Theory of Networks

1. Theory of Networks Course Announcement Dmitri Krioukov dima@caida.org June 1st, 2005, syslunch

2. 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

3. 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

4. Fiber 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 past Encapsulation Routing devices IP Routers ATM ATM switches SONET DCS

5. IP MPLS SONET Lambda path Lambda path Lambda path The real picture is even worse:fiber-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

6. 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

7. 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?

8. 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

9. 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)

10. 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

11. 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 examples

12. 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

13. 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

14. 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

15. 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 • Is their topological similarity coincidental or is there an explanation?

16. 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

17. 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

18. 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

19. Connection between the equilibrium and growing network models • ... in works by Dorogovtsev, Newman, Krzywicki, and Burda

20. 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.

21. 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