Complex Networks
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Transcript of Complex Networks
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Complex Networks
Albert Diaz GuileraUniversitat de Barcelona
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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
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Presentation
2 hours per session approxhomework– short exercises: analytical calculations– computer simulations– graphic representations
What to do with the homework?– BSCW: collaborative network tool
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BSCW
Upload and download documents (files, graphics, computer code, ...)Pointing to web addressesAdding notes as commentsDiscussionsInformation about access bscw.ppt
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1. INTRODUCTION
Complex systemsRepresentations– Graphs– Matrices
Topological properties of networksComplex networks in nature and societyTools
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Physicist out their land
Multidisciplinary researchReductionism = simplicityScaling propertiesUniversality
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Multidisciplinary research
Intricate web of researchers coming from very different fieldsDifferent formation and points of viewDifferent languages in a common frameworkComplexity
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Complexity
Challenge: “Accurate and complete description of complex systems”Emergent properties out of very simple rules– unit dynamics– interactions
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Why is network anatomy important
Structure always affects functionThe topology of social networks affects the spread of informationInternet + access to the information - electronic viruses
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Current interest on networks
Internet: access to huge databasesPowerful computers that can process this informationReal world structure:– regular lattice?– random?– all to all?
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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
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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
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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)
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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
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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 -
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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
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Bipartite graph
Board of directors
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Directed relationships
Sometimes relational data has a directionThe adjacency matrix is not symmetricExamples:– links to web pages– information – cash flow
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Topological properties
Degree distributionClusteringShortest pathsBetweennessSpectrum
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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
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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
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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
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Clustering
Cycles in social network analysis languageCircles of friends in which every member knows each other
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Clustering coefficient
Clustering coefficient of a node
Clustering coefficient of the network2/)1(
ii
i
kkE
iC
N
iiCN
C1
1
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What happens in real networks?
The clustering coefficient is much larger than it is in an equivalent random network
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Directedness
The flow of resources depends on directionDegree– In-degree– Out-degree
Careful definition of magnitudes like clustering
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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
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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
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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
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Global centrality of the whole network?
Mean shortest path = average over all pairs of nodes in the network
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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
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Pair dependency
Pair dependency of point i on point kSum of betweenness of k for all points that involve iRow-element on column-element
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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
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Spectrum of the adjancency matrix
Set of eigenvalues of the adjacency matrixSpectral density (density of eigenvalues)
N
jjN 1
1
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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
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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
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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
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UCINET
General purposeCompute basic conceptsExercises:– How to compute the quantities we have defined so
far– Other measures (cores, cliques, ...)
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PAJEK
Drawing package with some computationsExercises:– Draw the networks we have used– Check what can be computed– Displaying procedures
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Complex networks in nature and society
NOT regular latticesNOT random graphsHuge databases and computer power
“simple” mathematical analysis
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Networks of collaboration
Through collaboration actsExamples:– movie actor – board of directors– scientific collaboration networks (MEDLINE,
Mathematical, neuroscience, e-archives,..)=> Erdös number
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Communication networks
Hyperlinks (directed)
Hosts, servers, routers through physical cables (directed)
Flow of information within a company: employees process informationPhone call networks (=2)
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Networks of citations of scientific papers
Nodes: papersLinks (directed): citations=3
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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
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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)
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Engineering networks
Power-grid networks: generators, transformers, and substations; through high-voltage transmission linesElectronic circuits: electronic components (resistor, diodes, capacitors, logical gates) - wires
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Average path length
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Clustering
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Degree distribution