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= M-Diagrams =

Introduction
M-Diagrams (also known as Martingale) can be used to compare the performance of two or more random processes over time, such as clocks or more complicated functions like centrality functions. M-Diagrams plot the difference between the expected value of the two processes at each time step, and the point at which the M-Diagram deviates from a straight line indicates the point at which the processes begin to diverge. This method can be used to compare the performance of clocks or other functions.

= Time Diagrams = Time diagrams are used to compare the performance of two or more clocks by plotting their inverse functions against each other. M-Diagrams are a useful tool for comparing clocks because they capture the variance of one clock with respect to another. To use M-Diagrams to compare clocks, the clocks must first be normalized and the first clock must be removed from the second clock. The x-coordinate in an M-Diagram represents the first clock, while the y-coordinate shows the difference between the two clocks.

Formally:

= Examples =

The Case of Felix Baumgartner
Felix Baumgartner set several world records on October 14, 2012 when he traveled to the stratosphere and jumped back down to Earth. His jump broke several skydiving records and lasted a total of 9 minutes and 9 seconds, 4 minutes and 19 seconds of which were in free fall. Baumgartner's jump could potentially save lives and money as a cheaper method for entering space. M-Diagrams can be used to analyze Baumgartner's experience of time during the jump, which can be seen to tick faster due to fear.

The next figure (Fig. 13.2) displays the timing of Baumgartner's jump from midnight to midnight. In Part A, the standard clock is shown in blue and Baumgartner's psychological clock during the jump is shown in red. The psychological clock slows from the time Baumgartner enters the balloon until he lands and returns to normal speed afterwards. Part B shows a normalized clock of Baumgartner's psychological clock. Both clocks start at 0 and end at 1. Part C shows the Martingale diagram comparing Baumgartner's psychological clock to the standard clock, with the maximum at the entry into the balloon and the minimum at the safe return to Earth. The second figure (Fig. 13.3) illustrates the time from midnight before Baumgartner's jump until he lands. The return to normalcy is not shown. In Part A, the standard clock is shown in blue and Baumgartner's psychological clock during the jump is shown in red. The psychological clock slows from the time Baumgartner enters the balloon until he lands and returns to normal speed afterwards. Part B shows a normalized clock of Baumgartner's psychological clock. Both clocks start at 0 and end at 1. Part C shows the M diagram comparing Baumgartner's psychological clock to the standard clock, with the minimum at the safe return to Earth.

In terms of the narrative, in Fig. 13.2, there is a return to normalcy following the action. The action builds, occurs, and life returns to normal. In Fig. 13.3, the narrative cuts right as the action ends. In the book "Analyzing Narratives in Social Networks" by Zvi Lokter, Lotker said "This is mirrored in many great works, such as Macbeth, Julius Caesar, and Romeo and Juliet. Action is more exciting than real life, which explains why authors frequently end narratives as action ceases".

= Uses of M-Diagrams =

M-Diagrams for General Functions
Martingale diagrams, which are useful for comparing monotonic increasing functions like clocks, can also be applied to general functions. The main advantage of using Martingale diagrams is that the y-coordinate rescales from measuring the range of the functions to the degree to which the two functions deviate from each other, which is a useful measurement when comparing two functions. To create an M-Diagram, Again, we rescale by linear operation the x- and y-coordinate for both functions and then take the difference.

= Conclusion = M-Diagrams are useful tools to use for comparing two clocks. M-Diagrams are an extremely powerful tool for analyzing performance texts. M-Diagrams are used heavily throughout the book "Analyzing Narratives in Social Networks" by Zvi Lokter.

= References =