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Ecological Applications, 23(1), 2013, pp. 239–254� 2013 by the Ecological Society of America

Modeling spatial decisions with graph theory: logging roads andforest fragmentation in the Brazilian Amazon

ROBERT WALKER,1,2,8 EUGENIO ARIMA,3 JOE MESSINA,1 BRITALDO SOARES-FILHO,4 STEPHEN PERZ,5 DANTE VERGARA,6

MARCIO SALES,7 RITAUMARIA PEREIRA,1 AND WILLIAMS CASTRO1

1Department of Geography, Michigan State University, 116 Geography Building, East Lansing, Michigan 48823 USA2Nucleo de Meio Ambiente, Univesidade Federal do Para, Rua Augusto Correa, 01–Cidade Universitaria, Bairro Guama,

CEP66075-900, Belem, Brazil3Department of Geography and the Environment, University of Texas, GRG 334, Mailcode A3100, Austin, Texas 78712 USA

4Centro de Sensoriamento Remoto, Avenida Antonio Carlos, 6627, Belo Horizonte, MG, CEP 31270 900 Brazil5Department of Sociology and Criminology and Law, University of Florida, 3219 Turlington Hall, P.O. Box 117330,

Gainesville, Florida 32611-7330 USA6School of Environmental Science and Management, University of the Philippines Los Banos College, Laguna 4031 Philippines

7Instituto do Homem e Meio Ambiente da Amazonia (IMAZON), Rua Domingos Marreiros, 2020–Fatima, Belem, Para,CEP 66.060-160 Brazil

Abstract. This article addresses the spatial decision-making of loggers and implicationsfor forest fragmentation in the Amazon basin. It provides a behavioral explanation forfragmentation by modeling how loggers build road networks, typically abandoned uponremoval of hardwoods. Logging road networks provide access to land, and the settlers whotake advantage of them clear fields and pastures that accentuate their spatial signatures. Inshaping agricultural activities, these networks organize emergent patterns of forestfragmentation, even though the loggers move elsewhere. The goal of the article is to explicatehow loggers shape their road networks, in order to theoretically explain an important type offorest fragmentation found in the Amazon basin, particularly in Brazil. This is accomplishedby adapting graph theory to represent the spatial decision-making of loggers, and byimplementing computational algorithms that build graphs interpretable as logging roadnetworks. The economic behavior of loggers is conceptualized as a profit maximizationproblem, and translated into spatial decision-making by establishing a formal correspondencebetween mathematical graphs and road networks. New computational approaches, adaptedfrom operations research, are used to construct graphs and simulate spatial decision-makingas a function of discount rates, land tenure, and topographic constraints. The algorithmsemployed bracket a range of behavioral settings appropriate for areas of terras devolutas,public lands that have not been set aside for environmental protection, indigenous peoples, orcolonization. The simulation target sites are located in or near so-called Terra do Meio, once amajor logging frontier in the lower Amazon Basin. Simulation networks are compared toempirical ones identified by remote sensing and then used to draw inferences about factorsinfluencing the spatial behavior of loggers. Results overall suggest that Amazonia’s loggingroad networks induce more fragmentation than necessary to access fixed quantities of wood.The paper concludes by considering implications of the approach and findings for Brazil’smove to a system of concession logging.

Key words: Amazonia; forest fragmentation; graph theory; logging.

INTRODUCTION

Forest fragmentation has emerged as an important

environmental issue in Amazonia, given the spatial

pattern of deforestation can compromise ecological

processes (e.g., Ferreira and Laurance 1997, Aldrich

and Hamrick 1998, Laurance et al. 2000, Cochrane

2001). Despite substantial research addressing such

consequences, less is known about fragmentation’s

underlying drivers, although land change science has

begun addressing the issue (e.g., Nagendra et al. 2004,

Pan et al. 2004, Soares-Filho et al. 2004, Walsh et al.

2002, 2003). The present article contributes to our

knowledge in this regard by gaining insights into the

spatial decision-making of loggers with respect to the

design and implementation of logging road networks.

Such networks provide access to land, and the settlers

who follow clear fields and pastures that consolidate the

spatial signature of the roads. In this manner, logging

roads organize the local pattern of forest fragments,

even as loggers move on to new areas (Fig. 1).

Research has linked roads to deforestation (e.g.,

Nelson and Hellerstein 1997, Soares-Filho et al. 2004,

Manuscript received 6 October 2011; revised 13 July 2012;accepted 20 August 2012. Corresponding Editor: V. C.Radeloff.

8 E-mail: [email protected]

239

2006, Walker 2004), but the interest here resides in how

they structure landscapes, a critical aspect of their roadecology (Forman et al. 2002, Freitas et al. 2010). In

pursuing this interest, the article presents a computa-tional approach that simulates road network formationby translating the economic behavior of loggers into

spatial outcomes. The simulation of road networks hasoften focused on the minimization of road building costs

(e.g., Tomlin 1990, Soares-Filho et al. 2004, 2009, Jiang2007). Arima et al. (2005, 2008) build on this literature

for the Amazonian case by addressing harvest revenuesin addition to costs. Merry et al. (2009) also contributeto understanding Amazonian fragmentation processes

by modeling the formation of new logging centers,which are sensitive to revenues and created as a function

of proximity to older but defunct centers as well ascommercial wood.The present article moves beyond these recent

advances in three substantial ways. First, it provides aformal, graph theoretic statement of the economic

decisions that loggers make when they build roadnetworks to access tropical hardwoods. Second, it

develops a new computational approach to simulatingnetworks, which it uses together with existing spatialsoftware to study road building in an important

Amazonian frontier. Third, it subjects simulation resultsto performance assessment via inference. This enables

identification of a ‘‘best’’ model configuration, which isthen interpreted in light of the theoretical formulation togain insight into the human drivers of forest fragmen-

tation associated with the spatial behavior of loggers.Thus, the article provides a modeling approach with the

potential for management applications, as Brazil moves

to the concession system in harvesting its national

forests. Knowing how loggers design road networks isessential to finding policy levers that can minimize forest

fragmentation. Although simulation results are inter-preted in light of a formal statement of the behavior ofan economic agent, namely the logger, the computa-

tional approach is not ‘‘agent-based’’ as the modelingcommunity has come to understand the term (Parker et

al. 2003). Nor does it rely strictly on the identification ofleast-cost pathways, a functionality available in current

GIS software (Tomlin 1990, Jiang 2007). Rather, thesoftware builds on classical optimization algorithms inoperations research designed to identify networks that

achieve explicit goals, like profit maximization (Merry etal. 2009). By definition, profit maximization considers

revenues in addition to costs, which complicates botheconomic decision-making and simulation.The graph theory and computational algorithms

address logging road networks that produce ‘‘dendriticfragmentation,’’ so named for its visual similarity to the

trellis structure of stream hierarchies (Chorley andHaggett 1967, Arima et al. 2005). Dendritic fragmenta-

tion, quite similar in form to the independent settlementpattern of spontaneous colonization (Oliveira Filho andMetzger 2006), is observed in poorly monitored

indigenous areas and in terras devolutas, public landsthat have not been designated for specific uses or subject

to colonization programs (Treccani 2001). Dendriticlandscapes provide a sharp contrast with fishbonefragmentation, a pattern found in official settlements

(Oliveira Filho and Metzger 2006). Although loggerscollaborate with small-holders in extending the frontiers

of fishbone colonization (Perz et al. 2007), this orderly

FIG. 1. Road network as initial fragmentation template. Logging road networks provide access to land, and the settlers whofollow clear fields and pastures that consolidate the spatial signature of the roads.

ROBERT WALKER ET AL.240 Ecological ApplicationsVol. 23, No. 1

pattern is not likely to maximize profits given road

segments are constrained to be linear and they do not

necessarily lead to high revenue harvest sites (Arima et

al. 2005). Thus, the spatial outcome differs significantly

when loggers operate in isolation, or in advance of the

agricultural frontier. Then, their spatial decision-making

leads to dendritic fragmentation.

Dendritic fragmentation has impacted large parts of

the Amazon basin, such as the study area in so-called

Terra do Meio, an informal designation for a region of

;80 000 km2 in central Para State, Brazil (Pinto 2005;

Fig. 2). By way of comparison, the Altamira Polygon,

appropriated for colonization in the 1970s, and which

displays the lion’s share of the lower basin’s fishbone

fragmentation, covers 64 000 km2. Evidently, dendritic

and fishbone patterns of forest fragmentation occupy

similar extents across the reaches of the basin. Once a

major logging frontier, the Brazilian government placed

FIG. 2. Terra do Meio study area in Para State in the Brazilian Amazon Basin. The bottom panel shows the simulation sitesused in this study.

January 2013 241GRAPH THEORY AND SPATIAL DECISIONS

much of Terra do Meio under protection in 2004,

motivating the sector to migrate west (Merry et al.

2009). Prior to that, grileiros (land grabbers), loggers,

miners, and ranchers viewed the terras devolutas of

Terra do Meio as theirs for the taking (Pinto 2005).

Amerindians have long considered the region home,

particularly those belonging to Ge-speaking tribes, and

along the Xingu River over 30 000 km2 of indigenous

reserves have been demarcated for several groupings of

the Kayapo peoples (Robert 2010; see Plate 1). Terra do

Meio reveals a patchwork of vegetative covers including

open and closed tropical forest, and cerrado woodlands.

High concentrations of mahogany sustained a predatory

logging sector, with several (approximately five) large

companies dominating production from the mid 1980s

until ;2000, when they abandoned the region to smaller

operations ( personal communications, forest engineer in

Novo Progresso, 2009 and colonist in Sao Felix do

Xingu, 2011; Pinto [2005]). These include Maginco,

Perrachi, IMPAR, Pelegrino, Pau D’Arco, and Angelim

(Pinto 2005). The simulations to be presented address

road networks associated with large, well-capitalized

companies because they possess sufficient capital to

build roads.

GRAPH THEORY

Before proceeding to the methods and results sections,

it is necessary to discuss the article’s adaptation of graph

theory, which enables us to formally represent the

economics of spatial decision-making. Mathematical

graph theory possesses considerable potential for

ecological applications, given its explicit representation

of spatial phenomena such as connectivity (e.g., Rayfield

et al. 2011). For the present case, graphs function as the

conceptual link between economic behavior and spatial

outcomes. As indicated, a prime contribution of the

article resides in adapting mathematical graphs to reflect

both economic decisions and the empirics of wood

exploitation, within a spatial context. What mathema-

ticians call a complete graph may be represented as G(N,

E), with N ¼ n nodes and number of edges, E,

accounting for all possible combinations of n nodes

taken two at a time, or n!/2!(n� 2)! combinations (West

1996). A logging road network becomes a graph by

defining roads as edges and logging sites as nodes.

Suppose a logger has access to n logging sites and can

build roads between all pairs. The complete network is

depicted for four sites (n ¼ 4) in Fig. 3A. The

combinatorial calculation identifies six edges, each

providing a potential route, in which case a logger with

four available logging sites builds a road network using

six potential routes. The network actually constructed,

given in Fig. 3B, is referred to as a subgraph, Gi, because

it includes only a subset of the edges of the complete

graph, G. Loggers are likely to build road networks that

are subgraphs, given unnecessary costs incurred by

connecting all pairs of sites with redundant segments.

The graphs in the figure are highly schematic, only

suggestive of what actual road networks look like. Since

our objective is to simulate empirical networks, the

article adapts graph theory in order to represent profits;

this enables us to build on existing approaches that are

primarily focused on cost minimization. With this

modification, it is then possible to gain insight into the

spatial behavior of loggers via formal statements. We

start with the representation of profits, then follow with

the identification of optimal networks from an economic

perspective. The term ‘‘optimality’’ in the context of the

article refers only to economic optimality, and does not

imply that an ‘‘optimal’’ road network is the best for the

environment.

Network profits

Our graph theory formulation assumes loggers are

price takers who maximize profits defined as the

difference between revenues and costs. It weights

logging sites (or nodes) by revenues and edges (or

roads) by road construction costs, which are distance

sensitive. To simplify the exposition, logging site

‘‘revenues’’ are taken as the total harvest revenues net

of extraction and processing costs, which the logger can

anticipate in advance for all locations. That is, the

money earned by selling wood from each site (i.e., the

revenue) is the net of all costs except those associated

with building a road to access the site. Formulated this

way, ‘‘nodal’’ revenue minus ‘‘edge’’ costs yields

network profit, which is what the logger is assumed

to maximize. A time dimension is added, with networks

emerging by the addition of one edge per time period.

This brings important realism to the problem, for

loggers do not make static decisions in a timeless world.

Finally, no distinction is drawn between fixed and

variable costs; depreciation for the logger’s capital

equipment can be added to whatever variable costs are

borne in each time period.

Spatial behavior

A logger’s spatial behavior can be revealed by

establishing theoretical claims about profit maximizing

networks, under the micro-economic assumption the

logger will try to build such a network (Varian 1992).

These claims show the economic conditions under which

different networks maximize profits. Thus, if it is

observed empirically that a specific network has been

built, graph theory can be used via these claims to

identify some of the economic circumstances that

produce them. The empirical identification follows from

spatial simulation representing the possible networks. A

network of special interest is the subgraph referred to as

a tree; which links all sites without redundant connec-

tions (West 1996). The minimum spanning tree (MST)

comprises the set of edges, or roads, yielding the cost-

minimizing network, where cost is road-building cost.

The MST can be found via spatial simulation using a

classical algorithm referred to as the Prim algorithm, our

conceptual and computational start point (Prim 1957).


Economic interpretation of the simulated networks is

based on four claims involving identification of the

optimal network (Table 1). Claim 1 establishes that

profit-maximizing networks are always trees. Claim 2

shows that the MST maximizes profits when a logger

draws little distinction between today’s and tomorrow’s

rewards, or when the logger has a very low discount rate,

possibly equal to zero (Baumol 1968, Hirshleifer 1970).

The computational implication is that empirical net-

works should be similar to networks generated by Prim’s

algorithm. Claim 3 states that the MST may not produce

the profit-maximizing network when discount rates are

high, in which case the Prim algorithm no longer

effectively simulates observed networks. Claim 4 dem-

onstrates that insecure property rights can have the same

effect as high discount rates in leading to the construc-

tion of profit-maximizing networks different from the

MST. An important contribution of the article is to

accommodate the implications of Claims 3 and 4 by

implementing an algorithm that generates a profit

maximizing network alternative to the MST. This is

the ordered tree (OT).

Claim 1.—Let a logging road network graph be G(V,

E), where V is the set of nodes (i.e., logging sites), and

E is the set of edges (i.e., potential road segments); p(A)

is profit for some arbitrary graph, A, composed by the

difference of revenues and costs. Revenues net of

harvesting costs are obtained at the individual nodes

(logging sites), while distance-sensitive costs are in-

curred in road construction. Assume that G is not a

tree, which implies it possesses at least one ‘‘cycle,’’ or

redundant connecting segment between two nodes

(West 1996). Without loss of generality, consider the

case of only one. By implication, there exists some

segment (i.e., edge) that adds only road construction

costs without adding revenue; call this segment e. If

removed, the new graph minus the edge, or graph G –

e, remains ‘‘connected’’ in the sense that all the nodes

(or sites) can be visited and revenues taken. Thus, p(G� e) � p(G), given road construction costs are less. But

this contradicts the assumption of optimality, which

establishes the claim.

Claim 2.—Assume a fixed point of origin, o, for

accessing K logging sites. In each time period starting at

t ¼ 0, the logger builds a road to one of the sites to

extract wood, continuing until all K sites have been

visited. Thus, the logger chooses from K ! temporal

sequences, each of which produces a unique tree graph.

Write the kth graph as Gk, and the revenue (net of wood

harvesting costs at the site) it generates at time t as rtk. As

defined, t also identifies a specific logging site in an

ordered sequence, namely the site to be visited at time t.

TABLE 1. Graph theory claims about road networks.

Number Claim

1 Profit-maximizing road networks are always treenetworks.

2 The minimum spanning tree (MST) is always theprofit-maximizing road network in the absence oftime discounting.

3 The MST may not be the profit-maximizing roadnetwork with high discount rates.

4 The MST may not be the profit-maximizing roadnetwork in the case of open access resources.

FIG. 3. Linking mathematical graphs to road networks: (A) a complete set of potential routes (complete graph, G), and (B) areduced set for minimizing cost (subgraph, Gi ).


The present value for the revenues of Gk, or the sum of

yearly revenues discounted by time, is

Rk ¼XK�1

t¼0

1

ð1þ dÞtrt

k

where d is the discount rate (Hirshleifer 1970). If road

building costs are ct�1,t (for the segment built between

time t � 1 and t), then the present value of profit given

multiple time periods is

p ¼XK�1

t¼0

1

ð1þ dÞtrt

k � ct�1;t

� �:

Here, c�1,0 is the cost of building to the first harvestable

site in the initial period, which is not discounted. In the

case of a zero discount rate (d¼ 0), the quotient term in

the summation is 1, and the present value of revenue is

Rk ¼XK�1

t¼0

rtk:

But this sum is identical for all logging site sequences, in

which case the MST maximizes the present value of

profits since it minimizes the present value of road-

building costs; by definition, the MST has the shortest

sum of road (edge) lengths. An implication is that the

Prim algorithm identifies the optimizing road network; it

does so by selecting in each period the nearest site (node)

from those that remain.

Claim 3.—When discount rates are high, the MST

may not yield the maximum present value of profits, in

which case networks generated by Prim’s algorithm will

diverge from empirical networks, under the assumption

that loggers are profit maximizers. In the extreme case (d

! ‘), as is likely in a high-risk frontier environment, the

logger is best off scanning opportunities in a time-

myopic fashion, building roads in each period that

maximize profits only for that period. This may be seen

by considering a simple case involving two logging sites

subscripted as v and w, with path o! v! w generating

the MST. As the discount rate grows increasingly large,

the present value of MST profits is pMST ! rv – c0,v,

since 1/(1þ d )frw – cv,wg ! 0. If profits from exploiting

site w are greater in the initial period (po,w . pMST), then

the optimal path prescribes an initial visit to a site that

would be visited later in constructing a road network by

the guidelines of the MST.

Claim 4.—Insecure property rights can have the effect

of high discount rates. Again, for the MST sequence o

! v ! w, the logger may question the likelihood of

being able to exploit site w, which is more distant than v.

The expected present value of MST profit is then pMST¼rv – c0,v, if the logger thinks it possible that a competitor

(or government) will close access to w by the second

period. On the other hand, if the logger builds first to the

more distant location, w, in order to preemptively

exploit it, the present value of profit is p0,w ¼ rw – c0,wþ 1/(1þ d )frv – cw,vg. The discounted term on the right-

hand side reflects an assumption that the logger thinks a

second period harvest close to the point of origin, o, is

feasible. The logger pursues this course when

rw � c0;w þ1

ð1þ dÞ rv � cw;v

� �. rv � c0;v:

Claims 3 and 4 demonstrate conditions under which

the MST may not maximize the present value of profits.

In such a situation, empirical networks differ from those

generated by Prim’s algorithm. This is illustrated in Fig.

4, with the MST given in Fig. 4A. Fig. 4B shows the case

when a logger seeks the big pay-off first, and conse-

quently builds a network different from the MST. This

carries significant implications for our computational

FIG. 4. Two types of road networks for connecting an original site o with a nearer, less resource-rich, site v and a more distant,more resource-rich, site w. (A) The minimum spanning tree (MST) comprises the set of edges, or roads, yielding the cost-minimizing network, where cost is road-building cost. (B) The ordered tree (OT) results when road-builders seek out sitesgenerating the most revenue at each step (profit-maximizing network).


application, by necessitating an algorithm that can

mimic spatial behavior when discount rates are high,

or when property rights are insecure. The algorithm,

referred to as the ordered tree (OT), seeks out sites

generating the most revenue at each step. The OT

maximizes the present value of profits when revenues are

large relative to road-building costs, a reasonable

assumption for the study area. Most of the logging for

the period under investigation involved mahogany, the

most valuable of Amazonia’s hardwoods ( personal

communications, Kayapo Chief in Sao Felix do Xingu,

2010 and colonist in Sao Felix do Xingu, 2011).

Claims 3 and 4 also possess implications for the

broader discussion about forest fragmentation in the

Amazon Basin. In particular, they identify conditions

under which logging generates more fragmentation than

necessary to harvest a fixed quantity of wood. Since the

MST always reflects the minimum length network to

exploit a set of logging sites, any other network

possesses additional kilometers of roadway. Thus, if

Amazonian loggers build networks more consistent with

an OT than with an MST, it can be concluded that the

incentives they face have generated ‘‘excess’’ fragmenta-

tion.

METHODS

The methodologies implemented by the research are

primarily computational. However, a prime objective is

to reflect actual decision-making behavior, in which case

model structures and assumptions are based, when

possible, on field work using social-science research

methodology. The social-science data comprises infor-

mation gleaned from three field campaigns conducted in

Para State in 2009, 2010, and 2011. All major logging

centers were visited along BR-163 and BR-230; also

visited was Sao Felix do Xingu, gateway to Terro do

Meio from the east. An attempt was made to apply a

scientific survey to sawmill managers, but the sector was

in crisis from the global recession and from environ-

mental law enforcement operations and many mills had

closed. The teams resorted to key informant interviews

of knowledgeable individuals, including, but not limited

to, sawmill managers; these interviews were aimed

primarily at obtaining information on road-building

costs, and at the way in which loggers spatially organize

their productive activities. Such ethnographic informa-

tion does not possess the statistical properties of a

scientific sample, but it is used here to justify model

assumptions that might otherwise appear arbitrary. It is

also used to shed light on results.

The primary analytical method involves simulations

of road networks, followed by comparisons to the actual

networks they attempt to reproduce. A raster data

structure is used, with grid cell sizes of 90 m, and extents

on the order of 100�1000 km2. Interactive Date

Language (IDL) was coded to build two-dimensional

road networks, issuing from an origin and reaching a

series of logging sites, taken as target destinations. The

origin points were identified by visual inspection of the

actual road networks, observable on satellite images; the

logging sites were defined computationally. The software

expands on existing approaches in operations research,

starting with Prim’s algorithm (Prim 1957). Specifically,

two software codes are implemented to identify net-

works conforming to the MST and OT. Simulation

outputs following model implementation are compared

to actual road networks, and the parameterization and

code structure that yield the closest approximation are

discussed with respect to implicated economic behaviors

and likely consequences for forest fragmentation. We

now consider three computational adaptations necessary

to reflect real world networks. This is followed by

consideration of the data and discussion of the site

selection process.

From theoretical to computational graphs

Our graph theory formulation and associated claims

establish that the classical MST solution may not be

optimal. To bracket the extreme cases of high discount-

ing and/or insecure land tenure, we developed an

alternative configuration referred to as the OT. To

simulate both the MST and the OT, we adapt our

computational algorithms in order to transform abstract

mathematical graphs into ones with empirical features

(Table 2). These adaptations involve the representation

of edges (road routes) and nodes (logging sites and

branch points).

Edge weights.—The computational models weight

edges by construction costs. This adaptation replaces

the straight edges of theoretical graphs with routes that

are cheaper to build (Fig. 5). Computational identifica-

tion of least-cost paths stems from a well-known GIS

problem solved by Dijkstra with the algorithm that

bears his name (Dijkstra 1959). In our case, distance-

sensitive costs involve only road construction, as

transportation expenditures to and from the sawmill

are negligible by comparison ( personal communication,

logger in Novo Progresso and Itaituba, 2009). This

empirical circumstance allows us to consider only road-

building as important to the emergence of the network,

on the cost side. On a homogenous landscape, such costs

are proportional to straight-line, Euclidean distance.

Topographic relief compromises the equivalence be-

tween minimum cost and minimum distance. Thus, a

real-world, profit-maximizing graph possesses edges like

the meandering pathways of Fig. 5A.

TABLE 2. Graph theory adaptations to reflect empiricalnetworks.

Number Adaptation

1 Edge weights are based on least-cost pathways, notEuclidean distances (Dijkstra algorithm).

2 Nodes are weighted, in addition to edges.3 Nodes can branch from edges (Steiner rectilinear

problem).


Defining nodes.—A second computational adaptation

produces logging site revenues, in effect weighting the

nodes to be accessed for logging. Together with edge

weights, this enables the identification of profit-maxi-

mizing networks. Nodal weighting involves data inten-

sive calculations which are discussed in the subsection

Wood revenues. The third adaptation allows new roads to

branch from previously constructed ones, and not just

logging sites. Graph theory assumes that all nodes are

given in advance, as per graph theory symbolism, G(V,

E). The present application also assumes preexisting

nodes, namely those identified as sites the logger intends

to access for wood extraction. However, new ones may

arise when it proves less costly to build from an existing

road segment. In such a situation, the new road’s point of

origin provides an ex post network node not given in the

original V set, as indicated in Fig. 5B. Here, if it is less

costly to access logging site c from the segment, ab, than

from node b, a new node appears in the network at node

b*. Such a theoretical adaptation necessitates adjusting

our software to approximate solutions to the so-called

Steiner rectilinear problem (Garey et al. 1977, Ivanov

and Tuzhilin 1994, Dean 1997, Promel and Steger 2002).

Data used in the analysis

Our computational approach requires two types of

data input, as well as classified satellite images for

evaluating model performance. The inputs include

elevation maps for representing road-building costs,

and geo-coded biomass density for producing surrogate

measures of wood revenue. The classified satellite images

yields maps of logging roads against which the

simulations can be compared.

Road building costs.—Road building costs are taken

as weight functions reflecting elevation differences

between neighboring grid cells (Arima et al. 2005,

2008). Features such as streams and rivers are given

higher weights because bridging a river is more costly

than opening a road. Since little is known about how

loggers actually evaluate road-building costs, we model

them using different functions, as discussed in Results.

Elevations are produced by a digital elevation model

(DEM), whose resolution derives from the 90-m posting

resolution of the Shuttle Radar Topography Mission

(SRTM). Although some behaviorally meaningful ele-

vation gradients may be missed, the data allow for broad

geographic coverage, necessary for the application.

Wood revenues.—Revenues are taken to be propor-

tional to wood density, a proxy for harvest volume.

Thus, logging sites are weighted for revenues using data

from the RADAM project, which conducted extensive

forest inventories for 2328 Amazonian locations (Oli-

veira 1999). For use here, the RADAM data are

spatially allocated across the basin to provide grid

coverage of wood density, using kriging with an external

drift. This predicts wood biomass density for unsampled

locations by using the information at sampled RADAM

sites, together with variables likely to affect density such

as soil type, forest type, and elevation (Goovaerts 1997,

Sales et al. 2007). To identify logging sites, or nodes, we

applied a neighborhood function over the density

surface raster using a moving window kernel of 16 3

16 cells and weights equal to one. This function assigns

the sum of the 16 3 16 neighbors to each cell and

essentially identifies continuous regions with high

density. We then divided the resulting raster into equal

interval isopleths and delineated the polygons around

the largest intervals, which were taken as the logging

areas. Large polygons therefore indicate a large contin-

uous concentration of high timber volume, and presum-

ably more revenue. Finally, the centroid of each area

was located to serve as a road network destination (i.e.,

node) in the simulations.

Remote sensing of road networks.—We compare our

simulated networks to maps compiled in an existing

digital library maintained by Instituto do Homem e

Meio Ambiente da Amazonia, or IMAZON (Brandao

and Souza 2006). The library contains maps of

Amazonian logging roads identified by visual interpre-

tation of bands 3 and 5 (scale 1:50 000) using LandSat

imagery as early as 1985 and as recently as 2002,

covering most of Para State (Fig. 2). Such roads can be

ephemeral, disappearing then reappearing at a later

date; they can also manifest gaps on satellite imagery

given their modest size. The visual interpretation

appears to work well when remotely sensed roads are

compared to known ones (Greenpeace 2001, Brandao

and Souza 2006). For the study region in Terra do Meio,

attributes in the IMAZON map library indicate road

segments had all been identified by 2001, but there is no

way of knowing at what point in time they first became

visible. We obtained a 1996 Landsat Image PATH/

ROW 225/65 for ourselves to gain insight into network

FIG. 5. (A) In the real world, topography may make theleast-cost pathway from the origin to a point one thatmeanders. (B) New nodes may arise when it proves less costlyto build from an existing road segment. Here, if it is less costlyto access logging site c from the segment, ab, than from node b,a new node appears in the network at node b*.


emergence and dynamics. Earlier scenes had a large

percentage of cloud cover and were deemed inappropri-

ate for road identification. The path/row combination

covers the simulation sites, which are now discussed.

Simulation sites

The spatial simulations were conducted to replicate

two logging road networks found in or near Terra do

Meio; one of these was previously considered by Arima

et al. (2008). The two simulation sites are found

upstream, on the left and right banks of the Xingu

River (LBX and RBX; Fig. 2). Our field work implicates

a single enterprise, which was able to gain exclusive

access to the indigenous territories of the Kayapo tribe.

The point of origin for simulating roads in the RBX site

was taken as the network intersection with the Xingu,

the likely point of transshipment to sawmills down-

stream (in Sao Felix do Xingu); the network is isolated

and Mahogany floats. For LBX, the origin is taken as

the closest point of the network to a north-south road

connecting to the Xingu River. This is the only apparent

outlet to Sao Felix do Xingu. The sites were determined

by visual inspection of the IMAZON digital map library

(see Fig. 2). The selection process sought dendritic

networks that appeared autonomous, so that we could

reasonably attribute them to individual logging opera-

tions. We also searched for networks with large extents

in order to capture the spatial activity of enterprises

sufficiently capitalized to vertically integrate down to

harvest and road-building activities. A multi-product

facility processing ;80 000 m3/yr (Verıssimo et al. 2002,

Lentini et al. 2005) requires on the order of 3200

hectares a year, assuming 25 m3/ha of marketable timber

( personal communications, loggers in Novo Progresso

and Uruara, 2009). On a decadal time cycle, which

approximates the mahogany boom in Terra do Meio

(1985–2000; Pinto 2005), the land required is ;30 000

ha, yielding a territorial domain approaching 20 3 20

km, probably an underestimate given patchy distribu-

tions of mahogany. These calculated areas are in order-

of-magnitude agreement with the size of the networks

used as simulation sites. Although large, such holdings

are common in the region (Almeida 1995, Fernandes

1999). Modeling with large extents has the added

advantage of mitigating any measurement issues asso-

ciated with the ‘‘poor’’ resolution of the RADAM and

SRTM DEM data sets.

In practice, it may be difficult to separate one logger’s

roads from another’s in the map library, given the

possibility of simultaneous activity. Another difficulty

arises from the nature of the data. Although the

IMAZON archive uses satellite imagery that spans a

significant time period (1985–2002), the attributes for

the individual road segments only give an entry of one

year, in which case it is not possible to fully record the

dynamics of road construction. This impacts our ability

to observe the actual emergence of networks in the study

region. To mitigate this lack of information, we

examined a 1996 Landsat image to see how much, if

any, of the road networks selected for simulation existed

then. The image shows that none of the network

segments had been built at that time. This is consistent

with the region’s history, where logging began in ;1985

with the opening of Estrada Canopus, which traversed

Terra do Meio west, from the left bank of the Xingu

(across from Sao Felix do Xingu) to a cassiterite mine

near the Iriri River (Pinto 2005). It presumably took

time for logging to diffuse south to the selected logging

sites. An exploitation start date of 1996 or shortly

thereafter suggests a rapid rate of construction. In that

roads through primary forest are built at 5–10 km per

week (20–40 km per month), a five-year period (e.g.,

1996–2001) gives more than enough time to exploit the

areas in question, given network lengths do not exceed

150 km. These estimates come from surveys conducted

in 2003 and 2004 of 113 individuals along the Trans-

amazon Highway, who provided estimates of extension

lengths and construction times for 23 unofficial roads, or

travess~oes. For 78 observations on extensions equal to or

greater than 10 km, the average construction rate is 5.67

km per week. These include all road-building agents

(colonists, fazendeiros, municipal government, Instituto

de Colonizacao e Reforma Agraria [INCRA], and

loggers). In a few cases of loggers building long

extensions (40 to 100 km), the rate reaches 10 km per

week. The logging firm associated with the selected sites

operated in the vicinity of Sao Felix do Xingu, as did

other large interests ( personal communications, Kayapo

Chief in Sao Felix do Xingu, 2010, colonists in Vila

Clariano and Ramal Xada, 2010, and colonist in Sao

Felix do Xingu, 2011). Terra do Meio has experienced

land conflicts, but large logging operations, mainly

responsible for the local road networks, can be assumed

to have controlled their territorial domains, despite lack

of land titles and little-to-no law enforcement ( personal

communication, colonist in Sao Felix do Xingu, 2011;

Pinto 2005). Thus, fragmentation in Terra do Meio

received an important initial push from just a handful of

logging firms, despite the larger number of smaller

operations found in and around Sao Felix do Xingu

(;20) into the first millennial decade ( personal commu-

nication, colonist in Sao Felix do Xingu 2011; see also

Pereira et al. [2010]).

RESULTS

To summarize, the MST and OT algorithms produce

networks bracketing a range of behavioral settings

affecting the spatial decision-making of loggers. The

MST has been previously implemented (Arima et al.

2008), while the OT is new. We implement these models

for two sites in central Para State, with three possible

functions reflecting road-building costs, including linear

(i.e., constant), increasing, and decreasing marginal

costs in gradient (let hi and hj be the elevation at two

adjacent cells in a grid; the linear cost of traversing those

cells is jhi � hjj, jhi � hjj2 for the increasing case and


jhi� hjj1/2 for the decreasing case). Crossing three cost

functions with two behavioral settings (MST and OT)

yields six possible models for simulation. We compare

the results for these six models to real road networks

using the IMAZON archive, and assess performance to

see which of them works ‘‘best.’’ On the basis of the best

model, we draw conclusions about the spatial behavior

of loggers, and implications for forest fragmentation.

Our performance assessment expands on the method

suggested by Goodchild and Hunter (1997) and Arima

et al. (2008), which is designed for linear features as

opposed to two-dimensional maps with categoric spaces

(see also Tveite and Langaas 1999). The method we use,

described in another manuscript currently under review,

is based on computational inference. In essence, it

determines the probability of observing a simulated

network under the null hypothesis of a random

generation process. If extremely unlikely, we conclude

that the simulation has explanatory power. The like-

liness or unlikeliness is determined by reference to a

probability histogram for randomized networks pro-

duced computationally, as depicted in Fig. 6 for the site

on the left bank of the Xingu (LBX). The measure

implemented to operationalize the procedure is the

percentage of the network falling within some buffer of

the empirical network being modeled. The rationale for

the buffer length, elaborated in the manuscript under

review, evolves from the principle of ensuring model

accuracy, measured by the percentage of the simulated

network found near the empirical network. For the

histogram presented in Fig. 6, it was determined to be

1026 m. In comparing more than one simulation, we

select the ‘‘best’’ model (or parameter setting) by viewing

the percentage of network found within the buffer,

together with significance probabilities. Higher percent-

ages are taken to reflect greater accuracy and better

performance; a low-significance probability (or a) means

the model has explanatory power. This approach is

analogous to the examination of regression coefficients

and their levels of significance in regression modeling.

The results, presented in Table 3, show that the OT

algorithm performs best across the two sites, with a 75%

within buffer given for both cases, and uniformly high

significance levels (low a probabilities). This can be

visually confirmed for the LBX site by reference to Fig.

6, where it can be seen that the value 0.75 falls far to the

right, in the upper tail of the probability histogram.

Accuracies for the simulations using the MST algorithm

tend to drop considerably, down to 56% for the site on

the right bank of the Xingu (RBX), and 58% for LBX.

Although the superiority of the OT algorithm is robust

across the sites, the road-building cost shows a variable

performance. For LBX, a linear function (constant

marginal cost in gradient) proves most effective, while

one with increasing marginal costs (in gradient) per-

forms best for RBX. Sample simulation results are given

in Fig. 7 for the LBX site, where the wood inventory

TABLE 3. Model results for the proportion of simulationnetworks within buffer and significance (P values are inparentheses).

Algorithm andcost structure

Sites

RBX LBX

Ordered tree

Constant 0.740 (0.100) 0.750 (,0.001)Increasing 0.750 (0.007) 0.694 (,0.001)Decreasing 0.655 (0.030) 0.737 (,0.001)

Spanning tree

Constant 0.668 (0.027) 0.583 (0.003)Increasing 0.592 (0.130) 0.635 (,0.001)Decreasing 0.563 (0.343) 0.586 (0.003)

FIG. 6. The null distribution for hypothesis testing of simulation results (LBX site).


data identified eight target nodes for modeling; as the

figure shows, segments of the OT tend to locate closer

than the MST. By coincidence, eight nodes were also

determined for the RBX site. Given the time period

involved, roads were built to more than one node in

some of the years. This is not a problem empirically, as

extraction and road-building crews may function

independently (Verıssimo et al. 1995). The theoretical

development could be adapted for such empirical

specificities, although without producing much new

insight.

DISCUSSION

The results indicate that loggers prefer the architec-

ture of ordered tree (OT) networks. This has implica-

tions for the extent of forest fragmentation, in that the

OT requires more roads to be built than the MST in

accessing a fixed quantity of wood. For the simulation

results, this ranges from 1% to 8%, depending on the site

and the cost function assumption. Thus, logging in the

Amazon basin may have produced an excessive amount

of forest fragmentation. The secondary effects of

excessive road building probably aggravate the immedi-

FIG. 7. A map showing simulated road networks vs. actual roads (LBX site).


ate impacts on biodiversity and ecosystem function

brought by the physical disturbance of the road. In

implementing an OT architecture, loggers are more

likely to build long segments early in the exploitation

cycle than they would in constructing an MST. We are

aware of a road extension of about 100 km built in the

mid-1980s in order to gain access to mahogany in Terra

do Meio. This road allowed the occupation of lands

outside the federal government’s original plan, and

intensified conflict between farmers and indigenous

peoples. Its construction may never have occurred had

loggers pursued a different spatial strategy. Although we

can assert that loggers maximize profits, they do so on

the basis of economic and institutional factors that lead

to different spatial outcomes. As our theoretical claims

suggest, either discount rates or the strength of property

rights can tip the balance one way or the other (Table 1).

For the results in question, the conclusion is that OT

networks indicate either high discount rates, weak

property rights, or some combination. Despite this

ambiguity, we can interpret based on empirical context.

Since the patterns were largely observed prior to 2000 in

what was a remote logging frontier, the architecture of

the road networks observed probably reflects high

discount rates. Our personal experience in the region is

that loggers had little to fear from government

enforcement of any kind until after 2000. Thus, there

was no incentive to make deep penetrations of public

lands to avoid the opportunity cost of a forfeited harvest

due to possible government intervention. As for

competitive and pre-emptive logging by rivals, it also

appears that the larger interests acted cooperatively to

lay out general logging areas, in order to avoid conflicts

among themselves ( personal communication, colonist in

Sao Felix do Xingu, 2011).

The discount rate is not the only behavioral factor

that underlies Amazonian fragmentation patterns asso-

ciated with loggers. Very recently, strong incentives have

emerged to access one of the few remaining areas of

Terras Devolutas in central Para State. This has led to

long penetrations north from the Transamazon High-

way, and produced a conflict referred to as the Guerra

de Macapuxi, or the War of Macapuxi (Diario do Para

2009). Loggers in the region are very much aware of the

Brazilian government’s new willingness to enforce

environmental law, and are fearful that it will continue

to put formerly accessible lands under protected status.

One large interest operating recently along the Trans-

amazon Highway stored logs at the sawmill several years

in advance of processing, presumably to avoid restric-

tions on future access to land ( personal communication,

colonists in Uruara, 2009) This is consistent with the

preemptive behavior implicit in Claim 4 and Fig. 4. The

maintenance of large stocks of unsawn wood adds

considerably to inventory costs in the form of degraded

raw material inputs (Kamien and Schwartz 1981). This

is a waste of valuable hardwood that need not have

occurred, and it adds to the ledger of environmental

costs associated with OT networks.

Although the result on tree structure is robust, the

finding on road construction costs is not. A constant

function gives the best performance in one case (LBX),

and an increasing function, in the other (RBX). One

possible explanation is that the spatial behavior of

loggers could be linked to topography, given elevation

gradients do not impede road-building until a threshold

of 308 ( personal communication, forest engineer in Novo

Progresso, 2009). Thus, with little topographic relief,

loggers remain relatively unconcerned about gradient

and do their cost calculations accordingly. Alternatively,

broken topography (with many slopes of 308 or more)

sensitizes loggers to steep terrain. The spatial implica-

tion is a network architecture showing slope avoidance,

consistent with treating costs as increasing in gradient.

Although we do not test this for the two simulation sites,

we show the impact of alternative cost functions on

network structure in Fig. 8 for the LBX site. From left

to right and from top to bottom, the panels depict

changes in the MST in response to increasing marginal

costs in slope. The simulated networks (in yellow) grow

increasingly complex as the algorithm seeks pathways in

avoidance of slope. While it is evident that road-building

with extreme sensitivity to slope induces networks with

extra road mileage (lower panel, to the right), constant

or decreasing marginal costs (upper panel, to the left)

create networks that pay less attention to topographic

relief, possibly leading to erosional impacts.

The results overall suggest that Amazonia’s emergent

dendritic networks possess suboptimal features and lead

to more fragmentation than necessary to access fixed

quantities of wood. Thus, some Amazonian logging

practices, particularly the illegal ones of the recent past,

do not reflect a sustainable use of Amazonia’s forest

resources. This is hardly a controversial finding, as many

have implicated the logging sector as a prime culprit in

environmental changes affecting the region (Asner et al.

2005). Perhaps the most significant accusation to date

points to the extent of illegal logging operations, with

the majority of harvest being of dubious legality,

particularly with incursions onto lands that have been

declared off limits (e.g., ecological and indigenous

reserves [Monteiro and Souza 2006]). In such a

situation, the logger acts as if ‘‘there is no tomorrow,’’

given the ever-looming potential of government reaction

(Schneider et al. 2002).

This outcome is essentially identical to what happens

with high discount rates (Claim 3). It is therefore

analogous to the inferior asset problem in resource

economics, a circumstance in which economic agents

liquidate renewable resources given they place little-to-

no value on future harvests (Clark 2010). By the

arguments presented, such a response magnifies the

extent of fragmentation, and also raises rates of

deforestation, in which case a key policy question is

how to create incentives to encourage a view to the long


run on the part of the logging sector. In other words,

how can loggers be encouraged to make spatial decisions

in accord with the social rate of discount, which would

be both forest conserving and fragmentation reducing

(Baumol 1968). Brazil’s prohibition of mahogany

exploitation in 2005 has no doubt reduced the intensity

of forest fragmentation processes, as has coverage by the

Convention on International Trade in Endangered

Species of Wild Fauna and Flora (CITES), as of 2003.

Stricter enforcement of environmental laws has proba-

bly contributed to recent declines in rates of deforesta-

tion, as well (Nepstad et al. 2009). These various actions

may ultimately encourage loggers to stretch their time

horizons, and thereby improve their spatial decision-

making.

Although the theoretical results explain how specific

landscape patterns emerge, two caveats are in order. The

first is that our analysis is restricted to one type of

fragmentation, and does not apply to fishbone or large-

property patterns found in Amazonia (Oliveira Filho

and Metzger 2006). Those patterns reflect different

economic behaviors, and different actors or combina-

tions of actors. Hence, more theoretical work is

necessary to fully account for the various ways in which

the Amazonian forest is fragmenting. Our second caveat

is that the contribution as it stands remains partial given

human behavior is embedded in social realities that both

FIG. 8. The impact of increasing road building costs on network architecture (LBX site). From left to right and from top tobottom, the panels depict changes in the MST in response to increasing marginal costs in slope. The simulated networks are shownin yellow; actual roads are shown in red.


enable and constrain it, such as the distribution of

economic resources and the extent of prior government

investments in infrastructure. Full treatment of the

forces driving fragmentation, in dendritic landscapes or

any other, must also address the broader social

processes that bring factors of production, especially

capital but also labor, to bear on the tropical forest

landscapes of Amazonia.

For the dendritic case, such a treatment must

ultimately tie logging to the behavioral, social, and

political circumstances impacting the sector. These

circumstances include the intensity of time discounting,

which influences the results considerably and is linked to

interest rates (Baumol 1968). Also important is the

constitutional codification of Brazilian property institu-

tions, which has enabled loggers to exploit terras

devolutas, given general societal acceptance of the

creation of rights in land by posse, or physical

occupation. Although such lands have historically been

exploited with little fanfare, current uncertainty of

government response has probably encouraged and

intensified interest in the ‘‘here and now,’’ at the expense

of tomorrow. Market signals have no doubt intensified

this interest. The premium placed on high-value

hardwoods, especially mahogany (Swietenia macrophyl-

la), has encouraged the emergence of wide-reaching

dendritic networks that would have been spatially

concentrated under lower price regimes. This draws

attention to the impact of the demand side on

Amazonia. Forest fragmentation here stems from the

deployment of factors of production, but such factors

would hardly find their way to forest frontiers without

market pressures.

CONCLUSIONS

The analytical and computational approaches provide

tools of potential use to forest managers, given Brazil’s

move to a system of concession logging (Rohter 2007).

Those tasked with concession oversight should stay on

the lookout for OT networks, with long segments and

PLATE 1. The research for this article involved social-science survey methods and key informant interviews. An important partof the modeling exercise was to gain direct insight into how loggers make spatial decisions, and in particular how they collaborateamong each other to territorialize their operations so as to avoid conflicts. Initial information indicated that one large logginginterest had gained nearly exclusive access to the upper Xingu Valley in the lands of the Kayapo Indians. About 9000 Kayapo livein small settlements scattered across 30 000 square kilometers of indigenous homelands, in southern Para State. Some arecompletely isolated and refuse contact with Brazilian society. This photo was taken when authors Robert Walker (kneeling) andEugenio Arima (standing) visited the Kayapo Indians in Sao Felix do Xingu in order to interview several caciques (chiefs) abouttheir experience with the logging company in question. The Kayapo no longer allow loggers onto their territories. Photo credit:Tarcisio Feitosa.


excessive length overall. Modeling can help them

collaborate with loggers to identify and implement

compact networks more in line with the MST. The

increased sectoral capacity likely to result with industry

rationalization will probably intensify incursions into

terras devolutas over the short run, particularly as such

lands become increasingly scarce. Thus, the Brazilian

government should also continue with its admirable

effort to extend its protected areas program (Soares-

Filho et al. 2009).

Our application of mathematical graph theory

provides insight into an important type of Amazonian

fragmentation associated with logging road networks.

The behavioral formulation identifies conditions under

which ‘‘excessive’’ fragmentation occurs, high discount

rates and uncertain property rights; these lead to the

overexploitation of renewable resources more generally

(Gordon 1954, Clark 2010). Thus, the article establish-

es a correspondence between spatial and aspatial

modes of resource degradation. The simulations shed

light on a disturbance regime that has affected very

large parts of Amazonia, and suggests that excessive

fragmentation has occurred. The impact of logging on

the forest, as bad as it had been, is worse than it had to

be.

As both the theoretical formulation and the simula-

tion results make clear, the spatial behavior of loggers

expresses economic objectives. Consequently, the proper

combination of carrots and sticks, that is, of economic

incentives, can encourage them to treat concession areas

as properties to be managed, not exhausted (Schmithu-

sen 1980, Walker and Smith 1993). Logging is an

important part of Amazonia’s economic base. Helping

loggers place long-run value on the resources that have

enriched them is one way to keep Amazonian jobs afloat

without adding unacceptable environmental costs to

society’s bottom line.

ACKNOWLEDGMENTS

We acknowledge support from the National Science Foun-dation under two research grants: Territorializing ExploitationSpace and the Fragmentation of the Amazon Forest (NSF-GSS-0822597) and Socio-spatial Processes of Road Extensionsand Forest Fragmentation in the Brazilian Amazon (NSF-GSS-0243102). We also thank the National Geographic Society forsupport under the grant Expedition on the Iriri (#7219-02). Ourmanuscript benefited greatly from Tarcısio Feitosa whoprovided us with useful documents and information, as wellas his penetrating insights. Finally, we are indebted to ThaısBarbosa Morais for help in conducting interviews with loggersand other individuals knowledgeable about the sector.

LITERATURE CITED

Aldrich, P. R., and J. L. Hamrick. 1998. Reproductivedominance of pasture trees in a fragmented tropical forestmosaic. Science 281:103–105.

Almeida, A. W. B. 1995. Carajas: a guerra dos mapas.Falangola, Belem, Brazil.

Arima, E., R. Walker, S. Perz, and M. Caldas. 2005. Loggersand forest fragmentation: behavioral models of road buildingin the Amazon basin. Annals of the Association of AmericanGeographers 95(3):525–541.

Arima, E. Y., R. Walker, M. Sales, C. Souza, Jr., and S. G.Perz. 2008. Emergent road networks and the fragmentationof space in the Amazon Basin. Photogrammetric Engineeringand Remote Sensing 74(6):699–709.

Asner, G. P., D. E. Knapp, E. N. Broadbent, P. J. C. Oliveira,M. Keller, and J. N. Silva. 2005. Selective logging in theBrazilian Amazon. Science 310:480–482.

Baumol, W. J. 1968. On the social rate of discount. AmericanEconomic Review 58(4):788–802.

Brandao, A., and C. Souza, Jr. 2006. Mapping unofficial roadswith Landsat images: a new tool to improve the monitoringof the Brazilian Amazon rainforest. International Journal ofRemote Sensing 27(1):177–189.

Chorley, R. L., and P. Haggett. 1967. Models in geography,Madingley lecture 2. Methuen, London, UK.

Clark, C. W. 2010. Mathematical bioeconomics. Third edition.Wiley and Sons, New York, New York, USA.

Cochrane, M. A. 2001. Synergistic interactions between habitatfragmentation and fire in evergreen tropical forests. Conser-vation Biology 15(6):1515–1521.

Dean, D. J. 1997. Finding optimal routes for networks ofharvest site access roads using GIS-based techniques.Canadian Journal of Forest Research 27 (1):11–22.

Diario do Para. 2009. Madeira: Minc participa de apreensaohistorica. Diaro do Para 27 June 2009. http://www.diariodopara.com.br/N-49560-MADEIRAþþMINCþPARTICIPAþDEþAPREENSAOþHISTORICA.html

Dijkstra, E. W. 1959. A note on two problems in connectionwith graphs. Numerische Mathematik 1:269–271.

Fernandes, M. 1999. Donos de Terra: Trajetorias da UniaoDemocratica Ruralista—UDR. Universidade Federal doPara/Nucleo de Altos Estudos Amazonicos, Belem, Brazil.

Ferreira, L. V., and W. F. Laurance. 1997. Effects of forestfragmentation on mortality and damage of selected trees incentral Amazonia. Conservation Biology 11:797–801.

Forman, R. T., et al. 2002. Road ecology: science and solutions.Island Press, Washington, D.C., USA.

Freitas, S., and 2010. Effects of roads, topography, and landuse on forest cover dynamics in the Brazilian Atlantic Forest.Forest Ecology and Management 259:410–417.

Garey, M. R., R. L. Graham, and D. S. Johnson. 1977. Thecomplexity of computing Steiner minimal trees. SIAMJournal of Applied Mathematics 32:835–859.

Goodchild, M. F., and G. J. Hunter. 1997. A simple positionalaccuracy measure for linear features. International Journal ofGeographical Information Science 11(3):299–306.

Goovaerts, P. 1997. Geostatistics for natural resources evalu-ation. Oxford University Press, New York, New York, USA.

Gordon, S. H. 1954. The economic theory of a commonproperty resource. Journal of Political Economy 62(2):124–142.

Greenpeace. 2001. Partners in mahogany crime. GreenpeaceInternational, Amsterdam, The Netherlands. http://www.greenpeace.org/raw/content/international/press/reports/partners-in-mahogany-crime.pdf

Hirshleifer, J. 1970. Investment, interest, and capital. PrenticeHall, Englewood Cliffs, New Jersey, USA.

Ivanov, A. O., and A. A. Tuzhilin. 1994. Minimal networks: theSteiner problem and its generalizations. CRC Press, BocaRaton, Florida, USA.

Jiang, Z. 2007. The road extension model in the land changemodeler for ecological sustainability of IDRISI. Pages 1–8 inH. Samet and C. Shahabi, editors. Proceedings of the 15thInternational Symposium on Advances in GeographicInformation Systems, ACM GIS 2007, Seattle, 7–9 Novem-ber 2007. Association for Computing Machinery (ACM),New York, New York, USA.

Kamien, M. I., and N. L. Schwartz. 1981. Dynamic optimiza-tion: the calculus of variations and optimal control ineconomics and management. Elsevier, New York, NewYork, USA.


Laurance, W. F., P. Delamonica, S. G. Laurance, H. L.Vasconcelos, and T. E. Lovejoy. 2000. Rainforest fragmen-tation kills big trees. Nature 404:836.

Lentini, M., D. Pereira, D. Celentano, and R. Pereira. 2005.Fatos florestais da Amazonia 2005. Imazon, Belem, Brazil.

Merry, F., B. Soares-Filho, D. Nepstad, G. Amacher, and H.Rodrigues. 2009. Balancing conservation and economicsustainability: the future of the Amazon timber industry.Environmental Conservation 44:395–497.

Monteiro, A., and C. Souza, Jr. 2006. Imagens de satelite paraavaliar planos de manejo florestal. Imazon, Belem, Para,Brazil. http://www.imazon.org.br/publicacoes/o-estado-da-amazonia/imagens-de-satelite-para-avaliar-planos-de-manejo

Nagendra, H., D. K. Munroe, and J. Southworth. 2004. Frompattern to process: landscape fragmentation and the analysisof land use/land cover change. Agriculture, Ecosystems andEnvironment 101:111–115.

Nelson, G. C., and D. Hellerstein. 1997. Do roads causedeforestation? Using satellite images in econometric analysisof land use. American Journal of Agricultural Economics79:80–8.

Nepstad, D., et al. 2009. The end of deforestation in theBrazilian Amazon. Science 326:1350–1351.

Oliveira, A. A. B. 1999. Projeto Radam Brasil, Rio de Janeiro,Brazil. Brazilian Institute of Geography and Statistics(IBGE), Rio de Janiero, Brazil. www.projeto.radam.nom.br/ingles/history.html

Oliveira Filho, B., and J. P. Metzger. 2006. Thresholds inlandscape structure for three common deforestation patternsin the Brazilian Amazon. Landscape Ecology 21:1061–1073.

Pan, W. K. Y., S. J. Walsh, R. E. Bilsborrow, B. Frizzelle,C. M. Erlien, and F. Baquero. 2004. Farm-level models ofspatial patterns of land use and land cover dynamics in theEcuadorian Amazon. Agriculture, Ecosystems and Environ-ment 101:117–134.

Parker, D. C., S. M. Manson, and M. Janssen et al. 2003.Multi-agent systems for the simulation of land use and landcover change: a review. Annals of the Association ofAmerican Geographers 93(2):314–337.

Pereira, D., D. Santos, M. Vedoveto, J. Guimaraes and A.Verıssimo. 2010. Fatos florestais da Amazonia 2010. Imazon,Belem, Brazil.

Perz, S., C. Overdevest, M. Caldas, R. Walker, and E. Y.Arima. 2007. Unofficial Road Building in the BrazilianAmazon: Dilemmas and Models for Road Governance.Environmental Conservation 32(2):1–10.

Pinto, J. N. A. 2005. Terra do Meio: estudo de caso na novafronteira amazonica. Thesis. Universidade Federal do Para,Para, Brazil.

Prim, R. C. 1957. Shortest connection networks and somegeneralizations. Bell System Technical Journal 36:1389–1401.

Promel, H. J., and A. Steger. 2002. The Steiner tree problem: atour through graphs, algorithms, and complexity. Advancedlectures in mathematics. Vieweg, Braunschweig, Germany.

Rayfield, B., M.-J. Fortin, and A. Fall. 2011. Connectivity forconservation: a framework to classify network measures.Ecology 92:847–858.

Robert, P. 2010. Conflitos, aliancas e recomposic~oes territoriaisem projetos de desenvolvimento sustentavel: experiencias daterra Indıgena Kayapo (Sul do Para). Pages 333–351 in R.Araujo and P. Lena, editors. Desenvolvimento sustentavel esociedades na Amazonia. Museu Paraense Emilio Goeldi,Belem, Brazil.

Rohter, L. 2007. Brazil gambles on monitoring as loggersadvance in Amazon. New York Times, 14 January 2007.

Sales, M., C. Souza, Jr., P. C. Kyriakidis, D. Roberts, and E.Vidal. 2007. Improving spatial distribution estimation offorest biomass with geostatistics: a case study of Rondonia,Brasil. Ecological Modelling 205:221–230.

Schmithusen, F. 1980. Forest utilization contracts—a key issuein forest policy and in the development of the tropicalhardwoods sector. FAO Paper 5. FAO, Pattaya, Thailand.

Schneider, R., E. Y. Arima, A. Verissimo, C. Souza, Jr., and P.Barreto. 2002. Sustainable Amazon: limitations and oppor-tunities for rural development. World Bank, Washington,D.C., USA.

Soares-Filho, B., A. Alencar, D. Nepstad, G. Cerqueira, D. C.Vera Diaz, S. Rivero, L. Solorzano, and E. Voll. 2004.Simulating the response of land-cover changes to road pavingand governance along a major Amazon highway: theSantarem-Cuiaba corridor. Global Change Biology 10:745–764.

Soares-Filho, B. S., D. C. Nepstad, L. M. Curran, G. C.Cerqueira, R. A. Garcia, C. A. Ramos, E. Voll, A.McDonald, P. Lefebvre, and P. Schlesinger. 2006. Modellingconservation in the Amazon basin. Nature 440:520–523.

Soares-Filho, B. S., H. Rodrigues, and W. Costa. 2009.Modeling environmental dynamics with Dinamica EGO.Universidade federal de Minas Gerais, Belo Horizonte,Brazil. http://www.csr.ufmg.br/dinamica/tutorial/Dinamica_EGO_guidebook.pdf

Tomlin, D. C. 1990. Geographic information systems andcartographic modeling. Prentice Hall, Englewood Cliffs, NewJersey, USA.

Treccani, G. D. 2001. Violencia e grilagem: instrumentos deaquisicao da propriedade da terra no Para. UniversidadeFederal do Para, Belem, Brazil.

Tveite, H., and S. Langaas. 1999. An accuracy assessmentmethod for geographical line data sets based on buffering.International Journal of Geographical Information Science13(1):27–47.

Varian, H. R. 1992. Microeconomic analysis. Third edition.Norton, New York, New York, USA.

Verıssimo, A., P. Barreto, R. Tarifa, and C. Uhl. 1995.Extraction of a high-value natural resource in Amazonia:the case of mahogany. Forest Ecology and Management72:39–60.

Verıssimo, A., M. A. Cochrane, C. Souza, Jr., and R. Salomao.2002. Priority areas for establishing national forests in theBrazilian Amazon. Conservation Ecology 6(1):4.

Walker, R. 2003. Evaluating the performance of spatiallyexplicit models. Photogrammetric Engineering and RemoteSensing 69(11):1271–1278.

Walker, R. 2004. Theorizing land cover and land use change:the case of tropical deforestation. International RegionalScience Review 27:247–270.

Walker, R., and T. E. Smith. 1993. Tropical deforestation andforest management under the system of concession logging: adecision-theoretic analysis. Journal of Regional Science33(3):387–419.

Walsh, S. J., R. E. Bilsborrow, J. J. McGregor, B. Frizzelle,J. P. Messina, W. K. Pan, K. A. Crews-Meyer, G. N. Taff,and F. Baquero. 2003. Integration of longitudinal surveys,remotes sensing time series, and spatial analysis. Pages 91–130 in J. Fox, R. R. Rindfuss, S. J. Walsh, and V. Mishra,editors. People and the environment: approaches for linkinghousehold and community surveys to remote sensing andGIS. Kluwer Academic Publishers, Boston, Massachusetts,USA.

Walsh, S. J., J. Messina, K. A. Crews-Meyer, R. E. Bilsborrow,and W. K. Pan. 2002. Characterizing and modeling patternsof deforestation and agricultural extensification in theEcuadorian Amazon. Pages 187–214 in S. J. Walsh andK. A. Crews-Meyer, editors. Linking people, place, andpolicy: a GIScience approach. Kluwer Academic Publishers,Boston, Massachusetts, USA.

West, D. B. 1996. Introduction to Graph Theory. Prentice-Hall,New Jersey.
