Social Graphs – An Application of Graphs

1. Specification

The concept of Social Graph was introduced by M. Zuckerberg at the first Facebook F8 conference in 2007 when he introduced the Facebook platform to model relationships among internet users

Part 1

Consider the attached file Graph.java which defines a generic undirected Graph class. Design and implement a class SocialGraph (in a separate source file SocialGraph.java) which extends the class Graph. In a social graph, the vertices (graph nodes) represent people names while the un-directed edges represent the acquaintance relationships between them. Class SocialGraph should define constructor(s) and enhance the behavior of the class Graph by defining the following social graphs’ specific methods

normalizedDegreeOfCentrality calculates and returns the normalized degree of centrality for a given

vertex v. The required value is calculated as:

degree(v) / (n-1)

where degree(v) represents the number of vertex incident edges and n represents the number of graph vertices.

For social graphs, a high degree of centrality for a person v reflects his/her dominant position in the group or

his/her social interaction skills.

numberOfTrianglesIncidentToVertex, calculates and returns the number of triangles incident to vertex v. The

algorithm below calculates the number of triangles incident for all graph vertices (V is the set of vertices, E is the

set of edges):

foreach v in V

foreach pair of vertices (p, q) in AdjacencyList(v)

if (p, q) is in E then add 1 to triangles[v]

listOfTrianglesIncidentToVertex calculates and returns the list of triangles incident to vertex v.

A triangle should be specified by its vertices.

clusterIndividual for a given vertex v, calculates and returns the percentage indicating how close its neighbors

are to make a complete graph and is calculated as:

[(number of edges connecting v’s neighbor vertices) / (number of edges, potentially connecting v’s neighbor

vertices)] * 100

where:

– the number of edges connecting v’s neighbor vertices is calculated as:

(number of triangles incident to vertex v)

– the number of edges, potentially connecting v’s neighbor vertices is calculated as:

[degree(v) * (degree(v) – 1)] / 2

This value measures how close wrapped are the persons in the social graph around the given person.

averageClustering for the social graph is calculated as (the sum applies to all vertices v in V):

(1 / n) * ∑ clusterIndividual (v)

This value indicates the overall density of the social graph.

isAcquaintance determines whether two persons supplied as parameters can establish social contact direct or

through a chain of transitive acquaintance relationships (in terms of graphs it means that there is a path

between the two nodes representing the two persons).

Part 2

Design and implement a driver program TestSocialGraph (in a separate source file TestSocialGraph.java) for testing the methods implemented in Part 1. The driver program should build a social graph from an input file data.txt. After building the social graph, in a loop, the program should invite the user to select for execution one of the following operations: (1) normalizedDegreeOfCentrality, (2) numberOfTrianglesIncidentToVertex, (3) listOfTrianglesIncidentToVertex, (4) clusterIndividual, (5) averageClustering, (6) isIndirectAcquaintance, (7) addVertex, (8) addEdge, (9) printEdges and (0) exit the loop and the program. As a result of each operation execution, relevant information should be displayed to the user. For example, as a result of invoking clusterIndividual method, the cluster individual value for the given person should be displayed.

An example of the data input file content, its format and the corresponding social graph layout is shown in the attached file SocialGraph_Example.pdf.

The programs should compile without errors.

Notes.

1. You may consider that there are no errors in the input file structure.

2. If an operation requires additional information, the user will be prompted to enter it.

3. The input file (a simple .txt file) should be generated by the students using a simple text editor such as Notepad.

4. Person names (instead of indices) should be used in I/O operations involved by the user interface