Theory of Networks

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Theory of Networks

• Course Announcement
• Dmitri Krioukov
• dima_at_caida.org
• June 1st, 2005, syslunch

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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

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Provocation kcs cant 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

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The real picture is even worsefiber-cutting
experiment in the past
Encapsulation
Routing devices
IP
Routers
ATM
ATM switches
SONET
DCS
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The real picture is even worsefiber-cutting
experiment now/future
Fiber bundle
Fiber strand
Lambda path
Encapsulation
Routing devices
IP
Routers
VPN LSP
Routers
LDP LSP
Routers
RSVP-TE LSP
Routers
SONET/TDM LSP
DCS
Optical/LSC LSP
OXCs
Fiber/FSC LSP
FXCs
Fiber strand
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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 gtgt X
• dependence of routing on topology
• generic topologies stretch 1, RT O(n)
  stretch 3, RT O(n1/2)
• trees stretch 1, RT O(1)
• Network robustness, resilience under attack,
  speed of virus spreading
• Traffic engineering, capacity planning, network
  management
• Network measurements both topology and traffic
• Network evolution

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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?

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Empirical observationreview of available
literature

• Numbers of important topology papers
• CS lt10
• math 10
• physics gt100, 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

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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 gt100 year old history of practically useful
  results used in our every-day life (e.g. material
  science)

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Statistical mechanicsproblem 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

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Statistical mechanicstwo examples

• Ideal gases
• given gas consists of molecules
• given N, V, T, equilibrium
• find P, S, CV, CP, ...

• Erdos-Rényi graphs
• given network consists of nodes and links
• given n, m,maximally random
• find P(k), P(k1,k2), C(k), d(x), ...

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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

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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

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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

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Other real-world networkswith similar topologies

• Description and basic properties of
• engineered networks
• WWW
• e-mail
• phone calls
• power grids
• electronic circuits
• social networks
• paper citations
• movie collaborations
• acquaintance networks
• sexual contacts
• language networks
• word webs
• biological networks
• metabolic reactions
• protein interactions
• food webs
• phylogenetic trees

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Basic facts fromstatistical 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

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Equilibrium networks

• Ensembles of random networks
• Classical Erdos-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

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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

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Connection between the equilibrium and growing
network models

• ... in works by Dorogovtsev, Newman, Krzywicki,
  and Burda

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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.

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Source material

• S. N. Dorogovtsev and J. F. F. Mendes,Evolution
  of Networks,http//www.amazon.com/exec/obidos/ASI
  N/0198515901/
• R. Pastor-Satorras and A. Vespignani,Evolution
  and Structure of the Internet,http//www.amazon.c
  om/exec/obidos/ASIN/0521826985/
• D. Aldous, From Random Graphs to Complex
  Networks,UC Berkeley, STAT 206,http//www.stat.b
  erkeley.edu/users/aldous/Networks/
• Statistical mechanics