Alessandro Vespignani (CNRS, LPT-Paris)

•Alain Barrat (CNRS, LPT-Paris)

•Yamir Moreno (University of Saragoza)

•Alexei Vazquez (University of Notre Dame)

•Romualdo Pastor-Satorras (UPC -Barcelona)

•Roberto Percacci (INFN)

•Marc Barthelemy (CEA-Paris)

•Luca Dall’Asta (CNRS, LPT-Paris)

•Ignacio Alvarez Hamelin (CNRS, LPT-Paris)

The Physical Internet

•Satellites•Computers (routers)•Modems (??)•Phone cables•Optic fibers•EM waves

Technological Heterogeneity

A network is a system that allows its abstract/mathematical representation as a graph

Vertices (nodes) = elements of the system

Edges (links) = interactions/relations among the elements of the system

Internet tomographyInternet tomography

•Multi-probe reconstruction (router-level)•Use of BGP tables for the Autonomuos System level (domains)

•CAIDA•NLANR•RIPE•IPM •PingER

Claffy et al (1999).

Topology and performance measurements

Graph representation

different granularities

CAIDA AS cross section map

Shortest paths = minimum (# hops) between two nodes

Regular lattice with N= 104 d ~ 102

Small world with N= 104 d ~ ln N

Average fraction of nodes within a shortest path of lenght d

Distribution of Shortest paths (# hops) between two nodes

Haphazard set of points and linesRandomness

This does not imply complexity!!

Erdös-Rényi model (1960)

Poisson distribution

With probability p an edge is established among couple of vertices

<k> = p N

Where “the complications” are ??

(Late 90s large networks graphs and data become available)

Clustering coefficient = connected peers will likely know each other

C =# of links between 1,2,…n neighbors

n(n-1)/2

1

2

3

n

Higher probability to be connected

Connectivity distribution

Scale-free properties

P(k) = probability that a node has k links

P(k) ~ k -P(k) ~ k - ( 3)

• <k>= const• <k2> • <k>= const• <k2>

Diverging fluctuations

Faloutsos et al. 1999

•Router level & AS level

Pastor Satorras, Vazquez &Vespignani, PRL 87, 258701 (2001)

Scale-free topology generators

INET (Jin, Chen, Jamin)

BRITE (Medina & Matta)

Classical Internet topology generators

•Waxman generator

•Structural generatorsTransit-stubTiers

ExponentiallyBoundedDegree distributions

Modeling of the Internet structure with ad-hoc algorithmstailored on the properties we consider more relevant

The Internet growth

1997 2000AS 3112 9107

1997 2000AS 3112 9107

In 1999:•3410 new AS•1713 lost AS

Pastor Satorras, Vazquez &Vespignani, PRL 87, 258701 (2001)Qian, Govindan et al. (2002)

Main Features of complex networks

•Many interacting units•Dynamical evolution•Self-organization

•Many interacting units•Dynamical evolution•Self-organization

Non-trivial architectureUnexpected emergent properties

Cooperative phenomena

Complexity

Supervising entity Project/blueprint

StatisticalStatistical physics approach to physics approach to network modelingnetwork modeling

Microscopic processes of the many component units

Macroscopic statistical and dynamical properties of the system

Cooperative phenomenaComplex topology

Natural outcome of the dynamical evolution

Shift in focus : Dynamical processes

Modeling starts from the understanding of the basic mechanisms underlying the networks’ growth

Complex topology is spontaneously generated in the models (opposite to ad-hoc constructions)

Richer understanding of the interplay among dynamics,traffic and economical requirements.

Preferential attachmentPreferential attachment mechanismmechanism

Networks expand by the addition of new nodes

Examples:

WWW : addition of new documents Internet : connection of new routers

Nodes are wired with higher probabibility to highly connected nodes

Examples:

WWW : links to well known web-pages Internet : links to well connected ISP

How to generate scale-free graphHow to generate scale-free graph

Growth : at each time step a new node is added with m links to be connected with previous nodes

Preferential attachment: The probability that a new link is connected to a given node is proportional to the number of node’s links.

by Barabasi & Albert (1999)

The BA modelThe BA model

The preferential attachment is following the probability distribution :

The generated connectivity distribution is

P(k) ~ k -P(k) ~ k -

BA network

Degree distribution

Preferential Attachment in InternetPreferential Attachment in Internet

Probability that a link connects to a node with connectivity k

(k) ~ k p(k) k

-1.2

1

.0 Linear preferential attachment

Pastor Satorras, Vazquez &Vespignani, PRL 87, 258701 (2001)Qian, govindan et al. (2002)Jeong, Neda and Barabasi (2003)

Shift of focus:

Static construction Dynamical evolution

Direct problem

Evolution rules Emerging topology

Inverse problem

Given topology Evolution rules

More models

•Generalized BA model

(Redner et al. 2000)

(Mendes & Dorogovstev 2000)

(Albert et al.2000)

j jj

iii k

kk

)(

Non-linear preferential attachment : (k) ~ k

Initial attractiveness : (k) ~ A+k

Rewiring

•Highly clustered

(Eguiluz & Klemm 2002)

•Fitness Model (Bianconi et al. 2001)

•Multiplicative noise (Huberman & Adamic 1999)

Heuristically Optimized Trade-offs (HOT)

Papadimitriou et al. (2002)

New vertex i connects to vertex j by minimizing the function Y(i,j) = d(i,j) + V(j)d= euclidean distanceV(j)= measure of centrality

Optimization of conflicting objectives

What else……

•Hierarchies and correlations (architecture)

•Robustness and resilience

•Spreading phenomena

•Routing and database updating

The Hierarchy of the InternetThe Hierarchy of the Internet

•Stub AS : has only one connection to another AS•Multi-homed AS: two or more connections to other ASs but does not carry transit traffic•Transit AS: Two or more connections to other ASs and carries transit traffic

HIERARCHICAL DECOMPOSITION

• Four level hierarchy (linear scale) (Govindan and Reddy 1994)• Three-tier hierarchy (log scale) (Chang et al. 98)• Jellyfish hierarchy (connectivity shells) (Tauro et al. 2001)

Connectivity correlations Connectivity correlations

Degree correlation function< knn(k)> = k’ k’ p(k’|k)

Average nearest neighbors degree

Pastor Satorras, Vazquez &Vespignani, PRL 87, 258701 (2001)

Connectivity Hierarchy Connectivity Hierarchy

Degree correlation function< knn(k)> = k’ k’ p(k’|k)

Average nearest neighbors degree

Highly degree ASs connect to low degree ASs

Low degree ASs connect

to high degree ASs

No hierarchy for the router map

Clustering HierarchyClustering Hierarchy

Clustering coefficient as a function of the vertex degree

Highly degree ASs bridge not connected regions of the Internet Low degree ASs have links with highly interconnected regions of the Internet

No hierarchy at the router level

•Scale-free hierarchy•Continuum of levels

Modular construction

Small groups of networks organized in larger groups which act as the modules at the next level and so on “ad libitum”

Models validation (part II)

Clustering hierarchyDegree hierarchy

•Absence of any epidemic threshold (critical point)

•Active state for any value of

•The infection pervades the system whatever spreading rate

•In infinite systems the infection is infinitely persistent (indefinite stationary state)

Density of infected individuals

Scale-free connectivityScale-free connectivity

Pastor-Satorras &Vespignani, PRL 86, 3200(2001)

Rationalization of computer virus data

•Lack of healthy phase = standard immunization cannot drive the system below thershold!!!

Distributed database updating

Broadcast = each elements passes the update to neighbors

Epidemics = the update is spread as an infective agent

E= efficiency = # updated databases

# of messages sent

Warning => not deterministicNot all elements are contacted!!

Trade-offs between efficiency and reliability

Moreno, Nekovee,Vespignani(03)

Internet is ever changing at all levels

Is it too ambitious the attempt to have a dynamical theoryof the Internet at the large scale ??

The lesson of statistical physics and cooperative phenomena:Basic symmetry and principles win over the microscopic details when

we look at emergent properties

One step back….

Deployement of measurement tools•Active •Passive

Netscan (traceroute based tool) maps the paths to selected IP addressfrom a testing host (single probe).

Testing host

= directed graph spanning tree

• One path to each node

• NO cross-paths

Burch & Cheswick (1999)

Interconnected level mapsInterconnected level maps

Heuristic methods (Govindan et al.)•Router level maps

Very effective for intranetwork

Measurements infrastructures

Merging partial spanning tress from multiple sources

Sampling is incompleteLateral connectivity is missed (edges are underestimated)Finite size sample

Govindan et al 2002

Introduction of Biases

Vertices and edges best sampled in the proximity of sources

Number of sources and target (total traceroute probes)

Statistical properties of the sampled graph sharply different from the original one

Crovella et al. 2002

Clauset & Moore 2004

De Los Rios & Petermann 2004

Mean-Field Theory of traceroute-like exploration

= Ns Nt

N Nt= # targets (t -> density of targets)

Ns = # sources

pij = 1 –exp ( - bij )

pi = 1 – (1- t ) exp ( - bi )

= t Ns

k*i = 2 +2 bi )

Edge detection probability

Vertex detection probability

Effective degree observed

bi , bij Betweenness

Betweenness centrality = # of shortest paths traversing a vertex or edge (flow of information ) if each individuals send a message to all other

individuals

Scale-free graph are better discriminated

Tail is sampled very effectively

Homogeneous graphs give riseto spurious effects

Average connectivity always dominate

Heavy tails properties are a genuine feature of the Internet

Quantitative analysis might be strongly biased

however

What else….

•Router level very limited maps

•Optimized strategies

•Massive deployement traceroute@home

The dark side of the moon……Traffic and weights

•The internet is a weighted networks bandwidth, traffic, efficiency, routers capacity

•Data are scarse and on limited scale

•Interaction among topology and traffic

•Traffic and routing