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ESE 680 Special topics in ESE Distributed Dynamical Systems

ESE 680 Lecture 2 01/11/2007. ESE 680 Special topics in ESE Distributed Dynamical Systems. Ali Jadbabaie. Department of Electrical and Systems Engineering and GRASP Laboratory University of Pennsylvania. 365 GRW jadbabai@seas.upenn.edu.

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ESE 680 Special topics in ESE Distributed Dynamical Systems

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  1. ESE 680 Lecture 2 01/11/2007 ESE 680 Special topics in ESEDistributed Dynamical Systems Ali Jadbabaie Department of Electrical and Systems Engineering and GRASP Laboratory University of Pennsylvania 365 GRW jadbabai@seas.upenn.edu http://www.seas.upenn.edu/~jadbabai/ESE680/ese680.html

  2. Course Info • This is a RESEARCH SEMINAR • Requires a lot of INDEPENDENT, critical reading of literature • You are expected to actively PARTICIPATE in discussions • A LOT of reading is required and you need to be able to present papers • There are very few didactic lectures • Here is a brief summary of last time

  3. Nonlinear/uncertain hybrid/stochastic etc. Complex networked systems Single Agent Flocking/synchronization consensus Multi-agent systems Networked dynamical systems Complexity of dynamics Complexity of interconnection

  4. Networked dynamical systems System size State dimensionality

  5. Statistical Physics and emergence of collective behavior Simulations and conjectures but few “proofs’

  6. Working systems but few “proofs’

  7. Overview Nonlinear/uncertain hybrid/stochastic etc. Complex networked systems ? Complexity of dynamics Single Agent ? Flocking/synchronization consensus Multi-agent systems Complexity of interconnection

  8. r neighbors of agent i agent i Multi-agent setting: Vicsek’s kinematic model • How can a group of moving agents collectively decide on direction, based on nearest neighbor interaction? How does global behavior emerge from local interactions?

  9. Synchronization Fireflies Flashing From D. Attenborough “Trials of Life – Talking to strangers”

  10. KuramotoModel All-to-all interaction This is the Kuramoto model We assume throughout homogeneous coupling.

  11. Kuramoto model & graph topology 2 1 3 6 4 5

  12. Ubiquity of Dual Decompositions • Dual Decomposition is THE key idea that makes the internet protocols “work” in a distributed asynchronous fashion • What is the connection between flocking, oscillator synchronization, and the internet? • But, how DO “these internets” work? • Senator Ted Stevens (R-Alaska), (the architect of the 280 million dollar bridge to nowhere: “ …and again, the Internet is not something you just dump something on. It's not a big truck. It's a series of tubes. And if you don't understand those tubes can be filled and if they are filled, when you put your message in, it gets in line and it's going to be delayed by anyone that puts into that tube enormous amounts of material, enormous amounts of material.”

  13. The Internet hourglass Applications Web FTP Mail News Video Audio ping napster Transport protocols TCP SCTP UDP ICMP IP Ethernet 802.11 Power lines ATM Optical Satellite Bluetooth Linktechnologies

  14. The Internet hourglass Applications Web FTP Mail News Video Audio ping napster TCP IP Ethernet 802.11 Power lines ATM Optical Satellite Bluetooth Linktechnologies

  15. IP on everything The Internet hourglass Applications IP under everything Web FTP Mail News Video Audio ping napster TCP IP Ethernet 802.11 Power lines ATM Optical Satellite Bluetooth Linktechnologies

  16. Congestion Control for the Internet Aims to avoid congestion collapse • Congestion collapse appeared in the late 80’s because of the lack of congestion control • Intuitive congestion control design alleviated the problem… • Until a few years ago when it was shown to be unstable! New designs aim to achieve: • Optimal sharing of available resources at equilibrium; • Scalable stability for • arbitrary topologies (size and connectivity) • arbitrary links’ capacities • arbitrary, inhomogeneous round trip times (time delays) Last property has been proven only for the linearized system – here we provide a proof for the nonlinear case

  17. Routers Mesh-like core of fast, low degree routers Hosts

  18. High degree nodes are at the edges. Routers Hosts

  19. Power Laws and Internet Topology A few nodes have lots of connections Source: Faloutsos et al (1999) Observed scaling in node degree and other statistics: • Autonomous System (AS) graph • Router-level graph How to account for high variability in node degree? number of connections rank rank Most nodes have few connections

  20. Los Alamos fire 6 Data compression (Huffman) WWW files Mbytes (Crovella) 5 4 Cumulative 3 Frequency Forest fires 1000 km2 (Malamud) 2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Decimated data Log (base 10) Size of events

  21. 18 Sep 1998 Forest Fires: An Example of Self-Organized Critical Behavior Bruce D. Malamud, Gleb Morein, Donald L. Turcotte 4 data sets

  22. -1 -1/2 6 Web files 5 Codewords 4 Cumulative 3 Frequency Fires 2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Size of events Log (base 10)

  23. Los Alamos fire 6 >1e5 files Data compression (Huffman) WWW files Mbytes (Crovella) 5 4 >4e3 fires Cumulative 3 Frequency Forest fires 1000 km2 (Malamud) 2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Decimated data Log (base 10) Size of events

  24. 20th Century’s 100 largest disasters worldwide 2 10 Technological ($10B) Natural ($100B) 1 10 US Power outages (10M of customers) 0 10 -2 -1 0 10 10 10

  25. Technological ($10B) Natural ($100B) US Power outages (10M of customers) 20th Century’s 100 largest disasters worldwide 2 10 1 10 0 10 -2 -1 0 10 10 10

  26. 2 10 Log(Cumulative frequency) 1 10 = Log(rank) 0 10 -2 -1 0 10 10 10 Log(size)

  27. 100 80 Technological ($10B) rank 60 Natural ($100B) 40 20 0 0 2 4 6 8 10 12 14 size

  28. 100 10 3 2 1 2 10 Log(rank) 1 10 0 10 -2 -1 0 10 10 10 Log(size)

  29. Slope = -1 (=1) 20th Century’s 100 largest disasters worldwide 2 10 Technological ($10B) Natural ($100B) 1 10 US Power outages (10M of customers) 0 10 -2 -1 0 10 10 10

  30. -1 -1/2 6 Data compression WWW files Mbytes 5 4 Cumulative 3 Frequency Forest fires 1000 km2 2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Decimated data Log (base 10) Size of events

  31. 6 Data compression WWW files Mbytes 5 exponential 4 -1 Cumulative 3 Frequency Forest fires 1000 km2 2 -1/2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Size of events

  32. 6 Data compression WWW files Mbytes 5 exponential 4 Cumulative All events are close in size. 3 Frequency Forest fires 1000 km2 2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Size of events

  33. Most events are small But the large events are huge 6 Data compression WWW files Mbytes 5 4 -1 Cumulative 3 Frequency Forest fires 1000 km2 2 -1/2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Size of events

  34. But most packets are in huge files But most trees are in huge fires 6 Most files are small Data compression WWW files Mbytes 5 4 -1 Cumulative 3 Frequency Forest fires 1000 km2 Most fires are small 2 -1/2 1 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Size of events

  35. Most events are small But the large events are huge 6 Data compression WWW files Mbytes Robust 5 4 -1 Cumulative 3 Frequency Forest fires 1000 km2 2 -1/2 1 Yet Fragile 0 -1 -6 -5 -4 -3 -2 -1 0 1 2 Size of events

  36. Large scale phenomena is extremely non-Gaussian • The microscopic world is largely exponential • The laboratory world is largely Gaussian because of the central limit theorem • The large scale phenomena has heavy tails (fat tails) and power laws

  37. Power Laws in Topology Modeling • Recent emphasis has been on whether or not a given topology model/generator can reproduce the same types of macroscopic statistics, especially power law-type degree distributions • Lots of degree-based models have been proposed • All of them are based on random graphs, usually with some form of preferential attachment • All of them are connectivity-only models and tend to ignore engineering-specific system details • Examples: BRITE, INET, Barabasi-Albert, GLP, PLRG, CMU-generator

  38. Models of Internet Topology • These topology models are merely descriptive • Measure some feature of interest (connectivity) • Develop a model that replicates that feature • Make claims about the similarity between the real system and the model • A type of “curve fitting”? • Unfortunately, by focusing exclusively on node degree distribution, these models that get the story wrong • We seek something that is explanatory • Consistent with the drivers of topology design and deployment • Consistent with the engineering-related details • Can be verified through the measurement of appropriate system-specific details

  39. Hosts Heuristically Optimal Network Mesh-like core of fast, low degree routers Cores High degree nodes are at the edges. Edges

  40. Intermountain GigaPoP U. Memphis Indiana GigaPoP WiscREN Northern Lights OARNET Great Plains Front Range GigaPoP U. Louisville Merit NYSERNet OneNet StarLight Arizona St. NCSA Qwest Labs Iowa St. U. Arizona UNM Oregon GigaPoP WPI Pacific Wave Kansas City Indian- apolis Denver Pacific Northwest GigaPoP SINet Chicago Seattle SURFNet ESnet New York MANLAN U. Hawaii GEANT Rutgers U. Wash D.C. UniNet Sunnyvale MREN WIDE MAGPI CENIC Los Angeles Northern Crossroads TransPAC/APAN AMES NGIX Tulane U. Atlanta Houston LaNet SOX PSC North Texas GigaPoP U. Delaware Drexel U. DARPA BossNet Texas GigaPoP Mid-Atlantic Crossroads Texas Tech SFGP/ AMPATH Miss State GigaPoP UT Austin NCNI/MCNC U. Florida UMD NGIX UT-SW Med Ctr. U. So. Florida Florida A&M Abilene Backbone Physical Connectivity (as of December 16, 2003) Internet router-level topology 0.1-0.5 Gbps 0.5-1.0 Gbps 1.0-5.0 Gbps 5.0-10.0 Gbps

  41. 2 10 1 10 0 10 0 1 2 3 10 10 10 10 Low degree mesh-like core identical power-law degrees Completely different networks can have the same node degrees.

  42. 2 10 1 10 0 10 0 1 2 3 10 10 10 10 High degree hub-like core Low degree mesh-like core identical power-law degrees Completely different networks can have the same node degrees.

  43. Low degree core • High degree edge routers • Failure and attack tolerant High degree edge routers Rare Likely Mainstream “Physics” view Space of graphs Completely opposite • High degree hubs • Failure tolerant • Attack fragile

  44. High variability Power law CLT Marginalization Maximization Mixtures Power laws are ubiquitous, not just the internet More normal than Normal Low variability Exponential Gaussian Marginalization (Markov property) Central Limit Theorem (CLT)

  45. Power laws are unexceptional More normal than Normal High variability Low variability Exponential Power law Gaussian Marginalization (Markov property) CLT Marginalization Maximization Mixtures Central Limit Theorem (CLT)

  46. Demo 2 10 median 1 10 Reality 0 10 -2 -1 0 10 10 10

  47. Robust 2 10 median 1 10 Yet Fragile 0 10 -2 -1 0 10 10 10

  48. Lessons learnt • You cant just analyze graphs of complex networks without domain knowledge. A network is much more than a graph. • Degree distributions DO NOT tell us everything • Need to couple GRAPH with DYNAMICS • This is the essential message of the course

  49. Course Road Map • Theme is dynamics+ graph theory • Tentative flow of presentations: • Linear algebra of non-negative matrices and basics of graph theory • Markov chains and Perron Frobenius Theory • Graph Laplacians • Synchronization, agreement and consensus • control theory and robotics • Networking • Physics • Complex networks, power laws • Kleinberg’s model, Barabassi’s preferential attachment • Newman’s survey papers • Small world networks, Watts Strogatz Model • Google’s PageRank • Reaction rate equations, metabolic networks, systems biology • Internet, degree distributions, internet topology • Random graph models • Dual decomposition theory • Beyond graphs • Simplicial complexes and algebraic topology • Coverage problems • Distributed optimal control

  50. Need volunteers for each section • We have about 12 weeks, 24 sessions, we could read about 18-20 papers

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