Complex Networks

Albert Diaz GuileraUniversitat de Barcelona

Complex Networks0 Presentation1 Introduction

– Topological properties– Complex networks in nature and society

2 Random graphs: the Erdos-Rényi model3 Small worlds4 Preferential linking5 Dynamical properties

– Network dynamics– Flow in complex networks

Presentation

2 hours per session approxhomework– short exercises: analytical calculations– computer simulations– graphic representations

What to do with the homework?– BSCW: collaborative network tool

BSCW

Upload and download documents (files, graphics, computer code, ...)Pointing to web addressesAdding notes as commentsDiscussionsInformation about access bscw.ppt

1. INTRODUCTION

Complex systemsRepresentations– Graphs– Matrices

Topological properties of networksComplex networks in nature and societyTools

Physicist out their land

Multidisciplinary researchReductionism = simplicityScaling propertiesUniversality

Multidisciplinary research

Intricate web of researchers coming from very different fieldsDifferent formation and points of viewDifferent languages in a common frameworkComplexity

Complexity

Challenge: “Accurate and complete description of complex systems”Emergent properties out of very simple rules– unit dynamics– interactions

Why is network anatomy important

Structure always affects functionThe topology of social networks affects the spread of informationInternet + access to the information - electronic viruses

Current interest on networks

Internet: access to huge databasesPowerful computers that can process this informationReal world structure:– regular lattice?– random?– all to all?

Network complexity

Structural complexity: topology

Network evolution: change over time

Connection diversity: links can have directions, weights, or signs

Dynamical complexity: nodes can be complex nonlinear dynamical systems

Node diversity: different kinds of nodes

Scaling and universality

MagnetismIsing model: spin-spin interaction in a regular latticeExperimental models: they can be collapsed into a single curveUniversality classes: different values of exponents

Representations

From a socioeconomic point of view: representation of relational dataHow data is collected, stored, and prepared for analysisCollecting: reading the raw data (data mining)

Example

People that participate in social eventsIncidence matrix:

A B C D E

1 1 1 1 1 0

2 1 1 1 0 1

3 0 1 1 1 0

4 0 0 1 0 1

Adjacence matrix: event by event

1 2 3 4

1 - 3 3 1

2 3 - 2 2

3 3 2 - 1

4 1 2 1 -

Adjacence matrix: person by person

- 2 2 1 1

2 - 3 2 1

2 3 - 2 2

1 2 2 - 0

1 1 2 0 -

Graphs (graphic packages: list of vertices and edges)

Persons

person person Strength

A B 2

D E 0

Events

event event Strength

1 2 3

3 1

Bipartite graph

Board of directors

Directed relationships

Sometimes relational data has a directionThe adjacency matrix is not symmetricExamples:– links to web pages– information – cash flow

Topological properties

Degree distributionClusteringShortest pathsBetweennessSpectrum

Degree

Number of links that a node hasIt corresponds to the local centrality in social network analysisIt measures how important is a node with respect to its nearest neighbors

Degree distribution

Gives an idea of the spread in the number of links the nodes haveP(k) is the probability that a randomly selected node has k links

What should we expect?

In regular lattices all nodes are identicalIn random networks the majority of nodes have approximately the same degreeReal-world networks: this distribution has a power tail

kkP )( “scale-free” networks

Clustering

Cycles in social network analysis languageCircles of friends in which every member knows each other

Clustering coefficient

Clustering coefficient of a node

Clustering coefficient of the network2/)1(

ii

i

kkE

iC

N

iiCN

C1

1

What happens in real networks?

The clustering coefficient is much larger than it is in an equivalent random network

Directedness

The flow of resources depends on directionDegree– In-degree– Out-degree

Careful definition of magnitudes like clustering

Ego-centric vs. socio-centric

Focus is on links surrounding particular agents (degree and clustering)Focus on the pattern of connections in the networks as a whole (paths and distances)Local centrality vs. global centrality

Distance between two nodes

Number of links that make up the path between two points“Geodesic” = shortest pathGlobal centrality: points that are “close” to many other points in the network. (Fig. 5.1 SNA)Global centrality defined as the sum of minimum distances to any other point in the networks

Local vs global centrality

A,C B G,M J,K,L All other

Local 5 5 2 1 1

global 43 33 37 48 57

Global centrality of the whole network?

Mean shortest path = average over all pairs of nodes in the network

BetweennessMeasures the “intermediary” role in the networkIt is a set of matrices, one for ach node

Comments on Fig. 5.1

kijBRatio of shortest paths bewteen i and j that go through k

10 kijB

There can be more than one geodesic between i and j

Pair dependency

Pair dependency of point i on point kSum of betweenness of k for all points that involve iRow-element on column-element

Betweenness of a point

Half the sum (count twice) of the values of the columnsRatio of geodesics that go through a pointDistribution (histogram) of betweennessThe node with the maximum betweenness plays a central role

Spectrum of the adjancency matrix

Set of eigenvalues of the adjacency matrixSpectral density (density of eigenvalues)

N

jjN 1

1

Relation with graph topology

k-th moment

N*M = number of loops of the graph that return to their starting node after k stepsk=3 related to clustering

1

1

3221 ,,..,

,,11

1

1 ... iiii

iiiiN

k

N

kN

jjNk k

k

AAAATrM

A symmetric and real => eigenvalues are real and the largest is not degenerateLargest eigenvalue: shows the density of linksSecond largest: related to the conductance of the graph as a set of resistancesQuantitatively compare different types of networks

ToolsInput of raw dataStoring: format with reduced disk space in a computerAnalyzing: translation from different formatsComputer tools have an appropriate language (matrices, graphs, ...)Import and export data

UCINET

General purposeCompute basic conceptsExercises:– How to compute the quantities we have defined so

far– Other measures (cores, cliques, ...)

PAJEK

Drawing package with some computationsExercises:– Draw the networks we have used– Check what can be computed– Displaying procedures

Complex networks in nature and society

NOT regular latticesNOT random graphsHuge databases and computer power

“simple” mathematical analysis

Networks of collaboration

Through collaboration actsExamples:– movie actor – board of directors– scientific collaboration networks (MEDLINE,

Mathematical, neuroscience, e-archives,..)=> Erdös number

Communication networks

Hyperlinks (directed)

Hosts, servers, routers through physical cables (directed)

Flow of information within a company: employees process informationPhone call networks (=2)

Networks of citations of scientific papers

Nodes: papersLinks (directed): citations=3

Social networks

Friendship networks (exponential)Human sexual contacts (power-law)Linguistics: words are connected if– Next or one word apart in sentences– Synonymous according to the Merrian-Webster

Dictionary

Biological networks

Neural networks: neurons – synapsesMetabolic reactions: molecular compounds – metabolic reactionsProtein networks: protein-protein interactionProtein folding: two configurations are connected if they can be obtained from each other by an elementary moveFood-webs: predator-prey (directed)

Engineering networks

Power-grid networks: generators, transformers, and substations; through high-voltage transmission linesElectronic circuits: electronic components (resistor, diodes, capacitors, logical gates) - wires

Average path length

Clustering

Degree distribution