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Title: Sequential Dynamical Systems, Socio-Technical Simulations and Interaction Based Computing

Author: Madhav M. Marathe

Submitted to: IMA, Minneapolis, MN

Los Alamos National Laboratory

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Form 836(8/00)

Sequential Dynamical Systems, Socio-Sequential Dynamical Systems, Socio-Technical Simulations and Interaction Technical Simulations and Interaction

Based ComputingBased Computing

Madhav V. MaratheBasic and Applied Simulation Science (CCS-5)

Los Alamos National Laboratorymarathe@lanl.gov

Joint work with: H.B. Hunt III, S.S. Ravi, D.J. Rosenkrantz and R.E. Stearns (SUNY-Albany)

and group members in CCS-5

Urban Infrastructure Suite: Interdependent Infrastructure Simulations

Sequential Dynamical Systems (SDS)

nor3(x1,x2,x3) = (x1 x2 x3),

–Local functions (e.g. Boolean): correspond to agent or entity decision rules.

–Dependency graph: represents inter-agent communication capabilities.

–Partial orders (permutation): capture the update order of agent updates

GraphGraph

Phase space Phase space withwith

Л= Л= (1,2,3,4)(1,2,3,4)

Phase space withPhase space with

Л= (1,3,2,4)Л= (1,3,2,4)

Why Computational SDS (cSDS)

• Motivation:

– Size: 107 travelers, 109 nodes, 109 transceivers & 1012pkts/hr

• Efficiently scalable on HPC. Usual agent based concepts do not

scale

– Composed of smaller heterogeneous, inter-operable simulations

– Need to formalize simulation methods (esp. ST simulations)

• cSDS: Enabling Idea-- local functions can be viewed procedurally

– Computational cost and semantics of implementing simulations

– A formal framework for design, analysis and specifications of socio-

technical simulations

– A bridge between a mathematical theory of simulation and HPC algorithm design and implementation

Examples: CA Based Simulation of Roadway Traffic

7.5 meter 1 lane cellularautomaton grid cells

intersection with multipleturn buffers (not internallydivided into grid cells)

single-cell vehicle

multiple-cell vehicle

ResultsComputational Complexity and

Tractability

Modeling & Computational Power of SDS

• Simple SDS are Universal Computing devices

– SDS as formal systems encompass such models as Hopfield networks, cellular automata, communicating finite state machines, etc

– capable of simulating Turing machines for all “natural” complexity classes

• Are simulations optional : NO ( often no shorter computation possible)

• Simple interactions with simple local functions are intractable (Global algorithms are intractable: e.g. Reachability)

Example: REACHABILITY Problem (RP)

• REACHABILITY Problem : Starting from a configuration P can SDS S reach

a configuration T in less than r steps ?

• Dichotomy based on local functionsDichotomy based on local functions

– RP for SDS is PSPACE-hard when

• all nodes have identical symmetric Boolean function

• A fixed permutation is used to update the local states

• graph is constant degree & bandwidth bounded (fixed radius size)

– RPT for SDSs is in P when each local function is

• Boolean Symmetric and Monotone (threshold):

• individual nodes need no have same local function.

• Ordering need not be strict so far as it is fair

• Interaction graph can be arbitrary

– Corollary:Transient lengths < O(# of edges), no limit cycles > 1 threshold

SDS

Basic Technique: Local Simulations

SS

SS11

Local Transformation of the interaction Local Transformation of the interaction graphgraph

•Structure Preserving Local Transformations

• Very Efficient and Distributed

•Phase Space of S is embedded in the phase space of S1

c & F (c): configurations of Sc & F (c): configurations of S

c’ & F(c’): configurations of Sc’ & F(c’): configurations of S11

Each node replaced by a Each node replaced by a constant # of nodesconstant # of nodes

Local Inter-simulations: SDS Compliers

SynchronousSynchronous

Finite arityFinite arity

Simple pathSimple path

SynchronousSynchronous

Boolean SymmetricBoolean Symmetric

Bandwidth boundedBandwidth bounded

SequentialSequential

Boolean SymmetricBoolean Symmetric

Bandwidth boundedBandwidth bounded

SequentialSequential

Boolean SymmetricBoolean Symmetric

Identical functionsIdentical functions

Bounded degreeBounded degree

Bandwidth boundedBandwidth bounded

DeterministicDeterministic

Space BoundedSpace Bounded

LBALBA

Results

Parametric Local Algorithms

Example: Local Provable Algorithm for

Contention Resolution at MAC layer

Distance-2 Interference in 802.11

• Two way transmission needed in 802.11

• s to t transmission received at x y cannot transmit

• w to a transmission received at t w cannot transmit

• No transmission on edges within distance-2 of (s,t) in interference graph

Network Capacity: Distance-2 Matching

Transceivers on a plane with identical power levels: capacity of this network is 2

Red edges are not within distance 2. Similarly nodes a and f are not within distance 2.

Network Capacity and MAC layer scheduling

• Efficient provable methods for computing the instantaneous media-

access layer capacity of ad-hoc networks

– Concentration results for computing capacity in Erdos Renyi

random graphs and geometric random graphs.

– Alternative proof of network capacity

• Distributed provable protocols for MAC layer scheduling

– Extension to MAC-aware routing protocols (designing unified

protocols for routing and scheduling)

Theorem: A local algorithm running in O(log n) rounds can compute a

constant factor approximation for the disance-2 matching in a unit

disk graph

Distributed vs Sequential Algorithm: Performance Comparison

• Large fraction of edges selected in the first few roundsLarge fraction of edges selected in the first few rounds

• Size of matching is within 4 times the optimalSize of matching is within 4 times the optimal

• Number of rounds required is quite smallNumber of rounds required is quite small

Network Capacity and Topology

• MAC layer capacity critically depends on topology

• Protocols need to be optimized for specific topologies

ILLUSTRATIONAdHopNet: Simulation Based

Analytical Tool for 3G+ Telecommunication Networks

Urban Infrastructure Suite: Interdependent Infrastructure Simulations

Schematic of a Hybrid Communication Network

System MobilityUPMoST Technology

Radio Packet Network:SORSRER

Wireline/BasestationNetwork

Satellite Network

Functional Design of AdHopNet

Device Generator

Session Generator

PacketSimulator

TopologicalGraph Module

UPMoST Module

UPMoST Entities

Filter

UPMoSTData File

Survey Data

DeviceMobility

Device Data

Device Data

PolygonDefinitionsOptional

Device Status

Session File

Graph File

Packet Stream

Packet Data

ANALYSIS

Survey Data

Topography Data

Occlusion Data

Graph Data

Packet Twist

PacketDuplication

NetworkDynamics QoS

RESTORED

NetworkAnalysis

DynamicsAnalysis

Vulnerability Analysis

Packet Stream

Packet Data

Graph Data

LARGE-SCALE ANALYSIS AND MEASUREMENTS

AdHopNet

UPMoST

QuickTime™ and a Cinepak decompressor are needed to see this picture.

Module 1: Device Assignment

Cell phone/PDA

Activity variation in time

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Mobile Entities Colored by Age

Module 2: Session Generation

John DoeJohn Doe

•In carIn car

•Age = 34Age = 34

•Income > $26kIncome > $26k

Jane SmithJane Smith

•At WorkAt Work

•Age = 57Age = 57

•Income > $100kIncome > $100k

Video stream, 14.5 kbps, 3.48 minutes

Cell Assignment

Coverage region

Base Station

Locations at which devices begin active sessions

Connections between devices

Active connections in each cell (BTS load)

Handoffs per second per cell

Module 3: Dynamic Construction of (interaction) Network

Timestep: 200

Transceiver connectivity at an instant in time after executing MAC layer protocol.

radio range

radioOcclusionNo No connection

Enlarged view of the focused area

Simulated Cars using TRANSIMS

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Dynamic Ad-hoc Network of Transceivers on Cars

Ad-hoc Network as broadcast radius increases

Variation in Clustering Coefficient (CC) with Node degree in TRANSIMS and Random Way Point Generated Networks

Mobility Models Matters for Ad-hoc NetworksMobility Models Matters for Ad-hoc Networks

Mobility Models Matter: Degree Distributions

SD: Structured DistributionSD: Structured Distribution

RD: Random DistributionRD: Random Distribution

Mobility Affects Protocol Performance: Packet Delivery Success Rate

Module 4: Packet Simulator

• Connecting in and out-channels in a network of three transceivers. Left: Network

- Middle: Channel connections.

• Internal functions f1, f2 and f3.

• Each function is a composition of functions representing the MAC layer, the

routing layer and the transport layer.

• Resulting function: f3 o f2 o f1

Parametric Routing and Scheduling in AdHopNETF

lood

ing

Des

tinat

ionA

ttra

ctor

Dire

cted

Tra

nsm

issi

on

RESTORED: Receiver Oriented Stochastic Re-generation of Data

Module 5: RESTORED: Constructing Smaller Monte-Carlo Simulations

kk

kkk

kkk

tj

iij

arctan

1

0

1

(index distortion)

(time and index distortion)

(phase angle)

Packets sent at rate 1/0, ik is index of kth packet arriving at destination

Unified view of Computing and Simulations

A Unified view of Computing and ST systems: Interaction based Representation

Natural Questions with this perspective

• Algorithmic Semantics: What are we computing ? Traditional View: How to

compute

– What does a market Compute ?

– What are the semantics of TCP ?

• Computational Complexity: How hard is it to compute/design global system

properties

– Who will be sick after 5 days of an epidemic

– How can we design local functions so that a given algorithmic semantic

is implemented

• HPC Implementation: How can we implement our abstraction efficiently

(distributed algorithms)

– Paralel (parametric, approximate, local, efficient) Algorithms

A Unified theory of Computing and Simulations ?

• Unified approach for the entire spectrum of computing

– Nanoscale computing to Grid Computing

– Next generation computing systems are formally speaking simulations

• Unified approach for specifying and analyzing large scale socio-technical

simulations

• (Prof. Abramsky):There is a unified science of information embracing both

``artificial'' and ``natural'' computation

Modified Thesis: There is a unified science of computing and simulations

Summary

• First Steps in a Computational theory of SDS (and Simulations)

– Computational Complexity

– Designing Efficient HPC capable local algorithms

– Algorthmic Semantics & Formal Specifications of Simulations

• SDS and their generalizations can be used to design, specify and

analyze the socio-technical simulations that such as AdHopNET

• Open Questions:

– Complexity of REACHABILITY problem for NOR SDS

• Is NOR Universal local function ? Conjecture: PSPACE-hard.

– Complete Characterization of complexity of Predecessor

existence Dichotomy result

Interdependent Infrastructure and Social Systems: Interdependent Infrastructure and Social Systems:

Transport NetworkTransport Network

Wireless Ad-hoc Network

Social Contact Network

Portland Power Grid