Optimization-Based Reverse Engineering for Complex Networks

1. Optimization-Based Reverse Engineeringfor Complex Networks David Alderson Operations Research Naval Postgraduate School ONR MURI: NexGeNetSci

2. David Alderson • Operations Research Dept, NPS Assistant Professor, arrived September 2006 • Caltech: Postdoctoral Scholar, 2003-2006 • “complex yet fragile” nature of the Internet • Stanford University PhD (2003) Management Science & Engineering Advisors: William J. Perry and Nicholas Bambos • Diverse professional and research experience: Goldman Sachs, Xerox PARC, IPAM (UCLA), Santa Fe Institute • Princeton University BSE (1993) Civil Eng & Operations Research Research Agenda: management and operation of network infrastructure systems, so as to ensure efficient and reliable performance while protecting these infrastructures from large-scale disruptions resulting from accidents, failures, or attacks. ONR MURI: NexGeNetSci

3. challenges in studying complex network systems • the term “network” is ambiguous • a Rorschach test • a discrete approximation to any continuous relationship • “graph”  mathematical definition e.g., G = (N, A) • “network” = graph + data (annotation) • network dynamics are fundamental to many systems: behavior on top of a (fixed) graph structure dynamics ON networks dynamics OF networks evolution of the graph structure itself Many systems of interest involve the interaction of the two ONR MURI: NexGeNetSci

4. NETWORK SCIENCE January, 2006 Some Basic Questions Q1. To what extent does there exist a “network structure” that is responsible for large-scale properties in complex systems? Q2. To what extent are there “universal laws” governing the structure (and resulting behavior) of complex networks? To what extent is self-organization responsible for the emergence of system features not explained from a traditional (i.e., reductionist) viewpoint? Q3. How can one assess the vulnerabilities or fragilities inherent in these complex networks in order to avoid “rare, yet catastrophic” disasters? More practically, how should one design, organize, build, and manage complex networks? ONR MURI: NexGeNetSci

5. STRUCTURE FUNCTION • constraints • uncertainties • components • interactions purposeful behavior of interacting components a fundamental question in the study of complex systems • one approach: study the system of interest as an artifact • assume no prior knowledge about system • Q1: What is the system structure? • Q2: What is the system function? • Q3: How does structure support function? • hard to know what “matters” from outside looking in • modeling choices: affect the outcome • different assumptions lead to different (opposite!) results • a view incompatible with traditional engineering design • design of components/interactions to ensure system function • assumes knowledge of relationship: structure and function ? ONR MURI: NexGeNetSci

6. Network Science Approach (current): a graph theoretic foundation descriptive models graph connectivity (structure) graph evolution (dynamics) null hypothesis: random graphs STRUCTURE FUNCTION • constraints • uncertainties • components • interactions purposeful behavior of interacting components a fundamental question in the study of complex systems ? • large data samples, uncertainty  random ensembles • dynamics, statistical properties  statistical mechanics • emphasis: “likely” configurations • Common theme: • self-organization and “emergent” structure (i.e., “emergent complexity”) ONR MURI: NexGeNetSci

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8. STRUCTURE FUNCTION • constraints • uncertainties • components • interactions purposeful behavior of interacting components an “engineering view” of complex systems Forward engineering = Design of components/interactions to insure system function ? Reverse engineering = Model the structure to explain observed function what “matters” for the given system under study? ONR MURI: NexGeNetSci

9. STRUCTURE FUNCTION • constraints • uncertainties • components • interactions purposeful behavior of interacting components an alternate approach to complex network research Null hypothesis: the structure of the network has been “designed” to achieve the existing function ? Basic idea: use an optimization-based framework to reverse-engineer the objectives, constraints, tradeoffs shaping the system design Solving this type of inverse problem is highly underconstrained. Key question: what else to bring into the model? ONR MURI: NexGeNetSci

10. Examples • Internet topology modeling • reverse engineering the (implicit, ad hoc) design of a single, centralized decision maker (i.e., the ISP) • the behavior of TCP/AQM • reverse engineering a theory (i.e., primal-dual optimization algorithm) to explain the successes and failures of a decentralized, asynchronous protocol • network formation games • exploring the collective behavior of self-interested agents who cooperate and/or compete ONR MURI: NexGeNetSci

11. Example: Internet Topology Modeling • Who builds real router-level topologies? • How do technology and cost influence deployment? • How does one evaluate a “good” design? • What drives their structure? • What about power laws? the “decision makers” are individual ISPs they provide CONSTRAINTS on what the ISP can do network PERFORMANCE can be measured in terms of traffic some form of an (implicit) OPTIMIZATION problem, although actual “design” may be decentralized and heuristic to the first order, they should be a non-issue ONR MURI: NexGeNetSci

12. example: Internet topology modeling Existing router-level topology: a solution to a DESIGN problem • customer demands • geographic dispersion • variations in size • primary source of uncertainty • physical constraints on components • distance/delay, capacity • functional constraints on the system as a whole • throughput, delay, cost • robustness to input uncertainty, component loss modeling approach: constrained optimization • not the language of random graphs • problem driven by graph annotations, not graph connectivity • domain-specific, not generic • transforms network modeling from an exercise in data fitting to an exercise in reverse-engineering ONR MURI: NexGeNetSci

13. Step 1: Constrain to be feasible Step 2: Compute traffic demand Bj Step 3: Compute max flow xij Bi tradeoff: number of connections (degree) vs connection speed Toy Example: Evaluating Network Throughput Given realistic technology constraints on routers, how well is the network able to carry traffic? ONR MURI: NexGeNetSci

14. (Forward) Optimization Inverse Optimization Reverse-Engineering via Optimization (example: a simple linear program) Given cost vector c feasible region X= { x: Ax = b, x  0 } Given feasible point x0 X Solve Minimize c x subject to x X Solve Minimize ĉ – c  subject to x0 = argmin{ ĉ x: x X } Result x* = minimum cost solution Result ĉ* = cost vector minimized by x0 Reference: R.K. Ahuja and J.B. Orlin. 2001. Inverse Optimization. Operations Research 49(5): 771-783. ONR MURI: NexGeNetSci

15. (Forward) Optimization “Inverse Optimization” Reverse-Engineering via Optimization (example: a general mathematical program) Given system performance f(x) feasible region X= {x: g(x) 0, h(x) = 0} Given feasible point x0 Solve Maximize f(x) subject to x X Solve Find f, X subject to x0is a “good” solution to Maxf(x) s.t. x X Result x* = best system “design” Result a design problem solved by x0 ONR MURI: NexGeNetSci

16. “Inverse Optimization” Reverse-Engineering via Optimization (case study: the router-level Internet) Given feasible point x0 Empirical evidence (measurement studies) Solve Find f, X subject to x0is a “good” solution to Maxf(x) s.t. x X Use of “first principles” Heuristically optimal Result a design problem solved by x0 ONR MURI: NexGeNetSci

17. Hosts Heuristically Optimal Topology Sparse, mesh-like core of fast, low-degree routers. Relatively uniform connectivity within core. Core High cost of links drives traffic aggregation at network edge Edges Possibly high variability in connectivity at edge. High degree nodes are at the edges. ONR MURI: NexGeNetSci

18. alternate approaches yield OPPOSITES in terms of engineering Optimization-based: • Focus: engineering design • Uses domain-specific details • Sparse network core • High performance and robustness Degree-based: • Focus: matching statistics • Ignores domain-specific details • High degree central “hubs” • Poor performance and robustness These stark differences are independent of the actual statistics ONR MURI: NexGeNetSci

19. reverse engineering: transport layer protocol web server my computer router router AQM TCP AQM TCP ONR MURI: NexGeNetSci

20. reverse engineering: transport layer protocol web server my computer router router AQM TCP AQM TCP ONR MURI: NexGeNetSci

21. Primal: Dual: reverse engineering: transport layer protocol Kelly/Low Formulation: TCP/AQM as a Primal-Dual Algorithm my computer Ref: S. Low. A Duality Model of TCP and Queue Management Algorithms. IEEE/ACM Trans. on Networking 11(4):525-536, 2003. router source algorithm (TCP) iterates on rates link algorithm (AQM) iterates on prices AQM TCP • Reverse-Engineering: • Theoretical support for existing protocols • Insight for new/improved protocols • Major TCP schemes • Maximize aggregate source utility • With different utility functions ONR MURI: NexGeNetSci

22. reverse engineering: the Internet protocol stack my computer router APP General Approach: An engineering design perspective to understand, explain the complex structure observed. Take a single layer in isolation and assume that the other layers are handled near optimally. TCP/AQM IP PHY/LINK ONR MURI: NexGeNetSci

23. reverse engineering: the Internet protocol stack my computer router APP TCP/AQM If the current router-level Internet is the answer, what is the question? IP ? PHY/LINK ONR MURI: NexGeNetSci

24. reverse engineering: the Internet protocol stack my computer router APP ? If TCP/AQM is the answer, what is the question? TCP/AQM IP PHY/LINK ONR MURI: NexGeNetSci

25. reverse engineering: the Internet protocol stack The entire protocol stack as a decentralized, asynchronous, layered solution to a global resource allocation problem? my computer router APP Network Utility Maximization (NUM) as a unifying framework for design of network protocols TCP/AQM IP Ref: Chiang, Low, Calderbank, and Doyle. Layering as Optimization Decomposition. Proc. of the IEEE 95:255–312, 2007. PHY/LINK ONR MURI: NexGeNetSci

26. network formation games • Let N denote the set of players, |N|=n. • Each player i=1,2,…n chooses a strategy si which defines the connections to build to other players (nodes). • connection “cost” is borne by one/both of the players • Let s=(s1,s2,…sn) denote the collective player strategies • Let A(s) be the set of all edges resulting from strategy s; G(s)=(N,A(s)) is the resulting graph • Unilateral Connection Game (UCG) • Bilateral Connection Game (BCG) • Each player optimizes local utility (combines connection cost with benefit of being connected to network) ONR MURI: NexGeNetSci

27. AS-topology as a hybrid network formation game Two types of business relationships among ASes: • customer-provider relationship • unilateral: customer as price taker pays provider • peering relationship • bilateral: both ASes agree to share traffic at low cost How do the collective decisions of selfish players lead to a global network structure • stability • sustainable economics • compare with social optimum: “price of anarchy” ONR MURI: NexGeNetSci

28. common themes • many complex networks can be understood in terms of design problems: • tradeoffs: what is desirable vs. what is feasible • modeling: constrained optimization • reverse engineering: identify the key objectives and constraints shaping design and operation • Internet as a canonical case study • new mathematics: • inverse optimization • decentralized, asynchronous, myopic decisions • integrated controls, communication, computation • successful reverse engineering invites new (and important) forward engineering problems ONR MURI: NexGeNetSci

29. Optimization-Based Reverse Engineeringfor Complex Networks David Alderson dlalders@nps.edu 831.656.1814 ONR MURI: NexGeNetSci