Graphs

Graphs are one of the most common data structures in computer science; graph-based modeling of problems is at the heart of many systems we use every day, such as routing in Google Maps or recommending friends on Facebook.

Social Network Analysis (SNA) is a branch of sociology that explores social structures through the use of analytical tools, such as graphs. In this assignment, you will implement basic social network analysis algorithms on a graph extracted from GitHub.

You can find the graph data at this link. The data looks like this (CSV format):

follower,followed
1570,9236
9236,1570
13256,9236
9236,13256
13256,1570
1570,13256

If we take the first line, it means that user 1570 follows user 9236.

Loading the graph

T (10 points): Write a function that takes as input a file name and loads the data into an adjacency list representation.

def load(graph_file='github.csv'):
    """
    Loads the data from the file in the provided argument into an in-memory
    graph (as an adjacency list)
      """
    graph_data = open(graph_file)
    
    graph = list()
    
    for line in graph_data:
        pair = line.split(',')
        
        existing_node = [(node, neighbours) for (node, neighbours) in graph if node == int(pair[0].strip())]
        if len(existing_node) == 1:
            existing_node[0][1].append(int(pair[1].strip()))
        else:
            graph.append((int(pair[0].strip()), [int(pair[1].strip())]))
            
        existing_node = [(node, neighbours) for (node, neighbours) in graph if node == int(pair[1].strip())]
        if len(existing_node) == 1:
            existing_node[0][1].append(int(pair[0].strip()))
        else:
            graph.append((int(pair[1].strip()), [int(pair[0].strip())]))
    
    return sorted(graph)

Test your implementation:

graph = load()
for node in graph:
    print(node)

(92, [34575, 2907, 59570, 34655, 4140, 24553, 59570, 1567, 4140, 1569, 150, 17888, 10967, 465, 7580, 26457, 5810])
(93, [2907, 2812, 6145, 44451, 3029, 2618])
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(344, [2907, 7580, 61272, 61272, 35469, 35469, 6569, 187852])
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(416, [2653, 24452, 2653])
(418, [7182, 7182, 2983, 28117, 28117, 5274, 2983, 17800, 28328, 18567, 6145, 13803, 91474, 193606, 17800, 7580, 28328, 47427, 13803, 2560])
(431, [160992])
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(1601, [4140])
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(12549692, [9236, 1570, 13256, 1563, 1564, 7547, 7547, 93466, 93466, 72824])

From now on, you must use the graph returned by load in all the assignments below.

Basic graph metrics

T (10 points): Who are the 10 most connected users?

def in_degree(graph, node_id):
    count = 0
    for (node, neighbours) in graph:
        for outgoing in neighbours:
            if outgoing == node_id:
                count = count + 1
            
    return count

def most_connected(graph, n = 10):
    """Returns the ids of the top-n most connected users"""
    most_connected = list()
    
    for (node, neighbours) in graph:
        connectedness = in_degree(graph, node)
#         connectedness += len(neighbours)
        
        if len(most_connected) < n:
            most_connected.append((connectedness, node))
            
        
        elif connectedness > most_connected[0][0]:
            del(most_connected[0])
            most_connected.append((connectedness, node))
            
        most_connected.sort()
    
    return [(node, connectedness) for (connectedness, node) in most_connected]
    

graph = load()
n = len(graph)
most_connected_nodes = most_connected(graph, n)

total_sum = 0
for (node, connectedness) in most_connected_nodes:
    total_sum += connectedness

    
print('mean: ', total_sum / n)
median = (most_connected_nodes[int(n/2)][1] + most_connected_nodes[int(n/2+1)][1]) / 2
print('median: ', median) 

Hint: it helps if you first define a method called in_degree that calculates the number of incoming edges in to a node.

T (10 points): What is the mean and what is the median number of connections?

mean: 7.060330578512397 median: 2.0

Computing shortest paths

Shortest paths are the basis for many network measures. You will need to implement Dijkstra algorithm.

T (20 points): Write a function that computes the shortest paths between all node pairs in the graph.

Hint: Choose the appropriate shortest path algorithm for undirected graph with no edge weights. A pair of nodes $(n_1, n_2)$ is, for our purposes, equivalent to the pair $(n_2, n_1)$.

Hint: How to find all unique node pairs? Given that you create a non-duplicate list of all your nodes, you can use Python's itertools.combinations function like so:

{python}
from itertools import combinations
a = [1,2,3,4,5]
pairs = list(combinations(a, 2))
print pairs

def shortest_path(graph, source):
    graph_dict = dict(graph)
    
    distances = dict([(node, float('inf')) for (node, neighbours) in graph])
    prev = dict()
    distances[source] = 0
    
    Q = dict(distances)
    
    while len(Q) != 0:
        u = min(Q, key=Q.get)
        Q.pop(u)
        
        for v in graph_dict[u]:
            alt = distances[u] + 1
            
            if alt < distances[v]:
                distances[v] = alt
                prev[v] = u
    
    paths = dict([(node, None) for (node, neightbours) in graph])
    for (node, _) in paths.items():
        path = list()
        current = node
        if prev.get(current, None) != None:
            while current != source:
                path.insert(0, current)
                current = prev[current] 
            path.insert(0,source)
        
        paths[node] = path
    
    paths_distances = dict()
    for (x,y) in distances.items():
        paths_distances[(source, x)] = (y, paths.get(x, None))
        
    return paths_distances

def shortest_paths(graph):
    """
    Computes the shortest paths between any pair of nodes in the graph
  
    @return A dictionary whose keys are node pairs and values are sequences
    indicating the shortest path between the node pair.
    """

    shortest_paths = dict()
    count = 0
    for (node, neighbours) in graph:
        
        for key, value in shortest_path(graph, node).items():
            shortest_paths[key] = value
            
        count += 1
        if count % 100 == 0:
                print(count)
        
    return shortest_paths

shortest_paths = shortest_paths(load())

count = 0
for key, value in shortest_paths.items():
    count += 1
    if count > 100:
        break
    

100
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800
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1500

Ranking important users

One of the primary uses of SNA is to identify important/influencial nodes. A typical metric we use to quantify the importance of a node is centrality. Several centrality measures exist; for our purposes it is enough to calculate the Betweeness Centrality of each node. The pseudocode to calculate it is given below.

To compute the betweenness of a node $n$

1. For each pair of nodes $(v1, v2)$, compute the shortest paths between them
2. For each pair of nodes $(v1, v2)$ determine the fraction of shortest paths that include $n$
3. Sum this fraction over all pairs of vertices $(v1, v2)$

T (30 points): Write a function that computes the Betweenness centrality for all nodes in the provided network

def betweenness(graph):
  
    betweenness = dict()
    paths = shortest_paths
    
    amount_of_paths = 0
    for ((_, _),(_,path)) in paths.items():
        if len(path) != 0:
            amount_of_paths += 1
    
    progress = 0
    for (node, _) in graph:
        
        if progress % 100 == 0:
            print(100*progress/len(graph),'%')
        
        progress += 1
        count = 0
        for ((source, end),(length,path)) in paths.items():
            if node in path:
                count += 1
        betweenness[node] = count/amount_of_paths
        
    return betweenness

T (10 points): Use the function above to rank the nodes (users) in terms of importance and print 10 most important users with their importance.

importance = betweenness(load()).items()
importance2 = sorted(importance, key=lambda x: x[1], reverse=True)

print(importance2[:10])

[(7580, 0.13881338073376218), (1570, 0.11327569872098532), (1563, 0.07304915949831103), (465, 0.07186377984478277), (2907, 0.055715057309951875), (1063, 0.05350699717102671), (3892, 0.0526121517463044), (24452, 0.04830228398641739), (1564, 0.04587286443506864), (2556, 0.04354416341637263)]