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Optimization-Based Reverse Engineering for Complex Networks

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## Optimization-Based Reverse Engineering for Complex Networks

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### Optimization-Based Reverse Engineeringfor Complex Networks

### Optimization-Based Reverse Engineeringfor Complex Networks

David Alderson

Operations Research

Naval Postgraduate School

ONR MURI: NexGeNetSci

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

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

January, 2006

Some Basic QuestionsQ1. 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

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

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

FUNCTION

- constraints
- uncertainties

- components
- interactions

purposeful behavior of interacting components

an “engineering view” of complex systemsForward 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

FUNCTION

- constraints
- uncertainties

- components
- interactions

purposeful behavior of interacting components

an alternate approach to complex network researchNull 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

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

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

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

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 ThroughputGiven realistic technology constraints on routers, how well is the network able to carry traffic?

ONR MURI: NexGeNetSci

(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

(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

“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

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

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

reverse engineering: transport layer protocol

web

server

my

computer

router

router

AQM

TCP

AQM

TCP

ONR MURI: NexGeNetSci

reverse engineering: transport layer protocol

web

server

my

computer

router

router

AQM

TCP

AQM

TCP

ONR MURI: NexGeNetSci

Dual:

reverse engineering: transport layer protocolKelly/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

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

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

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

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

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

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

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

David Alderson

dlalders@nps.edu

831.656.1814

ONR MURI: NexGeNetSci

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