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Structure and models of real-world graphs and networks

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### Structure and models of real-world graphs and networks

Jure Leskovec

Machine Learning Department

Carnegie Mellon University

jure@cs.cmu.edu

http://www.cs.cmu.edu/~jure/

Examples of networks

(b)

(c)

(a)

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

(d)

(e)

- 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 can’t 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

Pick a random node, count how many nodes are at distance 1,2,3... hops

Number of nodes

Distance (Hops)

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 = 3*number of triangles / number of connected triples

- Alternative definition:

Ci = triangles connected to vertex i / number triples centered on vertex i

- Clustering coefficient:

Ci=1, 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:

C=O(1) as n→∞

In Erdos-Renyi random graph: C=O(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/(α-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

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 > 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

Internet

- 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:

- a=1: linear growth – constant out-degree (assumed in the literature so far)
- a=2: quadratic growth – clique

E(t)

a=1.2

N(t)

Citations

E(t)

a=1.7

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 γ=const over time, we obtain densification only for 1<γ<2, then γ=a/2
- (b) For γ<2 degree distribution has to evolve according to:
- Power-law: y=b xγ, for γ<2 E[y] = ∞

pk=kγ

(a)

γ(t)

a=1.1

(b)

γ(t)

a=1.6

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

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 M=0.5n(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 k=2m/n:
- k=1-ε: all components are of size log n
- k=1+ε: 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

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 εi 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 εi

Exponential random graphs

- Includes Erdos-Renyi as a special case
- Assume parameters βi 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:
- Can’t 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 (p=0):
- Clustering coefficient C=(3k-3)/(4k-2)=3/4
- Mean distance L/4k
- Almost random graph (p=1):
- Clustering coefficient C=2k/L
- Mean distance log L / log k
- No power-law degree distribution

Rewiring probabilityp

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 Simon’s 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 β select k random vertices and link to them
- With prob 1-β edges are copied from a randomly chosen node
- Generates power-law degree distributions with exponent 1/(1-β)
- 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 Attachmentleads to Densification Power Lawwith exponent
- a … densification exponent
- b … community tree branching factor
- c … difficulty constant, 1 ≤ c ≤ b
- If c = 1: easy to cross communities
- Then: a=2, quadratic growth of edges – near clique
- If c = b: hard to cross communities
- Then: a=1, 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 – 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

N*K x M*L

Stochastic Kronecker Graphs

- Create N1N1probability matrixP1
- 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

flip biased

coins

P1

P2

Fitting Kronecker to Real Data

- Given a graph G and Kronecker matrix P1 we can calculate probability that P1 generated G: P(G|P1):

P1

G

Pk

P(G|P1)

σ… node labeling

Fitting Kronecker: 2 challenges

- Invariance to node labeling σ (there are N! labelings)
- Calculating P(G|P1) takes O(N2) (since one needs to consider every cell of adjacency matrix)

P1

G

Pk

P(G|P1)

1

2

3

4

==

2

1

4

3

Fitting Kronecker: Solutions

- Node Labeling: can use MCMC sampling to average over (all) node labelings
- P(G|P1) takes O(N2): Real graphs are sparse, so calculate P(Gempty) and then “add” the edges. This takes O(E).

σ… node labeling

P=

G=

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

Epidemiological processes

- The simplest way to spread a virus over the network
- S: Susceptible
- I: Infected
- R: Recovered (removed)
- SIS model: 2 parameters
- β… virus birth rate
- δ… virus death (recovery) rate
- SIR model: as one gets cured, he or she can not get infected again

SIS model

ζitdepends onβand 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 s>1 virus prevails
- For s<1 virus dies

λ1… largest eigen value of graph adjacency matrix

Navigation in small-world networks

v

- Milgram’s experiment showed:
- (a) short paths exist in networks
- (b) humans are able tofind 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) ~ dα)
- Can be show that only for α=2 delivery time is poly-log in number of nodes n

Deliver timeT < nβ

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 let’s 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

IN44 M

OUT44 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

α 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

If α is too low, then the Euclidian distances become unimportant, and the network resembles a star:

Fabrikant&al

But if α grows at least as fast as √n, 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 α is anywhere in between, we have a power law

Through a rather complex and elaborate proof, Fabrikant&al prove this initial assumption will produce a power law distribution – I’ll save you the math!

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