Emergence of connectedness in Social Networks
Prerequisite: Basics of NetworkX
The emergence of connectedness is to check whether the graph is connected or not. It says that in a graph of N nodes, we just need NLogN edges to make graph connected.
Approach:
The following algorithm to be followed:
- Take any random graph with N number of nodes.
- Now choose any node as a starting node.
- Now add random edges to the graph and check every time if the graph is connected or not.
- If the graph is not connected then repeat steps 2 and 3.
- If the graph is connected then check the number of edges that are added and this will be equal to NLogN.
- Now check both the plots will be almost similar.
Code for Checking connectedness:
Python3
import networkx as nximport matplotlib.pyplot as pltimport random # Add N number of nodes in graph g # and return the graphdef add_nodes(N): g = nx.Graph() g.add_nodes_from(range(N)) return g # Add 1 random edgedef add_random_edge(g): z1 = random.choice(g.nodes()) z2 = random.choice(g.nodes()) if z1 != z2: g.add_edge(z1, z2) return g # Continue adding edges in graph g till # it becomes connecteddef continue_add_connectivity(g): while(nx.is_connected(g) == False): g = add_random_edge(g) return g # Creates an object of entire process.# Input- number of nodes and Output- # number of edges required for graph# connectivity.def create_instance(N): g = add_nodes(N) g = continue_add_connectivity(g) return g.number_of_edges() # Average it over 100 timesdef create_average_instance(N): l = [] for i in range(0, 100): l.append(create_instance(N)) return numpy.average(l) # Plot the graph for different number # of edgesdef plot_connectivity(): a = [] b = [] # j is the number of nodes j = 10 while (j <= 1000): a.append(j) b.append(create_average_instance(j)) i += 10 plt.xlabel('Number of nodes') plt.ylabel('Number of edges required') plt.title('Emergence of connectivity') plt.plot(a, b) plt.show() a1 = [] b1 = [] j = 10 while (j <= 400): a1.append(j) b1.append(j*float(numpy.log(j)/2)) j += 10 plt.plot(a1, b1) plt.show() # mainplot_connectivity() |
Output:
10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
Plot of Emergence of Connectedness
