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= Direct Attack, Voronoi Partitions in social networks = In general a direct attack is a strategic move or action taken by one party to gain an advantage over the other while Voronoi partitions are a graphical representation of the regions in space that are closest to a given set of points.

Definition
Direct Attack - a direct attack in general is a strategic move or action taken by one party to gain an advantage over the other. It may involve the use of force or other forms of coercion to achieve a specific goal, such as capturing territory or defeating the enemy.

In the context of a Voronoi conflict function (It's a mathematical model that allows us to calculate the probability of success for a direct attack), a direct attack can be represented as a change in the state of the system, which is described by a set of variables in a mathematical model. For example, a direct attack may involve an increase in the military strength or resources of one party, which is reflected in the values of the variables in the model.

Direct Attack as Metric Space
In the book Analyzing Narratives in Social Networks (Lotker, 2021) the formally definition of direct attack is as follow-

A conflict function F describes a direct attack if for all metric space (V, dis) there exists a set of anchors A = {A1,..., Ak } ⊂ V                                                                                                       and a party member function Pm|A such that F (Pm|A ,[disi,j]) is the outcome of the direct conflict which is determined by the distances between the anchor sets A and the non-anchor sets V \ $$\bigcup_{Aj\in A}$$Aj. We denote the outcome of direct attack conflict functions as-

F (Pm|A ,[disi,j])DACF = PPF(ΨiVoroni, >)

Definition
Voronoi partition - Voronoi partitions, also known as Voronoi diagrams, are a graphical representation of the regions in space that are closest to a given set of points. They are named after the mathematician Georgy Voronoi, who developed the concept in the late 19th century.

A voronoi partition consists of a set of points, called seeds, and a corresponding set of regions, called cells, that are defined based on the distances between the seeds and other points in space. Each cell consists of all points that are closer to its corresponding seed than to any other seed. Voronoi partitions are used in a variety of applications, including image processing, computer graphics, and spatial analysis. They are particularly useful for visualizing the distribution of points in space and for analyzing the relationships between different objects or locations.

In a social network, a Voronoi partition can be used to represent the relationships between individuals or groups. For example, if a social network is represented as a graph, with nodes representing individuals and edges representing relationships between them, a Voronoi partition can be used to divide the graph into regions based on the relationships between the nodes.

Each region in the Voronoi partition corresponds to a node in the graph, and consists of all nodes that are closer to that node than to any other node. In this way, the Voronoi partition can be used to visualize the structure of the social network and the relationships between different individuals or groups.

One application of Voronoi partitions in social networks is to study the influence or centrality of different individuals or groups within the network. For example, if a node has a large and densely connected region in the Voronoi partition, it may be considered to be highly influential or central within the network.

Voronoi Partition as a Greedy Algorithm
"It turns out that there is a connection between Voronoi partitions and greediness. In particular, it is possible to compute Voronoi partitions using minimum spanning trees".

In the book Analyzing Narratives in Social Networks (Lotker, 2021) which is quoted above, the writer describes an algorithm that demonstrates how to compute the Voronoi partition using minimum spanning tree.

First we need to use Prim's algorithm in order to find the minimum spanning tree, and then we use the algorithm decribed in the book for finding the Voronoi Graph when given a metric space. For more details it's recommended to read the book mentioned.

Mathematical Definition of Voronoi Partitions
In the book Analyzing Narratives in Social Networks (Lotker, 2021) the mathematical definition is as follow-

Voronoi partitions contain sites (also called anchors in our context) which are a set of k points in some metric space (V, d), i.e., Q = (p1, p2,... pk ) where pi ∈ V for all i = 1,..., k.             The Voronoi partition divides the metric space V into k + 1 parts. For each of the points pi, the points receive the set ρi ⊂ M of all nearest points x to pi. Formally ρi = {x ∈ M : dis(pi, x) < dis(pj, x), j $$\neq$$ i}. The remaining points which are at minimum equal distance to more than one point are ρk+1 = V \ $$\bigcup_{j=1}^k$$ρj.