complex networks luis miguel varela cost meeting, lisbon march 27 th 2013

Complex Networks

Luis Miguel VarelaCOST meeting, Lisbon

March 27th 2013

Complex Networks


March 27th 2013

Complex Networks

Luis Miguel Varela Cabo

Introduction

Main properties of complex networks

Computational algorithms

Applications- Market models

Regional trade and development

- Other social network models of interest

Suggested trends

Introduction

Complex networks analysis in socioeconomic models

Statistical mechanics of complex networks


Applications - Market models - Regional trade and development - Other social network models of interest

Suggested trends

Complex Networks


Introduction






Suggested trends


Graph: an undirected (directed) graph is an object formed by two sets, V and E, a set of nodes (V={v1,…,vN}) and an unordered (ordered) set of links (E={e1…eK}).

Adjacency matrix:

- Contains most of the relevant information about the graph- A symmetric: undirected graph (a)- A non symmetric: directed acyclic graph (b)

Complex Networks


Introduction






Suggested trends

Measures on networks

Applications

social networks

financial networks

Economic networks

Small-worlds:Erdös-Bacon number

Degree distribution- Assortativeness- Preferential attachment

Clustering coefficient

Betweenness


Objects kindly provided by G. Rotundo

Complex Networks


Introduction





- Other social networks models of interest

Suggested trends


Small-worlds relatively short path between any two nodes, defined as the number of edges along the shortest path connecting them. The connectedness can also be measured by means of the diameter of the graph, d, defined as the maximum distance between any pair of its nodes. Networks do not have a “distance” : no proper metric space. Chemical distance between two vertices lij: number of steps from one point to the other following the shortest path.

In most real networks, < l > is a very small quantity (small-world)

In a square lattice of size N:

In a complex network of size N:

lji

ij llplNN

l )()1(

2

Nl

Nl log47.3

200

l

N

Complex Networks


Introduction






Suggested trends

The diameter of the network is small if compared to the number of nodes ~(log(N))

Examples (social networks):• sociological experiment of Stanley Milgram (1967):

anybody can be contacted through at most d=6 intermediaries

• Hollywood actors (mean) d=3.65

• Co-authors in maths (mean) d =9.5

Photo of a poster in the metro station of Paris, advertising on music events

small world

Slide kindly provided by G. Rotundo

Kevin Bacon number:

Kevin Bacon.

Nick Nolte

CAPE FEAR Robert De Niro

GOODFELLAS Joe Pesci

JFK

Degree of separation of Nick Nolte: 3

Val Kilmer Tom Cruise Kevin Bacon

Number of intermediaries of Hollywood actors to have worked with Kevin Bacon (social game popular in 1994)

Degree of separation of Val Kilmer: 2

TOP GUN A FEW GOOD MAN

Example 1

Example 2

small world-Bacon number (analogously to Erdos for mathematicians)


Statistical mechanics of complex networksCentrality, To go from one vertex to other in the network, following the shortest path, a series of other vertices and edges are visited. The ones visited more frequently will be more central in the network. Betweenness, number of shortest paths that passes through a given node for all the possible paths between two nodes. Measures the “importance” of a node in a network.

Number of shortest paths including vC(v)=

Total number of shortest paths

Betweenness (red=0,blue=max)

Example: node has the most high C(v)

Objects kindly provided by G. Rotundo

Complex Networks


Introduction






Suggested trends

Statistical mechanics of complex networksClustering coefficient, ratio between the number Ei of edges that actually exist between these ki nodes and the total number ki (ki-1)/2 gives the value of the clustering coefficient of node i. The clustering coefficient provides a measure of the local connectivity structure of the network

Clustering spectrum: Average clustering coefficient of the vertices of degree k

Average clustering coefficient

c Low c Large)1(

2

2

ii

i

i

ii kk

EkE

c

i

icNc

1

ii

kk ckNpkc

i )(

1)(

Clustering coefficient of real networks and random graphsR. Albert, A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

Complex Networks


Introduction






Suggested trends

Counting triangles: friends of friends are friends of mine, too?

clustering coefficient

Clustering coefficient =

N. triangles

N. Tringles less one links + +=

AB

C

D

A is friend of B, that is friend of A and C, but A and C are not friends

Transitivity property


A network is called sparse if its average degree remains finite when taking the limit N>> . In real (finite) networks, <k> <<N

Average degree

Degree distribution: p(k) probability that a node has a definite amount of edges. In directed networks the in-degree and out-degree are defined.

AB

WS

random)1(

)(

k

e

ppk

N

kp k

kNk

i k

i kkpkN

k )(1


degree 1 degree 2 degree 3

Object kindly provided by G. Rotundo

Complex Networks


Introduction






Suggested trends

Fundamental concepts for network topology description

Two-vertex correlations Real networks are usually correlated: degrees of the nodes at the ends of a given vertex are not in general independent. P(k’ | k)= probability that a k-node points to a k’-node.

Uncorrelated network: independent of k

Correlated network: p(k’|k) depends on both k’ and k

Degree of detailed balanced condition: P(k) and P(k’ | k) are not independent, but are related by a degree detailed balance condition.

Consequence of the conservation of edges

Number of edges k k’ = number of edges k’ k


k

kpkkkp

)'(')|'(

)'|(')'()|'()(

)'|(')'()|'()(

kkpkkpkkkpkp

kkpkkNpkkkpkNp

Complex Networks


Introduction






Suggested trends

Fundamental concepts for network topology descriptionCorrelation measures

Average degree of the nearest neighbors of the vertices of degree. Alternative to p(k’|k)

knn(k) dependent on k: correlations

Assortative: knn(k) increasing function of kDisassortative: knn(k) decreasing function of k


Two-vertex correlations

'

)|'(')(k

nn kkpkkk

Complex Networks


Introduction






Suggested trends

Fundamental concepts for network topology description

- Motifs: A motif M is a pattern of interconnections occurring either in a undirected or in a directed graph G at a number significantly higher than in randomized versions of the graph, i.e. in graphs with the same number of nodes, links and degree distribution as the original one, but where the links are distributed at random.

- Community (or cluster, or cohesive subgroup) is a subgraph G(N,L), whose nodes are tightly connected, i.e. cohesive.

S. Boccaletti et al. Physics Reports 424 (2006) 175 –308


Complex Networks


Introduction






Suggested trends

Nodes are divided into groups - high internal connection - low connection to the other groups

community structure

Example:

Network of friends at school

Divided by younger age,

Older age,

White and black


Bocaletti et al. Physics Reports 424, 175 – 308 (2006).

Directed networks and weighted networks

Weighted networks: strong and weak ties between individuals in social networks

NodesLinksweights

Weighted degree

Weighted clustering coeff.


Complex Networks


Introduction






Suggested trends


Ising model in networks: paradigm of order-disorder transitions in agent-based models

2D spin system

1D regular lattice (Ising, 1925)

2D regular lattice (onsager, 1944)

No phase transition!

Complex Networks


Introduction






Suggested trends



2D spin system

1D WS network (Viana-Lopes et al., 2004)

Complex Networks


Introduction






Suggested trends

Phase transition in 1D! (long-distance correlations dramatic increase in connectivity)


Complex Networks


Introduction






Suggested trends


2D spin system

Mean-field for 1D AB network (Bianconi, 2004)

Phase transition in 1D! (long-distance correlations dramatic increase in connectivity)

Watts-Strogatz algorithm

Start with order Randomize rewiring with probability p excluding self-connections and duplicate edges

Complex Networks


Introduction






Suggested trends


Duncan J. Watts, Steven H. Strogatz, Nature 393, 440 (1998)

Albert-Barabási algorithm

1. Network growth: start with a small number of nodes and at each time step add a new node that links to m already existing nodes

2. Preferential attachment (evolving network): the probability that a new node links to node i depends on the degree of the already existing node:

Albert-Barabási

Dorogovtsev- Mendes-Samukhin

Complex Networks


Introduction






Suggested trends


1. Potential degree distribution (extreme events, superspreader, hierarchies):

P(k) ~ k-g

2. Average path length shorter than in exponentially distributed networks.

3. Degree of correlation of the degree of the different nodes

4. Clusterization degree ~ 5 times greater than that of random networks.

Scale free networks (e.g. Albert-Barabasi)

75.0N

kCrand

Complex Networks


Introduction






Suggested trends


preferential attachment

New nodes are attached to the hubs (preferentially)

Scale-free networks

Social networks models

Some objects kindly provided by G. Rotundo

Real or hypothetical. Depends on the amount of data:

Intrinsic characteristics (e. g. classes). Full description (e. g. contact tracing). Building algorithm (e. g. Barabási-Albert). Sampling of the degree distribution (e. g. polls). Tools:

Standard statistical methods and software.Analysis and visualization interactive programs.

POPULATION ANALYSIS

Complex Networks


Introduction






Suggested trends


Pajek: http://pajek.imfm.si/doku.php

Others: Cytoscape (http://www.cytoscape.org/), UCINet.

Complex Networks


Introduction






Suggested trends


Much more time-consuming than ODE-based methods Automatization need:

Long unattended runs. Parallelization

Most convenient option: high-level language + network algorithm libraries.

Python (http://www.python.org)

NumPy: array treatment (MATLAB-like).Scipy: scientific functions on NumPy.RPy : integrates R in Python with NumPy.Parallelism, access to databases, text processing and binary files, user graphic interfaces, 2D/3D plots, geographical information systems...Windows distribution: Python(x,y) (http://www.pythonxy.com)

Complex Networks


Introduction






Suggested trends


General methods: Calculus sheets [catastrophic precission: G. Almiron et al., Journal of Statistical Software 34 (2010)]. Analysis environments: MATLAB, Mathematica, Octave, etc. Specific: R+statnet, Python+NetworkX

POSTPROCESSING

Complex Networks


Introduction






Suggested trends


SIMULATION

NetworkX: http://networkx.lanl.gov/,Included in Python(x,y).Generators, algebra, input/output, representation...Optimized algorithms, programmed in low level languages.Nodes can contain any type of data.Integration with NumPy.

Complex Networks


Introduction






Suggested trends


Complex Networks


Introduction






Suggested trends


SIMULATION

NetworkX: http://networkx.lanl.gov/ ,Included in Python(x,y).Generators, algebra, input/output, representation...Optimized algorithms, programmed in low level languages.Nodes can contain any type of data.Integration with NumPy.

Complex Networks


Introduction






Suggested trends


Complex Networks


Introduction






Suggested trends

Applications: market models

GDP and other macroeconomic indicators

Miskiewicz and Ausloos

Gligor and Ausloos

Lambiotte and Ausloos

Redelico, Proto and Ausloos

Complex Networks


Introduction






Suggested trends

Applications: market models

Market correlations and concentrations. Tax evasion.

Rotundo and coauthors

Westerhoff and coauthors

Complex Networks


Introduction






Suggested trends

Applications: market modelsSpreading of innovations

Complex Networks


Introduction






Suggested trends

Applications: regional trade and development

Complex Networks


Introduction






Suggested trends

Applications: other social networks

Complex Networks


Introduction






Suggested trends

Suggested trends

- Directed diffusion of commodities and people: complex networks based description

- Nonlinear production processes: emergence of time and space patterns (enzyme model of production…)

J. D. Murray, Mathematical Biology, Springer, 2001.

Complex Networks


Introduction






Suggested trends

Suggested trends-Go beyond spin ½ systems (Potts models) (richness of decision).

-Apply known statistical physics models (phase transitions, percolation, non-Markovian processes, linear response theory…

- Combine with dynamic processes for: a) Spreading of innovations and market models

b) Financial models

c) Companies/banks networks

d) ETC.

MANY THANKS FOR

YOUR ATTENTION

Complex Networks


March 27th 2013

complex networks luis miguel varela cost meeting, lisbon march 27 th 2013

Documents

real finite networks

social network models

undirected graph

undirected directed

set of nodes v

relevant information

adjacency matrix