Visualiser et fouiller des réseaux - Méthodes et exemples dans R

Visualiser et fouiller des réseaux
Méthodes et exemples dans R
Nathalie Villa-Vialaneix - nathalie.villa@toulouse.inra.fr
http://www.nathalievilla.org
Unité MIA-T, INRA, Toulouse
AG PEPI IBIS - 1er avril 2014
AG PEPI IBIS (01/04/2014) Network & R Nathalie Villa-Vialaneix 1 / 39

Outline
1 Import data
2 Visualization
3 Global characteristics
4 Numerical characteristics calculation
5 Clustering

What is a network/graph? réseau/graphe
Mathematical object used to model relational data between entities.

The entities are called the nodes or the vertexes (vertices in British)
n÷uds/sommets

A relation between two entities is modeled by an edge
arête

(non biological) Examples
Social network: nodes: persons - edges: 2 persons are connected
(friends), as in my facebook network:
In the following: this network will be used to illustrate the presentation.
Note: if you want to test the script with your own facebook network, you can extract it
at http://shiny.nathalievilla.org/fbs.
AG PEPI IBIS (01/04/2014) Network R Nathalie Villa-Vialaneix 4 / 39

Modeling a large corpus of medieval documents
Notarial acts (mostly baux à ef, more
precisely, land charters) established in a
seigneurie named Castelnau Montratier,
written between 1250 and 1500, involving
tenants and lords.
a
a
http://graphcomp.univ-tlse2.fr

Modeling a large corpus of medieval documents
• nodes: transactions and individuals
(3 918 nodes)
• edges: an individual is directly involved
in a transaction (6 455 edges)

Standard issues associated with networks
Inference
Giving data, how to build a graph whose edges represent the direct links
between variables?
Example: co-expression networks built from microarray data (nodes = genes; edges =
signicant direct links between expressions of two genes)

Inference
between variables?
Graph mining (examples)
1 Network visualization: nodes are not a priori associated to a given
position. How to represent the network in a meaningful way?
Random positions
Positions aiming at representing
connected nodes closer

Inference
between variables?
Graph mining (examples)
1 Network visualization: nodes are not a priori associated to a given
position. How to represent the network in a meaningful way?
2 Network clustering: identify communities (groups of nodes that are
densely connected and share a few links (comparatively) with the other groups)

More complex relational models
Nodes may be labeled by a factor

... or by a numerical information. [Villa-Vialaneix et al., 2013]

... or by a numerical information. [Villa-Vialaneix et al., 2013]
Edges may also be labeled (type of the relation) or weighted (strength of
the relation) or directed (direction of the relation).

Before we start...
Available material
• this slide (on slideshare or on my website, page seminars);
• data: my facebook network with two les: fbnet-el.txt (edge list)
and fbnet-name.txt (node names and list)
• script: a R script with all command lines included in this slide,
fb-Rscript.R

Before we start...
Available material
• this slide (on slideshare or on my website, page seminars);
• data: my facebook network with two les: fbnet-el.txt (edge list)
and fbnet-name.txt (node names and list)
• script: a R script with all command lines included in this slide,
fb-Rscript.R
Outline
Introduce basic concepts on network mining (not inference)
Illustrate them with R (required package: igraph) on my facebook
network

Import data
Outline
1 Import data
2 Visualization
5 Clustering

Import data
Import a graph from an edge list with igraph
edgelist - as.matrix(read.table(fbnet -el.txt))
vnames - read.table(fbnet -name.txt)
vnames - read.table(fbnet -name.txt, sep=,,
stringsAsFactor=FALSE ,
na.strings=)
The graph is built with:
# with ` graph . edgelist '
fbnet0 - graph.edgelist(edgelist , directed=FALSE)
fbnet0
# IGRAPH U --- 152 551 --
See also help(graph.edgelist) for more graph constructors (from an
adjacency matrix, from an edge list with a data frame describind the nodes,
...)

Import data
Vertexes, vertex attributes
The graph's vertexes are accessed and counted with:
V(fbnet0)
# Vertex sequence :
# [1] 1 2 3 4 5...
vcount(fbnet0)
# [1] 152
Vertexes can be described by attributes:
# add an attribute for vertices
V(fbnet0)$initials - vnames [,1]
V(fbnet0)$list - vnames [,2]
fbnet0
# IGRAPH U --- 152 551 --
# + attr : initials (v/c), list (v/c)

Import data
Edges, edge attributes
The graph's edges are accessed and counted with:
E(fbnet0)
# [1] 11 -- 1
# [2] 41 -- 1
# [3] 52 -- 1
# [4] 69 -- 1
# [5] 74 -- 1
# [6] 75 -- 1
# ...
ecount(fbnet0)
# 551
igraph can also handle edge attributes (and also graph attributes).

Import data
Settings
Notations
In the following, a graph G = (V , E, W ) with:
• V : set of vertexes {x1, . . . , xp};
• E: set of edges;
• W : weights on edges s.t. Wij ≥ 0, Wij = Wji and Wii = 0.

Import data
Settings
Notations
The graph is said to be connected/connexe if any node can be reached
from any other node by a path/un chemin.

Import data
Settings
Notations
The graph is said to be connected/connexe if any node can be reached
from any other node by a path/un chemin.
The connected components/composantes connexes of a graph are all its
connected subgraphs.

Import data
Settings
Notations
Example 1: Natty's FB network has 21 connected components with 122
vertexes (professional contacts, family and closest friends) or from 1 to 5
vertexes (isolated nodes)

Import data
Settings
Notations
Example 2: Medieval network: 10 542 nodes and the largest connected
component contains 10 025 nodes (giant component / composante
géante).

Import data
Connected components
is.connected(fbnet0)
# [1] FALSE
As this network is not connected, the connected components can be
extracted:
fb.components - clusters(fbnet0)
names(fb.components)
# [1] membership csize no
head(fb.components$membership , 10)
# [1] 1 1 2 2 1 1 1 1 3 1
fb.components$csize
# [1] 122 5 1 1 2 1 1 1 2 1
# [11] 1 2 1 1 2 3 1 1 1 1
# [21] 1
fb.components$no
# [1] 21

Import data
Largest connected component
fbnet.lcc - induced.subgraph(fbnet0 ,
fb.components$membership ==
which.max(fb.components$csize))
# main characteristics of the LCC
fbnet.lcc
# IGRAPH U --- 122 535 --
# + attr : initials (v/x)
is.connected(fbnet.lcc)
# [1] TRUE

Visualization
Outline
1 Import data
2 Visualization
5 Clustering

Visualization
Visualization tools help understand the graph
macro-structure
Purpose: How to display the nodes in a meaningful and aesthetic way?

Visualization
macro-structure
Standard approach: force directed placement algorithms (FDP)
algorithmes de forces (e.g., [Fruchterman and Reingold, 1991])

Visualization
macro-structure
• attractive forces: similar to springs along the edges

Visualization
macro-structure
• repulsive forces: similar to electric forces between all pairs of vertexes

Visualization
macro-structure
• repulsive forces: similar to electric forces between all pairs of vertexes
iterative algorithm until stabilization of the vertex positions.

Visualization
Visualization
• package igraph1
[Csardi and Nepusz, 2006] (static
representation with useful tools for graph mining)
1
http://igraph.sourceforge.net/
2
http://gephi.org, http://tulip.labri.fr/TulipDrupal/, http://www.cytoscape.org/

Visualization
Visualization
• package igraph1
[Csardi and Nepusz, 2006] (static
representation with useful tools for graph mining)
• free interactive software: Gephi, Tulip,
Cytoscape, ...
2
1
http://igraph.sourceforge.net/
2
http://gephi.org, http://tulip.labri.fr/TulipDrupal/, http://www.cytoscape.org/

Visualization
Network visualization
Dierent layouts are implemented in igraph to visualize the graph:
plot(fbnet.lcc , layout=layout.random ,
main=random layout, vertex.size=3,
vertex.color=pink, vertex.frame.color=pink
vertex.label.color=darkred,
edge.color=grey,
vertex.label=V(fbnet.lcc)$initials)
Try also layout.circle, layout.kamada.kawai,
layout.fruchterman.reingold... Network are generated with some
randomness. See also help(igraph.plotting) for more information on
network visualization

Visualization

Visualization
Plot attributes
igraph integrates a pre-dened graph attribute layout:
V(fbnet.lcc)$label - V(fbnet.lcc)$initials
and label and color node attributes
V(fbnet.lcc)$label - V(fbnet.lcc)$initials
V(fbnet.lcc)$color - rainbow(length(unique(
V(fbnet.lcc)$list )))[
as.numeric(factor(V(fbnet.lcc)$list ))]
plot(fbnet.lcc , vertex.size=3,
vertex.label.color=black, edge.color=grey)

Visualization

Global characteristics
Outline
1 Import data
2 Visualization
5 Clustering

Density / Transitivity Densité / Transitivité
Density: Number of edges divided by the number of pairs of vertexes. Is
the network densely connected?

Examples
Example 1: Natty's FB network
• 152 vertexes, 551 edges ⇒ density = 551
152×151/2
4.8%;
• largest connected component: 122 vertexes, 535 edges ⇒ density
7.2%.
Example 2: Medieval network (largest connected component): 10 025
vertexes, 17 612 edges ⇒ density 0.035%.
Projected network (individuals): 3 755 vertexes, 8 315 edges ⇒ density
0.12%.

Transitivity: Number of triangles divided by the number of triplets
connected by at least two edges. What is the probability that two friends of
mine are also friends?
Density is equal to
4
4×3/2
= 2/3 ; Transitivity is equal to 1/3.

Examples
Example 1: Natty's FB network
• density 4.8%, transitivity 56.2%;
• largest connected component: density 7.2%, transitivity 56.0%.
Example 2: Medieval network (projected network, individuals): density
0.12%, transitivity 6.1%.

graph.density(fbnet.lcc)
# [1] 0.0724834
transitivity(fbnet.lcc)
# [1] 0.5604524

Numerical characteristics calculation
Outline
1 Import data
2 Visualization
5 Clustering

Extracting important nodes
1 vertex degree degré: number of edges adjacent to a given vertex or di = j
Wij .
Vertexes with a high degree are called hubs: measure of the vertex
popularity.
Number of nodes (y-axis) with a given degree (x-axis)

Wij .
popularity.
Two hubs are students who have been hold back at school and the
other two are from my most recent class.

Wij .
popularity.

Wij .
The degree distribution is known to t a power law loi de puissance
in most real networks:
1 2 5 10 20 50 100 200 500
1550500
Names
Transactions
This distribution indicates preferential attachement attachement
préférentiel.

Wij .
2 vertex betweenness centralité: number of shortest paths between all pairs of
vertexes that pass through the vertex. Betweenness is a centrality measure
(vertexes that are likely to disconnect the network if removed).
The orange node's degree is equal to 2, its betweenness to 4.

Wij .

Wij .
Vertexes with a high be-
tweenness ( 3 000) are
2 political gures.

Wij .
Example 2: In the medieval network: moral persons such as the
Chapter of Cahors or the Church of Flaugnac have a high
betweenness despite a low degree.

Degree and betweenness
fbnet.degrees - degree(fbnet.lcc)
summary(fbnet.degrees)
# Min . 1 st Qu . Median Mean 3 rd Qu . Max .
# 1.00 2.00 6.00 8.77 15.00 31.00
fbnet.between - betweenness(fbnet.lcc)
summary(fbnet.between)
# Min . 1 st Qu . Median Mean 3 rd Qu . Max .
# 0.00 0.00 14.03 301.70 123.10 3439.00
and their distributions:
par(mfrow=c(1,2))
plot(density(fbnet.degrees), lwd=2,
main=Degree distribution, xlab=Degree,
ylab=Density)
plot(density(fbnet.between), lwd=2,
main=Betweenness distribution,
xlab=Betweenness, ylab=Density)

Degree and betweenness distribution

Combine visualization and individual characteristics
par(mar=rep(1,4))
# set node attribute `size ' with degree
V(fbnet.lcc)$size - 2*sqrt(fbnet.degrees)
# set node attribute ` color ' with betweenness
bet.col - cut(log(fbnet.between +1),10,
labels=FALSE)
V(fbnet.lcc)$color - heat.colors (10)[11 - bet.col]
plot(fbnet.lcc , main=Degree and betweenness,
vertex.frame.color=heat.colors (10)[ bet.col],
vertex.label=NA, edge.color=grey)

Clustering
Outline
1 Import data
2 Visualization
5 Clustering

Clustering
Vertex clustering classication
Cluster vertexes into groups that are densely connected and share a few
links (comparatively) with the other groups. Clusters are often called
communities communautés (social sciences) or modules modules
(biology).

Clustering
(biology).
Example 1: Natty's facebook network

Clustering
(biology).
Example 2: medieval network

Clustering
(biology).
Several clustering methods:
• min cut minimization minimizes the number of edges between clusters;
• spectral clustering [von Luxburg, 2007] and kernel clustering uses
eigen-decomposition of the Laplacian/Laplacien
Lij =
−wij if i = j
di otherwise
(matrix strongly related to the graph structure);
• Generative (Bayesian) models [Zanghi et al., 2008];
• Markov clustering simulate a ow on the graph;
• modularity maximization
• ... (clustering jungle... see e.g., [Fortunato and Barthélémy, 2007,
Schaeer, 2007, Brohée and van Helden, 2006])

Clustering
Find clusters by modularity optimization modularité
The modularity [Newman and Girvan, 2004] of the partition
(C1, . . . , CK) is equal to:
Q(C1, . . . , CK) =
1
2m
K
k=1 xi ,xj ∈Ck
(Wij − Pij)
with Pij: weight of a null model (graph with the same degree distribution
but no preferential attachment):
Pij =
didj
2m
with di = 1
2 j=i Wij.

Clustering
Interpretation
A good clustering should maximize the modularity:
• Q when (xi, xj) are in the same cluster and Wij Pij
• Q when (xi, xj) are in two dierent clusters and Wij Pij
(m = 20)
Pij = 7.5
Wij = 5 ⇒ Wij − Pij = −2.5
di = 15 dj = 20
i and j in the same cluster decreases the modularity

Clustering
Interpretation
(m = 20)
Pij = 0.05
Wij = 5 ⇒ Wij − Pij = 4.95
di = 1 dj = 2
i and j in the same cluster increases the modularity

Clustering
Interpretation
• Modularity
• helps separate hubs (= spectral clustering or min cut criterion);
• is not an increasing function of the number of clusters: useful to
choose the relevant number of clusters (with a grid search: several
values are tested, the clustering with the highest modularity is kept)
but modularity has a small resolution default (see
[Fortunato and Barthélémy, 2007])

Clustering
Interpretation
• Modularity
• helps separate hubs (= spectral clustering or min cut criterion);
• is not an increasing function of the number of clusters: useful to
choose the relevant number of clusters (with a grid search: several
values are tested, the clustering with the highest modularity is kept)
but modularity has a small resolution default (see
[Fortunato and Barthélémy, 2007])
Main issue: Optimization = NP-complete problem (exhaustive search is
not not usable)
Dierent solutions are provided in
[Newman and Girvan, 2004, Blondel et al., 2008,
Noack and Rotta, 2009, Rossi and Villa-Vialaneix, 2011] (among
others) and some of them are implemented in the R package igraph.

Clustering
Node clustering
One of the function to perform node clustering is multilevel.community
(probably not the best with large graphs):
fbnet.clusters - multilevel.community(fb.lcc)
modularity(fbnet.clusters)
# Graph community structure calculated with the mul
# Number of communities ( best split ): 7
# Modularity ( best split ): 0.566977
# Membership vector :
# [1] 3 2 2 2 2 3 5 3 5 5 4 3 3 2 ...
table(fbnet.clusters$membership)
# 1 2 3 4 5 6 7
# 2 18 26 32 12 8 24
See help(communities) for more information.

Clustering
Combine clustering and visualization
# create a new attribute
V(fbnet.lcc)$community - fbnet.clusters$membership
fbnet.lcc
# IGRAPH U --- 122 535 --
#+ attr : layout (g/n), initials (v/c), list (v/c),
# label (v/c), color (v/c), community (v/n),
# size (v/n)
Display the clustering:
par(mfrow=c(1,1))
par(mar=rep(1,4))
plot(fbnet.lcc , main=Communities,
vertex.frame.color=
rainbow (9)[ fbnet.clusters$membership],
vertex.color=
rainbow (9)[ fbnet.clusters$membership],
vertex.label=NA, edge.color=grey)

Clustering
Combine clustering and visualization

Clustering
Export the graph
The graphml format can be used to export the graph (it can be read by
most of the graph visualization programs and includes information on node
and edge attributes)
write.graph(fbnet.lcc , file=fblcc.graphml,
format=graphml)
see help(write.graph) for more information of graph exportation formats

Clustering
Further references...
on clustering...
• overlapping communities communautés recouvrantes;
• hierarchical clustering [Rossi and Villa-Vialaneix, 2011] provides an approach;
• organized clustering (projection on a small dimensional grid) and
clustering for visualization [Boulet et al., 2008,
Rossi and Villa-Vialaneix, 2010, Rossi and Villa-Vialaneix, 2011];
• ...
on R
CRAN Task View: gRaphical Models in R
on network inference
you can start with the course An introduction to network inference and
mining

Clustering
References
Blondel, V., Guillaume, J., Lambiotte, R., and Lefebvre, E. (2008).
Fast unfolding of communites in large networks.
Journal of Statistical Mechanics: Theory and Experiment, P10008:17425468.
Boulet, R., Jouve, B., Rossi, F., and Villa, N. (2008).
Batch kernel SOM and related Laplacian methods for social network analysis.
Neurocomputing, 71(7-9):12571273.
Brohée, S. and van Helden, J. (2006).
Evaluation of clustering algorithms for protein-protein interaction networks.
BMC Bioinformatics, 7(488).
Csardi, G. and Nepusz, T. (2006).
The igraph software package for complex network research.
InterJournal, Complex Systems.
Fortunato, S. and Barthélémy, M. (2007).
Resolution limit in community detection.
In Proceedings of the National Academy of Sciences, volume 104, pages 3641.
doi:10.1073/pnas.0605965104; URL: http://www.pnas.org/content/104/1/36.abstract.
Fruchterman, T. and Reingold, B. (1991).
Graph drawing by force-directed placement.
Software, Practice and Experience, 21:11291164.
Newman, M. and Girvan, M. (2004).
Finding and evaluating community structure in networks.
Physical Review, E, 69:026113.
Noack, A. and Rotta, R. (2009).
Multi-level algorithms for modularity clustering.

Clustering
In SEA '09: Proceedings of the 8th International Symposium on Experimental Algorithms, pages 257268,
Berlin, Heidelberg. Springer-Verlag.
Rossi, F. and Villa-Vialaneix, N. (2010).
Optimizing an organized modularity measure for topographic graph clustering: a deterministic annealing
approach.
Neurocomputing, 73(7-9):11421163.
Rossi, F. and Villa-Vialaneix, N. (2011).
Représentation d'un grand réseau à partir d'une classication hiérarchique de ses sommets.
Journal de la Société Française de Statistique, 152(3):3465.
Schaeer, S. (2007).
Graph clustering.
Computer Science Review, 1(1):2764.
Villa-Vialaneix, N., Liaubet, L., Laurent, T., Cherel, P., Gamot, A., and SanCristobal, M. (2013).
The structure of a gene co-expression network reveals biological functions underlying eQTLs.
PLoS ONE, 8(4):e60045.
von Luxburg, U. (2007).
A tutorial on spectral clustering.
Statistics and Computing, 17(4):395416.
Zanghi, H., Ambroise, C., and Miele, V. (2008).
Fast online graph clustering via erdös-rényi mixture.
Pattern Recognition, 41:35923599.

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