Lectures on Complex Networks - Home — dibernardodibernardo.tigem.it/courses-and-openings/files/... · ijikjk ii i aaa kk C 1/2 1 CLUSTERING DIRECTED (or even worse WEIGHTED) CLUSTERING

Lectures on Complex Networks


G. CaldarelliCNR-INFM Centre SMC Dep. Physics University “Sapienza” Rome, Italyhttp://www.guidocaldarelli.com


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INTRODUCTION

1.1 Networks Ubiquity

6From TechnologicalSystems to Social andBiological ones, similarstructures appear.


GRAPH THEORY

1.5 The Properties of Complex Networks

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•Scale Invariant Degree Distribution P(k)•Distribution of distances P(l) peaked around small values•Clustering (with respect to random connections)•Assortativity


GRAPH THEORY


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•Scale Invariant DegreeDistribution P(k)

b) Actors

d) Neuroscientists

a) WWW

c) Physicists


GRAPH THEORY


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•Distribution of distances p(l) peaked around small values

A.Vazquez, R Pastor-Satorras, A.Vespignani Phys.Rev.E 65 066130 (2002)


GRAPH THEORY


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•C l u s t e r i n g(with respect tor a n d o mconnections)

M.E.J. Newman, M. Girvan, Phys.Rev.E 69 026113 (2004)


GRAPH THEORY


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•Assortativity C(k)~k-! or average degree of neighbours <Knn>

a, In assortative networks, well-connected nodes tend to join to other well-connected nodes, as in many social networks —here illustrated by friendship links in a school in the United States6. b, In disassortative networks, by contrast, well-connected nodes join to a much larger number of less-well-connected nodes. This is typical of biological networks; depictedhere is the web of interactions between proteins in brewer's yeast, Saccharomyces cerevisiae7. Clauset and colleagues'hierarchical random graphs2 provide an easy way to categorize such networks.

S. Redner, Nature 453, 47-48 (2008)


GRAPH THEORY

1.1.2 Basic Quantities (degree)

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! Degree k (In-degree kin and out-degree kout ) = number of edges (oriented) per vertex

! Distance d = number of edges amongst two vertices ( in the connected region !)

! Diameter D = Maximum of the distances ( in the connected region !)

A Graph is an objectcomposed by verticesand edges


GRAPH THEORY

1.1 Definition and Adjacency Matrix

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GRAPH THEORY

Through the adjacency matrix we can write11.1

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!=

=nj

iji ak,1

!=

=nj

ij

in

i ak,1

!=

=nj

ji

out

i ak,1

DEGREE

WEIGHTED DEGREE (Strength) !=

=nj

w

ij

w

i ak,1


SIMPLE ORIENTED WEIGHTED


GRAPH THEORY

One way to visualize the behaviour of the degree in anetwork (especially for large ones) is to check the behaviourof the degree frequency distribution P(k)

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GRAPH THEORY

1.1.3 Basic Quantities (distance)

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Through the adjacency matrix we can write

!"

!#$

!%

!&'

= () ij Pk,l

klij ad minDISTANCE

WEIGHTED DISTANCE!"

!#$

!%

!&'

= () ij Pk,l

w

klij ad min

!"

!#$

!%

!&'

= () ij Pk,l

w

kl

ija

d1

min

DISTANCE "INVERSE OFWEIGHTS

DISTANCE " SUM OF WEIGHTS


GRAPH THEORY

1.1.3 Basic Quantities (distance)

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Similarly to the degree, one usually plots the histogram ofthe frequency density P(d) of distances d


GRAPH THEORY

1.1.4 Basic Quantities (Clustering)

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GRAPH THEORY

1.1.4 Basic Quantities (Clustering)

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Through the adjacency matrix we can write

( ) !"

=jk

jkikij

ii

i aaakk

C2/1

1CLUSTERING

DIRECTED (or even worse WEIGHTED) CLUSTERING

•J Saramäki et al. Phys Rev E 75 027105 (2007).


GRAPH THEORY

1.2 Trees

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GRAPH THEORY

1.2 Trees

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Trees are particularly important, since they appear in

•Girvan-Newman Algorithm for communities•Taxonomy of species•Taxonomy of information


GRAPH THEORY

1.2 Trees

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Trees can also be viewed as a way to filter information. In astock exchange the prices of all stocks are correlated, butrestricting only to a Minimum Spanning Tree is a way tovisualize the stronger correlations

In order to build a MinimumSpanning Tree

1) Compute the correlationbetween the N(n-1)/2 vertices

2) Define a distance out ofcorrelation.

3) Rank the distances4) Draw the vertices of the

shortest distance5) Run on the ranking, whenever

you find a new (or two)vertex(-ices), draw it (them)if you do not close a cycle.

6) Stop whenever all the verticeshave been drawn


GRAPH THEORY

1.2 Trees

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One way to visualize the behaviour of a tree (on top of thedegree) is to check the behaviour of the basin size frequencydistribution P(n).


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GRAPH THEORY

1.3 Vertex Correlation: Assortativity

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A way to check the Conditioned Probability P(k|k’) that avertex whose degree is k is connected with another vertexwith degree k’ is given by the measure of the average degreeof the neighbours.


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GRAPH THEORY


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Similarly, one can define a measure of the assortativity bystarting from the correlation function for degrees of vertices i,j

If we introduce the variance

We have the ASSORTATIVITY COEFFICIENT r given by


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GRAPH THEORY


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2.6 M.E.J. Newman, Phys.Rev.Lett 89 208701 (2002)


GRAPH STRUCTURE: COMMUNITIES

2.1 Introduction

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The presence of communities in a graph is one of the mostimportant features.•Communities are important for Amazon, to run their businesses•For biologists to detect proteins with the same function•For physicians to detect related diseases


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2.2 Motifs

Motifs are the smallest version of communities, it is currentlyunder debate if their presence is important or not in the area ofComplex networks.

Mangan, S. and Alon, U. PNAS 100, 11980–11985 (2003).Mazurie et al. Genome Biology 6 R35 (2005)


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2.3 Classes of Vertices

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Some communites must be defined as made by vertices withsimilar properties (they could share no edge at all).More often the communities are made by vertices sharing manyedges


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2.4 Centrality Measures Betwenness and Robustness

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The concept of centrality is at the basis of the Newman-Girvanmethod for the analysis of communities.

Girvan, M. and Newman, M.E.J. PNAS, 99, 7821–7826 (2002).


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!Starting from a graphWe iteratively•Compute betweenness•Cut the edges with thelargest value of it

For the graph above the "result is given by the seriesof deletionsE-F, E-I, B-D, A-D, C-F,C-L,D-H, G-L, D-I, B-E, A-B, H-I,F-L, C-G.


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2.6Guimerà, R., Danon, L., Díaz-Guilera, A., Giralt, F., and Arenas, A..Physical Review E, 68, 065103 (2002).


Biological Networks

6.1 Protein Interaction Network

Protein interact in various ways during cell life.

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11Uetz et al. Nature 403 623 (2000)H. Jeong et al. Nature, 411, 41 (2002)Giot et al. Science 302, 1727 (2003)Rual et al. Nature 437, 1173 (2005)Ramani et al. Molecular Biology 4, 180 (2008)


Biological Networks

6.2 Metabolic Networks

Multinetwork analysis of acarbon/light/nitrogen-responsive metabolicnetwork.

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Most of the reactions arenot reversible and thenetwork is oriented.

M.L. Gifford et al. Plant Physioloogy on-line Essay 12.2H. Jeong et al. Nature 407 651 (2000).D. Segré et al. PNAS 99 15112 (2002).R. Gumerà et al. Nature 433, 895 (2005)


INTRODUCTION

6.3 Gene Regulatory Networks

A particular class ofreactions in which thereis the expression of agene is at the basis ofGRN.

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11J. Hasty et al. Nat. Rev. Gen., 2, 268 (2001).

T.I. Lee et al. Science 298, 799 (2002)


Geophysical Networks

Trees and Basin distributions

The case of rivernetworks presents ascale invariance inthe basin sizedistributions

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11Rodrıguez-Iturbe, I. and Rinaldo, A.Fractal River Basins: Chance and Self-Organization.Cambridge University Press, Cambridge (1996).

Dodds, P.S. and Rothman, D.H. Physical Review E,63, 016115, 016116, 016117 (2000).


Ecological Networks

Food Webs data

Most of theactivity onecologicalnetworks isrelated tothestatisticalpropertiesof foodwebs

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11PEaCE LAB http://www.foodwebs.org

R.J. Williams N.D. Martinez Nature 404 180 (2000)


Ecological Networks

Statistical properties of food webs

J.M. Montoya, R.V. Solé, Journal ofTheoretical Biology 214 405 (2002)

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11D.B. Stouffer et al. Ecology, 86, 1301–1311 (2005).

Small-world but notscale-free


Technological Networks

9.1 Internet

Traditionally the analysisof the Internet structure ismade by means oftraceroutes. That is to say,by exploring all the pathsfrom a given point to allthe possible destinations.

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A. Vazquez, R. Pastor-Satorras and A.Vespignani PRE 65 066130 (2002)

R. Pastor-Satorras and A. VespignaniEvolution and Structure ofInternet: A Statistical Physics Approach.Cambridge University Press (2004)



9.2 World Wide Web

WWW is probably thelargest networkavailable of the orderof billions of elements.Different centralityproperties arise in suchstructure.

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11 A. Broder et al. Computer Network, 33, 309 (2000).

D. Donato et al. Journal of Phys A, 41, 224017 (2008).


Social Networks

Wikipedia66.1

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Social Networks

Wikipedia66.1

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11in–degree(empty) and out–degree(filled)Occurrency distributions for the Wikgraphin English (o) and Portuguese (#).

The degree shows fat tailsthat can be approximated

by a power-law function ofthe kind P(k) ~ k-!

Where the exponent isthe same both for

in-degree and out-degree.

V. Zlatic et al. Physical Review E, 74, 016115 (2006).

A. Capocci et al. Physical Review E, 74, 036116 (2006).


Financial Networks

Stock Correlation Network66.1

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Correlation based minimal spanning trees of real data from daily stock returns of 1071 stocks for the 12-yearperiod 1987-1998 (3030 trading days). The node colour is based on Standard Industrial Classification system.The correspondence is:red for mining cyan for construction yellow for manufacturinggreen for transportation,, light blue for public electric,gas and sanitary services administration magenta wholesale tradeblack for retail trade purple for finance and insurance orange for service industries

Bonanno et al. Physical Review E 68 046130 (2003).


Financial Networks

Stock Ownership Network66.1

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Garlaschelli et al. Physica A, 350 491 (2005).


Financial Networks

Trading Webs66.1

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11M. Angeles Serrano, M. Boguna Phys Rev. E 68 015101 (2003)D. Garlaschelli et al. Physica A, 350 491 (2005). C.A. Hidalgo, et al. Science 317 482 (2007)


Financial Networks

Bank’s Credit Network66.1

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G. De Masi et al. PRE, 98 208701 (2007)

Banks exchange moneyovernightly, in order tomeet the customer needsof liquidity as well asECB requirements

The network shows a rather peculiar architecture

The banks form a disassortative network where largebanks interact mostly with small ones.


http://www.scale-freenetworks.com

INTRODUCTION


INTRODUCTION

Free Resources on the web

•R. Albert, A.-L. Barabasi STATISTICAL MECHANICS OF COMPLEX NETWORKS Review of Modern Physics 74, 47 (2002) http://arxiv.org/abs/cond-mat/0106096

•M.E.J. Newman THE STRUCTURE AND FUNCTION OF COMPLEX NETWORKS, SIAM Review 45, 167-256 (2003) http://arxiv.org/abs/cond-mat/03030516

•R. Diestel GRAPH THEORY, Springer-Verlag (2005) http://www.math.ubc.ca/~solymosi/443/GraphTheoryIII.pdf

•Networks Visualization and Analysis PAJEK, Springer-Verlag (2005) http://pajek.imfm.si/doku.php

•Bibliography I collected # http://www.citeulike.org/user/gcalda/tag/book


INTRODUCTION

Dissemination on Networks

•A.-L. Barabási Linked, Perseus (2001)

•M. Buchanan Nexus, W.W. Norton (2003)

•D. Watts Six Degrees: the science of a Connected Age W.W.Norton (2004)

•M. Buchanan The Social Atom, W. W. Norton (2007)


INTRODUCTION

Resources in your library

•S.N. Dorogovtsev, J.F.F. Mendes Evolution of Networks, OxfordUniversity Press (2003)

•S. Bornholdt, H.G. Schuster(eds) Handbook of Graphs andNetworks, Wiley (2003)

•R.Pastor Satorras, M. Rubi, A. Diaz-Guilera Statistical Mechanics

of Complex Networks Lecture Notes in Physics, Vol. 625 (2003)

•R. Pastor Satorras, A. Vespignani Evolution and Structure of theInternet Cambridge University Press (2004)


INTRODUCTION

Resources in your library

•G. Caldarelli, A. Vespignani (eds) Handbook of Graphs andNetworks, Wiley (2003)

•U Brandes T Erlebach, Network Analysis - MethodologicalFoundations, LNCS Tutorial 3418, Springer Verlag, (2005)

•M.E.J Newman, A.-L. Barabási, D. Watts The Structure andDynamics of Networks, Princeton University Press (2006)

•G. Caldarelli, A.Vespignani (eds) Large Scale Structure andDynamics of Complex Networks, World Scientific Press (2007)

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