COMP2521: Assignment 2
Social Network Analysis
A notice on the class web page will be posted after each major revision. Please check the class notice board and this assignment page
frequently (for Change Log). The specification may change.
FAQ:
You should check Ass2 FAQ, it may offer answers to your queries!
Change log:
[5pm Tue 03/Nov] Wasserman and Faust formula is useful for graphs with more than one connected components. However, if a node is not
connected to any other node (isolated), its closeness value should be zero (0).
Objectives
to implement graph based data analysis functions (ADTs) to mine a given social network.
to give you further practice with C and data structures (Graph ADT)
Admin
Marks 20 marks
Individual Assignment This assignment is an individual assignment.
Due 05:00pm Friday, 20 November, 2020 (Friday of Week-10)
Late
Penalty
2 marks per day off the ceiling.
Last day to submit this assignment is 8pm Monday 23 November 2020, of course with late penalty.
Submit TBA
Aim
In this assignment, your task is to implement graph based data analysis functions (ADTs) to mine a given social network. For example, detect
say "influenciers", "followers", "communities", etc. in a given social network. You should start by reading the Wikipedia entries on these topics.
Later I will also discuss these topics in the lecture.
Social network analysis
Centrality
The main focus of this assignment is to calculate measures that could identify say "influenciers", "followers", etc., and also discover possible
"communities" in a given social network.
Dos and Don'ts !
Please note that,
For this assignmet you can use source code that is available as part of the course material (lectures, exercises, tutes and labs). However,
you must properly acknowledge it in your solution.
All the required code for each part must be in the respective *.c file.
You may implement additional helper functions in your files, please declare them as "static" functions.
After implementing Dijkstra.h, you can use this ADT for other tasks in the assignment. However, please note that for our testing, we will
use/supply our implementation of Dijkstra.h. So your programs MUST NOT use any implementation related information that is not available
in the respective header files (*.h files). In other words, you can only use information available in the corresponding *.h files.
Your program must not have any "main" function in any of the submitted files.
Do not submit any other files. For example, you do not need to submit your modified test files or *.h files.
If you have not implemented any part, must still submit an empty file with the corresponding file name.
.
Provided Files
We are providing implementations of Graph.h and PQ.h . You can use them to implement all three parts. However, your programs MUST NOT
use any implementation related information that is not available in the respective header files (*.h files). In other words, you can only use
information available in the corresponding *.h files.
Also note:
all edge weights will be greater than zero.
we will not be testing reflexive and/or self-loop edges.
we will not be testing the case where the same edge is inserted twice.
Download files:
Ass2_files.zip
Ass2_Testing.zip
Part-1: Dijkstra's algorithm
In order to discover say "influencers", we need to repeatedly find shortest paths between all pairs of nodes. In this section, you need to
implement Dijkstra's algorithm to discover shortest paths from a given source to all other nodes in the graph. The function offers one important
additional feature, the function keeps track of multiple predecessors for a node on shortest paths from the source, if they exist. In the
following example, while discovering shortest paths from source node '0', we discovered that there are two possible shortests paths from node '0'
to node '1' (0->1 OR 0->2->1), so node '1' has two possible predecessors (node '0' or node '2') on possible shortest paths, as shown below.
We will discuss this point in detail in a lecture. The basic idea is, the array of lists ("pred") keeps one linked list per node, and stores multiple
predecessors (if they exist) for that node on shortest paths from a given source. In other words, for a given source, each linked list in "pred"
offers possible predecessors for the corresponding node.
Node 0
Distance
0 : X
1 : 2
2 : 1
Preds
0 : NULL
1 : [0]->[2]->NULL
2 : [0]->NULL

Node 1
Distance
0 : 2
1 : X
2 : 3
Preds
0 : [1]->NULL
1 : NULL
2 : [0]->NULL

Node 2
Distance
0 : 3
1 : 1
2 : X
Preds
0 : [1]->NULL
1 : [2]->NULL
2 : NULL

The function returns 'ShortestPaths' structure with the required information (i.e. 'distance' array, 'predecessor' arrays, source and no_of_nodes in
the graph)
Your task: In this section, you need to implement the following file:
Dijkstra.c that implements all the functions defined in Dijkstra.h.
Part-2: Centrality Measures for Social Network Analysis
Centrality measures play very important role in analysing a social network. For example, nodes with higher "betweenness" measure often
correspond to "influencers" in the given social network. In this part you will implement two well known centrality measures for a given directed
weighted graph.
Descriptions of some of the following items are from Wikipedia at Centrality, adapted for this assignment.
Closeness Centrality
Closeness centrality (or closeness) of a node is calculated as the sum of the length of the shortest paths between the node ( ) and all other
nodes ( ) in the graph. Generally closeness is defined as below,
where is the shortest distance between vertices and .
However, considering we may have more than one connected components, for this assignment you need to use Wasserman and Faust
formula to calculate closeness of a node in a directed graph as described below:
where is the shortest-path distance in a directed graph from vertex to , is the number of nodes that can reach, and denote the number
of nodes in the graph.
Wasserman and Faust formula is useful for graphs with more than one connected components. However, if a node is not connected to any other
node (isolated), its closeness value should be zero (0).
For further explanations, please read the following document, it may answer many of your questions!
Explanations for Part-2
Based on the above, the more central a node is, the closer it is to all other nodes. For for information, see Wikipedia entry on Closeness
centrality.
Betweenness Centrality
The betweenness centrality of a node is given by the expression:
where is the total number of shortest paths from node to node and is the number of those paths that pass through .
For this assignment, use the following approach to calculate normalised betweenness centrality. It is easier! and also avoids zero as
denominator (for n>2).
where, represents the number of nodes in the graph.
For further explanations, please read the following document, it may answer many of your questions!
Explanations for Part-2
Your task: In this section, you need to implement the following file:
CentralityMeasures.c that implements all the functions defined in CentralityMeasures.h.
For more information, see Wikipedia entry on Betweenness centrality
Part-3: Discovering Community
In this part you need to implement the Hierarchical Agglomerative Clustering (HAC) algorithm to discover communities in a given graph. In
particular, you need to implement Lance-Williams algorithm, as described below. In the lecture we will discuss how this algorithm works, and
what you need to do to implement it. You may find the following document/video useful for this part:
Hierarchical Clustering (Wikipedia), for this assignment we are interested in only "agglomerative" approach.
Brief overview of algorithms for hierarchical clustering, including Lance-Williams approach (pdf file).
Three videos by Victor Lavrenko, watch in sequence!
Agglomerative Clustering: how it works
Hierarchical Clustering 3: single-link vs. complete-link
Hierarchical Clustering 4: the Lance-Williams algorithm
Distance measure: For this assignment, we calculate distance between a pair of vertices as follow: Let represents maximum edge weight of
all available weighted edges between a pair of vertices and . Distance between vertices and is defined as . If and are not connected,
is infinite.
For example, if there is one directed link between and with weight , the distance between them is . If there are two links, between and w,
we take maximum of the two weights and the distance between them is . Please note that, one can also consider alternative
approaches, like take average, min, etc. However, we need to pick one approach for this assignment and we will use the above distance
measure.
You need to use the following (adapted) Lance-Williams HAC Algorithm to derive a dendrogram:
Calculate distances between each pair of vertices as described above.
Create clusters for every vertex , say .
Let represents the distance between cluster and , initially it represents distance between vertex and .
For k = 1 to N-1
- Find two closest clusters, say and . If there are multiple alternatives, you can select any one of the pairs of closest
clusters.
- Remove clusters and from the collection of clusters and add a new cluster (with all vertices in and ) to the
collection of clusters.
- Update dendrogram.
- Update distances, say , between the newly added cluster and the rest of the clusters ( ) in the collection using
Lance-Williams formula using the selected method ('Single linkage' or 'Complete linkage' - see below).
End For
Return dendrogram
Lance-Williams formula:
where , , , and define the agglomerative criterion.
For the Single link method, these values are: , , , and . Using these values, the formula for Single link method is:
We can simplify the above and re-write the formula for Single link method as below
For the Complete link method, the values are: , , , and . Using these values, the formula for Complete link method is:
We can simplify the above and re-write the formula for Complete link method as below
Please see the following simple example, it may answer many of your questions!
Part-3 Simple Example (MS Excel file)
Your task: In this section, you need to implement the following file:
LanceWilliamsHAC.c that implements all the functions defined in LanceWilliamsHAC.h.
Assessment Criteria
Part-1: Dijkstra's algorithm (20% marks)
Part-2:
Closeness Centrality (22% marks),
Betweenness Centrality (23% marks)
Part-3: Discovering Community (15% marks)
Style, Comments and Complexity: 20%
Testing
Please note that testing an API implementation is very important and crucial part of designing and implementing an API. We offer the
following testing interfaces (for all the APIs you need to implement) for you to get started, however note that they only test basic cases.
Importantly,
you need to add more advanced test cases and properly test your API implementations,
the auto-marking program will use more advanced test cases that are not included in the test cases provided to you.
Instructions on how to test your API implementations are available on the following page:
Testing your API Implementations
Submission
You need to submit the following five files:
Dijkstra.c
CentralityMeasures.c
LanceWilliamsHAC.c
Submission instructions on how to submit the above five files will be available later.
As mentioned earlier, please note that,
all the requried code for each part must be in the respective *.c file.
you may implement addition helper functions in your files, please declare them as "static" functions.
after implementing Dijkstra.h, you can use this ADT for the other tasks in the assignment. However, please note that for our testing, we will
use/supply our implementations of Graph.h, PQ.h and Dijkstra.h. So your programs MUST NOT use any implementation related
information that is not available in the respective header files (*.h files). In other words, you can only use information available in the
corresponding *.h files.
your program must not have any "main" function in any of the submitted files.
do not submit any other files. For example, you do not need to submit your modified test files or *.h files.
If you have not implemented any part, you need to still submit an empty file with the corresponding file name.
.
Plagiarism
This is an individual assignment. Each student will have to develop their own solution without help from other people. You are not permitted to
exchange code or pseudocode. If you have questions about the assignment, ask your tutor. All work submitted for assessment must be entirely
your own work. We regard unacknowledged copying of material, in whole or part, as an extremely serious offence. For further information, read
the Student Conduct.
-- end --
CWF
x
y ∈ V ∧ y ≠ x
C(x) = .
1
∑y d(y, x)
d(y, x) x y
CWF (u) = ∗ .
n− 1
N − 1
n− 1
∑all v reachable from u d(u, v)
d(u, v) u v n u N
CWF
v
g(v) = ∑
s≠v≠t
σst(v)
σst
σst s t σst(v) v
normal(g(v)) = ∗ g(v)
1
((n− 1)(n− 2))
n
wt
v w d v w d = 1/wt v w
d
v w wt 1/wt v
1/max(wtvw,wtwv)
i ci
Dist(ci, cj) ci cj i j
ci cj
ci cj cij ci cj
Dist(cij, ck) cij ck
Dist(cij, ck) = αi ∗Dist(ci, ck) + αj ∗Dist(cj, ck) + β ∗Dist(ci, cj) + γ ∗ abs(Dist(ci, ck) −Dist(cj, ck))
αi αj β γ
αi = 1/2 αj = 1/2 β = 0 γ = −1/2
Dist(cij, ck) = 1/2 ∗Dist(ci, ck) + 1/2 ∗Dist(cj, ck)− 1/2 ∗ abs(Dist(ci, ck)−Dist(cj, ck))
Dist(cij, ck) = min(Dist(ci, ck),Dist(cj, ck))
αi = 1/2 αj = 1/2 β = 0 γ = 1/2
Dist(cij, ck) = 1/2 ∗Dist(ci, ck) + 1/2 ∗Dist(cj, ck) + 1/2 ∗ abs(Dist(ci, ck)−Dist(cj, ck))
Dist(cij, ck) = max(Dist(ci, ck),Dist(cj, ck))

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