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Alessandro Vespignani (CNRS, LPT-Paris)
•Marc Barthelemy (CEA-Paris)
•Alain Barrat (CNRS, LPT-Paris)
•Romualdo Pastor-Satorras (UPC -Barcelona)
•Yamir Moreno (University of Saragoza)
•Alexei Vazquez (University of Notre Dame)
•Roberto Percacci (INFN)
•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 tomography
Claffy et al (1999).
•Multi-probe reconstruction (router-level)
•Use of BGP tables for the Autonomuos System level (domains)
•CAIDA
•NLANR
•RIPE
•IPM
•PingER
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
Distribution of Shortest paths (# hops)
between two nodes
Average fraction of nodes
within a shortest path of lenght d
Haphazard set of points and lines
Randomness
This does not imply complexity!!
Erdös-Rényi model
(1960)
With probability p an edge is
established among couple of
vertices
Poisson distribution
<k> = p N
(Late 90s large networks graphs and data become available)
Where “the complications” are ??
Clustering coefficient = connected peers will likely know each other
n
3
Higher probability to be connected
2
1
# of links between 1,2,…n neighbors
C=
n(n-1)/2
Connectivity distribution
P(k) = probability
that a node has k
links
•Router level & AS level
P(k) ~ k -g
(2 < g  3)
• <k>= const
• <k2>  
Faloutsos et al. 1999
Pastor Satorras, Vazquez &Vespignani,
PRL 87, 258701 (2001)
Scale-free properties
Diverging fluctuations
Classical Internet topology generators
•Waxman generator
•Structural generators
Transit-stub
Tiers
Exponentially
Bounded
Degree distributions
Scale-free topology generators
INET (Jin, Chen, Jamin)
BRITE
(Medina & Matta)
Modeling of the Internet structure with ad-hoc algorithms
tailored on the properties we consider more relevant
The Internet growth
AS
1997
3112
2000
9107
Pastor Satorras, Vazquez &Vespignani,
PRL 87, 258701 (2001)
Qian, Govindan et al. (2002)
In 1999:
•3410 new AS
•1713 lost AS
Main Features of complex networks
•Many interacting units
•Dynamical evolution
•Self-organization
Supervising entity
Project/blueprint
Non-trivial architecture
Unexpected emergent properties
Cooperative phenomena
Complexity
Statistical physics approach to
network modeling
Microscopic processes of the
many component units
Macroscopic statistical and dynamical
properties of the system
Cooperative phenomena
Complex 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 attachment mechanism
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 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 model
The preferential attachment is
following the probability
distribution :
The generated connectivity distribution is
P(k) ~ k -3
Degree distribution
BA network
Preferential Attachment in Internet
Pastor Satorras, Vazquez &Vespignani,
PRL 87, 258701 (2001)
Qian, govindan et al. (2002)
Jeong, Neda and Barabasi (2003)
Probability that a link
connects to a node with
connectivity k
m(k) ~ k a p(k)  k a-g
a-g = -1.2
a  1.0
Linear preferential attachment
Shift of focus:
Static construction
Dynamical evolution
Direct problem
Evolution rules
Emerging topology
Inverse problem
Given topology
Evolution rules
More models
•Generalized BA model
Non-linear preferential attachment : (k) ~ ka
(Redner et al. 2000)
Initial attractiveness : (k) ~ A+ka
(Mendes & Dorogovstev 2000)
Rewiring
(Albert et al.2000)
•Highly clustered
(Eguiluz & Klemm 2002)
 ( ki ) 
i k i
 j j k j
•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) = a d(i,j) + V(j)
d= euclidean distance
V(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 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
Pastor Satorras, Vazquez &Vespignani,
PRL 87, 258701 (2001)
Average nearest neighbors degree
< knn(k)> = Sk’ k’ p(k’|k)
Degree correlation function
Connectivity Hierarchy
Average nearest neighbors degree
< knn(k)> = Sk’ k’ p(k’|k)
Degree correlation function
Highly degree ASs
connect to low degree ASs
Low degree ASs connect
to high degree ASs
No hierarchy for the router map
Clustering 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)
Degree hierarchy
Clustering hierarchy
Scale-free connectivity
Density of infected individuals
•Absence of any
epidemic threshold
(critical point)
•Active state for any
value of l
•The infection pervades
the system whatever
spreading rate
•In infinite systems the
infection is infinitely
persistent (indefinite
stationary state)
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
Moreno, Nekovee,Vespignani(03)
E= efficiency =
# updated databases
# of messages sent
Warning => not deterministic
Not all elements are contacted!!
Trade-offs between efficiency and reliability
Internet is ever changing at all levels
Is it too ambitious the attempt to have a dynamical theory
of 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 address
from a testing host (single probe).
Testing host
• One path to each node
= directed graph
spanning tree
• NO cross-paths
Burch & Cheswick (1999)
Interconnected 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 incomplete
Lateral 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
e = Ns Nt = rt Ns
N
Ns = # sources
Nt= # targets (rt -> density of targets)
pij = 1 –exp ( -e bij )
Edge detection probability
pi = 1 – (1- rt ) exp ( -e bi )
Vertex detection probability
k*i = 2e +2 e bi )
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 rise
to spurious effects
Average connectivity
always dominate
Heavy tails properties are a genuine feature of the Internet
however
Quantitative analysis might be strongly biased
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