Models Slides

Classical Complex

Network Models

Sergio Gmez Universitat Rovira i Virgili, Tarragona (Spain)

Mediterranean School of Complex Networks

Salina 2014

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Outline

Models

Regular networks

Complex network models

Concluding remarks

Classical complex network models

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Outline

Models

Regular networks


Concluding remarks


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Models

Models

For networks

For processes on networks

For processes of networks

Why models?

We have data of real networks!

We know the details of real processes on real networks!

We know the evolution of real networks!

Models

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Why models? Because...

Data does not imply knowledge!

Models

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Models provide

Explanation

Prediction

Models

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Models provide

Explanation

Prediction

Models must be

Simple (Occams razor)

Accurate

Models

Ptolemy Tycho Brahe

Kepler

Copernicus

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Models provide

Explanation

Prediction

Models must be

Simple (Occams razor)

Accurate

Note: Agent-based models

Not simple, but more realistic

Difficult (even impossible) to identify relationships

Usually cannot provide explanations, even if accurate and

with good predictions

Models

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Network models

To explain the appearance of topological features

Power-law degree distribution, small-world property,

clustering, community structure, etc.

Models

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Network models




To investigate the relationship between topology and

function

Which features are needed for a certain phenomenon?

Models

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Network models





function


To understand processes on networks

Diffusion, epidemic spreading, rumors, innovations,

collaboration, competition, evolutionary games, routing,

congestion, synchronization, etc.

Models

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Network models





function


To understand processes on networks

Diffusion, epidemic spreading, rumors, innovations,

collaboration, competition, evolutionary games, routing,

congestion, synchronization, etc.

To understand processes of networks

Network formation, evolution, interaction, etc.

Models

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Example

Models

Network model

SF

Dynamics model

Kuramoto

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Example

Models

Real network

Star (proxy of SF)

Real dynamics

Rssler chaotic oscillator

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Outline

Models

Regular networks


Concluding remarks


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Regular networks

Traditional networks used in physics and

engineering

Sometimes allow analytical solutions

Discretization of continuous space

Regular networks

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Toy networks

Line

Star

Ring

Fully connected

Regular networks

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Lattices

Regular networks

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Lattices

Without and with periodic boundary conditions

Regular networks

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Other

Bethe lattices / Cayley trees (internal nodes of same k)

Ramanujan graphs (large spectral gap)

Regular networks

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Other

Fractal networks

Regular networks

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Other

Apollonian networks

Regular networks

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Outline

Models

Regular networks


Concluding remarks


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Models for these properties

Degree distribution

Average path length

Clustering

Communities

Other


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Models according to degree distribution

Erds-Rnyi model (ER)

Barabsi-Albert model (BA)

Configuration model (CM)

Interpolating model between ER and BA


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Model GN,K by Erds & Rnyi (1959)

N: number of nodes

K: number of edges (0 K N(N-1)/2)

Each edge connects a randomly selected (and not previously

connected) pair of nodes

Model GN,p by Gilbert (1959)

N: number of nodes

p: probability of having an edge (0 p 1)

Each pair of nodes has a probability p of having an edge



Model GN,p

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p=0.01

p=0.02 p=0.03

p=0.00

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Relationship between GN,K and GN,p

GN,p to GN,K

GN,K to GN,p

Property

Almost surely, connected network if


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Degree distribution

Binomial

Poisson, in the limit while constant


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Based on growth and preferential attachment (1999)

N: number of nodes

m0: number of initial nodes (m0 N)

m: number of edges for each new node (m m0)

The network begins with an initial small connected network

containing m0 nodes

New nodes are added until the network has the desired N

nodes (growth)

Each new node establishes m edges to the current available

nodes

The probability pi that each of the m edges connects to node i

is proportional to its current number of links ki (preferential

attachment)


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Degree distribution

Power-law (scale-free) with exponent = 3


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Note

Both growth and preferential attachment are needed to obtain

the SF degree distribution

History

Yule (1925): preferential attachment to obtain SF degree

distribution

Simon (1955): application of modern master equation method

Price (1976): application to the growth of networks

Barabsy & Albert (1999): rediscovery, name of preferential

attachment, popularity


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Many variations

Non-linear preferential attachment

Dynamic edge rewiring

Fitness models

Hierarchic growing

Deterministic growing

Local growing

Accelerating growth


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Build network given degree sequence

N: number of nodes

P(k): degree distribution

Assign a random degree ki to each node according to the

given degree distribution P(k)

Create a vector with all the slots (half edges)

Connect randomly pairs of slots

Could be used to rewire networks

Could be used to generate networks with scale-free (SF)

degree distribution for any value of the exponent > 2


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Algorithm


Random assignment

of degrees according

to P(k)

Vector of slots

Random permutation

of the vector of slots

Connection of slots

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Details

The number of slots must be even (2L)

Probability pij of connecting nodes i and j

Multi-edges and self-loops may appear


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Details

For large N, the number of multi-edges and self-loops is

negligible if and are finite

For scale-free degree distributions , the second

moment diverges if the exponent is (2,3]

Modify the algorithm to avoid the presence of multi-edges and

self-loops

Introduce a cut-off in P(k) scaling as

Algorithms to find a random permutation

FisherYates shuffle (1938)

Durstenfeld (1964) / Knuth (1969)


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Generalization

Configuration model to build networks with desired degree-

degree correlations P(k,k)

Main developers and theory

Bekessy, Bekessy & Komlos (1972)

Bender & Canfield (1978)

Bollobs (1980)

Wormald (1981)

Molloy & Reed (1995)


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Model by Gmez-Gardees & Moreno (2006)

N: number of nodes

m0: number of initial nodes (m0 N)

U(0): set of initially unconnected nodes

m: number of links for each selected node (m m0)

: probability of generating an ER edge

Initial small fully connected network containing m0 nodes

N- m0 randomly selected nodes in U(0) establish m edges to

the rest of the nodes

For each edge of the selected node

With probability the destination is chosen with uniform probability among the rest N-1 nodes (ER edge)

With probability 1- the destination is selected with preferential attachment (BA edge)


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Note

The preferential attachment (PA) probability depends only on

the links generated by the PA sub-procedure

Two different sub-models to define these PA probabilities

Degree distribution

= 1: ER network, binomial (Poisson) P(k)

= 0: BA network, SF with exponent = 3

0 < < 1: Interpolating network

BA ER

0 1


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Models according to average path length

Watts-Strogatz model (WS)


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Based on regular network and random rewirings (1998)

N: number of nodes

k: mean degree, even integer (k < N)

p: rewiring probability

Nodes initially in a regular ring lattice

Each node connects to its k nearest neighbors, k / 2 on each

side (clockwise and counterclockwise)

For each node, rewire each clockwise original edge with

probability p to a new random destination (multi-edges and

self-loops not allowed)

Regular ER

0 p 1


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Scheme


p=0.0

p=0.9 p=0.5

p=0.2 p=0.1

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Average path length and clustering


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Conclusion

Long range edges explain Small-World property

Variants

Adding long range edges instead of rewiring

Different initial regular networks


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Models according to clustering

Serrano-Bogu model

Hidden hyperbolic space model


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Serrano-Bogu model



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Serrano-Bogu model

Model based on Configuration Model (2005)

N: number of nodes


c(k): clustering distribution

Assignment of degrees to nodes according to P(k)

Assignment of number of triangles to each degree class

according to c(k)

Triangle formation

Closure of the network

The algorithm preserves both degree and clustering

distributions


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Serrano-Bogu model

Results

clustering distribution degree distribution


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Serrano-Bogu model



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Model by Krioukov, Papadopoulos, Vahdat & Bogu

(2009)

N: number of nodes


c(k): clustering distribution

The network is embedded in a hidden 2D hyperbolic space

The hidden space curvature affects the degree distribution

The temperature of the model affects the clustering

distribution

The model may be used for local routing


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Results


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Models according to communities

Planted partition model

Lancichinetti-Fortunato-Radicchi model (LFR)


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Model by Condon & Karp (2001)

n: number of communities

m: number of nodes per community

pin: probability of having an edge inside a community

pout: probability of having an edge between communities

The probability of having and edge between each pair of

nodes is

pin if they belong to the same community

pout if they belong to different communities

Based on ER model GN,p

zin = pin (m - 1): expected internal degree

zout = pout m (n - 1): expected external degree


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Girvan-Newman model (2002) is a particular case with

n = 4

m = 32

= zin + zout = 16

zin = 15 zin = 11


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Model based on planted partition model and

configuration model (2008)

1: exponent of the SF degree distribution

2: exponent of the SF distribution of community sizes

: mixing parameter, fraction of links to other communities

Generation of the community sizes according to 2

Generation of the degrees of the nodes according to 1 Creation and random assignment of the stubs for the internal

edges

Creation and random assignment of the stubs for the external

edges


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Example


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Variant

Extension by Lancichinetti & Fortunato (2009) for weighted

and directed networks with overlapping communities


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Other models

Models for hierarchical networks

Models for multiplex networks


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Other models




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Hierarchical model by Ravasz & Barabsi (2003)

RB25 and RB125


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Two-level planted partitions by Arenas, Daz-Guilera &

Prez-Vicente (2006)

H13-4-1


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Other models




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Multiplex networks

Edges of different classes: one layer per class of edge

The same nodes present in all layers

Models

Combinations of models for single layers (ER, BA, etc.)

Correlation between layers, e.g. degree correlation

(assortative, dissortative, random), community correlations,

etc.


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Outline

Models

Regular networks


Concluding remarks


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Concluding remarks

We need complex network models

Many models available to account for different

properties

Most important ones: ER, BA, CM, WS

New models needed for new network paradigms

(multiplex, interconnected, time-varying, etc.)


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Thank you for your attention!

Contact [email protected]

http://deim.urv.cat/~sergio.gomez


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