Complex Networks Measures and deterministic models Philippe Giabbanelli

Complex NetworksMeasures and deterministic models

Philippe Giabbanelli

IntroMeasures

(clustering, degree distribution)

Main courseDeterministic models

Side dish

Generalizing fractal graphs

LeftoverDiscussion

∙ clustering augmentation ….∙ fractal graphs

Complex networks

Motifs – Clustering – Average distance – Degree distribution

Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010

Given a graph G…

and a set S of random graphs of the same size and average degree,

a motif is a subgraph that appears at a ‘very’ different frequence in

G than in S.

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Milo et al., Science, 303, 2004 Milo et al., Science, 298, 2002

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For a given node i , we denote its neighborhood by Ni.

The clustering coefficient Ci of i is the edge density of its neighborhood.

Here, there are two edges between nodes in Ni.

At most, it’d be a complete graph with Ni.(Ni-1) edges.

Ci = 2.2/(5.4) = 0.2

Complex networks



For a given node i , we denote its neighborhood by Ni.

The clustering coefficient Ci of i is the edge density of its neighborhood.

If a graph has high clustering coefficient, then there are communities (i.e., cliques) in this graph.

People tend to form communities so it is common in social networks.

Complex networks



Average distance: average length of shortest path between all pairs of nodes

The average distance l is:

∙ small if l ln(n) ∝

∙ ultrasmall if l ln(ln(n))∝

← M.E.J Newman, The structure and

function of complex networks

Complex networks



Many measured phenomena are centered around a particular value.

Complex networks



Many measured phenomena are centered around a particular value.

There also exists numerous phenomena with a heavy-tailed distribution.

lets plot it on a log-log scale

Complex networks



There also exists numerous phenomena with a heavy-tailed distribution.

The equation of a line is p(x) = -αx + c.

Here we have a line on a log-log scale:

ln p(x) = -α ln x + c

apply exponent e

p(x) = ecxc -α

We say that this distribution follows a power-law, with exponent α.

Complex networks



We say that this distribution follows a power-law, with exponent α.

Keep in mind that this is quite common.

people’s incomes

computer files

moon craters visits on web pages

Complex networks


A network with high clustering and low average distance is small-world.

fast communications locally and globally

There are other definitions (e.g., network that you can navigate easily).

A network with power-law degree distribution is scale-free.

(Luckily, there aren’t other definitions, we’re already messy enough.)

See: Efficient measurement of complex networks using link queries (Tarissan, NetSciCom’09), aaaaReverse centrality queries in complex networks (Nielsen, MSc Thesis SFU dec. ’09)

Complex networks

Small world models (Watts Strogatz, Comellas et al., Giabbanelli et al.)

Ref.: Watts & Strogatz, « Collective dynamics of ‘small-world’ networks », Nature 393, 1998

Almost all examples you will find use a simplified version.

Get n nodes labelled from 0 to n.

A node i is connected to (i+1, i+2, …, i Δ/2) mod

n. Lets use Δ = 4.

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Change one endpoint for an edge with probability p

This scheme yields ‘good’ values for 0.01 < p 0.1

Small average distanceLarge clustering coefficient

Complex networks


Ref.: Comellas, Ozon, Peters « Deterministic small-world communication networks », 2000

Get n nodes labelled from 0 to n.

A node i is connected to (i+1, i+2, …, i Δ/2) mod n.

Lets use Δ = 6.

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A double step graph C(h; a,b) has h nodes, and i is connected to i a (mod h), i b (mod h)

Select h equidistant nodes, and connect them as C(h;a,b).

h=6, a=1, b=20

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Then, some deterministic fiddling to keep the degree unchanged…

Complex networks


Ref.: Giabbanelli & Peters, submitted to AlgoTel’10

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Intuition

Consider that we start with the cycle Cn.

The added edges should provide a good coverage of distances.

When we connect i to i 1,…,i (Δ/2), we create lots of short-range edges

Adding edges from a double-step graph mainly provides

medium-range edges

Complex networks



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Intuition

Consider that we start with the cycle Cn.

The added edges should provide a good coverage of distances.

As long as d(i)≠Δ, connect i to i 2 , …, i 20 k

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Intuition

We want high clustering coefficient.

What’s the graph with the highest clustering coefficient?

→ complete graph

If a node has degree Δ-1, we add to it a K Δ

Pretty artificial… but has the values required

for small-world.

Complex networks

Fractal graphs (Graph grammar, Zhang et al., Perspectives)

We’ll just introduce as much as we need.

Starting graph

A dotted edge is said to be active.

At each time step, all dotted edges get replaced by a pattern graph.

Pattern graph

Complex networks


We’ll just introduce as much as we need.

Starting graph

A dotted edge is said to be active.

At each time step, all dotted edges get replaced by a pattern graph.

Pattern graph

Complex networks


Ref.: Zhang, Rong, Guo, Physica A: Stat. Mech. And Appl., 363, 2006

Starting graph Pattern graph

Here’s the definition of ZRG using our graph grammar. t

This generates a (planar) small-world graph.

Giabbanelli, Mazauric, Pérennes, submitted to AlgoTel’10

L

0L

1L

It also has a simple labelling scheme.

Complex networks


Ref.: Miralles, Comellas, Chen, Zhang, Physica A, 389, 2010


Here’s the definition of M using our graph grammar.d,t

d

Example for d = 2

There is no triangle so the clustering coefficient is 0.

The result is scale-free, planar, with small average distance.

For d=1: Comellas, Mirales, Physica A, 388, 2009

Complex networks


Ref.: Comellas, Zhang, Chen, J. Physc. A., 42, 2009


We’re not limited to active edges. For example, lets have active cycles.

Given the active cycle and the pattern, how do we know which edge of the cycle gets replaced by which edge of the pattern?

We use a function that maps the active cycle in the pattern (= morphism)

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How many active cycles are there in the pattern graph?

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How many active cycles are there in the pattern graph?

This is NOT an active cycle.

4

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How many active cycles are there in the pattern graph? 4

The result is scale-free, planar, with small average distance.

Complex networks


So maybe we could discuss their properties a bit more generally.

We can define tons of patterns (people actually did and published them).

Lets have a look at active edges.

Complex networks


Pattern graph

Black box

Diameter D

N nodes

P

We start from a triangle with a pattern having two active edges.

P

For each box, we add Np-2 nodes.

We start with 3 nodes, add 3 boxes:

3(Np-2)+3 nodes

Diameter at most Dp2

Step t = 1

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Pattern graph

Black box

Diameter D

N nodes

P


P

The number of added nodes doubles at each step: we now add 3.2(Np-2) nodes.

The longest path is through 2t boxes

Diameter at most 2tDp

Complex networks


Pattern graph

Black box

Diameter D

N nodes

P


P

The average distance is small regardless of the pattern you choose.

The same conclusion holds for a pattern graph with at least two active edges, and

any starting graph.

P. Giabbanelli, Properties of fractal network models, submitted to Physica A

Complex networks

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Complex Networks Measures and deterministic models Philippe Giabbanelli