\rightarrow\{1,2,3,4\} $$ that assigns each vertex in V a unique integer value, with $$V(All)=1$$, $$V(1st Witch)=2$$, etc.

Next, we can define the link stream $$\gamma_L$$ as a function $$\gamma_L:T\rightarrow E$$, where E represents the set of edges in the link stream. For example, if the set of edges $$E $$ is $$E=\{(1,2),(2,3),(3,4)\}$$, then $$\gamma_L(1)=(1,2)$$, $$\gamma_L(2)=(2,3)$$, and $$\gamma_L(3)=(3,4)$$ This means that at time 1, the link stream consists of an edge from node 1 (representing the character "All") to node 2 (representing the character "1st Witch"), at time 2 the link stream consists of an edge from node 2 to node 3 (representing the character "2nd Witch"), and at time 3 the link stream consists of an edge from node 3 to node 4 (representing the character "3rd Witch").

Analyzing Narratives in Social Networks. (2022). Retrieved from https://link.springer.com/book/10.1007/978-3-030-68299-6