On the Boyd- Kuramoto Model : Emergence in a Mathematical Model for Adversarial C2 Systems

Transcription

1 On he oyd- Kuramoo Model : Emergence n a Mahemacal Model for Adversaral C2 Sysems Alexander Kallonas DSTO, Jon Operaons Dvson

2 C2 Processes: many are cycles! oyd s Observe-Oren-Decde-Ac Loop: Snowden s Cynefn Framework: Dfferen loops dependng on conex Ineracng OODA: Moon, Kruzns, Calber 2002 Operaonal Plannng Process Execue Sraegc Inen Operaonal Taccal Scope Elaboraons: Lawson s C2 Cycle; DOODA, Msson Analyss COA Dev COA Analyss Decde

3 The Kuramoo* Model Kuramoo Model --dm Oscllaor: K θ = ω + A sn θ θ θ aural Frequency Couplng ework adacency marx Phase Synchronsaon a Hgh K Incoherence a Low K * Kuramoo, Chemcal Oscllaons, Waves and Turbulence, Sprnger, erln, 984; Kallonas, Phys. ev. E 82, , 200 Also conrbuons by. Taylor and T. Dekker

4 Mappng Kuramoo o oyd θ = Pon of progress n decson cycle. K = Couplng = degree of ghness of conrol. ω = aural frequency of each node = nverse me perod for processng approprae nformaon accordng o envronmen n order o advance hrough cycle. A = nra-c2 ework = no us communcaons connecvy, bu also auhory, collaborave, socal, and vsual neworks. Who are my pons of reference for my decson cycle? Wh whom mus I muually adus o progress decsons? Perodcy of sne response funcon: rrelevance of sale nformaon or pas decsons: he curren decson cycle s all ha maers. O D θ O A

5 C2- Tme Perod Specrum p rob d ens y funco n ω accal ω operaonal Model n hs paper: only one such echelon ncluded. ω sraegc Modern mlary operaons nvolve dverse me scales and neworkng of processes days

6 The oyd- Kuramoo equaons cf Lancheser aron and Hughes salvo equaons Phase angles Inrnsc frequences β ω ρ ν Adacency marces M.. Ths s a Carcaure: Informal eworks n Tradonal Mlary Al 20; Herarchy n Insurgen eworks Memon e al 2008 β = ω + ρ = ν + = = sn β β + sn ρ ρ + = = M M F ρ β G β ρ. ω, ν [0,] unform random dsrbuon neracons only whn one echelon

7 Inellgence- Survellance- econnassance & OODA lue has oal IS F ρ β = sn ρ + λ β λ=π/ lue seeks o ge nsde adversary OODA loop G β ρ = sn β ρ exp β 2 / 2s IS srengh s= π.772 phase angle ed synchronses around lue `Acons wh narrow IS

8 Measures of Performance = Ψ e e r θ [ ] = ρ β = e r β = e r ρ Measure of nernal synchronsaon, Measure of exernal synchronsaon

9 asc Exreme ehavours r Dela_ Tme = = 0 0 = Tme lue focused exclusvely on ed; neher nernally coordnaes. r Dela_ Tme Tme.8.5 lue, ed focused exclusvely on nernal coordnaon bu no regard for each oher For hs nsance! ω.56 ν =

10 Emergence: he surprse Laughln: sysem quales or behavours no reducble o he sysem componens bu arse from her neracons. Inermedae Couplng Srong Couplng Eg. andom selecon of frequences makes nodes _ close even hough opologcally `far ; creaes affny for dynamcally formng a sub-cluser Ths canno be desgned for gven agen dfferenaon. Sub-clusers emerge a nermedae neracon srenghs. Each nsance s dfferen!

11 lue v ed a he Edge of Chaos r me r me =.075 = Dela_ Dela_ me me =.075 = lue becomes more synchronsed hrough sronger couplng o ed a cos of oscllaons n sayng nsde ed s OODA DID YOU EXPECT THAT?

12 r r r r me me me lue adaps o red me Anoher example: Only ed a he Edge =.08 =.08 =.08 =.08 = = = 2, a cos of zg-zag n sayng nsde ed s OODA Dela_ Dela_ Dela_ Dela_ me me me me

13 Conclusons There are more varables by whch radonal C2 srucures can acheve Agly; hey are subec o mahemacal modellng. Model enables fndng he balance pon for gven C2 srucures and me scales beween nernal coordnaon and responsveness o adversary. Emergence s nohng myscal: mahemacal models can capure such surprses n represenaons of C2. Applcaons: realsc nework daa, human facors daa, lmed/nerruped IS funcons also for lue. Mul-echelon, mul-me specrum: needs modfcaon of equaons. oyd-kuramoo can nermedae/cross-valdae beween hgh/low fdely models of C2 sysems. Thanks o Tony Dekker, chard Taylor, ran Hanlon, Sharon oswell and Paul Whbread for dscussons and encouragemen over he years.