What if? Incorporating uncertainty and contingency in archaeological network models Ray Rivers (Physics, ICL) Collaboration with Tim Evans (Physics, ICL):

What if? Incorporating uncertainty and contingency in archaeological network models Ray Rivers (Physics, ICL) Collaboration with Tim Evans (Physics, ICL): Institute of Complex Systems, Paris 2014

What if? Incorporating uncertainty and contingency in archaeological network models Ray Rivers (Physics, ICL) Collaboration with Tim Evans (Physics, ICL): Institute of Complex Systems, Paris 2014 R.J. Rivers & T.S. Evans, Les Nouvelles de l’archéologie 135, 21-28, 2014

© Imperial College LondonPage 3 Nodes/vertices = Major Population or Resource Sites Links/edges = ‘Exchange’ between sites - physical trade of goods - soft power and hard power/social cohesion - transmission of culture Exchange controlled by physical limitations of travel Question: Why do some sites become ‘important’ and others not? (Pre-)Historic Exchange networks

Theory modelling v. Data modelling NETWORK MODELS DATA

Theory modelling v. Data modelling Theory modelling! NETWORK MODELS DATA

Models adapted from Financial modelling - cost-benefit analysis Transport/migration modelling - generalised gravity - intermediate opportunities Theory modelling: Networks are ‘roughly’ optimal Equally appropriate for qualitative data

Thera Compare with record and then....

Thera Compare with record and then.... But is it as easy as that?

© Imperial College LondonPage 14 No laws: guaranteed ambiguity ! Wish to discriminate between I.Uncertainty quantification (UQ): largely a question of inputs! Incompleteness of data Uncertainty about model morphology - model inadequacy... II.Contingency: largely a question of outputs! Q.How susceptible are outcomes to ‘equally good’ choices? What if...? e.g. Nixon’s speeches for moon landing. Not black swan events! Issues are general, but applications have to be specific! Uncertainty and Contingency:

© Imperial College LondonPage 15 This Talk: Two problems/two data sets/two competing models Problems: A. Uncertainty induced by choice of ‘ease-of-exchange/deterrence’ function B.Contingency realised through ‘network landscapes’ or ‘social potentials’ Data sets: I.Greece in 9 th and 8 th C BCE - Emergence of the polis II.Middle Bronze Age (MBA) Aegean – Minoanisation Competing models: a.Wilson ‘retail’ (constrained gravity/entropy) model; uncertainty only? b.Cost-benefit ‘ariadne’ model (Evans/Rivers); uncertainty and contingency!

© Imperial College LondonPage 16 A. Uncertainty (model inadequacy) in ‘deterrence’ function Typically ‘exchange’ determined by ‘deterrence’ function V(x) = V(d/D) for travelling ‘distance’ d with distance scale D set by technology e.g. T ij = P i P j V(d ij /D) Question: How do we choose between ‘equal cost for equal distance’ i.e. exponential fall-off ‘so far and no further’ ? i.e. power behaviour fall-off with a shoulder V(x) is the main source of uncertainty/model inadequacy! Deterrence function – ‘ease of exchange’ function: V(x) – ‘physical function’ D – ‘calibration variable’

Cost-benefit models are generally not deterministic - allow for non-optimal behaviour! Contingency understood as reflecting the more or less equally good, but different, choices that can be made. ‘Satisficing’ strategy/bounded rationality - Look for the ‘best’ – be satisfied with the ‘good’ Not talking about ‘chaos’! Q. What if? How easy is it to make one choice rather than another? 17© Imperial College LondonPage 17 B. Contingency and the ‘Network Landscape’

Each point on ‘landscape’ corresponds to a network: look for ‘lowest’ point Not the geographical landscape! Q. What penalties are incurred by making different choices! ‘ Swiss valley’ landscape of networks - high penalties in crossing from one ‘valley’ to the next - low contingency ‘American mid-west’ landscape of networks - low penalties in roaming landscape - high contingency 18 Cost-benefit optimisation ≡ Minimising ‘altitude’ s in ‘network landscape’ (or social potential) © Imperial College LondonPage 18 ‘Cost’ of sustaining network: s = C{T ij } - B{T ij }

© Imperial College LondonPage 19 Emergence of the polis: Rihll and Wilson (1979, 1991)! I. Greece in 9 th and 8 th C BCE Urbanisation – emergence of dominant settlements Synoikism – surrendering of local sovereignty to a larger community In particular: Thebes Corinth Athens...... Argos Akraiphnion Kalyvia.....

© Imperial College LondonPage 20 Emergence of the polis: Rihll and Wilson (1979, 1991)! Urbanisation – emergence of dominant settlements Synoikism – surrendering of local sovereignty to a larger community In particular: Thebes Corinth Athens...... Argos Akraiphnion Kalyvia..... Distance scales: average distance d to n. neighbour ≈ 5km Journée (foot/mule) ≤ 30km; distance scale D ≈ 10 – 15 km > d I. Greece in 9 th and 8 th C BCE

© Imperial College London distance scale D ‘attractiveness’ ϒ - benefit of concentrated resources a. The Wilson ‘Retail’ model ‘Deterrence’ function V(x): Two ‘physical parameters: Designed to describe the dominance of supermarkets and shopping centres and the collapse of High Street shops! - latter day Synoikism Thebes, Athens, Carthage as the Tesco, Asda, Carrefour of geometric/archaic Greece! Not analogy, entropy/information is a ‘superconcept’ (Wilson)! Standard technique most recently used to describe Bronze Age Khabur triangle! (Davies et al., JAS 2014) Generalised gravity/ restricted entropy model

Generalised gravity/ restricted entropy model © Imperial College London distance scale D ‘attractiveness’ ϒ - benefit of concentrated resources Two ‘physical parameters: Basic assumptions: i.Interaction between two places is proportional to the size of the origin zone and the importance and distance from the origin zone of all the other sites which compete as destination zones ii. the ‘importance’ of a place is proportional to the interaction it attracts from other places iii.‘size’ of a site ~ site ‘importance’ a. The Wilson ‘Retail’ model ‘Deterrence’ function V(x): Choice of V(x) is main source of uncertainty!

Generalised gravity/ restricted entropy model © Imperial College London d ij = ‘distance’ from i to j (input) O i = outflow from i (now given as input) I j = inflow to j (now determined as output) f ij = deterrence function from i to j e.g. f ij = V(d ij /D) ϒ = ‘attractiveness’ coefficient (given as input) Then T ij = A i O i (I j ) ϒ V(d ij /D) Fixing ∑ j T ij = O i, ∑ i T ij = I j constrains A i, I j by self-consistent equations (A i ) -1 = ∑ j (I j ) ϒ V(d ij /D) and I j = ∑ I A i O i (I j ) ϒ V(d ij /D) distance scale D ‘attractiveness’ ϒ Two ‘physical parameters: ‘Deterrence’ function V(x): a. The Wilson ‘Retail’ model

A few important sites grow at the expense of small sites identifiable ‘regional structure’ Rhill & Wilson, Histoire & Mesure, 1979 Key sites are ‘in accord’ with historical record! © Imperial College LondonPage 24 Exponential deterrence function (blue) ! Athens, Thebes, Corinth almost inevitable as key sites A. Uncertainty

Power behaved deterrence function (red) ! Other key sites ‘roughly right’ in the sense that a key site can always be found in relevant neighbourhood! A. Uncertainty A few important sites grow at the expense of small sites identifiable ‘regional structure’ Rivers & Evans, Nouvelles de l’archéologie, 2014 Key sites in neighbourhoods A,B,C,... G, NOT in accord with historical record! - Thebes NO LONGER a significant site!

Can we use data to determine deterrence function? - good Bayesian question Yes! Thebes is crucial in that period – take exponential falloff! support this choice by looking at the data for subsidiary sites Milton Friedman approach: - all models are ‘wrong’ – what matters is agreement with data. 26 “A hypothesis is important if it "explains" much by little.... To be important, therefore, a hypothesis must be descriptively false in its assumptions; it takes account of, and accounts for, none of the many other attendant circumstances, since its very success shows them to be irrelevant for the phenomena to be explained.... Truly important and significant hypotheses will be found to have "assumptions" that are wildly inaccurate descriptive representations of reality, and, in general, the more significant the theory, the more unrealistic the assumptions.” Friedman 1953

Can we use data to determine deterrence function? - good Bayesian question No! Models designed to help our understanding of how the ‘real world’ works rather than demonstrate what happens in detailed reality. - two parameter fit for 109 sites, albeit with poor data a ‘good’ result is to get the data roughly correct “The purpose of a good model is to formulate simple concepts and hypotheses concerning them, and to demonstrate that, despite their simplicity, they give approximate accounts of otherwise complex behaviour of phenomena. If a model ‘works’ (faithfully represents the known evidence) then it shows that the assumptions and hypotheses built into the model contribute to an explanation of the phenomena” - Wilson (1981) - R & W model ‘accidentally too good to be true’ 27 - except for Thebes both choices give key sites in correct areas take lack of Thebes as statistically unimportant although historically disastrous! consider this ‘error’ to be due to factors beyond naive ‘retail’ effects e.g. micro-level social structure

Don’t expect too much - very broadbrush! Very few variables! © Imperial College London or Key point: Coarse-grain data and coarse-grain model!

© Imperial College LondonPage 29 b. Cost – benefit analysis: Aim to reduce network ‘cost’: s = C(T ij ) - B(T ij ) More like a construction kit than a black box! Tactic: Only use them in an environment that imposes structure e.g. Our model is ‘ariadne’ Some generalities but ultimately bespoke ‘ Goldilocks’ scenario: Treading the tightrope between ‘boom’ and ‘bust’

© Imperial College LondonPage 31 Reason: D » d too many ‘equally good’ destinations in a single journée - statistical fluctuations before the new pattern established! need path analysis/memory/brand loyalty which model does not possess ‘geography and technology’ not sufficient!

Roughly self- contained in space and time -c.2000 BC Distinct Minoan culture starts -c.1700 BC Knossos plays a dominant role Knossos II. Middle Bronze Age (MBA) Aegean

33 MBA Nodes/Site settings: 39 key sites: Knossos (1) Thera (10)

© Imperial College LondonPage 34 Regional groupings connect at sea distances of d ≈ 110km Sea distance from Knossos to Thera! Key point: MBA Marine technology matches distances: Distance scale d ≈ 110km crucial!

© Imperial College LondonPage 35 MBA Marine technology: Sail replaces\supplements oar for large distances Single journeys of 100km possible D ≈ d ! For the first time in the BA, technology is good enough to enable a fully connected exchange network to form This is what singles out MBA from any other network analysis on the same set of sites Should preferentially choose models whose dynamics are sensitive to geography!

© Imperial College LondonPage 36 a. Wilson retail model incapable of matching technology to geography insensitivity to choice of deterrence function irrelevant! Wilson model at D=75km!ariadne at D=75km!

Great ambiguity in how we choose and construct models! No rules! Models designed to help our understanding of how the ‘real world’ works rather than demonstrate what happens in detailed reality. Very few parameters – need to coarse-grain data and models Accept ‘uncertainty’ commensurate with coarse-graining at the cost of important history (e.g. Thebes) Low levels of contingency if difficult to roam social network ‘landscape’ Happens if D ≈ d - match of technology to geography (MBA Aegean) Potentially high levels of contingency if easy to roam social network ‘landscape’ Happens if D » d (Greece) - easy to make different choices with no penalty Need more sophisticated modelling e.g. ‘brand loyalty’ - in that case geography and technology are not enough! Conclusions: Theory modelling

References: R. Rivers and T. Evans 2014, ‘New approaches to Archaic Greek settlement structure’, Les Nouvelles de l’archéologie 135, 21-27 T. Rihll and A. Wilson 1987, ‘Spatial interaction and Structural Models in Historical Analysis: Some possibilities and an example’, Histoire and Mesure, 2: 5-32 -----, 1991, ‘Modelling settlement structures in Ancient Greece: New approaches to the Polis’, J. Rich and A. Wallace-Hadrill (eds), ‘City and Country in the Ancient World’. London, Routledge: 59-95 T. Davis, H. Fry, A. Wilson, A. Palmisano, M. Altaweel and K. Radner 2014, ‘ Application of an energy maximising and dynamics model for understanding settlement structure: The Khabur Triangle in the Middle Bronze and iron Ages’, Journ. Arch. Sci. 43, 143-154 R. Rivers, C. Knappett, T. Evans 2013, ‘Network Models and Archaeological Spaces’, Computational Approaches to Archaeological Spaces, Editor(s): Bevan, Lake, Left Coast Press, ISBN:978-1-61132-346-7 41

Thank you!

Geometric/Archaic Greece Significant, but acceptable model deficiency for Wilson model Potentially high levels of contingency if easy to roam social network ‘landscape’ Happens if D » d - easy to make different choices with no penalty Need more sophisticated modelling e.g. ‘brand loyalty Conclusions: Theory modelling MBA Aegean Technology and geography conspire to aid modelling! Low levels of contingency. D ≈ d - difficult to make different choices with no penalty Less model deficiency Retail model inappropriate! Moral: Horses for courses and use results as complementary to other approaches!

Don’t expect too much - very broadbrush! Delicate task of coarse-graining data In practice, BA data is poor - very incomplete - it relies on material objects as proxies for ‘exchange’ - very qualitative - makes coarse-graining easier! A: Coarse-graining Very few variables! © Imperial College London or

Don’t expect too much - very broadbrush! Delicate task of coarse-graining data E.g.cf. data model ‘Good’ fit with little coarse-graining! Very few variables! © Imperial College London or

Don’t expect too much - very broadbrush! Delicate task of coarse-graining data Howevercf. data model ‘Bad’ fit however much coarse-graining we adopt! Very few variables! © Imperial College London or

Optimisation: Simple division has a thermodynamical analogy: Most ‘likely’ networks (microcanonical – specify states) Most ‘beneficial’ networks (macrocanonical – specify averages) Relevant for describing ‘contingency’ Appendix B: Glossary of models © Imperial College London Page 47

1.Generalised gravity models (GGMs): Appendix B: Most likely networks: © Imperial College London Simple gravity model (SGM): O i = outflow from i I j = inflow to j f ij = deterrence function from i to j e.g. f ij = V(d ij /D) S i = population of i T ij = A i O i B j I j f ij No constraints: typically take A i O i ≡ S i, B j I ij ≡ S j ‘physical parameter: D

1.Generalised gravity models (GGMs): Appendix B: Most likely networks: © Imperial College London Doubly constrained gravity model (DCGM): O i = outflow from i (now given as input) I j = inflow to j (now given as input) f ij = deterrence function from i to j e.g. f ij = V(d ij /D) As before T ij = A i O i B j I j f ij Fixing ∑ j T ij = O i, ∑ i T ij = I j constrains A i B j by self-consistent equations (A i ) -1 = ∑ j B j I j f ij (B j ) -1 = ∑ I A i O i I j f ij ‘physical parameter: D

Generalised gravity/ restricted entropy model © Imperial College London d ij = ‘distance’ from i to j (input) O i = outflow from i (now given as input) I j = inflow to j (now determined as output) f ij = deterrence function from i to j e.g. f ij = V(d ij /D) ϒ = ‘attractiveness’ coefficient (given as input) Then T ij = A i O i (I j ) ϒ f ij Fixing ∑ j T ij = O i, ∑ i T ij = I j constrains A i, I j by self-consistent equations (A i ) -1 = ∑ j (I j ) ϒ f ij and I j = ∑ I A i O i (I j ) ϒ f ij distance scale D ‘attractiveness’ ϒ Wilson ‘Retail’ model One function V(x): Two ‘physical parameters: Devised to describe the rise in influence of supermarkets and shopping centres!

Generalised gravity/ restricted entropy model © Imperial College London T ij = A i O i (I j ) ϒ V(d ij /D) Fixing ∑ j T ij = O i, ∑ i T ij = I j constrains A i I j by self-consistent equations (A i ) -1 = ∑ j (I j ) ϒ V(d ij /D) I j = ∑ I A i O i (I j ) ϒ V(d ij /D) distance scale D ‘attractiveness’ ϒ Wilson ‘Retail’ model One function V(x): Two ‘physical parameters:

© Imperial College LondonPage 52 Appendix B: ‘Most beneficial’ Networks: ‘Cost – benefit’ analysis: s = C( T ij ) - B( T ij ) More like a construction kit than a black box! Aim: Only use them in an environment that imposes structure e.g. Our model is ‘ariadne’ Some generalities but ultimately bespoke ‘Goldilocks’ scenario: Treading the tightrope between ‘boom’ and ‘bust ’

© Imperial College LondonPage 53 ariadne: description of networks: Site Strength =  j (S i v i e ij ) = Total Trade Going Out S i, v i d ij, e ij S j, v j We find the values of site occupation (v i ) and trade levels (e ij ) that give us the most efficient use of resources (lowest energy) for given input of site size (S i ) and distances (d ij )

Look for the ‘best’ – be satisfied with the ‘good’ ‘Satisficing’ strategy Bounded rationality Stochastically Panglossian Contingency: Q. How susceptible are outcomes to equally good alternatives? 54 Q. When is ‘good’ good enough? Appendix C: Contingency - stochastic outcomes: © Imperial College LondonPage 54

I.Some models are deterministic e.g. maximum entropy models (incl. ‘gravity’) Contingency⁽ 1) understood through empirical output fluctuations that encode incomplete input data (passive) that encodes the stochastic behaviour of ‘agents’ (active) e.g. data-modelling SGM: T ij = P i P j V(d ij /D) ε ij ε ij - statistical noise – mean-square analysis Baysean viewpoint still possible: ensemble average = time average of single ‘history’ 55© Imperial College LondonPage 55 C. Contingency: Two aspects

II. Cost-benefit models not deterministic; allow for non-optimal behaviour Contingency⁽ 2) understood as reflecting the more or less equally good, but different choices that can be made. ‘Satisficing’ strategy/bounded rationality - Look for the ‘best’ – be satisfied with the ‘good’ Not talking about ‘chaos’! Concentrate on Contingency⁽ 2) ! 56© Imperial College LondonPage 56 C. Contingency: Two aspects

Frequentist: 1 2 © Imperial College LondonPage 57 Question: How do we measure contingency? Baysean: OR Answer: Baysean statistics! If variation low, use discrete differences If variation high, model is useless!

© Imperial College LondonPage 58 C: Contingency: ‘ariadne’ cost- benefit model In more detail: Run numberDominant centres listed numerically 1 22 25 26 32 38 41 81 83 106 2 18 24 27 40 41 49 77 89 98 3 54 55 70 71 Reason: D » d too many ‘equally good’ destinations in a single journée cost-benefit modelling based on geography doesn’t work, despite encoding homophily, etc. Resolution: need some form of ‘brand loyalty’

© Imperial College LondonPage 59 Uncertainty: 1.Incompleteness of data (settings) - intrinsic (where are the sites?) - extrinsic (where does the network end?) e.g. EBA Cyclades (3000 – 2000 BCE) This Talk: Address particular problems with modelling

I.R&W model is deterministic, but based on incomplete information. - even for chosen f(x)! Contingency⁽ 1) understood as the fluctuations that arise from averaging/tracing-out the information that we don’t possess. e.g. data-modelling : T ij = P i P j V(d ij /D) ε ij ε ij - statistical noise Only way to encode Contingency⁽ 1) in the R&W model(?) – mean-square analysis 60© Imperial College LondonPage 60 II. Contingency: Two aspects

II. Cost-benefit models not deterministic; allow for non-optimal behaviour Contingency⁽ 2) understood as reflecting the more or less equally good, but different choices that can be made. ‘Satisficing’ strategy/bounded rationality - Look for the ‘best’ – be satisfied with the ‘good’ Not talking about ‘chaos’! Concentrate on Contingency⁽ 2) ! 61© Imperial College LondonPage 61 II. Contingency: Two aspects

© Imperial College LondonPage 63 Contingency: ‘ariadne’ cost- benefit model -Very high contingency For all V(x)! Distance scale D Benefits in establishing links cf. benefits from local resources Costs in supporting links/ supporting population Identical inputs:

© Imperial College LondonPage 64 Contingency: ‘ariadne’ cost- benefit model In more detail: Run numberDominant centres listed numerically 1 22 25 26 32 38 41 81 83 106 2 18 24 27 40 41 49 77 89 98 3 54 55 70 71 Reason: D » d too many ‘equally good’ destinations in a single journée Cf. MBA Aegean: d ≈ D ≈ 100km for rigged sail matches distance scale for geographical connectivity! Low contingency! Very high contingency!

© Imperial College LondonPage 65 Reason: D » d too many ‘equally good’ destinations in a single journée - statistical fluctuations before the new pattern established! cf. MBA Aegean: Rivers, Evans & Knappett, 2013 d ≈ D ≈ 100km for rigged sail matches distance scale for geographical connectivity! Low contingency!

What if? Incorporating uncertainty and contingency in archaeological network models Ray Rivers (Physics, ICL) Collaboration with Tim Evans (Physics, ICL):

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What if? Incorporating uncertainty and contingency in archaeological network models Ray Rivers (Physics, ICL) Collaboration with Tim Evans (Physics, ICL):

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