Network Models

Social and Economic
Network Analysis
UNIT – II
NETWORK DYNAMICS

Overview
Network Models
Link Analysis
20-04-2021 VANI KANDHASAMY, PSGTECH 2

Facebook
Ego networks
# nodes: 4039
# edges: 88234
Average degree: 43.691
Diameter (longest shortest path): 8
Average Path length: 3.6925
Average clustering coefficient: 0.6055
https://snap.stanford.edu/data/#socnets

Facebook
Degree Distribution – Power law
(Skewed)
Size of Giant component - 4039
Hubs / Celebrity / Influencer

Key Network Measures –Real World
Network
Measure Description
Degree Distribution Power Law (Few Hubs)
Clustering Coefficient High
Average Path Length Small
Connectivity / Size of Giant component Large

Network Models
NETWORKS, CROWDS AND MARKETS - CHAPTER 18, 20

Why model?
•Definition: a miniature representation of real thing
•Simple representation of complex network
•Can derive properties mathematically
•Predict properties and outcomes

Summary

Erdös-Renyi / Random model
•Assumptions
• nodes connect at random
• network is undirected
•Key parameter (besides number of nodes N) : p or M
• p = probability that any two nodes share and edge (Gnp)
• M = total number of edges in the graph (Gnm)

Erdös-Renyi model

Key Network Measures – Erdös-Renyi
model
Measure Description
Degree Distribution ?
Clustering Coefficient ?
Average Path Length ?
Connectivity / Size of Giant component ?

Degree distribution
•Gnp-model: For each potential
edge we flip a coin
with probability p we add the edge
with probability (1-p) we don’t
•Degree distribution of Gnp is
binomial

Degree distribution
Avg. Degree:

Clustering Coefficient
𝐶𝑖 =
# 𝑝𝑎𝑖𝑟𝑠 𝑜𝑓 𝑖′𝑠 𝑓𝑟𝑖𝑒𝑛𝑑𝑠 𝑤ℎ𝑜 𝑎𝑟𝑒 𝑓𝑟𝑖𝑒𝑛𝑑𝑠
# 𝑝𝑎𝑖𝑟𝑠 𝑜𝑓 𝑖′𝑠 𝑓𝑟𝑖𝑒𝑛𝑑𝑠
ei is the number of edges between
i’s friends
ki is the number of i’s friends /
degree of i

Number of distinct pairs of
friends of node i of degree ki

Avg. Path Length
log-log plot
Avg. Path Length ~ O(log n)

Giant Component
Graph structure of Gnp as p (density) changes

Giant Component
ത
𝑘=1-ε -> all components are of
size Ω(log n)
ത
𝑘 =1+ε -> 1 component of size
Ω(n), others have size
Ω(log n)

Key Network Measures – Erdös-Renyi
model
Measures Description
Degree Distribution Binomial
Clustering Coefficient C = p = ҧ
𝐤 / n
Average Path Length O(log n)
Connectivity / Size of Giant component GCC exists
when ҧ
𝐤 >1

Real world network vs. Random Network
Parameters Facebook Gnp Match
Degree Distribution
Clustering Coefficient 0.6055 0.01
Average Path Length 3.6925 2.654
Connectivity / Size of
Giant component
4039 4039 (ത
k = 20.09)

Summary: Erdös-Renyi model
Giant connected component
Average path length
Clustering Coefficient – no local structure
Degree Distribution – absence of hubs

Small World Phenomenon - Milgram’s
experiment
NE
MA

Small World Phenomenon - Milgram’s
experiment
Six degrees of separation
Assume each human is connected to 100 other
people then:
Step 1: reach 100 people
Step 2: reach 100*100 = 10,000 people
Step 3: reach 100*100*100 = 1,000,000 people
Step 4: reach 100*100*100*100 = 100M people
In 5 steps we can reach 10 billion people

Small World Phenomenon

Network Models

high clustering
low average shortest path
Small world phenomenon
)
ln(
network N
l 
graph
random
network C
C 
Clustering implies edge “locality”
Randomness enables “short paths”

Watts-Strogatz / Small World model
Two components to the model:
1. Start with a regular lattice (High CC)
2. Rewire: (Low APL)
◦ Add edges to reach remote parts of the lattice
◦ For each edge with prob. p move the other end to a random node

Select a fraction p of edges
Reposition one of their endpoints
Add a fraction p of additional
edges leaving underlying lattice
intact
Watts-Strogatz / Small World model

1% of links rewired 10% of links rewired

Summary: Watts-Strogatz model
Average path length
Degree Distribution – absence of hubs

Poisson distribution

Power Law distribution

Power law distribution
Straight line on a log-log plot
Exponentiate both sides to get that p(k),
normalization
constant (probabilities
over all x must sum to 1) power law exponent a

Generating Power Law Networks
▪Ingredient # 1: growth over time
nodes appear one by one, each selecting m other nodes at
random to connect to
▪Ingredient # 2: preferential attachment
new nodes prefer to attach to well-connected nodes over less-
well connected nodes

Ingredient # 1: growth over time
•one node is born at each time tick
•at time t there are t nodes
•change in degree ki of node i (born at time i, with 0 < i < t)

Ingredient # 1: growth over time
•How many new edges does a node accumulate since it's birth at
time i until time t?

Degree distribution
•Let τ(100) be the time at which node with degree e.g. 100 is born
•Then the fraction of nodes that have degree <= 100 is (t – τ)/t

Ingredient # 2: preferential attachment
•Rich-get-richer phenomenon
•Cumulative advantage

Barabasi-Albert model
•the process starts with some initial subgraph
•each new node comes in with m edges
•probability of connecting to node i

To start, each vertex has an equal
number of edges (2)
◦ the probability of choosing any
vertex is 1/3
We add a new vertex, and it will
have m edges, here take m=2
◦ draw 2 random elements from the
array – suppose they are 2 and 3
Now the probabilities of selecting
1,2,3,or 4 are 1/5, 3/10, 3/10, 1/5

Properties of the BA graph
•The degree distribution is scale free with exponent α = 3
P(k) = 2 m2/k3
•The graph is connected
oEvery vertex is born with a link (m ≥ 1)
oIt connects to older vertices, which are part of the giant component
•The older are richer
oNodes accumulate links as time goes on
oPreferential attachment will prefer wealthier nodes

vertex introduced at time t=5
vertex introduced at time t=95
Barabasi-Albert
model
Age of node -> Degree of node
Degree of node -> Popularity of
node

Summary: Barabasi-Albert model
Average path length
Degree Distribution
Clustering Coefficient – no local structure

Decentralized Search
A GREEDY APPROACH

How to navigate a network?

▪Source s only knows locations of its friends and location of the Target t
▪s does not know links of anyone else but itself
▪Geographic Navigation: s “navigates” to a node geographically closest to t

ERDOS-RENYI MODEL WATTS-STROGATZ
Search time T

Kleinberg’s Model
▪Nodes still on a grid and connect to nearest neighbors
▪Additional links placed with

Kleinberg’s Model
Netlogo: DEMO

Kleinberg’s Model (α ≠ 2)

Kleinberg’s Model (α = 2)

Kleinberg’s Model

Diffusion in Networks
SIMPLE CONTAGION

Simple contagion
Random
• Density
• Netlogo:Demo
Scale
free
• Preferential
attachment
• Netlogo:Demo
Small
world
• Rewiring
probability
• Netlogo:Demo

Network Models

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