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Toward Validation and Control of Network Models

Toward Validation and Control of Network Models . Michael Mitzenmacher Harvard University. Internet Mathematics. Articles Related to This Talk. The Future of Power Law Research. A Brief History of Generative Models for Power Law and Lognormal Distributions. Motivation: General.

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Toward Validation and Control of Network Models

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  1. Toward Validation and Control of Network Models Michael Mitzenmacher Harvard University

  2. Internet Mathematics Articles Related to This Talk The Future of Power Law Research A Brief History of Generative Models for Power Law and Lognormal Distributions

  3. Motivation: General • Network Science and Engineering is emerging as its own (sub)field. • NSF : cross-cutting area starting this year. • Courses : Cornell (Easley/Kleinberg), Kearns (U Penn), many others. • For undergrads, not just grads! • In popular culture: books like Linked by Barabasi or Six Degrees by Watts. • Other sciences: Economics, biology, physics, ecology, linguistics, etc. • What has been and what should be the research agenda?

  4. My (Biased) View • The 5 stages of networking research. • Observe: Gather data to demonstrate a behavior in a system. (Example: power law behavior.) • Interpret: Explain the importance of this observation in the system context. • Model: Propose an underlying model for the observed behavior of the system. • Validate: Find data to validate (and if necessary specialize or modify) the model. • Control: Design ways to control and modify the underlying behavior of the system based on the model.

  5. My (Biased) View • In networks, we have spent a lot of time observing and interpreting behaviors. • We are currently very active in modeling. • Many, many possible models. • Perhaps easiest to write papers about. • We need to now put much more focus on validation and control. • Have been moving in this direction. • And these are specific areas where computer science has much to contribute!

  6. Models • After observation, the natural step is to explain/model the behavior. • Outcome: lots of modeling papers. • And many models rediscovered. • Example : power laws • Lots of history…

  7. History • In 1990’s, the abundance of observed power laws in networks surprised the community. • Perhaps they shouldn’t have… power laws appear frequently throughout the sciences. • Pareto : income distribution, 1897 • Zipf-Auerbach: city sizes, 1913/1940’s • Zipf-Estouf: word frequency, 1916/1940’s • Lotka: bibliometrics, 1926 • Yule: species and genera, 1924. • Mandelbrot: economics/information theory, 1950’s+ • Observation/interpretation were/are key to initial understanding. • My claim: but now the mere existence of power laws should not be surprising, or necessarily even noteworthy. • My (biased) opinion: The bar should now be very high for observation/interpretation.

  8. So Many Models… • Preferential Attachment • Optimization (HOT) • Monkeys typing randomly (scaling) • Multiplicative processes • Kronecker graphs • Forest fire model (densification)

  9. What Makes a Good Model… • New variations coming up all of the time. • Question : What makes a new network model sufficiently interesting to merit attention and/or publication? • Strong connection to an observed process. • Many models claim this, but few demonstrate it convincingly. • Theory perspective: significant new mathematical insight or sophistication. • A matter of taste? • My (biased) opinion: the bar should start being raised on model papers.

  10. Validation: The Current Stage • We now have so many models. • It is important to know the right model, to extrapolate and control future behavior. • Given a proposed underlying model, we need tools to help us validate it. • We appear to be entering the validation stage of research…. BUT the first steps have focused on invalidation rather than validation.

  11. Examples : Invalidation • Lakhina, Byers, Crovella, Xie • Show that observed power-law of Internet topology might be because of biases in traceroute sampling. • Pedarsani, Figueiredo, Grossglauser • Show that densification may also arise by sampling approaches, not necessarily intrinsic to network. • Chen, Chang, Govindan, Jamin, Shenker, Willinger • Show that Internet topology has characteristics that do not match preferential-attachment graphs. • Suggest an alternative mechanism. • But does this alternative match all characteristics, or are we still missing some?

  12. My (Biased) View • Invalidation is an important part of the process! BUT it is inherently different than validating a model. • Validating seems much harder. • Indeed, it is arguable what constitutes a validation. • Question: what should it mean to say “This model is consistent with observed data.”

  13. An Alternative View • There is no “right model”. • A model is the best until some other model comes along and proves better. • Greedy refinement via invalidation in model space. • Statistical techniques: compare likelihood ratios for various models. • My (biased) opinion: this is one useful approach; but not the end of the question. • Need methods other than comparison for confirming validity of a model.

  14. Time-Series/Trace Analysis • Many models posit some sort of actions. • New pages linking to pages in the Web. • New routers joining the network. • New files appearing in a file system. • A validation approach: gather traces and see if the traces suitably match the model. • Trace gathering can be a challenging systems problem. • Check model match requires using appropriate statistical techniques and tests. • May lead to new, improved, better justified models.

  15. Sampling and Trace Analysis • Often, cannot record all actions. • Internet is too big! • Sampling • Global: snapshots of entire system at various times. • Local: record actions of sample agents in a system. • Examples: • Snapshots of file systems: full systems vs. actions of individual users. • Router topology: Internet maps vs. changes at subset of routers. • Question: how much/what kind of sampling is sufficient to validate a model appropriately? • Does this differ among models?

  16. To Control • In many systems, intervention can impact the outcome. • Maybe not for earthquakes, but for computer networks! • Typical setting: individual agents acting in their own selfish interest. Agents can be given incentives to change behavior. • General problem: given a good model, determine how to change system behavior to optimize a global performance function. • Distributed algorithmic mechanism design. • Mix of economics/game theory and computer science.

  17. Possible Control Approaches • Adding constraints: local or global • Example: total space in a file system. • Example: preferential attachment but links limited by an underlying metric. • Add incentives or costs • Example: charges for exceeding soft disk quotas. • Example: payments for certain AS level connections. • Limiting information • Impact decisions by not letting everyone have true view of the system.

  18. My Related Work : Hash Algorithms • On the Internet, we need a measurement and monitoring infrastructure, for validation and control. • Approximate is fine; speed is key. • Must be general, multi-purpose. • Must allow data aggregation. • Solution : hash-based architecture. • Eventual goal: every router has a programmable “hash engine”.

  19. Vision • Three-pronged research data. • Low: Efficient hardware implementations of relevant algorithms and data structures. • Medium: New, improved data structures and algorithms for old and new applications. • High: Distributed infrastructure supporting monitoring and measurement schemes.

  20. The High-Level Pitch • Lots of hash-based schemes being designed for approximate measurement/monitoring tasks. • But not built into the system to begin with. • Want a flexible router architecture that allows: • New methods to be easily added. • Distributed cooperation using such schemes.

  21. What We Need Off-Chip Memory On-Chip Memory CAM(s) Memory Hashing Computation Unit Unit for Other Computation Programming Language Computation Control System Communication Architecture Communication + Control

  22. Lots of Design Questions • How much space for various memory levels? How to dynamically divide memory among competing applications? • What hash functions should be included? Openness to new hash functions? • What programming language and functionality? • What communication infrastructure? • Security? • And so on…

  23. Which Hash Functions? • Theorists: • Want analyzable hash functions. • Dislike standard assumption of perfectly random hash functions. • Hard to prove things about actual performance. • Practitioners • Want easy implementation, speed, small space. • Want simple analysis (back-of-the-envelope). • Will accept simulated results under right settings.

  24. Why Do Weak Hash Functions Work So Well? • In reality, assuming perfectly random hash functions seems to be the right thing to do. • Easier to analyze. • Real systems almost always work that way, even with weak hash functions! • Can Theory explain strong performance of weak hash functions?

  25. Recent Work • A new explanation (joint work with Salil Vadhan): • Choosing a hash function from a pairwise independent family is enough – if data has sufficient entropy. • Randomness of hash function and data “combine”. • Behavior matches truly random hash function with high probability. • Techniques based on theory of randomness extraction. • Extensions of Leftover Hash Lemma.

  26. What Functionality? • Hash tables should be a basic primitive. • “Best” hash tables: cuckoo hashing. • Worst case constant lookup time. • Simple to build, design. • How can we make them even better? • Move cuckoo hashing from theory to practice!

  27. Cuckoo Hashing [Pagh,Rodler] • Basic scheme: each element gets two possible locations. • To insert x, check both locations for x. If one is empty, insert. • If both are full, x kicks out an old element y. Then y moves to its other location. • If that location is full, y kicks out z, and so on, until an empty slot is found.

  28. Cuckoo Hashing Examples A B C E D

  29. Cuckoo Hashing Examples A B C F E D

  30. Cuckoo Hashing Examples A B C F E D

  31. Cuckoo Hashing Examples A B C F G E D

  32. Cuckoo Hashing Examples E G B C F A D

  33. Cuckoo Hashing Examples A B C G E D F

  34. Cuckoo Hashing Failures • Bad case 1: inserted element runs into cycles. • Bad case 2: inserted element has very long path before insertion completes. • Could be on a long cycle. • Bad cases occur with small probability when load is sufficiently low, but not low enough: • Theoretical solution: re-hash everything if a failure occurs. • For 2 choices, load less than 50%, n elements gives failure rate of Q(1/n); maximum insert time O(log n). • Better space utilization and rate for more choices, more elements per bucket.

  35. Recent Work : A CAM-Stash • Use a CAM (Content Addressable Memory) to stash away elements that would cause failure. • Joint with Kirsch/Wieder. • Intuition: if failures were independent, probability that s elements cause failures goes to Q(1/ns). • Failures not independent, but nearly so. • A stash holding a constant number of elements greatly reduces failure probability. • Implemented as a CAM in hardware, or a cache line in hardware/software. • Lookup requires also looking at stash.

  36. Modeling : Economic Principles • Joint work with Corbo, Jain, Parkes. • Exploration : what models make sense for AS connectivity. • Extending approach of Chang, Jamin, Mao, Willinger. • Entering nodes link according to business model, utility function. • Nodes revise their links based on new entrants. • Like the forest fire model. • Future considerations: how to validate such models.

  37. Conclusion : My (Biased) View • There are 5 stages of networking research. • Observe: Gather data to demonstrate power law behavior in a system. • Interpret: Explain the import of this observation in the system context. • Model: Propose an underlying model for the observed behavior of the system. • Validate: Find data to validate (and if necessary specialize or modify) the model. • Control: Design ways to control and modify the underlying behavior of the system based on the model. • We need to focus on validation and control. • Lots of open research problems.

  38. A Chance for Collaboration • The observe/interpret stages of research are dominated by systems; modeling dominated by theory. • And need new insights, from statistics, control theory, economics!!! • Validation and control require a strong theoretical foundation. • Need universal ideas and methods that span different types of systems. • Need understanding of underlying mathematical models. • But also a large systems buy-in. • Getting/analyzing/understanding data. • Find avenues for real impact. • Good area for future systems/theory/others collaboration and interaction.

  39. More About Me • Website: www .eecs.harvard.edu/~michaelm • Links to papers • Link to book • Link to blog : mybiasedcoin • mybiasedcoin.blogspot.com

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