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Structure and models of real-world graphs and

networks

- Jure Leskovec
- Machine Learning Department
- Carnegie Mellon University
- jure_at_cs.cmu.edu
- http//www.cs.cmu.edu/jure/

Networks (graphs)

Examples of networks

(b)

(c)

(a)

(d)

(e)

- Internet (a)
- citation network (b)
- World Wide Web (c)

- sexual network (d)
- food web (e)

Networks of the real-world (1)

- Information networks
- World Wide Web hyperlinks
- Citation networks
- Blog networks
- Social networks people interactios
- Organizational networks
- Communication networks
- Collaboration networks
- Sexual networks
- Collaboration networks
- Technological networks
- Power grid
- Airline, road, river networks
- Telephone networks
- Internet
- Autonomous systems

Florence families

Karate club network

Collaboration network

Friendship network

Networks of the real-world (2)

- Biological networks
- metabolic networks
- food web
- neural networks
- gene regulatory networks
- Language networks
- Semantic networks
- Software networks

Semantic network

Yeast protein interactions

Language network

XFree86 network

Types of networks

- Directed/undirected
- Multi graphs (multiple edges between nodes)
- Hyper graphs (edges connecting multiple nodes)
- Bipartite graphs (e.g., papers to authors)
- Weighted networks
- Different type nodes and edges
- Evolving networks
- Nodes and edges only added
- Nodes, edges added and removed

Traditional approach

- Sociologists were first to study networks
- Study of patterns of connections between people

to understand functioning of the society - People are nodes, interactions are edges
- Questionares are used to collect link data (hard

to obtain, inaccurate, subjective) - Typical questions Centrality and connectivity
- Limited to small graphs (10 nodes) and

properties of individual nodes and edges

New approach (1)

- Large networks (e.g., web, internet, on-line

social networks) with millions of nodes - Many traditional questions not useful anymore
- Traditional What happens if a node U is removed?

- Now What percentage of nodes needs to be removed

to affect network connectivity? - Focus moves from a single node to study of

statistical properties of the network as a whole - Can not draw (plot) the network and examine it

New approach (2)

- How the network looks like even if I cant look

at it? - Need statistical methods and tools to quantify

large networks - 3 parts/goals
- Statistical properties of large networks
- Models that help understand these properties
- Predict behavior of networked systems based on

measured structural properties and local rules

governing individual nodes

Statistical properties of networks

- Features that are common to networks of different

types - Properties of static networks
- Small-world effect
- Transitivity or clustering
- Degree distributions (scale free networks)
- Network resilience
- Community structure
- Subgraphs or motifs
- Temporal properties
- Densification
- Shrinking diameter

Small-world effect (1)

- Six degrees of separation (Milgram 60s)
- Random people in Nebraska were asked to send

letters to stockbrokes in Boston - Letters can only be passed to first-name

acquantices - Only 25 letters reached the goal
- But they reached it in about 6 steps
- Measuring path lengths
- Diameter (longest shortest path) max dij
- Effective diameter distance at which 90 of all

connected pairs of nodes can be reached - Mean geodesic (shortest) distance l

or

Small-world effect (2)

- Distribution of shortest path lengths
- Microsoft Messenger network
- 180 million people
- 1.3 billion edges
- Edge if two people exchanged at least one message

in one month period

7

Small-world effect (3)

- Fact
- If number of vertices within distance r grows

exponentially with r, then mean shortest path

length l increases as log n - Implications
- Information (viruses) spread quickly
- Erdos numbers are small
- Peer to peer networks (for navigation purposes)
- Shortest paths exists
- Humans are able to find the paths
- People only know their friends
- People do not have the global knowledge of the

network - This suggests something special about the

structure of the network - On a random graph short paths exists but no one

would be able to find them

Transitivity or Clustering

- friend of a friend is a friend
- If a connects to b, and b to c, then with

high probability a connects to c. - Clustering coefficient C
- C 3number of triangles / number of connected

triples - Alternative definition
- Ci triangles connected to vertex i / number

triples centered on vertex i - Clustering coefficient

Ci1, 1, 1/6, 0, 0

Transitivity or Clustering (2)

- Clustering coefficient scales as
- It is considerably higher than in a random graph
- It is speculated that in real networks
- CO(1) as n?8
- In Erdos-Renyi random graph CO(n-1)

Synonyms network

World Wide Web

Degree distributions (1)

- Let pk denote a fraction of nodes with degree k
- We can plot a histogram of pk vs. k
- In a Erdos-Renyi random graph degree distribution

follows Poisson distribution - Degrees in real networks are heavily skewed to

the right - Distribution has a long tail of values that are

far above the mean - Heavy (long) tail
- Amazon sales
- word length distribution,

Detour how long is the long tail?

This is not directly related to graphs, but it

nicely explains the long tail effect. It shows

that there is big market for niche products.

Degree distributions (2)

- Many real world networks contain hubs highly

connected nodes - We can easily distinguish between exponential and

power-law tail by plotting on log-lin and log-log

axis - In scale-free networks maximum degree scales as

n1/(a-1)

lin-lin

log-lin

pk

k

k

log-log

pk

k

Degree distribution in a blog network

Poisson vs. Scale-free network

Poisson network

Scale-free (power-law) network

(Erdos-Renyi random graph)

Degree distribution is Power-law

Function is scale free if f(ax) b f(x)

Degree distribution is Poisson

Degree distribution number of people a person

talks to on a Microsoft Messenger

Count

Highest degree

X

Node degree

Network resilience (1)

- We observe how the connectivity (length of the

paths) of the network changes as the vertices get

removed - Vertices can be removed
- Uniformly at random
- In order of decreasing degree
- It is important for epidemiology
- Removal of vertices corresponds to vaccination

Network resilience (2)

- Real-world networks are resilient to random

attacks - One has to remove all web-pages of degree gt 5 to

disconnect the web - But this is a very small percentage of web pages
- Random network has better resilience to targeted

attacks

Random network

Internet (Autonomous systems)

Preferential removal

Mean path length

Random removal

Fraction of removed nodes

Fraction of removed nodes

Community structure

- Most social networks show community structure
- groups have higher density of edges within than

accross groups - People naturally divide into groups based on

interests, age, occupation, - How to find communities
- Spectral clustering (embedding into a low-dim

space) - Hierarchical clustering based on connection

strength - Combinatorial algorithms
- Block models
- Diffusion methods

Friendship network of children in a school

MSN Messenger

Distribution of Connected components in MSN

Messenger network

Growth of largest component over time in a

citation network

Count

- Graphs have a giant component
- Distribution of connected components follows a

power law

Largest component

X

Size (number of nodes)

Network motifs (1)

- What are the building blocks (motifs) of

networks? - Do motifs have specific roles in networks?
- Network motifs detection process
- Count how many times each subgraph appears
- Compute statistical significance for each

subgraph probability of appearing in random as

much as in real network

3 node motifs

Network motifs (2)

- Biological networks
- Feed-forward loop
- Bi-fan motif
- Web graph
- Feedback with two mutual diads
- Mutual diad
- Fully connected triad

Network motifs (3)

Transcription networks

Signal transduction networks

WWW and friendship networks

Word adjacency networks

Networks over time Densification

- A very basic question What is the relation

between the number of nodes and the number of

edges in a network? - Networks are becoming denser over time
- The number of edges grows faster than the number

of nodes average degree is increasing - a densification exponent 1 a 2
- a1 linear growth constant out-degree (assumed

in the literature so far) - a2 quadratic growth clique

Internet

E(t)

a1.2

N(t)

Citations

E(t)

a1.7

N(t)

Densification degree distribution

Degree exponent over time

- How does densification affect degree

distribution? - Given densification exponent a, the degree

exponent is - (a) For ?const over time, we obtain

densification only for 1lt?lt2, then ?a/2 - (b) For ?lt2 degree distribution has to evolve

according to - Power-law yb x?, for ?lt2 Ey 8

pkk?

(a)

?(t)

a1.1

(b)

?(t)

a1.6

Shrinking diameters

Internet

- Intuition says that distances between the nodes

slowly grow as the network grows (like log n) - But as the network grows the distances between

nodes slowly decrease

Citations

Models of network generation and evolution

Recap (1)

- Last time we saw
- Large networks (web, on-line social networks) are

here - Many traditional questions not useful anymore
- We can not plot the network so we need

statistical methods and tools to quantify large

networks - 3 parts/goals
- Statistical properties of large networks
- Models that help understand these properties
- Predict behavior of networked systems based on

measured structural properties and local rules

governing individual nodes

Recap (2)

- We also so features that are common to networks

of various types - Properties of static networks
- Small-world effect
- Transitivity or clustering
- Degree distributions (scale free networks)
- Network resilience
- Community structure
- Subgraphs or motifs
- Temporal properties
- Densification
- Shrinking diameter

Outline for today

- We will see the network generative models for

modeling networks features - Erdos-Renyi random graph
- Exponential random graphs (p) model
- Small world model
- Preferential attachment
- Community guided attachment
- Forest fire model
- Fitting models to real data
- How to generate a synthetic realistic looking

network?

(Erodos-Renyi) Random graphs

- Also known as Poisson random graphs or Bernoulli

graphs - Given n vertices connect each pair i.i.d. with

probability p - Two variants
- Gn,p graph with m edges appears with probability

pm(1-p)M-m, where M0.5n(n-1) is the max number

of edges - Gn,m graphs with n nodes, m edges
- Very rich mathematical theory many properties

are exactly solvable

Properties of random graphs

- Degree distribution is Poisson since the presence

and absence of edges is independent - Giant component average degree k2m/n
- k1-e all components are of size log n
- k1e there is 1 component of size n
- All others are of size log n
- They are a tree plus an edge, i.e., cycles
- Diameter log n / log k

Evolution of a random graph

for non-GCC vertices

k

Subgraphs in random graphs

- Expected number of subgraphs H(v,e) in Gn,p is

a... of isomorphic graphs

Random graphs conclusion

Configuration model

- Pros
- Simple and tractable model
- Phase transitions
- Giant component
- Cons
- Degree distribution
- No community structure
- No degree correlations
- Extensions
- Configuration model
- Random graphs with arbitrary degree sequence
- Excess degree Degree of a vertex of the end of

random edge qk k pk

Exponential random graphs(p models)

- Comes from social sciences
- Let ei set of measurable properties of a graph

(number of edges, number of nodes of a given

degree, number of triangles, ) - Exponential random graph model defines a

probability distribution over graphs

Examples of ei

Exponential random graphs

- Includes Erdos-Renyi as a special case
- Assume parameters ßi are specified
- No analytical solutions for the model
- But can use simulation to sample the graphs
- Define local moves on a graph
- Addition/removal of edges
- Movement of edges
- Edge swaps
- Parameter estimation
- maximum likelihood
- Problem
- Cant solve for transitivity (produces cliques)
- Used to analyze small networks

Example of parameter estimates

Small-world model

- Used for modeling network transitivity
- Many networks assume some kind of geographical

proximity - Small-world model
- Start with a low-dimensional regular lattice
- Rewire
- Add/remove edges to create shortcuts to join

remote parts of the lattice - For each edge with prob p move the other end to a

random vertex - Rewiring allows to interpolate between regular

lattice and random graph

Small-world model

- Regular lattice (p0)
- Clustering coefficient C(3k-3)/(4k-2)3/4
- Mean distance L/4k
- Almost random graph (p1)
- Clustering coefficient C2k/L
- Mean distance log L / log k
- No power-law degree distribution

Rewiring probability p

Degree distribution

Models of evolution

- Models of network evolution
- Preferential attachment
- Edge copying model
- Community Guided Attachment
- Forest Fire model
- Models for realistic network generation
- Kronecker graphs

Preferential attachment

- Models the growth of the network
- Preferential attachment (Price 1965, Albert

Barabasi 1999) - Add a new node, create m out-links
- Probability of linking a node ki is

proportional to its degree - Based on Herbert Simons result
- Power-laws arise from Rich get richer

(cumulative advantage) - Examples (Price 1965 for modeling citations)
- Citations new citations of a paper are

proportional to the number it already has

Preferential attachment

- Leads to power-law degree distributions
- But
- all nodes have equal (constant) out-degree
- one needs a complete knowledge of the network
- There are many generalizations and variants, but

the preferential selection is the key ingredient

that leads to power-laws

Edge copying model

- Copying model
- Add a node and choose k the number of edges to

add - With prob ß select k random vertices and link to

them - With prob 1-ß edges are copied from a randomly

chosen node - Generates power-law degree distributions with

exponent 1/(1-ß) - Generates communities
- Related Random-surfer model

Community guided attachment

- Want to model/explain densification in networks
- Assume community structure
- One expects many within-group friendships and

fewer cross-group ones

University

Arts

Science

CS

Drama

Music

Math

Self-similar university community structure

Community guided attachment

- Assuming cross-community linking probability
- The Community Guided Attachment leads to

Densification Power Law with exponent - a densification exponent
- b community tree branching factor
- c difficulty constant, 1 c b
- If c 1 easy to cross communities
- Then a2, quadratic growth of edges near

clique - If c b hard to cross communities
- Then a1, linear growth of edges constant

out-degree

Forest Fire Model

- Want to model graphs that density and have

shrinking diameters - Intuition
- How do we meet friends at a party?
- How do we identify references when writing papers?

Forest Fire Model for directed graphs

- The model has 2 parameters
- p forward burning probability
- r backward burning probability
- The model
- Each turn a new node v arrives
- Uniformly at random chooses an ambassador w
- Flip two geometric coins to determine the number

in- and out-links of w to follow (burn) - Fire spreads recursively until it dies
- Node v links to all burned nodes

Forest Fire Model

- Simulation experiments
- Forest Fire generates graphs that densify and

have shrinking diameter

E(t)

densification

diameter

1.32

diameter

N(t)

N(t)

Forest Fire Model

- Forest Fire also generates graphs with

heavy-tailed degree distribution

in-degree

out-degree

count vs. in-degree

count vs. out-degree

Forest Fire Parameter Space

- Fix backward probability r and vary forward

burning probability p - We observe a sharp transition between sparse and

clique-like graphs - Sweet spot is very narrow

Clique-like graph

Increasing diameter

Constant diameter

Sparse graph

Decreasing diameter

Kronecker graphs

- Want to have a model that can generate a

realistic graph - Static Patterns
- Power Law Degree Distribution
- Small Diameter
- Power Law Eigenvalue and Eigenvector Distribution
- Temporal Patterns
- Densification Power Law
- Shrinking/Constant Diameter
- For Kronecker graphs all these properties can

actually be proven

Kronecker Product a Graph

Intermediate stage

Adjacency matrix

Adjacency matrix

Kronecker Product Definition

- The Kronecker product of matrices A and B is

given by - We define a Kronecker product of two graphs as a

Kronecker product of their adjacency matrices

N x M

K x L

NK x ML

Stochastic Kronecker Graphs

- Create N1?N1 probability matrix P1
- Compute the kth Kronecker power Pk
- For each entry puv of Pk include an edge (u,v)

with probability puv

Kronecker multiplication

Instance Matrix G2

P1

flip biased coins

P2

Fitting Kronecker to Real Data

- Given a graph G and Kronecker matrix P1 we can

calculate probability that P1 generated G

P(GP1)

P1

G

Pk

P(GP1)

s node labeling

Fitting Kronecker 2 challenges

P1

G

Pk

P(GP1)

- Invariance to node labeling s (there are N!

labelings) - Calculating P(GP1) takes O(N2) (since one needs

to consider every cell of adjacency matrix)

1

2

3

4

2

1

4

3

Fitting Kronecker Solutions

s node labeling

P

G

- Node Labeling can use MCMC sampling to average

over (all) node labelings - P(GP1) takes O(N2) Real graphs are sparse, so

calculate P(Gempty) and then add the edges.

This takes O(E).

Experiments on real AS graph

Degree distribution

Hop plot

Network value

Adjacency matrix eigen values

Why fitting generative models?

- Parameters tell us about the structure of a graph
- Extrapolation given a graph today, how will it

look in a year? - Sampling can I get a smaller graph with similar

properties? - Anonymization instead of releasing real graph

(e.g., email network), we can release a synthetic

version of it

Processes taking place on networks

Epidemiological processes

- The simplest way to spread a virus over the

network - S Susceptible
- I Infected
- R Recovered (removed)
- SIS model 2 parameters
- ß virus birth rate
- d virus death (recovery) rate
- SIR model as one gets cured, he or she can not

get infected again

SIS model

?it depends on ß and topology

Epidemic threshold for SIS model

- How infectious the virus needs to be to survive

in the network? - First results on power-law networks suggested

that any virus will prevail - New result that works for any topology
- For sgt1 virus prevails
- For slt1 virus dies

?1 largest eigen value of graph adjacency matrix

Navigation in small-world networks

v

- Milgrams experiment showed
- (a) short paths exist in networks
- (b) humans are able to find them
- Assume the following setting
- Nodes of a graph are scattered on a plane
- Given starting node u and we want to reach target

node v - A small world navigation algorithm navigates the

network by always navigating to a neighbor that

is closest (in Manhattan distance) to target node

v

u

Navigation in small-world networks

Network creation

- Start with random lattice
- Each node connects with their 4 immediate

neighbors - Long range links are added with probability

proportional to the distance between the points

(p(u,v) da) - Can be show that only for a2 delivery time is

poly-log in number of nodes n

Deliver time T lt nß

Navigation in a real-world network

- Take a social network of 500k bloggers where for

each blogger we know their geographical location - Pick two nodes at random and geographically

greedy navigate the network - Results
- 13 success rate (vs. 18 for Milgram)

Distribution of path lengths

Friendships vs. distance

Navigation in real-world network

- Geographical distance may not be the right kind

of distance - Since population is non-uniform lets use rank

based friendship distance - i.e., we measure the distance d(u,v) by the

number of people living closer to v than u does - Then

And the proof still works

- Some references used to prepare this talk
- The Structure and Function of Complex Networks,

by Mark Newman - Statistical mechanics of complex networks, by

Reka Albert and Albert-Laszlo Barabasi - Graph Mining Laws, Generators and Algorithms, by

Deepay Chakrabarti and Christos Faloutsos - An Introduction to Exponential Random Graph (p)

Models for Social Networks by Garry Robins, Pip

Pattison, Yuval Kalish and Dean Lusher - Graph Evolution Densification and Shrinking

Diameters, by Jure Leskovec, Jon Kleinberg and

Christos Faloutsos - Realistic, Mathematically Tractable Graph

Generation and Evolution, Using Kronecker

Multiplication, by Jure Leskovec, Deepayan

Chakrabarti, Jon Kleinberg and Christos Faloutsos

- Navigation in a Small World, by Jon Kleinberg
- Geographic routing in social networks, by David

Liben-Nowell, Jasmine Novak, Ravi Kumar,

Prabhakar Raghavan, and Andrew Tomkins - Some plots and slides borrowed from Lada Adamic,

Mark Newman, Mark Joseph, Albert Barabasi, Jon

Kleinberg, David Lieben-Nowell, Sergi Valverde

and Ricard Sole

Rough random materialthat did not make it into

the presentation

Bow-tie structure of the web

TENDRILS44M

SCC56 M

OUT44 M

IN44 M

DISC17 M

Broder al. WWW 2000, Dill al. VLDB 2001

Study of 3 websites

- study over three universities publicly indexable

Web sites

Australia

In- and out-degree distributions

New Zealand

In- and out-degree distributions

United Kingdom

In- and out-degree distributions

We assume this node would like to connect to a

centrally located node a node whose distances to

other nodes is minimized.

dij is the Euclidean distance hj is some measure

of the centrality of node j a is a parameter

a function of the final number n of points,

gauging the relative importance of the two

objectives

Fabrikant et al. define 3 possible measures of

centrality 1. The average number of hops from

other nodes 2. The maximum number of hops from

another node 3. The number of hops from a fixed

center of the tree

a is the crux of the theorem! Why? Here are

some examples

Fabrikantal

If a is too low, then the Euclidian distances

become unimportant, and the network resembles a

star

Fabrikantal

But if a grows at least as fast as vn, where n is

the final number of points, then distance becomes

too important, and minimum spanning trees with

high degree occur, but with exponentially

vanishing probability thus not a power law. if

a is anywhere in between, we have a power law

Through a rather complex and elaborate proof,

Fabrikantal prove this initial assumption will

produce a power law distribution Ill save you

the math!