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= Frequency graphs = Frequency graphs are a type of graph that is used to represent the frequency of occurrences of certain events or data within a given sample or population. In the context of social networks, frequency graphs are often used to visualize and analyze patterns of communication, interaction, and relationships among individuals or groups within a social network.

Frequency graphs can be useful in a variety of contexts, including marketing research, political analysis, and social network analysis. By visualizing the frequency of different types of interactions or relationships, researchers and analysts can gain insights into patterns of behavior and communication within a social network.

In social networks frequency graphs help the analyzer by visualizing the data of the network and allowing the analyzer to have more possibilities for further analyzations.

For example, a frequency graph might reveal that a particular group of individuals is more likely to engage in certain types of interactions, or that certain types of interactions are more common among certain groups, or to indicate the main characters in a group by visualizing the number of people they have spoken to and the frequency they spoke to other characters.

Definition
A weighted graph $$G = (V, E, W_F)$$ with n nodes is called a frequency graph

if the weighted adjacency matrix is $$W_f = [f_{i,j}] \text{ for all } i,j\in \{1,2,..,n\}\text{ }, v_{i},v_{j}\in V $$:

$$f_{i,j} = \begin{Bmatrix} \geq1 & \text{ if } (v_{i}, v_{j}) \in E \\ = 0 & \text{ else } \end{Bmatrix}$$

Kinds of frequency graphs
There are many kinds of frequency graphs in the context of social networks, such as:

Who spoke to whom graphs (WW graphs), AB graphs, ABA graphs and etc.

The who spoke to whom graph is a graph that each vertex is a character and the graph connects two characters that one spoke to the other in chronological order. The AB graphs are approximations of the WW, the approximation is done by connecting to characters in the graph only if they spoke one after the other.

And the ABA graphs are another approximation that connects two characters only if they had long conversations (the sequence of speakers must contain the two characters more for more than just one time of one of them speaks and the other response).

A way to create frequency graphs
Possible way to create frequency graphs are scripts. Scripts provide us with a natural way to mechanically represent text, since we can easily compute the frequency of pair of characters to speak one after the other.

In this case – the graphs are frequency graphs since the weights measure the frequency of communication (i.e., a high number means two characters speak to one another more frequently). From scripts we can derive all kinds of frequency graphs as described above and to use them in order to analyze the script and its subtext.

Identification of main character of a story
Determining the main characters of a story can be challenging, even for humans.

The frequency graphs visualize the relations between characters, but they also make it very simple to use the relations in the graph and use simple and existing algorithm to identify the main character.

In order to determine the main character mechanically, we can leverage established social network algorithms to locate the central node in a network on the AB graph.

For example, take the Degree centrality and PageRank centrality algorithms. These two algorithms were chosen since they are the most simple and useful of the centrality algorithms.

In general, degree centrality means that we can use the number of edges that are coming in to the vertex and the edges that are going out of the vertex to identify that the vertex with most edges coming in and out of him could be with good approximation the main character.

PageRank centrality uses a bit more complicated method, but at the end returns the main character according to its method.

Both of these algorithms operate on frequency graphs and are based on the way a frequency graph is defined.