Measures

The measures that have been implemented and presented in this tool rely on the mathematical formulas presented in this section. Some measures have been computed for the soccer datasets, while others have been implemented for the social network datasets. However, since both the datasets represent a network that evolves over time, it is possible to use all the formulas for both scenarios.

Heat maps

It is possible to compute several kinds of heat maps. An example is the radix tridimensional heatmap, where in the soccer dataset the peaks indicate an higher presence of players in the corresponding area:

Amortized degree

The amortized degree δ of a vertex i at the timestamp t_s is defined as:

For instance, in social network analysis, the GVR associated with amortized degree encodes which users are active and if their activity remains constant through time.

Activity peaks

also over a period of time:

In social network analysis, the activity peak measures the activity variation of a user, depending on the value of ρ or ρ_γ (a positive value denotes an increment in the user activity and viceversa).

Eccentricity

In soccer matches, it measures the accessibility degree of a level from the other vertices, for a given time frame. Eccentricity, therefore, describes the spread degree of the players on the soccer field and how central a player is if compared to the other players of the same team.

Density

For a specific moment:

and through a time interval:

where n_m is the number of edges and n_u is the number of vertices (thus, n_u ² is the maximum number of edges for a given timestamp)

Local clustering

For a specific moment:

and through a time interval:

The local clustering coefficient of a vertex in a directed graph represents the closeness of its neighbours to form a complete graph.