462
Documenta Praehistorica XLVII (2020)
Modelling in archaeology
“All history is ‘contemporary history’ ... Essentially
[we look at] the past through the eyes of the pre-
sent and its problems.” Benedetto Croce (1922.19)
The sentiment underlying this quote is arguably even
more valid when it comes to proto-historic and pre-
historic archaeology, for which we have even less
data than for other periods. However, we shall show
below that, rather than seeing this as a cause for
concern, some archaeologists have proposed that the
projection of contemporary society onto the past pro-
vides a new way to proceed. In this they directly re-
purpose economic models describing the attributes
of today’s society with a perceived historic counter-
part (e.g., urbanisation, mobility, migration, trade)
into historic contexts. A change of vocabulary is ne-
cessary, but the lexicon is often very transparent.
There is one immediate issue with this temporal and
linguistic translation, even when it is appropriate.
The 20th-century modelling that we mimic has its
own narrative formulation which may, or may not,
be realistic. In the worst case we are not only putting
an ancient gloss on a contemporary narrative, but
that narrative may itself be unreliable. The contem-
porary models that we shall discuss below have their
origins firmly in free-market economics. To quote the
free-market economist Milton Friedman (1953.14–
How do we avoid imposing the present on the past
when modelling spatial interactions|
Ray J. Rivers, Tim S. Evans
Department of Physics and Centre for Complexity Science, Imperial College London, London, UK
r.rivers@imperial.ac.uk< t.evans@imperial.ac.uk
ABSTRACT – Theoretical archaeological modelling for describing spatial interactions often adopts
contemporary socioeconomic ideas whose 20th-century language gets translated into historical beha-
viour with the simplest of lexicons. This can lead to the impression that the past is like the present.
Our intention in this paper is that, when this happens, we strip out as much of the contemporary con-
text as we can, to bring modelling back to basic epistemic propositions. We suggest that although the
underlying ontology may be specific to contemporary society the epistemology has much greater ge-
nerality, leading to essentially the same conclusions without the carapace of intricate economics.
IZVLE∞EK – Teoreti≠no arheolo∏ko modeliranje za opis prostorskih interakcij pogosto sprejema so-
dobne dru∫benoekonomske ideje, ki jih iz njihovega jezika 20. stoletja prevajajo v zgodovinska vede-
nja le z najpreprostej∏im besedi∏≠em. To daje vtis, da je preteklost podobna sedanjosti. Na∏ namen v
tem prispevku je zaznati ta pojav in ga v najbolj∏i meri o≠istiti sodobnega konteksta, da lahko mo-
deliranje vrnemo k osnovnim epistemolo∏kim nastavkom. Predlagamo, da ima epistemologija ve≠jo
generalizacijo, ≠eprav je osnovna ontologija zna≠ilna za sodobno dru∫bo, kar vodi v bistvu do ena-
kih zaklju≠kov brez ogrodja zapletene ekonomije.
KEY WORDS – networks; exchange; MaxEnt; city states; trade; Bronze Age Assyria
KLJU∞NE BESEDE – mre∫e; izmenjava; MaxEnt; mestne dr∫ave; trgovanje; bronastodobna Asirija
Kako se izognemo vsiljevanju sedanjosti v preteklost
pri modeliranju prostorskih interakcij|
DOI> 10.4312\dp.47.26
How do we avoid imposing the present on the past when modelling spatial interactions|
463
(Eaton, Kortum 2002; Anderson, van Windcoop
2003).
ii) Settlement structure in central Anatolia and the
Khabur Triangle in the Middle Bronze Age (Davis
et al. 2014; Palmisano, Altaweel (2015)) – ex-
plored in the framework of the 20th-century me-
thodology of dynamic ‘Retail Modelling’ (Harris,
Wilson 1978).
These models were chosen because they arguably
provide the most prescriptive application of free-
market 20th-century economic theory to the his-
toric past, although in very different ways.
The former is the simplest conceptually, almost a
straight transposition of the behaviour of contem-
porary traders onto that of their antecedents. Other
authors have noted the existence of free-market be-
haviour in historical systems, particularly the Roman
Empire, e.g., see Tom Brughmans and Jeroen Pob-
lome (2016) and Xavier Rubio-Campillo et al. (2017),
but what singles out the paper by Gojko Barjamovic
et al. (2017) is the very heavy economic machinery
(e.g., ‘Constant Elasticity of Substitution’, ‘Weibull
cost distributions’) which, as we shall discuss later,
they bring to make their case.
The second model is more subtle, concerned with
the building of centres of influence from local com-
munities, seeing synoikism in analogy with the cre-
ation of shopping centres. This is not to say that city-
state formation is predominantly an economic activ-
ity. The advantages of close interaction are as much
social and ‘political’ as economic. Unlike the case of
traders, the parallels are structural in mimicking the
way that dominant centres form but are not assumed
to be a direct translation. Nonetheless, the method
of solution is taken directly from the retail picture.
However, we shall see how both cases permit a ‘ma-
ximum entropy’ (‘MaxEnt’) representation (Jaynes
1957; 1979). This enables us to recast the models in
such a way as to de-emphasize the projection of free-
market narratives on the past, by looking for the
‘least surprising’ outcomes commensurate with our
limited knowledge. Although the authors of these
papers are well aware of this underlying duality
their concern is different from ours. In fact, Alan
Wilson (1970) provided an extensive discussion of
the issues that, to a large extent, we are paraphras-
ing here. One of our aims is to bring these ideas to
15) interpreting his use of ‘hypothesis’ to be synony-
mous with our use of ‘model’: “A hypothesis is im-
portant if it “explains” much by little ... To be im-
portant, therefore, a hypothesis must be descrip-
tively false in its assumptions ... Truly important
and significant hypotheses will be found to have
“assumptions” that are wildly inaccurate descrip-
tive representations of reality, and, in general, the
more significant the theory, the more unrealistic
the assumptions.” The result of these two narrative
mismatches could be a (pre)historic Just-so-Story of
homo economicus1, entertaining but little more.
Nonetheless, Friedman argued that what matters is
whether the model ‘works’. Rather than dismissing
any narratives because they all encode detail that
cannot be substantiated, we go for an approach as
lacking in narrative detail as possible. It can be
shown that in many contexts there is an epistemic-
ontic duality which permits a model reformulation
that brings modelling back to basic epistemic pro-
positions rather than imputing specific ‘economic’
agent-driven activity. This enables us to reduce the
need for agent narratives without losing our ability
to answer the major questions.
Spatial interaction modelling
We are specifically interested here in how archaeo-
logists borrow from the present in modelling the
patterns of ‘exchange’ in past society, particularly
between communities separated spatially. The nat-
ural framework for these questions is provided by
spatial networks, a pattern of nodes connected by
links. The nodes label the origins and targets of the
exchange, in our case sites, whereas the links de-
scribe the interactions among them. Within the
framework of such spatial interaction modelling con-
temporary economists, social geographers, transport
analysts and urban planners have provided tools
for modelling the key attributes of exchange. The
potential relevance to archaeology is clear and has
been realized successfully in many examples. In this
paper we shall consider the use of two different 20th-
century economic network models as templates for
describing different social hierarchies in the same
historical period (Middle Bronze Age) in the same
place (central Anatolia):
i)  Assyrian trade routes in the early Middle Bronze
Age (C19 BCE) in Anatolia (Barjamovic et al.
2017), explored in the framework of 20th-centu-
ry ‘Ricardian’ modelling of individual traders
1‘Homo economicus’, punning on Homo sapiens as ‘Economic man’, effectively first appears in the work of John Stuart Mill (1874)
on political economy, understood as an individual who acts to maximize their economic well-being within the opportunities and
constraints imposed by society.
Ray J. Rivers, Tim S. Evans
464
a wider audience of archaeologists than, from our
experience, are currently familiar with them. As we
have said, there is modelling that calls upon ideas
from contemporary society in less explicit ways than
these. However, the transparency of the models we
consider here gives clear insights into the mecha-
nisms of translating the present into the past that
other applications may lack. We stress again that the
analysis here is entirely within the framework of
exchange networks.
Irrespective of the model, there is a major difference
between contemporary economic and historic archa-
eological data. Economic data is typically ‘big data’.
Archaeological data, although voluminous, better
characterized as ‘lots of data’, is poor statistically.
Such data underdetermines the possible processes
that lead to its presence. On the other hand, theore-
tical modelling of the type discussed here, which
goes beyond the descriptive to have postdictive po-
wer, over-determines or overfits the data. This leads
to a delicate balancing act to which we shall return.
Of course, there is no reason why, for any particular
archaeological data set, quantitative models of the
type above should be satisfactory. Most simply, these
work best when ‘history is idling’, when we can as-
sume continuity of form, enabling us to go some way
to fill in the gaps in the data. That is, we describe
the periods of calm between the storms of war,
famine and general disturbance and we follow the
authors above in assuming that Middle Bronze Age
Assyria is such a time, although this is not to say that
there is stasis. We shall revisit this question later.
We are more concerned here with the underlying
nature of the modelling and shall only discuss the
data analysis of these models briefly. The reader is
referred to the original papers for details.
Assyrian trade as Ricardian economics
We take the examples of Assyrian trade and urbani-
sation in order, and treat the second with greater
brevity. We shall be as minimal in our use of equa-
tions as we can.
The underlying question posed by Barjamovic et al.
(2017) in their analysis of Assyrian trade is straight-
forward and of more general interest than just the
Assyrians: how do we locate communities/sites which
we know to have existed from the historical record,
but whose geographical position is uncertain? Their
suggestion is essentially to consider them as illumi-
nated trading ‘beacons’ and to triangulate their po-
sitions from their (trading) ‘intensity’ to sites whose
positions are known, on the assumption that they
will appear ‘dimmer’ the further away they are from
their trading partners. ‘Exchange’ here is limited to
the exchange of ‘goods’.
For the case in hand, Assyrian trade routes of the
Early/Middle Bronze Age (19th century BCE) are
identified from a cache of 12 000 deciphered cunei-
form tablets. Details do not matter for the discussion
here, but the basic data set comprises the origins
and destinations of the (only) 391 directed journeys
involving exchange between the sites listed in these
tablets. There are 26 named sites which are either
origins or destinations, the positions of 11 of which
are unknown. Because of incomplete exchange data
(e.g., a comparable number of untranslated tablets),
the absolute numbers of trips are a poor proxy for
site activity. Rather, for each site the fraction of trans-
actions from each of the other sites is estimated. The
aim is to use these ratios of flows to triangulate the
missing sites, using the known sites for calibration
purposes. The paper is clever and nuanced, but a
true understanding of its methods requires a crash
course in economics for most archaeologists (as well
as ourselves), and we present only the briefest ex-
planation. The reader is referred to the original pa-
per and its key references (most importantly Jona-
than Eaton and Samuel Kortum (2002) and James
Anderson and Eric van Windcoop (2003)) for a more
complete understanding.
A simple summary with as little algebra as possible
follows. Labelling sites by i, j, k ... with values 1, 2, ...
26 the assumptions made by Barjamovic et al.
(2017) are a 20th century version of Ricardian eco-
nomic theory in which we assume risk-taking inde-
pendent traders buying and exchanging goods (of
different types w) with the aim to make the best
deals subject to:
i)   No arbitrage (i.e. no guaranteed way to make a
‘profit’).
ii)  Identical ad valoram ‘iceberg melting’ for all
goods ω produced and acquired with different
efficiencies. That is, within a cargo a similar frac-
tion of goods of whichever type are needed to
‘pay’ (or are ‘melted’) for their cost of transpor-
tation and management.
iii) Ad valoram ‘melting’ is reciprocal i.e. there is
the same fractional penalty on goods going from
site i to j as from j to i.
iv) The outflow Oi at site i can be identified with the
‘site activity’ Si. This qualifies the simple ‘beacon’
How do we avoid imposing the present on the past when modelling spatial interactions|
465
simile above while leaving it essentially correct.
v)  Constant ‘Elasticity of Substitution’; this concerns
the ability of traders to substitute one good for
another if necessary.
vi) The cost of producing one unit of w in any city
i follows a Weibull distribution2 depending on
Ti, the efficiency of sourcing goods from i and a
parameter q > 0, a measure of the ease of distri-
bution (inverse of costs).
All modelling makes simplifying assumptions about
what constitutes exchange. Here, constant ‘Elasticity’
and ‘Weibull-costing’ allow us to aggregate different
types of goods and efficiencies of production, whe-
reas ad valoram ‘melting’ allows us to aggregate dif-
ferent modes of exchange with a common distance
scale beyond which exchange gets difficult. The cu-
mulative effect is that we can flatten ‘exchange’ from
site i to site j into a single integer-valued direction-
al label Tij which counts transactions (e.g., T27 mea-
sures the number of exchanges from site 2 to site
7). It is sufficient for the moment to observe that
these simplifying assumptions look anything but
simple. Given the discomfort that many archaeolo-
gists have with algebra we have intentionally only
partially decoded this language to highlight the dif-
ference in style and content between it and that of
more conventional archaeological modelling.
The details of identifying the missing sites are not
straightforward, and we refer the reader to the
original paper (Barjamovic et al. 2017). In partic-
ular, they involve extremising utility functions rep-
resenting consumer welfare/satisfaction subject to
constrained finances – hence the jibe homo econo-
micus for the actors in these pursuits. In defence,
their argument is that the historic Assyrian mer-
chants operated in a framework of trading contracts,
judicial taxation, trading colonies and ports which
make them as good a proxy for contemporary free
enterprise traders as we can get.
For all the sophistication of the modelling assump-
tions the outcome is relatively simple, although for
the reader not versed in algebra it may seem some-
what obscure. The resulting exchanges Tij from i to
j following from these assumptions take the form
of a generalised ‘gravity’ model (Erlander, Stewart
1990)
Tij = si (f avij)–q sj (2.1)
where f avij measures the ad valoram ‘melting’ as a
function of site ‘separation’ between site i and site
j. Increasing with distance, f avij measures the in-
creasing fraction of goods that are needed to pay for
the exchange, and it is this which makes distant sites
‘dim’. The si are derived from the site activity vari-
ables Si by3 (Σj denotes the sum over j = 1, 2, 3, ...
for fixed i)
Si = Oi = si Sj(f avij)–q sj
= si Sj(f avji)–q sj = Ii
(2.2)
To get an understanding of what this means we will
do a little further decoding. If we think of the Si (i =
1, 2, ... N = 26) as a measure of active population/
carrying capacity, then we think of the si more as a
measure of the strength or importance of the site,
something more akin to site activity (e.g., Gross Do-
mestic Product in a contemporary context). Suppose,
for example, that the Si are equal. Then those sites
which have more near neighbours with easier access
for exchange accrue larger values of si than those
whose neighbours are more remote or less numer-
ous and with which they exchange less by virtue of
the cost/effort doing so. We finally note that reci-
procity in melting leads to detailed balance; inflows
equal outflows at all sites (last equality in (2.2)), sug-
gesting that traders use the proceeds of exchange to
implement new exchanges in a simple way.
The data on ratios of transactions (Tij/Oi) as j varies
is sufficient, in principle, to triangulate the missing
sites. Assuming f avij grows as a power law with dis-
tance Barjamovic et al. (2017) get sensible answers
that accord with our historical understanding on in-
corporating further contextual information. We shall
return to this later. For the moment we shall also ig-
nore the fact that the data set is very small. The que-
stion that we wish to pose now is one of principle.
If we knew no economic theory could we reach re-
sults (2.1) and (2.2) by other means, in particular
means that do not require the explicit actions of
agents?
The answer is largely yes, as we shall now show.
Assyrian trade as the ‘most likely’ outcome
(‘MaxEnt’)
The alternative approach which enables us to evade
direct comparison with free-market economics can
be characterised as no more than making the ‘best
2 The Weibull probability distribution assumes that the cost ci of producing one unit of w in any city i takes the form: Pr[ci(w)
< c] = 1 – exp(–Tiwi–qcq).
3 In (2.2) the first equality is the repeat of the ansatz that identifies Si with the outflow Oi, the second equality is the definition
of Oi from (2.1). The third equality uses the reciprocity of melting and the final equality is the definition of the inflow Ii
Ray J. Rivers, Tim S. Evans
466
guess’. By that we mean what we would expect to
have happened, all other things being equal. This
is an old problem, the question of how to make best
use of partial information4. In principle we know
what to do. We list all the ‘worlds’ which are compa-
tible with our knowledge or, equivalently, ignorance,
and assume that each is equally likely, otherwise we
are withholding information. The most typical of
these is the way in which the system is most likely
to have behaved, and the question then devolves to
one of identifying this state and the extent to which
it more likely to be achieved than other competing
states of the system. For the moment we concentrate
on the first part of this question.
This is such a familiar approach that we can forget
that we are using it. A typical situation arises when
playing cards; we know our hand and those cards
which have been played and from that information
make plausible guesses as to the most likely hands
of our partners and opponents. Most simply, if we
have one ace and only half the pack is dealt, our
partner is unlikely to have three aces. The sugges-
tion is that we might think of using our limited
archaeological evidence in the same light.
It looks an almost impossible task to list all ‘worlds’
compatible with our limited knowledge. Remarkably,
this making best use of the limited information can
be quantified as the principle of maximum entropy
(‘MaxEnt’) (Jaynes 1957; 1979). Although we collo-
quially think of (Shannon) entropy as associated with
chaotic behaviour, it can be thought of as the num-
ber of questions with which we need to interrogate
the system to have complete knowledge of it. Think
of the popular game in which you have up to 20 que-
stions with yes/no answers with which to identify
what your opponent is thinking about. Entropy is
thus a measure of our ignorance about the system.
From this viewpoint the ‘most likely’5 state of the sys-
tem is the one with maximum entropy given our li-
mited knowledge, since systems with less entropy as-
sume more knowledge or have more implicit assum-
ptions. Edwin Jaynes (1957; 1973) has also rephrased
this as the Principle of ‘Maximum Ignorance’ or ‘Epi-
stemic Modesty’. The use of entropy in this way, to
identify the ‘least surprising’ of possible pasts, has
been termed a ‘superconcept’ by Alan Wilson (2010),
from whom much of the following is derived.
Implementing ‘MaxEnt’ is still problematic, but we
adopt the ‘law of parsimony or Occam’s Razor’6, the
principle that the simplest solution to a problem
tends to be the correct one. Although the principle
sounds straightforward it is difficult to formulate in
general. However, as happens here, when presented
with competing models of a similar form, we should
select the one with the fewest ‘significant’ unknown
variables and parameters7. Explicitly, for the models
of this paper the parameters of the ‘MaxEnt’ models
are a subset of those of the economic models, and
parsimony provides a simple marker for delineating
the different approaches. More generally, when the
models do not permit simple comparison, we fall
back on Bayesian analysis (e.g., Rubio-Campillo et
al. 2017). The assumption ‘all other things being
equal’ seems a flat Bayesian prior, but the extent to
which ‘MaxEnt’ is itself ‘Bayesian’ is disputed (e.g.,
see Cheeseman, Stutz 2004 and references therein),
and we will not take the discussion further.
As we have said, the analysis of Barjamovic et al.
(2017) was predicated on sources of trade behaving
as ‘beacons’ which become ‘dimmer’ the further we
move away from them, exemplary of Tobler’s First
Law of Geography, that “near things are more re-
lated than distant things” (Tobler 1970). As a first
step we show how Tobler’s law arises as the ‘most
likely outcome’ from a simple implementation of
‘MaxEnt’. We avoid explicit algebra where we can.
The interested reader can find a more mathematical
analysis in several chapters of Wilson (1970), in Sven
Erlander and Neil F. Stewart (1990) and in our recent
work (Rivers, Evans 2014; Evans, Rivers 2017).
Tobler’s ‘first law of geography’ as ‘MaxEnt’
As a first guess, our parsimonious approach, which
we take as our null model for exchange, assumes mi-
nimal ‘global’8 knowledge:
(a) Exchange takes place but it is (collectively) lim-
ited in scope.
4 There is an extended and tangled literature that we shall not attempt to reference here, encompassing the work of Jakob Bernoul-
li to Pierre-Simon Laplace (via Thomas Bayes) to Ludwig Boltzmann to John Maynard Keynes to Edwin Jaynes via Claude Shannon.
Details can be found in Jaynes (1979).
5 We use ‘most likely’, ‘most typical’ and ‘least surprising’ as synonymous.
6 Occam’s razor states that pluralitas non est ponenda sine necessitate, meaning “plurality should not be posited unnecessarily”
(William of Ockham (1285–1347/49)).
7 We distinguish between ‘control parameters’ such as known site positions and their separations, which are common to both eco-
nomic and entropy models, and ‘calibration parameters’ such as q, the measure of distribution costs, which act as model variables.
8   By ‘global’ we mean with reference to the system as a whole. By ‘local’ we mean information on a site by site basis.
How do we avoid imposing the present on the past when modelling spatial interactions|
467
(b) Exchange ‘costs’ or takes effort, but only so many
resources are available globally (i.e. collectively)
(c) The ‘cost’ or the effort required for exchange in-
creases with ‘distance’. In practice, the cost of
moving goods lies not just in the cost of their im-
mediate transport, but also in the costs of sus-
taining the network. This will include supporting
the agents and middlemen to enable the trans-
actions to take place.
The outcome of maximizing the entropy of the sys-
tem of exchange ‘flows’ subject to these global con-
straints is, indeed, ‘Tobler’s law’ applied to exchange:
that each site is connected to every other site and
exchange decreases with ‘distance’ between sites.
We do not have constant elasticity and ad valoram
costing to fall back upon. Nonetheless, we assume
parsimoniously that in the absence of further infor-
mation, as a null assumption, exchange can be crude-
ly characterised by a single number Tij whose value,
if large, suggests strong exchange from i to j and, if
small, weak exchange. For the case of Assyrian trade
this will just be the integer-valued number of trips
from i to j.
Then, in appropriate units, ‘MaxEnt’ gives the most
likely configuration of exchange flows as9
Tij = si fij sj (3.1)
where the input si are a measure of site activity, re-
lated to the active population of i, and fij is the so-
called deterrence or impedance function for flows
from i to j, a reflection of the cost/effort of exchange
from i to j, which decreases with increasing separa-
tion. We have recovered ‘Tobler’s law’ by replacing
‘cost/effort increases with distance’ with ‘exchange
decreases with distance’, a very plausible equiva-
lence10.
What we have here is the simplest of exchange mo-
dels, the ‘Simple Gravity Model’. As yet it is so sim-
ple that it does not incorporate networking. Re-
moving a site just erases its links without any need
for rearrangement of flows – the whole is just the
sum of the parts. This is as we would expect from
just implementing global constraints which make
no reference to individual sites. With this in mind,
as a second guess we introduce local constraints for
transactions. Most simply, we first adopt the idea
from ‘Proximal Point Analysis’ (PPA) that the total
exchange flowing from any particular site is limited,
with inflows unrestrained. ‘Proximal Point Analysis’
has had considerable success in archaeology (e.g.,
Broodbank 2000; Terrill 1986) in assuming most
simply that any site only has the resources/energy
to interact with a fixed number of nearest neigh-
bours11. We generalize this by extending our null
model in which we replace condition (b) above by:
(b) ‘only so many resources are available locally’,
constraining the local outflows Oi as in ‘PPA’. Typi-
cally, in the absence of any further information, we
take (in appropriate units) the total outflow equal to
the site’s local resources so Oi = Si, as in Barjamovic
et al. (2017).
In comparison to the simple gravity model this ad-
ditional constraint gives us a ‘Singly Constrained Gra-
vity Model’. The addition of this local constraint is
sufficient to network the model. For example, if we
double the outflows and inflows we get the sensible
scaling result that exchange flows double whereas,
for the ‘Simple Gravity Model’ of (3.1), doubling the
si leads to a quadrupling in flows.
In practice, a single constraint is not yet sufficient to
describe either Assyrian trade or, later, Assyrian city-
state formation. Each of these requires something
further.
Assyrian trade as the ‘Doubly Constrained Gra-
vity Model’
For the case in hand of Assyrian trade we make the
further constraint (repeating Barjamovic et al. 2017)
that, in the absence of more information, the deter-
rence function is reciprocal between sites; fij = fji for
all i, j.
Insofar that fij is a function of the ‘effective dis-
tance’12 dij between the sites i and j this becomes
the statement that these distances dij = dji are reci-
procal, our parsimonious choice in the absence of
further information.
9   We are being a little disingenuous here. What we are maximizing is the Shannon relative entropy (or Kullback-Liebler divergence),
the information loss on taking the penalties of exchange into account so that, in the absence of these penalties, there is uncon-
strained exchange between all interested parties.
10 However, ‘MaxEnt’ goes further than Tobler in specifying that exchange falls off with ‘distance’ exponentially with cost/effort (a
‘Boltzmann’ distribution).
11 Conventional ‘Proximal Point Analysis’ has equal unweighted on-off links so that restricting outflows restricts the number of sites
with which any site will interact. Here we allow for weighted links so that many more sites can interact in principle as long as
the total outflow is capped.
12 This may be geographical distance, on taking ‘friction’ due to different terrain into account or it may be travel time.
Ray J. Rivers, Tim S. Evans
468
This gives the ‘Doubly Constrained Gravity Model’,
for which the ‘MaxEnt’ solution is
Tij = si fij sj (3.2)
where 
Si = si S j fij sj (3.3)
Because of the reciprocity in deterrence the final
equation is also Oi = Si = Ii, for all i. As in (2.2), the
sk are now not independent, as in (3.1), but deter-
mined in terms of the input Si or Oi through the
constraints (3.3)! We stress that there is no need to
invoke individual agents behaving in particular ways.
In summary, on comparing (3.2) and (3.3) to (2.1)
and (2.2) we see what we have termed epistemic-
ontic duality. By this we mean that, once we accept
reciprocity between the exchange effort/cost between
sites, the ‘most likely’ outcome for finding missing
sites based on constrained local activity without hav-
ing to invoke agents directly is equivalent to the
technically much more sophisticated13 ‘Ricardian’
model of free 20th-century market traders with con-
stant elasticity of substitution and efficiencies satis-
fying a ‘Weibull’ distribution, and so on.
This is provided we identify (a) the outflows Oi = Si
in the two cases and (b) (f avij)–θ of Barjamovic et al.
(2017) with fij of the ‘Doubly Constrained Gravity
Model’ (up to a fixed scale factor).
It could be argued that we are being disingenuous
in downplaying the role of agents in ‘MaxEnt’. That
exchange occurs is a consequence of the presence of
agents, and that it costs something is because of the
efforts of agents. However, what we are saying from
our position of ignorance is generic with no refer-
ence to the type of good exchanged, the means of
exchange, the ease of production and access, let
alone assumptions about seeking ‘profit’. In fact, the
simple requirement that deterrence or impedance to
exchange increases with distance encodes no arbi-
trage. The model is to be thought of as a null model
in which our coarse-graining of activity and ‘cost’ is
taken as characterising some type of statistical aver-
aging over the detailed activities of these agents, in
this case in the framework of detailed balance.
Parsimony: primary and secondary problems
This comparison provides fertile ground for explor-
ing the utility of parsimony, although some care is
needed in its application. Trying to understand a
trading network (or any historical system) poses
several problems, often a primary problem which
characterises the analysis (here, the positions of the
missing sites) and a constellation of secondary con-
firmatory problems (e.g., the importance of these
sites) which set the details. These latter may require
more parameters which, given the uncertainties of
network modelling, are likely to be less justifiable.
There is an analogy with our understanding of the
Solar System which we find helpful. Essentially, the
geocentric Ptolemaic/Aristotelian world-view posi-
tioned the Earth at the centre of the universe with
the planets and the sun moving on circles embedded
in spheres around it, whereas the heliocentric Co-
pernican view had the sun at the centre with the
Earth and the other planets moving around it (also
in circles). The primary problem was whether the
geocentric or heliocentric viewpoint was correct. In
neither case were there ‘laws of nature’ to be in-
voked, in the way we understand the term today. At
best there was an argument for circles on symmet-
ric grounds as they permitted a Creator who could
be the ‘unmoved mover’ as the planets circulated.
It was as much because the heliocentric view pro-
vides a conceptually natural solution to the ‘wande-
rings’ of the planets that the geocentric view was
unable to do, rather than the data, that it prevailed.
Neither picture worked well quantitatively for the
secondary problems of how the individual planets
behaved. In the intellectual framework of the time,
in which the paradigm was circular motion, both pos-
sibilities required large numbers of epicycles (circles
on circles) to fit the data even approximately. We
know why this happens; Newton’s laws mean that the
planets move in ellipses, to which circles are a poor
approximation, although a heliocentric system of cir-
cles is still the better null model14. The analogy that
we would draw with archaeological modelling is that
there are no ‘laws’ of society so our major aim is to
identify the system as ‘heliocentric’ correctly (i.e.
‘solve’ the primary problem). Since epicycles are mis-
leading conceptually, only serving as a means to ‘save
the phenomena’ (Duhem 1969), we would argue that
we should not expect to have reliable solutions to the
secondary issues in the absence of hard data. This is
probably the best that we can hope for. Beyond that
we are back in the Just-So territory alluded to earlier.
13 Barjamovic et al. (2017) are well aware that they are describing a constrained gravity model, but their approach is very diffe-
rent to ours with regard to parsimony.
14 The argument is subtle. The symmetry of Newton’s law of gravity for orbits is, indeed, the symmetry of the circle but the solu-
tions (orbits) need not preserve this symmetry. Nonetheless, a simple circle is the natural null model solution.
How do we avoid imposing the present on the past when modelling spatial interactions|
469
The primary problem posed by Barjamovic et al.
(2017) was that of identifying the position of the
‘missing’ sites. What is surprising is that, as we have
seen, we get identical equations for the triangulation
of missing sites from ‘MaxEnt’ provided we make the
identification between the iceberg melting and de-
terrence functions stated above, and nothing more.
This outcome is independent of the N=26 efficien-
cies Ti of the paper and only dependent on q as an
exponent in the combination (f avij)–q. Since we do
not know either f avij or fij then q itself is a redundant
parameter. We stress that we imposed distance reci-
procity in our ‘Doubly Constrained Gravity Model’ as
the most parsimonious choice that we did not have
enough information to refute. If subsequent data
shows that reciprocity cannot be supported, from our
entropy viewpoint we just fall back to the ‘Singly
Constrained Gravity Model’ of (3.2) and (3.3) with
no symmetry. We are unaware of any corresponding
‘Ricardian’ counterpart (although see Ward et al.
2013).
We are not for the moment concerned with the suc-
cess of the enterprise in Barjamovic et al. (2017),
which calls upon supplementary historical data, his-
torical road systems, estimates of carrying capacity
and the like. Suffice to say, it seems to work ‘well’.
We consider the results of the paper a major contri-
bution to the field.
Uncertainty and robustness
We close this theoretical analysis with some brief
thoughts on the uncertainties of the estimated out-
comes that relate to the size of the network, which
is small by most network standards. Our ability to
predict missing sites is conditional on these uncer-
tainties.
Both the agent-related economic model and the
‘Doubly Constrained Gravity Model’ are determin-
istic in their (identical) expected values of exchange
events. However, the extent to which these estimates
are reliable differs in principle between the models.
Nominally, the ‘MaxEnt’ approach with its ‘greatest
likelihood’ stance seems at odds with a probabilis-
tic interpretation. However, building on the work of
Wilson (1970), Yee Leung and Jianping Yan (1997),
have shown that the uncertainty that we attribute to
the most likely ‘Doubly Constrained Gravity Model’
(‘MaxEnt’) flows is just what would be expected if,
as far as possible, individual exchanges occurred in-
dependently of each other (i.e. with no memory of
past events). That is, we have ‘Poisson statistics’. It
is not clear from our ‘MaxEnt’ viewpoint if the data
set is too small for us to be able to draw reliable
conclusions, particularly given the large number of
geographic links with no exchange. We have had a
related experience in applying cost-benefit analysis
to Greek city-state formation, for which the distance
scales were too small to prevent fluctuations that
were large enough to force us to abandon that par-
ticular model (Rivers, Evans 2014).
The situation for the economic modelling of Barja-
movic et al. (2017) is different by fiat. Their analy-
sis is closely related to that of Eaton et al. (2012)
and of João Santos Silva and Silvana Tenreyro
(2006) who adopt estimators which are ‘Poisson-
influenced’ but not exactly ‘Poisson’. In this way,
additional calibration parameters enable them to get
results that simple entropy prohibits. We are unable
to determine to what extent this choice of variance
is intrinsic to ‘Ricardian’ economic modelling, or is
just adding further epicycles.
Data fitting
This leads us briefly to consider the problems with
the data. For the case in hand we have N = 26 sites
and only data for order N2 links. With order N (‘Max-
Ent’) calibration parameters (largely the Si and the
coordinates of the missing sites) an acceptable match
to the data is possible, without being tested by non-
symmetric exchange. Better data would probably
make non-symmetric exchange untenable. It is not
clear how to generalise this ‘Ricardian model’ to ac-
commodate this (although see Ward et al. 2013).
Oversimplifying, in the first instance Barjamovic et
al. (2017) minimise the least square correlation
between the predicted ratios of transactions and
the ratios recorded from the tablets as they vary
the positions of the missing sites, constraining both
parameter values and the missing site positions.
They then do more, identifying multi-stop itineraries
which refer to missing sites to further constrain their
positions. As for site importance, they call upon sup-
plementary data, e.g., historic road-systems. To
check the robustness of the predictions they omit
known cities in a random way to check if their po-
sitions can be successfully reconstructed from the
data. As we said earlier, this is a subtle analysis not
really germane to our discussion, and we refer the
reader to the original paper.
As anticipated, a priori the two formalisms do not
give identical results for the secondary questions con-
cerning the ‘importance’ of the individual sites, since
it is difficult to compare the economic and ‘MaxEnt’
Ray J. Rivers, Tim S. Evans
470
models as they are used. Barjamovic et al. (2017),
largely with an economics background, do not ap-
proach networks in the same way as archaeologists
with a social networks background. While archaeolo-
gists adopt the conventional attributes of sites in net-
works such as ‘PageRank centrality’, ‘betweenness
centrality’, and so on (Newman 2010) to describe site
significance, Barjamovic et al. (2017) invoke ‘au-
tarky’, a measure of site self-sufficiency, the antithe-
sis of networking, to give importance to theirs. Whe-
reas the latter does make use of the hitherto redun-
dant parameters, the ‘MaxEnt’ results display the
emergent properties of the network with no further
parameters, parsimonious to a fault. Since we are not
comparing like with like the two methods cannot
agree in detail. Whether that matters in practice,
given the uncertainty of the historical record, is equal-
ly unclear. As yet there is no ‘Tycho Brahe’ to im-
prove the data.
However, from another viewpoint, this chimes with
our earlier observation that the contemporary mod-
elling may itself be unreliable. Indeed, it has been
argued that constant elasticity of substitution and
‘Weibull distributions’ are introduced for their ana-
lytic solvability, rather than their representation of
real systems (e.g., Spilimbergo et al. 2003). Further,
ad valoram ‘iceberg’ melting does not even work
when applied to the transportation of ice (Bosker,
Buringh 2018).
Assyrian settlement structure and city-state
formation
As we have said, Barjamovic et al. (2017) argued
that Assyrians are a good proxy for contemporary
free enterprise traders and that 20th-century models
should work in this case, but the argument for an
epistemic approach is more general. This duality is
present in our second example of city state forma-
tion in Bronze Age Assyria. It is sufficient to see how
the modelling fits into our general theme, and we
shall present it in less detail.
This example may seem surprising since, although
historic and pre-historic city-state formation have
some contemporary and near-contemporary paral-
lels, they seem to have little in common with the
models of trade exchange familiar to economists
along the lines of our earlier discussion. That a par-
allel can be drawn with 20th-century economics is
due to Wilson (1971; 1976), who repurposed the
free-market ‘shopping’ or ‘retail’ model of David L.
Huff (1964) and Tiruvarur R. Lakshmanan and Wal-
ter Hansen (1965) to this end. Wilson (1971; 1976)
and Britton Harris and Alan Wilson (1978) argued
that synoikism, the key ingredient of state-forma-
tion, has its counterpart in the patterns of depart-
ment stores incorporated in shopping centres.
The ‘Retail Model’
The basic assumptions of the retail model, in the ter-
minology of retail outlets, are that:
i)  In equilibrium, retailing ‘activity’ (e.g., cash flow)
is proportional to ‘capacity’ (e.g., floor space).
ii)  The aim is to maximise ‘consumer surplus’ sub-
ject to the constraint of fixed outflows. This
enables us to convert ‘capacity’ into site ‘attrac-
tiveness’, measured through the inflows.
iii) The inflows of the dominant sites partition space
into zones of influence.
There are variations in the way that the model can
be formulated but, most simply, the conversion of
capacity into attractiveness is effected by the intro-
duction of a further set of parameters Zi (one for
each site) which reflect site activity, converted into
site size. These are in addition to the flows Tij which
determine the inflows. The Zi are determined by ma-
ximising the ‘Marshall-Hotelling’ (Hotelling 1929)
consumer surplus. Again homo economicus looms
large.
The final step is to relate the Zi to the final attracti-
veness, identified through the inflow Ii, understood
as the Zi equilibrium values. This evolution of the
‘attractiveness’ of a site to its equilibrium value is
problematic. Most simply a linear response is adopt-
ed (Harris, Wilson 1978). More dramatically, it can
be understood as treating the agent ‘consumers’ as
‘prey’ to the outlets, as determined by a non-linear
‘Lokta-Volterra’ approach (Wilson 2008). However,
insofar as the required outputs are the equilibrium
values, the details of the approach to equilibrium
are not relevant as long as they avoid the ‘period-
doubling cascades’ that are a precursor to chaotic
behaviour (Osawa et al. 2017).
It is clear that the ‘retail’ model is of a particular
time and place, mainly late 20th-century Western na-
tions, for which it captures the ‘death of the High
Street’ and the creation of malls. The arrival of the
internet and online shopping has made the model
largely redundant. We might expect the archaeolo-
gical applications to be equally constrained in time
and space, but the model was subsequently trans-
lated by Tracey E. Rihll and Alan Wilson (1987;
1991) to describe the emergence of the polis in the
How do we avoid imposing the present on the past when modelling spatial interactions|
471
19th century BCE mainland Greek Iron Age city states
as a result of:
● Synoikism: Surrendering of local sovereignty to a
wider community.
● Urbanisation: Emergence of dominant settlements.
We would not be so crass as to pair15 Argos/Argos™,
for example, but the way in which dominant sites
arise which partition territory make the parallels be-
tween ancient and modern site-dominance plausible.
Its success in this case, despite some caveats (Evans,
Rivers 2017), has led to several successful subse-
quent applications: e.g., Bronze Age Crete (Paliou et
al. 2016; Bevan et al. 2016), La Tène West Europe
(Filet 2017) and Middle Bronze Age Anatolia (Davis
et al. 2014; Palmisano, Altaweel 2015), as discussed
below. Once urbanisation has been implemented, the
model is exhausted.
Assyrian settlement structure
The applications of the retail model that we con-
sider here are that of settlement formation in the
Middle Bronze Age and Iron Age Khabur triangle
(Davies et al. 2014), complemented by the work of
Alessio Palmisano and Mark Altaweel (2017) who ex-
tend this approach to settlements in Middle Bronze
Age Central Anatolia. That is, in part we are looking
at Assyrian society at approximately the same time
and place as Barjamovic et al. (2017) but at a diffe-
rent level of organisation, of settlement rather than
individual traders.
What interests us is that the equilibrium ‘consumer
surplus’ extremisation of the retail model permits a
re-interpretation as the maximisation of a constrain-
ed entropy (Wilson 1970), also invoked by the au-
thors above. To implement ‘MaxEnt’ we return to
the ‘Singly Constrained Gravity Model’ of the previ-
ous section with its local constraints on outflows
(also assumed in the retail model). As with the case
of Assyrian trade, we need to impose an additional
constraint on our generalised inflows. The creation
of zones of influence around dominant city-states is
an asymmetric process. Rather than the local con-
straint of detailed balance between inflows and out-
flows imposed on traders, we adopt the global con-
straint that the entropy of the inflows from the bur-
geoning city-states is fixed (Rihll, Wilson 1991). Suf-
fice to say that if we were to implement this final
constraint alone we would have (up to a multiplica-
tive constant)
Tij = Iig fij (5.1)
where, in the absence of any further information,
we have set all outflows equal (as in ‘Proximal Point
Analysis’). We see that, for ‘attractiveness’16 g > 1
sites with larger inflows become dominant at the
expense of the rest, commensurate with synoikism.
Imposing the other constraints makes (5.1) much
more complicated17. Nonetheless, the primary que-
stion of determining the dominant states has a so-
lution essentially replicating the equilibrium beha-
viour of the ‘retail’ approach, showing a few domi-
nant sites which partition space into zones of influ-
ence. We note that we can still preserve the sym-
metry fij = fj,i and dij = dji. The asymmetry in the out-
comes arises from the asymmetry in the way we
treat inflows and outflows.
Since city-state formation permits a ‘MaxEnt’ de-
scription with no direct reference to agents, the epi-
stemic-ontic duality is seen again, although parsimo-
ny is implemented differently here in two ways.
Most simply, the first accords with our earlier sim-
ple definition, in that the difference between ‘retail’
and ‘MaxEnt’ lies in doubling the number of vari-
ables in the ‘retail’ approach. These collapse to a sin-
gle set in equilibrium, those of ‘MaxEnt’. As a result
the equilibrium site ranking is the same in both ap-
proaches.
Secondly, the more profligate approach permitted
by the retail model lies in the way that it provides
narratives for the evolution of the system to its equi-
librium state (Harris, Wilson 1978). As for the im-
plementation of the model, Toby Davies et al. (2014)
adopt non-linear ‘Boltzmann-Lokta-Volterra’ preda-
tor/prey dynamics whereas Alessino Palmisano and
Mark Altaweel adopt linear ‘Boltzmann-Lokta-Vol-
terra’ dynamics. That is, with ‘Lokta-Volterra’ ‘time’
understood as historical time, in principle the retail
model allows us to address the diachronic ‘secon-
dary’ issues as to how site differentiation might arise,
unavailable to ‘MaxEnt’. How seriously we should
take these narratives is a separate issue, insofar as
they are not used in data comparison. There is a po-
tential problem in that if the retail model solution
is an ‘attractor’ to the deterministic ‘Lokta-Volterra
equations’ our narrative looks to be one of effective
historical determinism. This can be avoided by the
15 Where the first name is from Archaic Greece, the second a U.K. retail chain.
16 g is the Lagrange multiplier associated with the fixing of inflow entropy.
17 In fact, in the solution of the dynamical flows a further constraint between inflows and outflows is imposed which goes beyond
the original shopping model and its original archaeological applications. This does not change the nature of our argument.
Ray J. Rivers, Tim S. Evans
472
explicit inclusion of multiplicative noise in the ‘Lokta-
Volterra’ equations (Ellam et al. 2017), but simple
‘MaxEnt’ evades this problem by only providing equi-
librium site rankings.
Data
Insofar that it is the equilibrium values which are
used for data analysis, the clear separation into pri-
mary and secondary questions that we found so use-
ful for traders is not relevant because of the identi-
ty of the outputs.
Unlike the case for Assyrian traders, the input data
here is largely site populations and positions, and
the model outputs are not individual flows but site
inflows identified with site size. There are several
ways to correlate these outputs to the data that look
for effects that go beyond our expectations from
geography alone. To abbreviate a complex analysis
in each case, the primary comparison of Palmisano
and Altaweel (2015) is with conventional network
analysis. From site inflows they construct a ‘hierar-
chical Nystuen-Dacey’ network (Nystuen, Dacey
1961) that encodes synoikism. The resulting zonal
network is then analysed with conventional central-
ity measures. For Davies et al. (2014) stress is put
on site size distributions rather than on the indivi-
dual sites themselves. As in Barjamovic et al. (2017)
in each case robustness is demonstrated through
partial dataset sampling. Both papers do an excel-
lent job of making their cases, and we refer the read-
er to them for details.
Discussion
There is no doubt that 20th-century economic mod-
els have proved very useful in motivating archaeolo-
gical models with the same structure, adopted almost
unadorned by their historical context. In this paper
we have argued wherever possible for an ‘epistemi-
cally modest MaxEnt’ approach (Jaynes 1973), which
enables us to avoid an explicit narrative of agents
constrained by detailed behavioural rules whenever
possible. Although the examples discussed here are
very different, both economic and ‘MaxEnt’ approa-
ches rely on maximization in different ways:
● Economic models assume the maximization of be-
nefit to traders or sites, perhaps by the extremi-
zation of utility functions, the definition of homo
economicus adopting rational economic behav-
iour.
● ‘MaxEnt’ models make use of the more general ex-
tremization of entropy or, equivalently, make the
best use of limited information, usefully rephrased
as the ‘Principle of Maximum Ignorance’ (Jaynes
1957; 1973).
For the examples here we have seen that, in the
main, these different ways of looking at the same
primary problems (‘missing sites’ and ‘dominant
sites’, respectively) give the same key results.
We have made little reference so far to ‘Bayesian’
analysis, but it could be said that, insofar as we are
swapping homo economicus for a flat ‘Bayesian
prior’, we do not need to know economics to answer
the primary questions for the models given here.
However, from a viewpoint of parsimony this epis-
temic-ontic duality is not evenly balanced. Contrast
the list of assumptions made in ‘Ricardian’ and ‘Re-
tail’ modelling with those of the ‘Constrained Gra-
vity Models’. That these models can be put in corres-
pondence with ‘MaxEnt’ shows the redundancy in
the economic modelling assumptions when address-
ing the main questions for large enough systems.
This redundancy will not apply to secondary ques-
tions to which the models will give different answers.
As we have seen, the situation is different for small
systems, but our goal in this paper has been more
about generics.
Whether the data and the models are trustworthy
enough to give useful results to these secondary
questions is another matter in the light of what we
said earlier; assuming large enough data sets the
number of parameters is small, even for economic
modelling, such that we can only expect very broad
agreement with data from whichever viewpoint. The
example of Greek city-state formation (Rihll, Wil-
son 1987) is a case in point. Whereas there is very
good reason for Athens and Corinth to be dominant
states, within the modelling the significance of The-
bes is more equivocal (Rivers, Evans 2014; Evans,
Rivers 2017). That Thebes was as important as it
was, where it was, is due to factors that our simple
modelling cannot incorporate, such as the rise of
one socio-political ‘house’ over another. However, a
significant site somewhere in that region was to
have been expected.
It is because of these qualifications about secondary
issues that we reject the reverse engineering that –
since our ‘least surprising’ entropy results are, in
the first instance, commensurate with free-market
economic models – a free-market society is the ‘least
surprising’, if not ‘obvious’ outcome, for describing
exchange in this period. We would argue that the
How do we avoid imposing the present on the past when modelling spatial interactions|
473
devil lies in the secondary details, as we see in Eaton
and Kortum (2002) and Anderson and van Wind-
coop (2003). These are not the ‘least surprising’ out-
comes since they rely on subsidiary information.
This is why economists preserve their complicated
models rather than use ‘MaxEnt’. Without comment-
ing on the reliability of their models their data is
generally so good that the broad brush approach of
‘MaxEnt’ is inadequate.
For large enough systems this difference between
the economic and entropy-maximizing approaches
is seen most clearly in how they address temporal
change, to which we have referred. Whereas econo-
mic models are dynamic, ‘MaxEnt’ looks for equilib-
rium behaviour. Change can be accommodated in
‘MaxEnt’, e.g. an overall increased difficulty in travel
due to banditry/piracy affecting the exchange pat-
terns in the South Aegean (Knappett et al. 2011)
and in the more dramatic case of the eruption of
Thera (Rivers 2018). A similar parameter shift oc-
curs in Davis et al. (2014) to describe changes in set-
tlement patterns between Middle Bronze Age and
Iron Age sites. However, these are exogenous effects
unlike the endogenous behaviour encoded in the
‘Lokta-Volterra’ equations of the retail model (Wil-
son 2008; Ellam et al. 2017). We might argue that,
from an agent-related economic viewpoint, ‘history’
is an attempt to achieve ‘good’ functionality from a
non-optimal beginning whereas, from a ‘MaxEnt’
viewpoint, ‘history’ is an attempt to maintain ‘good’
functionality at all times as circumstances change.
These two models are not the only C20th economic
models that have been translated to the historic
past. In particular, there are ‘Intervening Opportu-
nity Models’ which assume that transactions between
two sites i and j are proportional to the number of
‘opportunities’ at destination site j and fall off in-
versely with the number of ‘intervening opportuni-
ties’. Introduced by Samuel Stouffer (1940) and de-
veloped by Morton Schneider (1959), this approach
was used to model commuting patterns, but it has
a natural extension in archaeology where, in an ex-
treme form, it occurs as ‘Proximal Point Analysis’, as
mentioned earlier. There is a ‘MaxEnt’ realization of
the general model, albeit in a slightly tortured way
(Wilson 1970). Beyond ‘Proximal Point Analysis’
(e.g., see Broodbank 2000; Terrill 1986) it has been
applied to Mediterranean maritime exchange by
coastal tramping in the Late Bronze Age (Rivers et
al. 2016).
Of course, there are models based on 20th-century
economics (e.g., cost-benefit analysis, in which we
look for the ‘best’ outcome) that seem to have no
direct epistemic counterpart in terms of making the
best use of information, but that is another story18.
Our aim here has been the more limited one of try-
ing to demystify unnecessarily complicated econo-
mic machinery that has been used to explain histo-
ric and prehistoric exchange.
One important class of models that we have not
addressed is that of agent-based models, which can
build on free-market behaviour (e.g., Brughmans,
Poblone 2016). Nominally, they fall outside our ana-
lysis in that our emphasis has been on avoiding bot-
tom-up narrative in favour of generic likelihood. In
fact, agent-based modelling could not replicate an
analysis as detailed as that of Barjamovic et al.
(2017), although it can be built upon entropy max-
imisation (Altaweel 2015), perhaps bringing the
best of both worlds. We shall not pursue this further.
So, in summary answer to the question in the title
of this paper, archaeologists often try to impose a
(free-market) present on to the past. However, for
the models we have discussed here we find that in
the first instance the significant results are nothing
more than the ‘least surprising’ results that follow
from maximising our ignorance (‘MaxEnt’) of free-
market behaviour under simple assumptions, the
most parsimonious approach. The situation is dif-
ferent for secondary questions when free-market
analogues are much less parsimonious (cf. ‘MaxEnt’)
and require a large amount of additional detail.
Given our poor understanding of the model parame-
ters and the ambiguity of the archaeological data to
which these models are applied, this can be spurious
or unable to be substantiated. Nonetheless, it has to
be said that the economic models, however suspect
their detailed assumptions, do provide a rationale as
to how different types of goods produced and tran-
sported differently can be aggregated. As such they
motivate the simple averaging that happens with
‘MaxEnt’ null modelling, even if the results are not
to be taken too seriously. However, that is the nature
of null models, and it is not clear in general that we
do any worse by continuing with ‘MaxEnt’ when pos-
sible. More case-by-case analysis is required but, ar-
guably, being simple can be enough “pluralitas non
est ponenda sine necessitate” (William of Ockham).
18 The difference may not be as great as it seems. For large systems there is a duality between the ‘most likely’ outcome and the ‘best’.
Ray J. Rivers, Tim S. Evans
474
Altaweel M. 2015. Settlement dynamics and hierarchy
from agent decision-making: a method derived from en-
tropy maximization. Journal of Archaeological Method
and Theory 22: 1122–1150.
http://doi.org/10.1007/s10816-014-9219-6
Anderson J. E. 1979. A theoretical foundation for the Gra-
vity Equation. American Economic Review 69: 106–116.
Anderson J. E., van Windcoop E. 2003. Gravity with Gra-
vitas: A solution to the border puzzle. American Econo-
mic Review 93: 170–192.
Barjamovic G., Chaney T., Cosar K. A., and Hortasu A.
2017. Trade, merchants, and the lost cities of the Bronze
Age. The Quarterly Journal of Economics 134: 1455–
1503. https://doi.org/10.1093/qje/qjz009
Bevan A., Wilson A. 2013. Models of settlement hierarchy
based on partial evidence. Journal of Archaeological Sci-
ence 40(5): 2415–2427.
https://doi.org/10.1016/j.jas.2012.12.025
Bosker E. M., Buringh E. 2018. Ice(Berg) transport costs.
CEPR Discussion Paper No. DP12660.
https://ssrn.com/abstract=3118287
Broodbank C. 2000. An island archaeology of the Early
Cyclades. Cambridge University Press. Cambridge.
Brughmans T., Poblome J. 2016. Roman bazaar or market
economy? Explaining tableware distributions through
computational modelling. Antiquity 90: 393–408.
https://doi.org/10.15184/aqy.2016.35
Cheeseman P., Stutz J. 2004. On the relationship between
Bayesian and Maximum Entropy Inference. AIP Confe-
rence Proceedings 735: 445.
https://doi.org/10.1063/1.1835243
Croce B. 1922. Historical materialism and the econo-
mics of Karl Marx. Transaction Publishers. New Jersey.
Davies T., Fry H., Wilson A., Palmisano A., Altaweel M.,
and Radner K. 2014. Application of an entropy maximis-
ing and dynamics model for understanding settlement
structure: The Khabur triangle in the Middle Bronze and
Iron Ages. Journal of Archaeological Science 43: 143–
154. http://dx.doi.org/10.1016/j.jas.2013.12.014
Duhem P. 1969. To save the phenomena, an essay on
the idea of physical theory from Plato to Galileo. Univer-
sity of Chicago Press. Chicago.
Eaton J., Kortum S. 2002. Technology, geography and
trade. Econometrica 70: 1741–1779.
Eaton J., Kortum S., and Sotelo S. 2012. International
trade: Linking micro and macro. NBER Working Paper
N.17864.
Ellam L., Girolami M., Pavliotis G. A., and Wilson A. 2017.
Stochastic model of urban structure. Proceedings of the
Royal Society A 474: 20170700.
http://dx.doi.org/10.1098/rspa.2017.0700
Erlander S., Stewart N. F. 1990. The gravity model in
transportation analysis. VSP. Utrecht.
Evans T. S., Rivers R. J. 2017. Was Thebes necessary? Con-
tingency in spatial modelling. Frontiers in Digital Huma-
nities 4(8). https://doi.org/10.3389/fdigh.2017.00008
Filet C. 2017. An attempt to estimate the impact of the
spread of economic flows on latenian urbanisation. Fron-
tiers in Digital Humanities 3: 10.
http://doi.org/10.3389/fdigh.2016.00010
Friedman M. 1953. Essays in positive economics. Uni-
versity of Chicago Press. Chicago.
Harris B., Wilson A. G. 1978. Equilibrium values and dy-
namics of attractiveness terms in production-constrained
spatial-interaction models. Environment and Planning
A 10: 371–388. https://doi.org/10.1068/a100371
Hotelling H. 1929. Stability in competition. Economic
Journal 39: 41–57.
Huff D. L. 1964. Defining and estimating a trading area.
Journal of Marketing 28: 37–38.
Jaynes E. T. 1957. Information theory and statistical me-
chanics. Physical Review 106: 620.
http://doi.org/10.1103/Phys Rev.106.620
1973. The well-posed problem. Foundations of Physics
3: 477–492. http://doi.org/10.1007/BF00709116
1979. Where do we stand on maximum entropy? In R.
D. Levine and M. Tribus (eds.), The Maximum Entropy
Formalism. M. I. T. Press, Cambridge, MA: 15–72.
Knappett C., Evans T. S., and Rivers R. 2011. Modelling
maritime interaction in the Aegean Bronze Age, II: The
eruption of Thera and the burning of the palaces. Anti-
quity 85: 1008–1023.
http://doi.org/10.1017/S0003598X00068459
Lakshmanan T. R., Hansen W. G. 1965. A retail market po-
tential model. Journal of the American Institute of Plan-
ners 31: 134–143.
References
How do we avoid imposing the present on the past when modelling spatial interactions|
475
Leung Y., Yang J. 1997. A note on the fluctuation of flows
under the entropy principle. Transportation Research
B31: 417–423.
Mill J. S. 1874. Essays on some unsettled questions of
political economy (2nd ed.). Longmans, Green, Reader &
Dyer. London.
Newman M. 2010. Networks: An Introduction. Oxford
University Press. Oxford.
Nystuen J. D., Dacey M. F. 1961. A graph theory interpre-
tation of nodal regions. Papers of the Regional Science
Association 7: 29–42.
Osawa M., Akamatsu T., and Takayama Y. 2017. Harris
and Wilson (1978) Model Revisited: The Spatial Period-
Doubling Cascade in an Urban Retail Model. Journal of
Regional Science 57: 442–466.
Paliou E., Bevan A. 2016. Evolving settlement patterns,
spatial interaction and the socio-political organisation of
late prepalatial south-central Crete. Journal of Anthropo-
logical Archaeology 42: 184–197.
http://doi.org/10.1016/j.jaa.2016.04.006
Palmisano A., Altaweel M. 2015. Landscapes of interaction
and conflict in the Middle Bronze Age: From the open
plain of the Khabur Triangle to the mountainous inland
of Central Anatolia. Journal of Archaeological Science:
Reports 3: 216–236.
https://doi.org/10.1016/j.jasrep.2015.06.015
Rihll T. E., Wilson A. G. 1987. Spatial interaction and
structural models in historical analysis: Some possibilities
and an example. Histoire & Mesure 2: 5–32.
Rihll T. E., Wilson A. G. 1991. Modelling settlement struc-
tures in Ancient Greece: New approaches to the Polis: I.
In J. Rich, A. Wallace-Hadrill (eds.), City and Country in
the Ancient World. Routledge. London: 59–95.
Rivers R. J. 2018. New approaches to the Theran erup-
tion. In J. Leidwanger, C. Knappett (eds.), Maritime net-
works in the ancient Mediterranean World. Cambridge
University Press. Cambridge: 39–60.
Rivers R., Evans T. S. 2014. New approaches to Archaic
Greek settlement structure. Les Nouvelles de l’archéolo-
gie 135: 21–27. http://doi.org/10.4000/nda.2316
Rivers R., Evans T. S., and Knappett C. 2016. From Oar to
Sail. In S. Ducruet (ed.), Maritime Networks; Spatial struc-
tures and time dynamics. Routledge. London and New
York: 63–76.
Rubio-Campillo X., Coto-Sarmiento M., Pérez-Gonzalez J.,
and Rodríguez J. 2017. Bayesian analysis and free market
trade within the Roman Empire. Antiquity 91(359): 1241–
1252. https://doi.org/10.15184/aqy.2017.131
Schneider M. 1959. Gravity Models and Trip Distribution
Theory. Papers of the Regional Science Association 5:
51–58.
Silva J. M. C. S., Tenreyro S. 2006. The log of gravity. The
Review of Economics and Statistics 88(4): 641–658.
Spilimbergo A., Vamvakidis A. 2003. Real effective ex-
change rate and the constant elasticity of substitution as-
sumption. Journal of International Economics 60: 337–
354.
Stouffer S. A. 1940. Intervening Opportunities: A Theory
Relating Mobility and Distance. American Sociology Re-
view 5: 845–867.
Terrell J. 1986. Prehistory in the Pacific Islands.
Cambridge University Press. Cambridge.
Tobler W. 1970. A computer movie simulating urban
growth in the Detroit region. Economic Geography 46
(Supplement): 234–240.
Ward M. D., Ahlquist M. S., and Rozenas A. 2013. Gravity’s
rainbow: A dynamic latent space model for the world
trade network. Network Science 1: 95–118.
http://doi.org/10.1017/nws.2013.1
Wilson A. G. 1970. Entropy in urban and regional mo-
delling. Pion Limited. London.
1971. A family of spatial interaction models and associa-
ted developments. Environment and Planning 3: 1–32.
1976. Retailers’ profits and consumers’ welfare in a spa-
tial interaction shopping model. In I. Masser (ed.), Lon-
don papers in regional science 6. Theory and practice
in regional science. Pion. London: 42–59.
2008. Boltzmann, Lokta and Volterra and spatial struc-
tural evolution: An integrated methodology for some
dynamical systems. Journal of the Royal Society In-
terface 5: 865–871.
2010. Knowledge power: Interdisciplinary education
for a complex world. Routledge. Abingdon.