Download Presentation

Theory of Networks
Course Announcement
Dmitri Krioukov
[email protected]
June 1st, 2005, syslunch
Purpose and motivation
Purpose of the presentation:



introduce the subject
describe the course skeleton
check if there is any interest
Purpose of the course



review results on the topological properties of large-scale networks
observed in reality, with an emphasis on the Internet
teach the most effective methods of massive network topology analysis
gain hands-on experience using these methods to obtain useful results
Motivation for the course


semantic intuition that networkers might be interested in networks
bridge the gap between islands of knowledge
Provocation: kc’s “can’t measure”
can't figure out where an IP address is
can't measure topology effectively in either direction, at any layer
can't track propagation of a routing update across the Internet
can't get router to give you all available routes, just best routes
can't get precise one-way delay from two places on the Internet
can't get an hour of packets from the core
can't get accurate flow counts from the core
can't get anything from the core with real addresses in it
can't get topology of core
can't get accurate bandwidth or capacity info

not even along a path, much less per link
can't trust whois registry data
no general tool for `what's causing my problem now?’
privacy/legal issues deter research
makes science challenging -- discouraging to academics
The real picture is even worse:
fiber-cutting experiment in the past
IP
ATM
IP
SONET
IP
ATM
IP
Encapsulation
IP
ATM
SONET
Fiber
IP
ATM
IP
IP
SONET
ATM
IP
Routing devices
Routers
ATM switches
DCS
The real picture is even worse:
fiber-cutting experiment now/future
IP
MPLS
SONET
Fiber strand
Lambda path
Lambda path
Fiber bundle
Lambda path
Lambda path
Fiber strand
Encapsulation
IP
VPN LSP
Routing devices
Routers
Routers
LDP LSP
RSVP-TE LSP
SONET/TDM LSP
Optical/LSC LSP
Fiber/FSC LSP
Routers
Routers
DCS
OXCs
FXCs
Why would I care?
Why topology is important?
“What-if” questions, like:

New routing and other protocol design, development, and
testing, e.g. of scalability/convergence properties:


new routing protocol might offer X-time smaller routing tables (RTs)
for today but scale Y-time worse, with Y >> X
dependence of routing on topology:




generic topologies: stretch = 1, RT = Ω(n); stretch = 3, RT = Ω(n1/2)
trees: stretch = 1, RT = Ω(1)
Network robustness, resilience under attack, speed of virus
spreading
Traffic engineering, capacity planning, network management
Network measurements: both topology and traffic
Network evolution
Picture summary
A lot of complexity
Large-scale system consisting of an
enormous number of heterogeneous elements
Fundamental impossibility to measure the
system completely
But we still need to study it
Is there any known way of how to do it?
Empirical observation:
review of available literature
Numbers of important “topology” papers



CS: <10
math: ~10
physics: >100, +1 book on the Internet, +several books on
scale-free networks
Example of important problem: given the degree
distribution, find the distance distribution



CS: 0
math: 2 papers on maximum and average distance
physics: 4 different approaches yielding distance distributions
Explanation of the observation
CS: does not have a well-established methodology
(every paper develops a new one)
math: the high level of rigor clashes with the high
level of complexity of the problems
physics: the methodology is well-established and
well-developed, and its effectiveness is verified by
>100 year old history of practically useful results
used in our every-day life (e.g. material science)
Statistical mechanics:
problem formulation
Given: a macroscopic system consisting of
a large number of microscopic elements
Given: an incomplete set of measurements
of some properties of the system
Find: probability distributions for other
properties of the system
Statistical mechanics:
two examples
Ideal gases



given: gas consists of
molecules
given: N, V, T,
equilibrium
find: P, S, CV, CP, ...
Erdős-Rényi graphs



given: network consists
of nodes and links
given: n, m,
maximally random
find: P(k), P(k1,k2),
C(k), d(x), ...
Ideal gas vs. the Internet
Two major differences
Size (1024 vs. 104)
 Complexity:

amount of information loss at the abstraction stage
 no way to tell what details do or do not “matter”

Statistical mechanics vs. kinetic theory
Skeleton of the course
Internet and its topology metrics
Other networks
Intro to statistical mechanics
Types of network models



Equilibrium networks
Non-equilibrium (growing) networks
Connection between the two
Applications (to the Internet)
and advanced topics
Internet and its topology metrics
Internet topology measurements
Metrics and why they are important










Size, average degree
Degree distribution
Degree correlations
Clustering
Rich club connectivity
Coreness
Distance, eccentricity
Betweenness
Spectrum
Entropy
Other real-world networks
with similar topologies
Description and basic properties of:

engineered networks






social networks





paper citations
movie collaborations
acquaintance networks
sexual contacts
language networks


WWW
e-mail
phone calls
power grids
electronic circuits
word webs
biological networks




metabolic reactions
protein interactions
food webs
phylogenetic trees
Is their topological similarity coincidental or is there an explanation?
Basic facts from
statistical mechanics
Elements of the probability theory
Elements of classical and quantum mechanics
Ensembles in statistical mechanics
Equilibrium and non-equilibrium systems
Entropy and the law of maximum uncertainty
Entropy and information
Statistical mechanics and thermodynamics
Equilibrium networks
Ensembles of random networks
Classical Erdős-Rényi random graphs as the canonical
ensemble
Power-law random graphs (PLRGs) as the microcanonical
ensemble
Correlations and clustering in the standard ensembles
Finite size and other constraints (of network being simple,
connected, etc.)
Equilibrium networks with arbitrary constraints (e.g.
longer-range correlations, clustering, etc.) and their
properties
Implications for topology generators
Watts-Strogatz, Kleinberg, and Fraigniaud models
Non-equilibrium (growing) networks
Exponential networks
Preferential attachment and its variations
Type of preference yielding scale-free networks
Correlations and clustering in growing networks
Deterministic networks with strong clustering
Network growth models equivalent to preferential
attachment (e.g. HOT)
Network growth models non-equivalent to
preferential attachment
Connection between the equilibrium
and growing network models
... in works by Dorogovtsev, Newman,
Krzywicki, and Burda
Applications (to the Internet)
and other advanced topics
Internet topology measurements: traceroute-like explorations, “hidden”
links, alias resolution, IP2AS mapping, sampling biases vs. betweenness
distributions, etc.
Internet topology generators and evolution models: Waxman, structural,
BRITE, Inet, PLRG, PFP, economy-based, etc.
Routing and searching in networks:



distance distribution in the microcanonical ensemble
compact routing in scale-free and Internet-like networks
greedy routing and searching in networks




embeddable in Euclidian spaces (P2P, geographical, etc.)
of the Kleinberg model (social networks)
with small treewidth, or low chordality, or strong clustering (the Fraigniaud model)
decomposability of a network into the local and global parts
Internet robustness: random failures and targeted attacks, percolation theory,
speed of virus spreading, epidemic threshold, network immunization
strategies, etc.
Spectral analysis: spectrum of the microcanonical ensemble, Internet
performance (conductance and congestion properties), Internet hierarchical
structure, etc.
Source material
S. N. Dorogovtsev and J. F. F. Mendes,
Evolution of Networks,
http://www.amazon.com/exec/obidos/ASIN/0198515901/
R. Pastor-Satorras and A. Vespignani,
Evolution and Structure of the Internet,
http://www.amazon.com/exec/obidos/ASIN/0521826985/
D. Aldous, From Random Graphs to Complex Networks,
UC Berkeley, STAT 206,
http://www.stat.berkeley.edu/users/aldous/Networks/
Statistical mechanics