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## Theory of Networks

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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

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

SONET

SONET

ATM

ATM

ATM

ATM

IP

IP

IP

IP

IP

IP

IP

IP

The real picture is even worse:fiber-cutting experiment in the pastEncapsulation

Routing devices

IP

Routers

ATM

ATM switches

SONET

DCS

MPLS

SONET

Lambda path

Lambda path

Lambda path

The real picture is even worse:fiber-cutting experiment now/futureFiber 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

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

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), ...

Statistical mechanics:two examplesIdeal 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 networkswith similar topologies

- Description and basic properties of:
- engineered networks
- WWW
- 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
- Is their topological similarity coincidental or is there an explanation?

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

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

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