Introduction to Complex Networks - Science South Tyrol,,UNIBZ,90,68.pdf ·...

Introduction toComplexNetworks

M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks

Complex withrespect to what?

Models

Wealthdistribution

Modeling theWTW

Recurrences

Introduction to Complex Networks:Theory and Applications

Maria I. Loffredo

Dipartimento Ingegneria dell’Informazione e Scienze MatematicheUniversita di Siena, http://www.dii.unisi.it

Complex Systems Community, http://csc.unisi.it

Bolzano, June 12, 2013

M.I. Loffredo (DIISM, UNISI) Introduction to Complex Networks 1 / 56

http://www.dii.unisi.it

http://csc.unisi.it


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Outlook

1 Introduction

2 Statistical Analysis

3 Generating Networks

4 Wealth distribution

5 Modeling the WTW

6 Recurrences in nonlinear dynamics



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Brief historical overview:

1736 Graph theory (Euler)

1937 Journal Sociometry founded

1959 Random graphs (Erdos-Renyi)

1967 Small-world (Milgram’s experiment)

late 1990s “Complex networks” (Strogatz, Watts, Barabasi )



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Konigsberg, at Euler’s time, and its bridges on the river Pregel:solution in terms of topological features



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

The map of the city becomes a graph

(a) A map of old Konigsberg, in which each area of the city is labeled with a different color point

(b) The graph: each of four land areas is represented as a node and each of the seven bridges as an edge



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Problem: is it possible to find a path that passes through all the edges exactly once?

Answer: Only “topology” plays a role, as an intrinsic property of the map

Euler shows that the existence of a walk in a graph, which traverseseach edge once, depends on the degrees of the nodes. The degree ofa node is the number of edges touching it.

An Eulerian path exists if and only if the vertices with odd degree arezero (starting and ending point coincide) or two (starting and endingpoint do not coincide).

Since the graph corresponding to historical Konigsberg has four nodesof odd degree, it cannot have an Eulerian path.



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Definition

Network is any real system that can be described by means of aGraph ⇐⇒ Set of N vertices (nodes) and L links (edges)

aij = adjacency matrix NxN

1

2

3 4

!!!!!

"

#

$$$$$

%

&

=

0101101101001100

a


Figure: Yeast Protein Interaction

Since the pioneeristic work ofBarabasi, Strogatz, Watts, the studyof networks has rapidly increasedacross various disciplines

Different kinds of systems share theproperty to be interconnectedstructures, obeying organizingprinciples not at all casual

Biology, Economics and SocialSciences provide a number of veryinteresting examples: neuralnetworks, food chains, metabolic andproteic webs, social networks


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Examples of real networks: we live in a connected world!

Biological networks Socioeconomic networks

Network Units ConnectionsBiochemical Cellular substrates Reactions (enzymes)Proteic Proteins InteractionsNeural Neurons SynapsesVascular Tissues Blood vesselsEcological Species Predations

Social People Social relationsEconomic Agents TransactionsFinancial Companies (Stocks) ShareholdingsInternet Computers CablesWorld Wide Web Web pages Hyperlinks



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Statistical Analysis

First order property: the degree of a vertex . . .

Degree ki of vertex i: number of links of vertex i(regular graphs =⇒ all vertices have the same degree)

. . . and its distribution P(K) ∼ probability of finding a vertex whosedegree is k

Universality:

Despite their different nature, many real networks display apower-law degree distribution: P(k) ∼ k−γ , (2 < γ < 3)

The system appears the same regardless of the level at which onelooks at it =⇒ Lack of a typical scale! (scale-free topology)



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Scale-free networks

Power Laws in real networks: log-log plots of P(k)

Many poorly connected vertices and Few highly connected vertices

real scale-free? =⇒ ... at least in a certain range ...

see also: Pasquale Cirillo, Are your data really Pareto distributed?, http://arxiv.org/abs/1306.0100 (June 2013)

Figure: Power Laws in real networks


http://arxiv.org/abs/1306.0100


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

ANND

Second order property, the Average Nearest Neighbour Degree: K nni of vertex i = mean

degree of the neighbours of i ≡ 1ki

∑j∈V (i) kj

(regular graphs =⇒ all vertices with the same ANND)

Correlation Spectrum: putting together vertices with the same degree −→ Knn(k)

Plots of K nn vs k in real networks: Assortativity (social networks) Disassortativity(technological and biological networks). “Similar” networks display “similar” behaviours

Consequences of assortativity:Resistance to attacks BUT Fast Epidemic spreading

also: robustness and self-healing of Internet as a complex network. Sept. 11, 2001 →switching station damaged close to the WTC. Immediately after the attack reachabilitydropped by about 9%. Within 30 minutes it had almost reached its old value again



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Assortativity: K nn and k arepositively correlated

Large sites connected to large sites

Network of members of

Italian boards of directors

Disassortativity: K nn and k arenegatively correlatedLarge sites connected to small sites

Hierarchical structure

World Trade Web (WTW)



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Clustering

Third order property: Clustering coefficient Ci of vertex i = fractionof interconnected neighbours of i , withki (ki − 1)/2 = maximum number of links between the neighbours of a vertex with degree ki .

Hierarchy =⇒ C (k) decreases with k

Global properties:

Average distance D of the network: average of the minimumdistances Dij between connected pairs of vertices

Connected clustersNumber nC and size si (i = 1 , . . . nC ) of connected clusters:nC sets of si vertices that can be reached from each other



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

A topological classification

From a topological point of view network can be divided into fourcategories:

unweighted, undirected networks (BUN:binary,undirected networks)

unweighted, directed networks (BDN: binary, directed networks)

weighted, undirected networks (WUN)

weighted, directed networks (WDN)

BUN BDN WUN WDN


Other properties: reciprocity

For each pair of links there are 4 possibilities:

aij=1, aji=0

aij=0, aji=1

aij=0, aji=0

aij=1, aji=1

i j

i j

i j

i j

Pair of “reciprocal” links D. Garlaschelli, M. I. Loffredo,

Phys. Rev. Lett. 93, 238701(2004)

aij=1 only if there is connection from i to j For directed networks the adjacency matrix is no longer symmetric: aij ! aji

Various definitions of degree: In-degree: Out-degree: Total degree:

!

iink = jia

j=1

N

"

!

ioutk = ija

j=1

N

"

!

itotk = i

ink + ioutk

In directed networks, the dyadic patterns are entirely determined byreciprocity, i.e. the tendency to form, or to avoid, mutual links


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Generating Network Models



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Complex with respect to what?

How can we study networks?

Statistical physics provides a number of tools to describe and analyze networks.

A complex network can be informally defined as a large collection of interconnected nodes.

What makes these networks complex is that they are generally so huge that it is impossible tounderstand or predict their overall behavior by looking into the behavior of individual nodesor links.

More specifically a network can be defined as “complex” only after a comparison with anull model

So the right question is:is the network more complex than this null model?

Null models are implemented by creating randomized versions of the original network(randomized graph ensemble) and preserving, at the same time, some topological properties.

Network too simple (regular) or too complicated (totally random) are not “complex”

Modeling: try to reproduce topological properties of real networksMatching with real networks =⇒ role of heterogeneity


Real networks are much more complex than regular graphs!

One possibility is to model the “disorder” by introducing randomness in the presence of the connections

Probabilistic models of networks

In a network with N vertices there are N(N-1)/2 pairs of vertices.

Therefore the expected number of links is < L> = p N (N-1) /2

The RANDOM GRAPH model (Erdos-Renyi 1959)

Each pair of vertices is connected with independent probability p (and not connected with probability 1-p).

p=0 p=0.1 p=0.5 p=1

To have an expected number of links equal to the observed one (L), one can choose p=2L/N(N-1)

The Small-World Model (Strogatz&Watts, 1998)

•  Start with a regular lattice

•  Each link of the lattice is rewired with probability p: one of its ends is moved to a randomly chosen vertex (introduction of disorder).

For intermediate values of p, 0<p<1, there exists a region with large average clustering and small average distance.

(small world region)

The model provides an interpolation between regular lattices and random graphs and successfully reproduces the small-world effect observed in real networks.

Degrees are peaked around mean value and, as p increases, the degree distribution broadens around the mean value, but no power-law tails are observed.

Examples: social networks, Internet, gene networks"

P(k)! k -" " =3

After a certain number of iterations, the degree distribution approaches a power-law

distribution:

P(k)! k -" " =3

Growth and preferential attachment are both necessary!

! Start with m0 vertices and no link;

! at each timestep add a a new vertex with m links, connected to m preexisting vertices chosen randomly with probability proportional to their degree k (preferential attachment).

The SCALE-FREE model (Barabási&Albert, Science 286 (1999)


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

BA model as an example of evolving models

The choice of the m partners is not random

High-degree vertices are more likely to attract future connections

Succeeds in reproducing the power-law with the only exponent -3

Limitations:Absence of degree-correlationsAbsence of clustering hierarchy

More refined Models:Growing networks with nonlinear preferential attachment (generalized BA models).

Well suited for WWW and collaboration networks

A-L. Barabasi and R. Albert, Statistical Mechanics of Complex Networks, Rev. Mod. Phys. 74, (2002) 47


! Each vertex i is assigned a fitness value xi drawn from a given distribution !(x) ;

! A link is drawn between each pair of vertices i and j with probability f(xi,xj) depending on xi and xj .

Power-law degree distributions can be obtained by choosing

!(x) " x-!

f(xi,xj) " xi xj

or

!(x)= e-x

f(xi,xj) " #(xi +xj –z)

The FITNESS model Caldarelli, Capocci, De Los Rios, Muñoz, Phys. Rev. Lett. 89 (2002)


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

HIDDEN-VARIABLE MODELS

Ensemble of networks constructed by keeping the fitness values {xi} , i = 1 , . . . ,N,fixed and repeating random assignment of links.

Topological properties depend on the fitness distribution and the functional form off (xi , xj )

Random Graph model recovered in the case of same fitness value for each vertex orconstant connection probability

Particularly suitable to detect the organizing mechanisms shaping the topology of realnetworks

Fitness x identified with empirical quantities: additional information, not topological innature, but intrinsically related to the role played by each vertex in the network

Test on World Trade WebHidden Variable ⇐⇒ Gross Domestic Product



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Wealth distribution on complex networks


Empirical wealth distributions Two typical forms (often combined):

Large wealth: Pareto’s law (power-law distribution):

Small wealth: Gibrat’s law (log-normal distribution):

!"

#$%

&=

0

222 w

wlog21-exp

2w1P(w)

'('Cumulative distribution ==> exponent -!

!

p(w)"w#(1+$ )

!

p(w) ="

w #exp $" 2 log2(w /w0)[ ]

Personal Income Distribution

Log-normal distribution with power-law tails (mixed form)

U.S.A. 1935-36 ($) Japan 1998 (M¥)

Badger 1980, Souma 2000

Empirical forms of “wealth” distributions:

The most general form of P(w) is “mixed”:

Combination of a power-law and a log-normal distribution

Search of theoretical models that can reproduce the mixed form

(the “pure” forms will be regarded as limiting cases).


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Independent agents models

Purely multiplicative stochastic process: wi (t) = ηi (t)wi (t)=⇒ Log-normal distribution

wi (t) = wealth of agent i at time t ηi (t) = Gaussian process

The purely multiplicative stochastic model explains the appearance ofGibrat’s law, but cannot reproduce the power-law tails of empiricalwealth distributions

On the contrary, multiplicative-additive stochastic process:wi (t) = ηi (t)wi (t) + ξi (t) =⇒ Pareto’s (power-law) distribution

ξi (t) = additive noise term


Wealth evolution with N agents:

wi (t) = wealth of agent i at time t

!i (t) = Gaussian process (mean m and variance 2"2 )

Jij = fraction of wealth flowing from j to i

No flow of wealth from j to i ==> Jij =0

Invariance under rescaling w --> ! w (money units are arbitrary)

Model of Bouchaud and Mézard (BM) Physica A 282 (2000) 536

Interactive multiplicative stochastic process: wealth evolution is determined by the interactions among economic agents

!

˙ w i(t) ="i(t)wi(t) + Jijw jj# i$ (t) % J jiwi

j# i$ (t)

Pajek

Now there is only one cluster

BM model on octopus networks Randomly connected core of M vertices plus N-M “tentacles”

BM model on octopus networks For intermediate values of M/N the mixed form appears!

Simulation parameters:

N=3000

T=10000

J=0.05

‹!›=1

‹!2›-‹!›2 = 0.1

Garlaschelli & Loffredo, Physica A 338 (2004) 113

Garlaschelli & Loffredo, J. Phys. A 41, 224018 (2008)


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

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GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

A crucial ingredient producing a realistic wealth distribution out of the Bouchaud-Mezardmodel is a heterogeneous link density in the network

In general, heterogeneity comes out not only from the degree distribution P(k), but onhigher-order properties, such as assortativity, hierarchy, etc.

In particular, in order to relate the empirical form of wealth distribution to higher-ordernontrivial topological properties would require knowledge of the corresponding transactionnetwork.

This is a difficult task, but it is possible for the Trade Network of World countries (WTW),whose analysis revealed the presence of degree correlation and hierarchical organization of thenetwork. These properties are expected to be responsible for the peculiar shape of thedistribution of the wealth of vertices (defined as the GDP of world countries).

Next: Interplay between network topology of the WTW and the GDP dynamics



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Modeling the World Trade Web



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

WTW

What

The World Trade Web (WTW) is the network representation of the international trade

system that allows a large scale analysis from an interdisciplinary approach. Links represent

the commercial relations between countries (nodes)

How

Tools and methodologies - inherited from statistical physics and network science - can be

exploited to extract information from the WTW, and to emphasize which properties signal a

nontrivial structural organization

Why

The network representation offers a new level of description that goes beyond the

country-specific analyses used in more traditional economic studies of trade



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

WTW

Two countries are linked if they have an IMPORT/EXPORT trade relationship

DATA: Trade flows (import/export) and GDP values (wealth) for all world countries from

1948 to 2000

Gleditsch, K. S. , Expanded Trade and GDP data, Journal of Conflict Resolution 46(5) (2002) 712



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Application of the fitness model to the WTW

the probability pij that two vertices {i , j} are connected depends on the values of anassociated “fitness” parameter x : pij = f (xi , xj )

Vertices = N World countries, N = N(t)

(Directed) Links ⇐⇒ flow of money between two trading countries: country i importsfrom j at time (year) t ⇐⇒ a link from i to j

Direction of link ⇐⇒ direction of wealth flow

Imported goods ⇐⇒ wealth flowing outExported goods ⇐⇒ wealth flowing in

Undirected version: the degree simply represents the number of trade partners

Different snapshots at different years



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Modeling the World Trade Web

Can the WTW topology be traced back to some intrinsic vertexproperty?

We assume that the Gross Domestic Product wi determines thepotential ability of developing trade connections of i

=⇒ f (xi , xj ) =δxi xj

1+δxi xjwhere xi ≡ wi

<w>is the relative GDP.

δ free parameter, fixed in order to reproduce the observed number of links.

Garlaschelli & Loffredo, Fitness-dependent topological properties of the World Trade Web, Phys. Rev. Lett. 93 (2004)188701

cited in the New Scientist Journal with the title: “The unique shape of global trade” (13 november 2004)


Cumulative Degree Distribution Average nearest neighbour degree

Clustering coefficient Hierarchy: highly connected vertices have poorly interconnected neighbours


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Interplay between topology and dynamics

In self-organizing networks, topology and dynamics coevolve in a continuous feedback,without exogenous driving

The World Trade Network (WTN) is one of the few empirically well documented examples ofself-organizing networks: its topology depends on the GDP of world countries, which in turndepends on the structure of trade

Therefore, understanding the WTN topological properties deviating from randomnessprovides direct empirical information about the structural effects of self-organization

D. Garlaschelli, T. Di Matteo, T. Aste, G. Caldarelli, M.I. Loffredo, Interplay between topology and dynamics in the World Trade

Web, European Physical Journal B 57 (2007) 159



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Complementary Research

D. Garlaschelli, M.I. Loffredo, Maximum likelihood: extracting unbiased information

from complex networks, Phys. Rev. E 78, 015101(R) (2008)

D. Garlaschelli, M.I. Loffredo, Generalized Bose-Fermi Statistics and Structural

Correlations in Weighted Networks, Phys. Rev. Lett. 102 (2009) 038701

M. Barigozzi, G. Fagiolo, D. Garlaschelli, The Multi-Network of International Trade:

A Commodity-Specific Analysis, Phys. Rev. E 81 (2010) 046104

G. Fagiolo, T. Squartini, D. Garlaschelli, Null models of economic networks: the case

of the world trade web, J. Econ. Interact. Coord. 8 (2013) 75 - 107

for a recent review on the WTW:

M. A. Serrano, D. Garlaschelli, M. Boguna, M. I. Loffredo

The World Trade Web: Structure, Evolution and Modeling, EOLSS - UNESCO, Oxford (2010)



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Recurrence Networks in Nonlinear Dynamics



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Recurrence Plots (RP)

Recurrence Plot(Eckmann, Kamphorst, Ruelle, 1987)

Given a trajectory {xi}Ni=1 (xi ∈ Rn) and a

threshold ε, a recurrence plot is definedstarting from the recurrence matrix

Ri,j (ε) = Θ (ε− ‖xi − xj‖) i , j = 1...N

Recurrence statement:

xi ≈ xj ⇔ ‖xi − xj‖ < ε ⇔ Ri,j (ε) = 1

0.2 sin(

2πtω1)

+ 0.8 cos(

2πtω2)

ω1 = 8Hz ω2 = 25Hz



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Large scale structures

Homogeneous Periodic Drift



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Small scale structures

Isolated points

Diagonal lines

Ri+k,j+k ≡ 1|l−1k=0

Vertical lines

Ri, j+k ≡ 1|v−1k=0



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Recurrence Quantification Analysis (RQA)

RQA (Webber&Zbilut,1992)

Quantify small scales structures

Recurrence density:

RR (ε) =1

N2

N∑i,j=1

Ri,j (ε) =1

N2

N∑l=1

l P (ε, l)

Webber & Zbilut, Embeddings and delays as derived from quantification of recurrence plots

Physics Letters A171 (1992)



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

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Clustering

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GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Recurrence Quantification Analysis (RQA)

Diagonal lines distribution

DET =∑N

l=lminl P(l)∑N

l=1 l P(l)

Lmean =∑N

l=lminl P(l)∑N

l=lminP(l)

Lmax = max {li}Nl

i=1 ; DIV = 1Lmax

ENTR = −∑N

l=lminp (l) ln (p (l))

p (l) = P(l)∑Nl=lmin

P(l)

TREND =∑Nτ=1(τ−N/2)(RRτ−〈RRτ 〉)∑N

τ=1(τ−N/2)2

Vertical lines distribution

LAM =∑N

v=vminv P(v)∑N

v=1 v P(v)

TT =∑N

v=vminv P(v)∑N

v=vminP(v)

Vmax = max {vi}Nv

i=1



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Entropy, information and Predictability

Lyapunov exponents

Correlation dimension and entropy

Shannon entropy

RQA obtained through the recurrence plots

=⇒ A way to quantify the complexity of the dynamics

Problems:

Optimal choice of parameters

Correct embedding (if any)

False recurrences (sojourn points)

Thiel, Romano, Kurths, How much information is contained in a recurrence plot?

Physics Letters A 330 (2004)

Letellier, Estimating the Shannon Entropy: Recurrence Plots versus Symbolic Dynamic

Physical Review Letters 96 (2006)



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

ApplicationsLogistic map: RP

Logistic map

xk+1 = r xk (1− xk )

r ∈ [3.5, 4]



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

ApplicationsLogistic map: RQA



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Applications (time series)Solar spots

Sunspot Index Data Center: http://sidc.oma.be/sunspot-data/

Yearly data (1700-2012) Monthly data (1955-2013)

RQA (yearly)DET 0.730Lmax 27.00ENTR 1.092

RQA (monthly)DET 0.878Lmax 102.00ENTR 1.593


http://sidc.oma.be/sunspot-data/


M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Applications (time series)Economic data

FTSE MIB

RQAReal Random

DET 0.948 DET 0.169Lmax 357.00 Lmax 5.00

ENTR 2.682 ENTR 0.311

DAX

RQAReal Random

DET 0.836 DET 0.110Lmax 114.00 Lmax 6.00

ENTR 1.777 ENTR 0.219



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Recurrence Networks

Recurrence Networks

Graph G = 〈N, L〉 defined by the adjacency matrix

Ai,j =

{1 if i and j are connected in G

0 otherwise

Ai ,j ≡ Ri ,j − δi ,j



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

Recurrence Networks



M.I. Loffredo

Intro

StatisticalAnalysis

Scale-freenetworks

Correlations

Clustering

Reciprocity

GeneratingNetworks


Models

Wealthdistribution

Modeling theWTW

Recurrences

References

[1] Boccaletti, Latora, Moreno, Chavez and Hwang,Complex Networks: Structure and Dynamics, Physics Reports 424 (2006) 175

[2] Caldarelli, Scale-free networks: complex webs in nature and technology,Oxford Univ. Press, 2007

[3] Caldarelli and Vespignani, Large scale structure and dynamics of complex networks:from information technology to finance and natural science, World Scientific, 2007

[4] Jackson, Social and Economic Networks Princeton Univ. Press, 2008

[5] Eckmann, Kamphorst, Ruelle,Recurrence Plots of Dynamical Systems, Europhysics Letters 4 (1987)

[6] Marwan, Romano, Thiel, Kurths,Recurrence plots for the analysis of complex systems, Physics Reports 438 (2007)

[7] Donner, Zou, Donges, Marwan, Kurths, Recurrence networks - A novel paradigm fornonlinear time series analysis, New Journal of Physics 12 (2010)


Introduction to Complex Networks - Science South Tyrol,,UNIBZ,90,68.pdf ·...

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Transcript of Introduction to Complex Networks - Science South Tyrol,,UNIBZ,90,68.pdf ·...