Chapitre 2: métriques de graphes et modèles...

Chapitre 2: metriques de graphes et modeles generatifs

Sophie Achard

GIPSA-lab, CNRS, Univ. Grenoble Alpes

M1 MIASHS

Sophie Achard (CNRS, Grenoble) Chapitre 2 26/09/2018 1 / 23

Graph features: sparsity

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low sparsity high sparsity

Density = number of edges that are present in the graph divided by thetotal number of possible edges.

Ds(G ) =|E |

N ∗ (N − 1)/2


Graph features: degree

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Degree = number of connections that node makes to other nodes.A = [Aij ]16i ,j6N is the adjacency matrix 1 6 i , j 6 N, Aij = 0 or 1.

di =∑j∈G

Aij .


Degree distribution

degree distribution

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degree sequence = {1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 6} 0 1 2 3 4 5 6

degree

Fra

ctio

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ver

tices

with

giv

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egre

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0.0

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Degree distribution

Given a degree sequence, the degree distribution is the set of discreteprobabilities {p(k)} for k = 0, . . . ,N − 1 such that

p(k) =#{i ∈ V , di = k}

N

Verify thatk=N−1∑k=0

pk = 1


Power law: scale-free graphs

P(k) = k−γ


Scale-free

Characterisation of graphs using the distribution of degrees.Representation of the cumulative distribution in a log-log plot.

0 1 2 3 4

−4

−3

−2

−1

log(k)

log(

cum

ulat

ive

dist

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ion)

+ data−− power law.. exponential law− truncated power law


Graph features: shortest path

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low average SP high average SP

A shortest path is a path between two vertices such that no shorter pathexists.

Average shortest path is the mean other all possible shortest paths.


Graph features: clustering

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C close to 0 C close to 1

Clustering = measure of information transfer in the immediateneighbourhood of each node. Let Ni denote the neighbourhood of node i .

We have: di = |Ni |.

Ci =2×#edges connecting nodes in Ni

di (di − 1).


Assortativity


Assortativity

r =|E |−1

∑Ni=1 jiki − [|E |−1

∑Ni=1

12(ji + ki )]2

|E |−1∑N

i=112(j2i + k2i )− [|E |−1

∑Ni=1

12(ji + ki )]2

(1)

where ji and ki are the degrees of the vertices at the ends of the ith edge.


Interpretation of graph metrics

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Interpretation of graph metrics

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sparse dense

Sparsity

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High Eglob Low Eglob

Shortest path

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Low Clust High Clust

Clustering

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Low Modularity High Modularity

Modularity


Regular graphs

Definition

A graph G is regular if each vertex has the same number of neighbours.We will denote a k-regular graph, a graph with k neighbours for eachvertex.

Lattice graph k-regular random graph

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Regular lattice graphs

Definition

A graph G is called a lattice graphs if it is defined on a grid.

Lattice graph Erdos-Renyi random graph

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Lattice graphs

Could you guess what will be the distribution of degree ? The minimumpath length ? The clustering ?


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Lreg ∼card(V )

2 < k >, and Creg =

3

4

where < k >= d(G) is the mean number of edges per vertex. (as shown inthe Watts and Strogatz model)Sophie Achard (CNRS, Grenoble) Chapitre 2 26/09/2018 13 / 23

Erdos-Renyi random graphs

Definition

GN,M model: a graph is chosen uniformly at random from thecollection of all graphs which have N vertices and M edges. Forexample, in the G3,2 model, each of the three possible graphs on threevertices and two edges are included with probability 1/3.

GN,p model: a graph is constructed by connecting nodes randomly.Each edge is included in the graph with probability p independentfrom every other edge. Equivalently, all graphs with N nodes and Medges have equal probability of

pM(1− p)L−M

where L = N(N − 1)/2.


Erdos-Renyi random graphs


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Properties of Erdos-Renyi random graphs

Theorem

The distribution of the degree of any particular vertex is binomial:

P(d(v) = k) =

(N − 1

k

)pk(1− p)N−1−k

The average degree per vertex is equal to (N − 1)p

GN,p should behave similarly to GN,M with M =(N2

)p as N increases.


Random graphs

Could you guess what will be the distribution of degree ? The minimumpath length ? The clustering ?


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Assuming card(V ) >< k >> ln(card(V )) > 1,

Lrand ∼log(card(V ))

log(< k >), and Crand =

< k >

N

where < k >= d(G) is the mean number of edges per vertex, andcard(V ) = N.Sophie Achard (CNRS, Grenoble) Chapitre 2 26/09/2018 16 / 23

Chung-Lu Model

Given a degree sequence d = (d1, d2, . . . , dN). Let us write 2ω =∑

i di .We suppose that maxi d

2i < 2ω. An instance is generated by connecting

all pairs (i , j) with probability:

pij =di dj2ω

, (2)


Watts and Strogatz Model

Regular Small−world Random

[Watts and Strogatz, Nature, 1998]


Watts and Strogatz model

A simple one: the Watts and Strogatz model.

How to move from a regular graph to a random one by rewiring the edges?

Regular Small−world Random



WS Model: Metrics

A simple one: the Watts and Strogatz model.



Barabasi-Albert model

Definition

The network begins with an initial connected network of m0 nodes. Newnodes are added to the network one at a time. Each new node isconnected to m ≤ m0 existing nodes with a probability that is proportionalto the number of links that the existing nodes already have. Formally, theprobability pi that the new node is connected to node i is

pi =ki∑j kj

,

where ki is the degree of node i and the sum is made over all pre-existingnodes j (i.e. the denominator results in the current number of edges in thenetwork). Heavily linked nodes (”hubs”) tend to quickly accumulate evenmore links, while nodes with only a few links are unlikely to be chosen asthe destination for a new link. The new nodes have a ”preference” toattach themselves to the already heavily linked nodes.

Note: in this case, P(k) = k−3Sophie Achard (CNRS, Grenoble) Chapitre 2 26/09/2018 21 / 23

Barabasi-Albert model

Theorem

Using preferential attachement, one can show that (asymptotically in t):

P(k) ∼ k−3

Note: this is shown using the calculation of time dependence of degree kiof vertex i .


An example on real data

Comparison of the human brain functional network with other networks:

Erdos-Renyi random graphs : randomly chosen connections

Scale-free graphs : distribution of the degree = power law(e.g. WWW)

Random

Scale-free

Brain


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