UNCONVENTIONAL COMPUTING - Babeș-Bolyai University

UNCONVENTIONAL

COMPUTING

Laura Dioşan Lecture 4

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Complex networks

Content

Basic concepts

Typology

Applications

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What is a complex network? A set of connected elements

A network (a graph) G=(N,M) is composed by a set of nodes N={n1,n2,…,nN} and

a set of links (edges) L= {l1,l2,…,lM}

A graph = a mathematical (abstract) representation of a network

Each element is represented by

location (physics) Node (computer science) Actor/agent (sociology) Vertex (graph theory)

Interaction of two elements is represented by:

Contact (physics) Link (computer science) Relation (sociology) Edge (graph theory)

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What is a complex network?

Nodes and links can be provided by various contexts

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What is a complex network?

A complex network is a network with

Topological attributes (features) non-trivial

Link patterns that are neither purely regular nor purely random

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Basic concepts

Description of a network (with N nodes and M kinks) by using matrix:

The matrix of link weights

Adjacency matrix

Laplaciane matrix L = A – K

K – diagonal matrix with

ijii ak

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Basic concepts Shortest path between 2 nodes (dij)

Shortest path (= number of edges) Shortest path( = sum of edge’s weights)

Average path l

(arithmetic or harmonic) mean of all the shortest paths between any 2 nodes of network

Diameter (D)

The longest (number of edges) path from all shortest path (as value)

Component

Set of all nodes that can be reached starting from a given node

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Basic concepts

Degree of a node (ki)

Number of links of a node

Strength of a node (si)

Sum of link weights of a node si = ∑wij

Betweenness (bi) of a node or of a link

Number of shortest paths that use that node/link

The node of largest degree is the strongest one?

Which is the node of largest betweenness?

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Basic concepts

Network motifs

Components (sub-graphs) that appear more frequently that we expect (randomly)

Each motif can code specific information

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Basic concepts

Clustering coefficient C are my friends,

friends of my friends?

Based on the number of triangles from the network

Coefficient of a node = proportion of

# links that connect the neighbours of the node

#possible links among these neighbours

C – mean over the coefficients of all the network’s nodes

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Basic concepts

Global efficiency

Harmonic mean of optimal paths between all the nodes from the network

Local efficiency (of a node) clustering

coefficient

Value of the shortest path among the neighbours of the node (without node)

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Basic concepts

Graph spectrum

Set of proper values of adjacency/Laplace matrix

A graph with N nodes has N proper values and N proper vectors

Important for topological features

Diameter

# cycles

Propagation of information

Important for connectivity features

Spectral density

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Basic concepts

Community structure

Community in a graph

A sub-graph with high connected nodes (more connected than random)

if the nodes of the network can be easily grouped into (potentially overlapping) sets of nodes such that each set of nodes is densely connected internally

Evaluation

Modularity M

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Basic concepts

Degree distribution

(cumulative) distribution of degrees

Proportion of network’s nodes of degree (greater or ) equal to a threshold

Can be

exponential distribution (random nets)

Power-law (scale-free nets)

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Basic concepts

Clustering distribution

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Basic concepts

Nearest neighbour degree knn(k) and assortativity

knn(k) – measures the degree of neighbours

quality of assortativity

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Typology Link orientation

Without orientation Oriented

Type of links

Free (without weights/coefficients) Weighted

Node type

Simple Bipartite

Topology Static Evolutionary

Node dynamic

Without dynamic With dynamic

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Typology

Link orientation dynamic processes in the network information propagation,

synchronisation,

robustness

Without orientation (symmetric links)

Oriented (asymmetric links)

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Typology

Link type dynamic processes in the

network

information propagation,

synchronisation,

robustness

Without weights (homogenous)

Weighted (eterogenous)

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Typology

Node type

Simple

Bipartite networks with nodes of 2 or more

types and links between nodes of the same type

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Typology

Topology network does not appear

instantly

Static (as structure)

Evolutionary (as structure)

Possible questions:

Which are the rules that govern/guide the evolution?

Which are the effects (consequences) of applying these rules over the net’s topology?

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Typology

Node dynamic Can be influenced by matrix of connections

Influence of net’s topology over dynamic process Sincronisations

Stocastic processes

And vice-versa (influence of process over net’s topology)

Without dynamic

With dynamic -> nodes are (coupled) dynamic systems: Regular oscillators – ex. Foucault oscillator,

Excitable systems

Chaotic oscillators – ex. Cryptology

Bi-stable systems – ex. semiconductor memory

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Applications

Social networks

Properties

Small-world

power-low distribution of nodes very high connected nodes (hubs)

Large clustering coefficient (relative to random networks)

Mix of assortativity

Ex: most connected nodes like to be together

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Applications

Social networks

Vilfredo Pareto (1848 – 1923)

Describe the allocation of wealth among individuals of different countries

All of them have a power-law distribution p(X≥ x) ~ x-β

Pareto principle (80-20 rule):

80% of effects are generated by 20% of causes

Consequence of cumulative distribution of power-law

Applied in various domains (Economy, ... Sociology )

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Applications

Social networks George K. Zipf (1902 – 1950)

Frequency of words in English language follows a rank distribution of type power-law N(r) ~rγ

Contexts (different scales) Words

Syllable

Phonemes

Distribution changes when semantic looses

Frequency and degree are similar

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Applications

Social networks

George K. Zipf (1902 – 1950)

Other domains

Linguistics

Population distribution

Income rankings

Relative to Pareto’s law

The rth largest city has n inhabitants (Zipf)

r cities have n or more than n inhabitants (Pareto)

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Applications

Social networks

Rumor spreading

Ignorant-Spreader-Stifler model (Daley and Kendal model)

Every time moment a random spreader i is selected; I contacts one of his neighbours j

If j is an ignorant, j becomes spreader

If j is a spreader or a stifler, i becomes stifler

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Applications

Social networks

Rumor spreading / epidemics in a small-world

Ignorant-Spreader-Stifler model (Daley and Kendal model)

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Applications

Social networks

Rumor spreading / epidemics in a small-world

How to fight?

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Applications

Social networks

Communities and their aim

Zachary (karate) club

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Applications

Social networks Communities and their

aim

Modular structure of complex networks

Existence of communities is not reflected by degree distribution, clustering coefficient or assortativity

Communities are related to the node functionalities

Community detection = a problem of multiple solutions

Algorithms

Link elimination

Agglomerative methods

Modularity-based methods

Spectral methods

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Applications

Social networks Communities and their aim Community detection

Link elimination

Eliminating weak links until the net brooks

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Applications


Agglomerative methods (Bottom-up)

Initially, every node belong to a community

Similar nodes

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Applications


Modularity-based methods

Modularity = # links among groups – estimated # links among groups in a random network

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Applications


Spectral methods

Split the network into

2 (or more) components

Spectral analysis of Laplacian matrix (contains all topological information about network)

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Applications

Biological networks

Properties

Small-world

Dissortative mixing

Many connected node are not preferentially connected each other

Organised in sub-modules

Large modularity

Community structures

Pioneers

Watts & Strogatz

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Applications

Biological networks

Metabolic nets

Protein nets

Yeast

6000 proteins

3 links/protein

20 000 links

Human body

~100 000 links

Genetically nets

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Applications

Biological networks

Ribonucleic acid (RNA) nets

Transformation sequence – structure is degenerated

The same structure can be obtained by various (multiple) chains

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Applications Biological networks

Brain functional nets RMN funcțional

Measures how different parts of brain answer to external stimulus

Electroencephalogram Recording the electrical activity waves

Magnetoencephalogram a functional neuro-imaging technique for mapping brain activity

by recording magnetic fields produced by electrical currents occurring naturally in the brain, using very sensitive magnetometers.

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Applications

Biological nets

Functional nets of brain

Alzheimer

Mild Cognitive Impairment

Schizophrenia

Epilepsy

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Applications

Musical nets

Note’s nets

Melody’s nets

Artist’s nets

User’s nets

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Applications

Rețele muzicale

Nets of musical notes

Note = node

Link = proximity (similarity) of notes

Nets of note duration

Nets of notes

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Applications


Zipf law

Context = a hierarchy of models (patterns) that appear in different tempos (harmonic progression, melody, tone, rithm, ...)

Note + duration

Note + note

Note’s nets assortative

Specific clustering distribution

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Applications


Why?

Create music starting from net’s properties and using guided random processes

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Applications

Musical nets

Melody nets

Node = melody

Links = different relations

Ex. Belong to a playlist

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Applications

Rețele muzicale ale melodiilor

De ce?

Analiza structurii

Drumuri, module

Detectarea celei mai influente melodii

Clasificare (etichetare)

Proiectarea unor sisteme automate de recomandare eficiente

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Applications

Rețele muzicale de melodii

Evoluează în timp

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Applications

Rețele muzicale de melodii

Gusturi muzicale

Apartenența melodiilor la playlist-urile utilizatorilor

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Applications

Rețele muzicale

Rețele ale artiștilor

Nod = artist

Legătură = diferite relații

Similaritate

Colaborare

Afinitate

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Applications

Rețele muzicale

Rețele ale artiștilor

Nod = artist


Similaritate

Colaborare

Afinitate

Au același număr de noduri

Au structuri diferite

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Applications

Rețele muzicale ale artiștilor

Detecția comunităților oferă informații despre rețea

Separarea rețelei pe baza modularității

În comunități

În sub-comunități

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Applications




Se pot identifica hub-uri

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Applications




Se pot identifica hub-uri

Se pot face cartografieri ale hub-urilor

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Applications

Rețele muzicale

Rețele ale utilizatorilor

Nod = utilizator (consumator)


De ce?

Folosite în sistemele de recomandare

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Applications

Rețele muzicale ale utilizatorilor

De ce e importantă topologia rețelei?

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Applications

Rețele muzicale

O diversitate de a proiecta muzica într-o rețea

Care model de rețea este cel mai potrivit?

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