Cascading Behavior in Social Networks

Prerequisite: Introduction to Social Networks, Python Basics

When people are connected in networks to each other then they can influence each other’s behavior and decisions. This is called Cascading Behavior in Networks.

Let’s consider an example, assume all the people in a society have adopted a trend X. Now there comes new trend Y and a small group accepts this new trend and after this, their neighbors also accept this trend Y and so on.

Example of Cascading Behavior( a=2,b=3 and p=2/5)

So, there are 4 main ideas in Cascading Behaviors:

1. Increasing the payoff.
2. Key people.
3. Impact of communities on the Cascades.
4. Cascading and Clusters.

Below is the code for each idea.

1. Increase the payoff.

Python3

# cascade pay off
import networkx as nx
import matplotlib.pyplot as plt


def set_all_B(G):
    for i in G.nodes():
        G.nodes[i]['action'] = 'B'
    return G

def set_A(G, list1):
    for i in list1:
        G.nodes[i]['action'] = 'A'
    return G

def get_colors(G):
    color = []
    for i in G.nodes():
        if (G.nodes[i]['action'] == 'B'):
            color.append('red')
        else:
            color.append('blue')
    return color

def recalculate(G):
    dict1 = {}

    # payoff(A)=a=4
    # payoff(B)=b=3
    a = 15
    b = 5

    for i in G.nodes():
        neigh = G.neighbors(i)
        count_A = 0
        count_B = 0

        for j in neigh:
            if (G.nodes[j]['action'] == 'A'):
                count_A += 1
            else:
                count_B += 1
        payoff_A = a * count_A
        payoff_B = b * count_B

        if (payoff_A >= payoff_B):
            dict1[i] = 'A'
        else:
            dict1[i] = 'B'
    return dict1

def reset_node_attributes(G, action_dict):
    for i in action_dict:
        G.nodes[i]['action'] = action_dict[i]
    return G

def Calculate(G):
    terminate = True
    count = 0
    c = 0

    while (terminate and count < 10):
        count += 1

        # action_dict will hold a dictionary
        action_dict = recalculate(G)
        G = reset_node_attributes(G, action_dict)
        colors = get_colors(G)

        if (colors.count('red') == len(colors) or colors.count('green') == len(colors)):
            terminate = False
            if (colors.count('green') == len(colors)):
                c = 1
        nx.draw(G, with_labels=1, node_color=colors, node_size=800)
        plt.show()
    if (c == 1):
        print('cascade complete')
    else:
        print('cascade incomplete')


G = nx.erdos_renyi_graph(10, 0.5)
nx.write_gml(G, "erdos_graph.gml")

G = nx.read_gml('erdos_graph.gml')
print(G.nodes())

G = set_all_B(G)

# initial adopters
list1 = ['2', '1', '3']
G = set_A(G, list1)
colors = get_colors(G)

nx.draw(G, with_labels=1, node_color=colors, node_size=800)
plt.show()

Calculate(G)

Output:

['0', '1', '2', '3', '4', '5', '6', '7', '8', '9']
cascade complete

2. Key people.

Python3

# cascade key people
import networkx as nx
import matplotlib.pyplot as plt

G = nx.erdos_renyi_graph(10, 0.5)
nx.write_gml(G, "erdos_graph.gml")

def set_all_B(G):
    for i in G.nodes():
        G.nodes[i]['action'] = 'B'
    return G

def set_A(G, list1):
    for i in list1:
        G.nodes[i]['action'] = 'A'
    return G

def get_colors(G):
    color = []
    for i in G.nodes():
        if (G.nodes[i]['action'] == 'B'):
            color.append('red')
        else:
            color.append('green')
    return color


def recalculate(G):
    dict1 = {}

    # payoff(A)=a=4
    # payoff(B)=b=3
    a = 10
    b = 5
    for i in G.nodes():
        neigh = G.neighbors(i)
        count_A = 0
        count_B = 0

        for j in neigh:
            if (G.nodes[j]['action'] == 'A'):
                count_A += 1
            else:
                count_B += 1

        payoff_A = a * count_A
        payoff_B = b * count_B

        if (payoff_A >= payoff_B):
            dict1[i] = 'A'
        else:
            dict1[i] = 'B'

    return dict1


def reset_node_attributes(G, action_dict):

    for i in action_dict:
        G.nodes[i]['action'] = action_dict[i]
    return G


def Calculate(G):
    continuee = True
    count = 0
    c = 0

    while (continuee and count < 100):
        count += 1

        # action_dict will hold a dictionary
        action_dict = recalculate(G)
        G = reset_node_attributes(G, action_dict)
        colors = get_colors(G)

        if (colors.count('red') == len(colors) or colors.count('green') == len(colors)):
            continuee = False
            if (colors.count('green') == len(colors)):
                c = 1

    if (c == 1):
        print('cascade complete')
    else:
        print('cascade incomplete')


G = nx.read_gml('erdos_graph.gml')

for i in G.nodes():
    for j in G.nodes():
        if (i < j):
            list1 = []
            list1.append(i)
            list1.append(j)
            print(list1, ':', end="")

            G = set_all_B(G)
            G = set_A(G, list1)
            colors = get_colors(G)
            Calculate(G)

Output:

['0', '1'] :cascade complete
['0', '2'] :cascade incomplete
['0', '3'] :cascade complete
['0', '4'] :cascade complete
['0', '5'] :cascade incomplete
['0', '6'] :cascade complete
['0', '7'] :cascade complete
['0', '8'] :cascade complete
['0', '9'] :cascade complete
['1', '2'] :cascade complete
['1', '3'] :cascade complete
['1', '4'] :cascade complete
['1', '5'] :cascade complete
['1', '6'] :cascade complete
['1', '7'] :cascade complete
['1', '8'] :cascade complete
['1', '9'] :cascade complete
['2', '3'] :cascade incomplete
['2', '4'] :cascade incomplete
['2', '5'] :cascade incomplete
['2', '6'] :cascade incomplete
['2', '7'] :cascade incomplete
['2', '8'] :cascade incomplete
['2', '9'] :cascade complete
['3', '4'] :cascade complete
['3', '5'] :cascade incomplete
['3', '6'] :cascade complete
['3', '7'] :cascade complete
['3', '8'] :cascade complete
['3', '9'] :cascade complete
['4', '5'] :cascade incomplete
['4', '6'] :cascade complete
['4', '7'] :cascade complete
['4', '8'] :cascade complete
['4', '9'] :cascade incomplete
['5', '6'] :cascade incomplete
['5', '7'] :cascade incomplete
['5', '8'] :cascade incomplete
['5', '9'] :cascade complete
['6', '7'] :cascade complete
['6', '8'] :cascade complete
['6', '9'] :cascade complete
['7', '8'] :cascade complete
['7', '9'] :cascade complete
['8', '9'] :cascade complete

3. Impact of communities on the Cascades.

Python3

import networkx as nx
import random
import matplotlib.pyplot as plt


def first_community(G):
    for i in range(1, 11):
        G.add_node(i)
    for i in range(1, 11):
        for j in range(1, 11):
            if (i < j):
                r = random.random()
                if (r < 0.5):
                    G.add_edge(i, j)
    return G

def second_community(G):
    for i in range(11, 21):
        G.add_node(i)
    for i in range(11, 21):
        for j in range(11, 21):
            if (i < j):
                r = random.random()
                if (r < 0.5):
                    G.add_edge(i, j)
    return G


G = nx.Graph()
G = first_community(G)
G = second_community(G)
G.add_edge(5, 15)

nx.draw(G, with_labels=1)
plt.show()

nx.write_gml(G, "community.gml")

Output:

Impact on clusters

4. Cascading on Clusters.

Python3

import networkx as nx
import matplotlib.pyplot as plt


def set_all_B(G):
    for i in G.nodes():
        G.nodes[i]['action'] = 'B'
    return G

def set_A(G, list1):
    for i in list1:
        G.nodes[i]['action'] = 'A'
    return G

def get_colors(G):
    color = []
    for i in G.nodes():
        if (G.nodes[i]['action'] == 'B'):
            color.append('red')
        else:
            color.append('green')
    return color

def recalculate(G):
    dict1 = {}
    a = 3
    b = 2
    for i in G.nodes():
        neigh = G.neighbors(i)
        count_A = 0
        count_B = 0

        for j in neigh:
            if (G.nodes[j]['action'] == 'A'):
                count_A += 1
            else:
                count_B += 1
        payoff_A = a * count_A
        payoff_B = b * count_B

        if (payoff_A >= payoff_B):
            dict1[i] = 'A'
        else:
            dict1[i] = 'B'
    return dict1

def reset_node_attributes(G, action_dict):
    for i in action_dict:
        G.nodes[i]['action'] = action_dict[i]
    return G

def Calculate(G):
    terminate = True
    count = 0
    c = 0
    while (terminate and count < 100):
        count += 1

        # action_dict will hold a dictionary
        action_dict = recalculate(G)
        G = reset_node_attributes(G, action_dict)
        colors = get_colors(G)

        if (colors.count('red') == len(colors) or colors.count('green') == len(colors)):
            terminate = False
            if (colors.count('green') == len(colors)):
                c = 1

    if (c == 1):
        print('cascade complete')
    else:
        print('cascade incomplete')
    nx.draw(G, with_labels=1, node_color=colors, node_size=800)
    plt.show()


G = nx.Graph()
G.add_nodes_from(range(13))
G.add_edges_from(
    [(0, 1), (0, 6), (1, 2), (1, 8), (1, 12),
     (2, 9), (2, 12), (3, 4), (3, 9), (3, 12),
     (4, 5), (4, 12), (5, 6), (5, 10), (6, 8),
     (7, 8), (7, 9), (7, 10), (7, 11), (8, 9),
     (8, 10), (8, 11), (9, 10), (9, 11), (10, 11)])

list2 = [[0, 1, 2, 3], [0, 2, 3, 4], [1, 2, 3, 4],
         [2, 3, 4, 5], [3, 4, 5, 6], [4, 5, 6, 12],
         [2, 3, 4, 12], [0, 1, 2, 3, 4, 5],
         [0, 1, 2, 3, 4, 5, 6, 12]]

for list1 in list2:
    print(list1)
    G = set_all_B(G)

    G = set_A(G, list1)
    colors = get_colors(G)
    nx.draw(G, with_labels=1, node_color=colors, node_size=800)
    plt.show()

    Calculate(G)

Output:

[0, 1, 2, 3]
cascade incomplete
[0, 2, 3, 4]
cascade incomplete
[1, 2, 3, 4]
cascade incomplete
[2, 3, 4, 5]
cascade incomplete
[3, 4, 5, 6]
cascade incomplete
[4, 5, 6, 12]
cascade incomplete
[2, 3, 4, 12]
cascade incomplete
[0, 1, 2, 3, 4, 5]
cascade incomplete
[0, 1, 2, 3, 4, 5, 6, 12]
cascade complete