characterizing connectivity relationships in freshwaters
TRANSCRIPT
RESEARCH ARTICLE
Characterizing connectivity relationships in freshwatersusing patch-based graphs
Tibor Er}os • Julian D. Olden • Robert S. Schick •
Denes Schmera • Marie-Josee Fortin
Received: 19 May 2011 / Accepted: 21 September 2011 / Published online: 4 October 2011
� Springer Science+Business Media B.V. 2011
Abstract Spatial graphs in landscape ecology and
conservation have emerged recently as a powerful
methodology to model patterns in the topology and
connectivity of habitat patches (structural connectiv-
ity) and the movement of genes, individuals or
populations among these patches (potential functional
connectivity). Most spatial graph’s applications to
date have been in the terrestrial realm, whereas the use
of spatially explicit graph-based methods in the
freshwater sciences has lagged far behind. Although
at first patch-based spatial graphs were not considered
suitable for representing the branching network of
riverine landscapes, here we argue that the application
of graphs can be a useful tool for quantifying habitat
connectivity of freshwater ecosystems. In this review
we provide an overview of the potential of patch-based
spatial graphs in freshwater ecology and conservation,
and present a conceptual framework for the topolog-
ical analysis of stream networks (i.e., riverscape
graphs) from a hierarchical patch-based context. By
highlighting the potential application of graph theory
in freshwater sciences we hope to illustrate the
generality of spatial network analyses in landscape
ecology and conservation.
Keywords Ecological networks � Spatial graphs �Graph theory � Stream network � Dendritic networks �Fragmentation
Introduction
Increased recognition that insight into complex sys-
tems can be gained from a network perspective has led
to the renaissance of a branch of mathematics called
graph theory (Harary 1969; Wasserman and Faust
1994; Aldous and Wilson 2000). In recent decades
scientists have acknowledged that ecological systems
can be represented as graphs or networks that contain
nodes depicting individual elements, and links repre-
senting relationships between the nodes (Bodin 2009).
Network or graph-based analyses [these terms are used
T. Er}os (&)
Balaton Limnological Research Institute of the Hungarian
Academy of Sciences, Klebelsberg K. u. 3, Tihany 8237,
Hungary
e-mail: [email protected]
J. D. Olden
School of Aquatic and Fishery Sciences,
University of Washington, Seattle, WA 98195, USA
R. S. Schick
Nicholas School of the Environment, Duke University,
Durham, NC 27708, USA
D. Schmera
Section of Conservation Biology, Department of
Environmental Sciences, University of Basel,
St. Johanns-Vorstadt 10, 4056 Basel, Switzerland
M.-J. Fortin
Department of Ecology and Evolutionary Biology,
University of Toronto, Toronto, ON M5S 3G5, Canada
123
Landscape Ecol (2012) 27:303–317
DOI 10.1007/s10980-011-9659-2
Author's personal copy
interchangeably following Jordan and Scheuring
(2004)] have been identified as an emerging tool for
studying ecological topology; defined here as the
collection of structures which give a mathematical
content to the intuitive notions of limit, continuity and
neighbourhood (see Bourbaki 1966 in Prager and
Reiners 2009). Traditional applications of graph-
based analyses in ecology have focused on modeling
species networks, such as predator–prey interactions
in food webs, plant-pollinator mutualistic relation-
ships or host-parasitoid webs (e.g., Proulx et al. 2005;
Bascompte et al. 2006; Jordan et al. 2006; Ings et al.
2009). Recently, however, spatially explicit graph-
based analyses have gained popularity in landscape
ecology and conservation biology (Urban and Keitt
2001; Garroway et al. 2008; Minor and Urban 2008;
Cumming et al. 2010; Dale and Fortin 2010; Galpern
et al. 2011). In these new applications, nodes can
represent species, assemblages, and habitat patches,
while links between these nodes can show the strength
of the interaction among species, the flow of energy,
and dispersal potential between populations/habitats
(Jordan et al. 2003, 2006; Urban et al. 2009).
Network analysis has now become an integral tool
in the ecologists’ toolbox, yet the application of
network techniques continues to vary substantially
among terrestrial and aquatic ecologists. This is
particularly true for patch-based spatial graphs—one
of the primary tools of terrestrial landscape ecologists
for modelling habitat connectivity. Here, spatial
graphs are defined as graphs in which the nodes have
locations and the links’ end points are defined by those
locations (Dale and Fortin 2010), whereas patch-based
graphs are spatial graphs that model the topological
structure of the landscape’s patches (Urban and Keitt
2001; Fall et al. 2007; Galpern et al. 2011). In patch-
based graphs, nodes represent habitat patches (at any
spatial scale), and links depict the functional connec-
tion between the nodes (Fall et al. 2007). Together the
nodes and links represent a patch-based graph of
interconnected habitat patches (Fig. 1). A graph is
connected if each node can be reached via some path
from any other node in the network, while an
unconnected graph consists of two or more compo-
nents (Urban et al. 2009; Rayfield et al. 2011; Fig. 1).
Since the terms and definitions of graph-based
Node
Link
Patch
Component
Fig. 1 Example of a simple terrestrial ‘‘lattice type’’ graph,
where the nodes indicate the habitat patches and the links
indicate direct connectivity relationships among the patches
based on e.g., population movement. This example graph
consists of two components, where a component is defined as a
subset of nodes where a path between all member nodes exists
[see Bodin (2009) for more details]
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analyses are used inconsistently in the ecological
literature, we use here the patch-based graph ‘‘con-
cept’’ in a broader context, which corresponds to the
habitat or landscape graph ‘‘concept’’ (i.e., habitat or
landscape graphs, see Urban et al. 2009; Bodin 2009;
and many examples in Galpern et al. 2011). We refer
the reader to the work of Urban and Keitt (2001),
Minor and Urban (2008), Bodin (2009), Urban et al.
(2009), Dale and Fortin (2010) and Galpern et al.
(2011) for basic terms, definitions, and useful infor-
mation about how to apply graphs in landscape
ecology and conservation.
Patch-based graphs have played a critical role in
terrestrial landscape ecology by enhancing our
knowledge of: (1) habitat patchiness of landscapes,
(2) habitat fragmentation, (3) the patterns of organ-
isms’ movements and home range extent, and (4)
ecological scaling issues, because networks can be
examined at different hierarchical levels and resolu-
tion (Wiens 1989; Turner 1989, 2005; Forman
1995). Patch-based spatial graphs are also useful
for quantifying the physical relations among land-
scape elements or patches (structural connectivity)
and how this topological structure affects the
movement of genes, individuals or populations
across the landscape (potential functional connectiv-
ity). They are often used to quantify the critical role
of individual habitat patches (nodes) or corridors
(links) in metapopulation dynamics (Jordan et al.
2003; Vasas et al. 2009; Galpern et al. 2011); and a
variety of indices are now available for evaluating
the different aspects of connectivity from this
viewpoint (Rayfield et al. 2011). Taken together,
graphs can help further ecological understanding of a
diverse array of ecological systems in a flexible,
iterative and exploratory manner with relatively little
data requirements (Urban and Keitt 2001; Calabrese
and Fagan 2004; Minor and Urban 2007; Zetterberg
et al. 2010). Patch-based graphs are especially ideal
for choosing among alternative plans in landscape
design and conservation (Anderson and Bodin 2009;
Vasas et al. 2009; Bodin and Saura 2010; Minor and
Lookingbill 2010).
Despite the fact that aquatic ecosystems may
represent a unique opportunity to test and advance
the use of patch-based graphs, to date these approaches
have only seen limited (but growing) use in research.
For example, the obvious patchy distribution of lentic
ecosystems (standing water bodies such as lakes and
ponds) at the landscape scale provides an excellent
opportunity for graph-based analyses of organisms
living within these habitats. Although, measuring
connectivity between seepage lakes is actually a
terrestrial landscape network problem (see below for
more details), we are aware of only a few studies that
have used a graph theoretic approach to model
ecological responses to the topological distribution
of lakes and ponds in the landscape, despite the
emphasized need (McAbendroth et al. 2005). For
example, Fortuna et al. (2006) examined the spatial
network of temporary ponds used as breeding sites for
amphibians, and using measures of graph theory they
modelled changes in the availability of suitable
breeding habitats in response to drought. Their study
showed that droughts not only reduced the number of
ponds suitable for reproduction and the recruitment of
tadpoles, but also changed the structural (topological)
properties of the pond networks with implications for
amphibian persistence. Similar applications of graph
theory have emphasised the importance of structural
landscape connectivity on the species richness pattern
of amphibian assemblages (Riberio et al. 2011) or on
the structure of pond turtle metapopulations (Pereira
et al. 2011). Examples from the marine sciences
have also demonstrated the importance of the net-
work approach for quantifying connectivity relation-
ships of habitats/populations (see e.g., Treml et al.
2008; Kininmonth et al. 2010; Jacobi and Jonsson
2011).
Lotic (running waters) ecosystems are seemingly
more problematic for patch-based graph analyses.
These systems were traditionally considered as linear
systems, where continuous changes in patterns and
processes could be observed along the longitudinal
profile of large river systems from source to mouth
(Vannote et al. 1980). Although this simplification
contributed substantially to our understanding of the
structure and functioning of many aspects of lotic
ecosystems, the predictions were often limited or even
failed. The most important of these predictions was the
effect of the hierarchical, patchy organization of
geomorphological patterns and processes on the biota
(Frissell et al. 1986; Poole 2002). Shifting away from
the linear system framework, in recent years, ecolog-
ical processes in running waters have been examined
from a network based context (Benda et al. 2004;
Grant et al. 2007). The network dynamics hypothesis
(Benda et al. 2004) emphasizes the importance of
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dendritic branching, the effects of tributaries on
mainstem hydrological and geomorphological pro-
cesses, and the combined effect of these on ecological
patterns and processes (e.g., flow of genes and material
and energy, movement and habitat use of species,
assemblage organization). Clearly, the importance of
such network level effects depend largely on the
topological characteristics of the tributaries (e.g., size,
abiotic features, number of inflows within a given
segment). Although graph-based analyses is a very
promising tool for better understanding the impact of
network topology on instream patterns and processes,
the number of such applications is still limited and the
full breadth of opportunities have not been considered.
Grant et al. (2007) posited that dendritic ecological
networks are different from typical ‘‘lattice type’’
ecological networks (i.e., nodes and links as illustrated
in Fig. 1), which can be modelled using patch-based
graphs. They argued that in dendritic networks both
the nodes and the links share ecological processes,
while in a more traditional ecological network, the
nodes are where the processes occur, and the links
support the transport of material and energy. They call
the need for ‘‘the development of a general conceptual
framework that encompasses both dendritic and lattice
networks’’. They also suggested, that ‘‘borrowing (and
modifying) components of lattice network theory may
prove a fruitful starting point’’.
We believe that the application of patch-based
graphs can be a useful tool for modelling stream
networks, because the patch-based concept is inherent
not only in terrestrial ecology, but also in stream
ecology (Townsend 1989; Schlosser 1991; Wiens
2002; Winemiller et al. 2010). Here we address the
research disparity between terrestrial and aquatic
systems by using riverine ecosystems (termed the
riverscape) to show how the graph-based approach can
be used in freshwater ecology and conservation.
Specifically, we develop a conceptual framework for
the topological analysis of stream networks (i.e.,
riverscape graphs), from a hierarchical patch-based
viewpoint. We provide practical examples of con-
structing graphs (i.e., nodes, links, weightings) that
both embrace the spatial complexity of these systems,
and allow for testing of basic and applied ecological
questions. We also summarize the application of graph
theoretic approaches to freshwater ecosystems to
further highlight the potential of this useful modelling
tool. It is our intention to advance our understanding of
the general (i.e., system independent) applicability of
network-based analyses in landscape ecology and
conservation.
Riverscape graphs: a patch-based approach
Graphs are models of real systems and as such their
advantage is to simplify both the visualization and
interpretation of ecological patterns and processes.
Therefore, graph models are flexible and can be
adjusted to the question of interest and can be
improved as new information on the real system
becomes available. The first step in viewing riverine
ecosystems as graphs is to define the nodes and links of
the graph. For this purpose, three scaling issues are
critically important: (1) the level of the spatial habitat
hierarchy at which the analyses are conducted; (2) the
resolution of the graph (i.e., the number and type of
‘patches’ that the nodes symbolize); and (3) the grain
at which the studied organisms perceive their envi-
ronment. In this work, nodes are defined as delineated
habitat patches, whereas links are elements that
comprise no habitat area but represent the possibility
of dispersal between two habitat patches (Saura and
Rubio 2010).
For the application of patch-based graphs in lotic
systems, the hierarchical habitat classification scheme
of Frissell et al. (1986) and riverscape paradigm
presented by Fausch et al. (2002) provide a logical
starting point. Riverscape graph models can be
depicted at the scale of the watershed or stream
network (105–103 m), segment (104–102 m), reach
(102–101 m), channel unit or riffle/pool (101–100 m)
and finally the microhabitat level (10-1 m). Note, that
the meanings of the terms micro, meso and macro-
habitat vary in the literature and may also depend on
how the study organism perceives and interacts with
its environment (Olden et al. 2004a). Consequently,
there is a context-dependent element that is unavoid-
able when using a hierarchical patch dynamic per-
spective for riverine systems.
From a topological viewpoint, the most detailed
graphs can be assembled at the microhabitat patch
level (Fig. 2). This is the level of the finest study grain
for any graph-based analyses, which cannot be
decomposed to smaller habitat elements with more
detailed analyses. Examples of this include ‘patches’
of periphyton on substrates, large woody debris, leaf
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packs in sand-silt habitats or gravel accumulated
patches at confluences that represent habitat for
macroinvertebrate and other benthic species (Lancas-
ter 1999; Palmer et al. 2000; Olden et al. 2004b).
These patches do not extend from bank to bank in most
cases, and can be scattered in the study area hetero-
geneously, similarly to vegetation patches in a terres-
trial landscape. Therefore, graphs constructed at this
level reflect features of the ‘conventional’ lattice type
graph, where the nodes symbolize two-dimensional
habitat patches. Here, animals may move from one
patch to another through a hostile or unfavourable
(i.e., patches with different quality) ‘‘landscape’’.
Further, depending on the size of the river system,
the spatial extent of such two-dimensional patches
may not necessarily relate to the ‘‘micro’’ scale. In
large rivers, for example, habitat patches for spawn-
ing, feeding or refuge areas may extend from some
meters to several hundred meters long and are large
enough for holding individual subpopulations or
patchy populations of fish (Schlosser 1991; Le Pichon
et al. 2006).
Riffle–pool level patches are built up of microhab-
itat patches of different number and type (Fig. 2). At
this level, individual channel units (e.g., riffle, pool,
run) are often delineated during the stream survey
(Er}os and Grossman 2005) and these units can be
considered as nodes for constructing the spatial graph.
The size and availability of riffle- and pool-level
patches can change drastically in time due to floods or
droughts, which can influence both population struc-
ture and assemblage organization of the biota, espe-
cially fish (Magoulick and Kobza 2003; Matthews and
Marsh-Matthews 2003). Therefore, graph analyses
conducted at this level could be of high conservational
importance for modelling the distribution and dynam-
ics of riffle- or pool-specialist species (Martin-Smith
1998). These dynamics could be a function of the
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3
4
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678 3
(a)
(b)
R1
R2R3
R4
R1: Natural section
R4: Vegetated section
Pool
R3: Reservoir
R2: Channelized section
Riffle
Riffle
Run
R4
R3
R2
R1
Pool
Riffle
Riffle
Run
Shallow habitat patch with sandy bottom and
low water velocity
Segment level nodes in the
stream network
Reach level nodes in the
segment
Riffle/pool level nodes in the
reach
Micro habitat level nodes in the
riffle/pool units
Subcatchment
Fig. 2 Hierarchical, patch
based view of lotic
ecosystems (a) and its
graph-based representation
(b). Numbers indicate
segments in the stream
network. Codes R1, R2, R3,
R4 represent different reach
level units. Riffle–pool level
units are indicated in R1
reach. Patches, delineated at
the microhabitat level are
indicated with different
‘textures’. For simplicity,
the graphs are simple
unweighted graphs, where
the nodes indicate the
habitat patches and the links
indicate binary adjacencies
(connected or not
relationships) among the
patches
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topological characteristics of riffle–pool patches (e.g.,
size of and distance between riffle patches and their
average abiotic characteristics) and changes in their
temporal availability.
Reaches and segments are usually considered as
one-dimensional longitudinal habitats in constrained
stream networks (Ward 1998) (Fig. 2); the reaches and
segments are distinguished by their hydrologic (e.g.,
flow, depth), geomorphologic (substrate composition)
or other abiotic characteristics (e.g. vegetation type
and coverage). Stream systems at this resolution form
the classical dendritic network structure, which is
frequently used for modelling hydrology (Rodriguez-
Iturbe and Rinaldo 2001; Paik and Kumar 2008),
population dynamics (i.e., demographic) (Fagan 2002;
Labonne et al. 2008), and very recently metacommu-
nity level processes (Auerbach and Poff 2011). Note,
that this does not mean that such graphs could not be
modified for incorporating lateral dimensions. For
example, additional nodes may be added to the graph
model for studying the network flow in large braided
(i.e., alluvial) river systems, where, beside the main
channel, alternative side arms connect upstream–
downstream reaches (Fig. 3). Dead arms and oxbow
lakes, situated in the floodplain, also form a significant
part of the riverscape at least in relatively natural
alluvial systems (Ward 1998). Although depicting
graph models for such systems may be challenging,
their application could facilitate conservation plan-
ning efforts. For example, the main channel, the side
arms and the associated oxbows are often utilized for a
variety of purposes (e.g., recreation, fishing, shipping,
flood control), some of which can have negative
ecological implications (Tockner and Schiemer 1997;
Jungwirth et al. 2002). Processes related to organismal
movement are especially critical for migrating fish
populations, which, ideally, use the side arms for
spawning, feeding and refuge from high flow condi-
tions (Schlosser 1991). Using graph models, we
envision that several alternative scenarios including
ephemeral links could be depicted and discussed
among stakeholder groups to optimize conservation
planning exercises.
Generally, the watershed or stream network levels
may be too coarse for the purpose of graph-based
analysis. Nevertheless, in very large river systems,
using the nodes to represent subcatchments, network
analyses can be fruitfully employed to better realize
network structure (Schick and Lindley 2007). Further,
by considering their connectivity relationships and
topological position at the regional scale, graph
indices (Rayfield et al. 2011) can help the selection
of individual subcatchments for conservation pur-
poses. This process can yield more effective conser-
vation area network designs.
Until now we have considered habitat patches (e.g.,
reaches, segments) as nodes, with stream junctions
(e.g., confluences) as links in assembling riverscape
graphs. There are, however, applications when it
might be useful to build the graph in the opposite way,
that is to have junctions as nodes and main channel
habitats as links. Such an approach is more often used
in the hydrological modelling of freshwaters. For
Rip-rap embankment
Channel creation
(a)
(b)
Fig. 3 Structural connectivity of habitats in a hypothetical,
large, floodplain river section. a The natural hydrosystem with
eu, para, plesio, and paleopotamal structure (see Tockner and
Schiemer 1997) and its graph-based representation. b The
modified system and its graph-based representation. Blackblocks indicate rip-rap embankments and are built for shoreline
protection and for enhancing navigation. Fish symbols indicate
the creation of new channels, which were built to connect
formerly separated water bodies in order to improve recreational
facilities (e.g., fishing, water sports). Note, that depending on
study purpose alternative graph forms are also possible (see
dashed lines for an example)
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example, Poulter et al. (2008) used network indices to
model hydrologic flowpath of a large artificial drain-
age system composed of canals, ditches, and streams
in coastal North Carolina, USA. They demonstrated
that simple indices, like the number of components,
the number of nodes and links, betweenness centrality,
etc. proved to be very useful to characterize and
visualize a drainage system, and identified nodes
(junctions, inflow and outflow points, water control
structures) that played critical role in the movement of
water in and out of the network.
In a similar study, a channel network design was
used to examine water flow in running water (streams
and rivers) and standing water (reservoirs, lakes and
wetland) habitats in the Xiaoqinghe River Basin,
China (Cui et al. 2009). Flow paths were optimized by
adding new nodes (junctions between newly added,
hypothetical and real water bodies) to the network or
deleting existing links. Overall, with graph-based
methods, it was easy to design and identify major flow
paths during floods and droughts, thereby providing
important management implications to mitigate river
risk, while still maintaining the river system in a
relatively natural state. In fact, hydrologists and
geomorphologists now routinely use different
watershed models to simulate hydrologic and biogeo-
chemical processes (Liu and Weller 2008), but the
more intensive application of graph-based approaches
has been applied only recently (Kent 2009).
The modelling of hydrologic flowpaths in time and
space can help ecologists better understand structural
connectivity, which in turn is necessary for formulating
hypotheses about potential functional connectivity
(Taylor et al. 1993). For example, knowledge of
hydrologic relationships and water movement is
needed to model the spread of invasive species in
canalized landscapes, where hydrologic flowpaths are
determined according to human needs (e.g., irrigation,
drought and flood control) (Cowley et al. 2007).
Furthermore, graph-based approaches can be used to
bridge terrestrial and aquatic networks by constructing
a graph (e.g., graph of subcatchments) of graphs (graph
of streams) (Dale and Fortin 2010). Hence the increas-
ing use of graph-based approaches can contribute to
enhanced collaborations between ecologists and hy-
drologists/fluvial geomorphologists to understand the
structure and functioning of freshwater ecosystems.
Overall, we propose that both the topology of habitat
patches and ecological data across a hierarchy of spatial
scales may be modelled successfully with graphs in
freshwater systems. Lattice-type graphs can be applied
when habitat patches can be reasonably delineated
based on geomorphic principles and/or related to
animal movement. In this case, several paths may be
identified between two distinct patches (nodes), due to
the two-dimensional structure of the patchy landscape
(Dale and Fortin 2010). Classical dendritic stream
networks may be modelled using a special type of
graphs called trees. By definition, a tree is an undirected
graph in which any two nodes are connected by exactly
one simple path. These graphs mimic the topological
structure of the stream network itself and emphasize
longitudinal (upstream–downstream) connectivity
relations in the system. In the simplest cases stream
networks can be described using minimum spanning
trees (see Er}os et al. 2011), which are used in terrestrial
landscape ecology to characterize the backbone of the
network, i.e., the links that connect all nodes with
minimum total link length (Urban and Keitt 2001; Fall
et al. 2007; Galpern et al. 2011). Below, we provide
introductory examples of how to construct and weight
riverscape graphs, focusing our examples to the level of
reach and segment habitat.
Weighting riverscape graphs
There are four alternative ways to build graph-based
models (Proulx et al. 2005) that are readily applicable
to riverine ecosystems. We stress that the choice of
method depends on the scale, the ecological question
of interest, and data availability. The first, perhaps
simplest, approach is the case where nodes and links
are unweighted. For these graphs the connection (i.e.,
links) between the habitat patches (nodes) are indi-
cated in a binary fashion and no qualitative or
quantitative differences between the patches are
considered (Fig. 4a; Model type I). Note, that even
this kind of relatively simple depiction can provide
considerable insight into the study system and indicate
a starting point for refining the ecological or conser-
vation aspects of the model with weighting nodes,
links, or both (Urban et al. 2009; Bodin 2009). For
example, in a recent study, Er}os et al. (2011) examined
the topological importance of stream segments in
maintaining connectivity in the stream network of the
Zagyva basin (Hungary), which is fragmented by a
series of large reservoir dams. Er}os et al. (2011)
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represented individual segments as unweighted nodes
and confluences as unweighted links in their spatial
graph model. Dams built on the segments were also
indicated and resulted in the segments being divided
into two or more subsegments (or reaches), thereby
increasing the number of nodes (but not of the links) in
the model. By characterizing the system by the
number of graph components (for definition see
above), the authors showed that the graph model of
the fragmented riverscape, i.e., dammed, consisted of
16 components (or subgraphs) compared with the
graph model of the system before reservoir construc-
tions. Prior to the addition of dams, the whole system
was traversable and the system consisted of a single
component. Er}os et al. (2011) used then an index
adapted from the study of sociobiological networks,
called betweenness centrality, to quantify the posi-
tional importance of individual segments in the stream
network from the viewpoint of connectivity (Fig. 5).
This study demonstrated that indices used in network
analysis—even with relatively simple unweighted
graphs—can provide important insight into patterns
and degrees of fragmentation and can help identify and
rank critical segments for network traffic in riverine
systems. Such analyses can help facilitate optimal
placement of reservoirs or conversely, the removal of
dams, such that the placement minimally impacts the
connectivity of the system, or the selective removal
maximally improves the connectivity of the system,
respectively. Other restoration efforts aim to increase
the permeability of riverine landscape by modifying
fish passage structure for dams and road culverts (Roni
et al. 2008) with the goal of minimizing the damage to
or loss of connectivity diversity.
A second, more data extensive approach includes
the weighting of connectivity intensity between
patches, which is depicted with arrows of different
lengths or thickness depending on the strength of the
inter-patch links (Fig. 4b; Model type II). For exam-
ple, small, artificial constructions (e.g., roads, weirs,
dykes) may serve as a semi-permeable barrier to
organism movements depending on the particular
species and seasonal changes in the water regime or
obstacle type (Warren 1998; Kerby et al. 2005;
Alexandre and Almeida 2010). Considering inter-
patch links with different weights in different seasons
may yield a more realistic understanding of connec-
tivity than the simple modelling with binary links
(presence/absence). Further, the direction of the
arrows can have strong implicative power. For exam-
ple, upstream versus downstream movement of stream
organisms can differ substantially, and incorporating
such variables (e.g., Olden et al. 2001) in models of
functional connectivity may increase prediction.
Distinguishing between different types of nodes
gives a third way for building more sophisticated
models (Fig. 4c; Model type III). This can be accom-
plished, for instance, by weighting nodes according to
habitat size (i.e., river length or lake area) or according
to the quality of the habitat patch (e.g., based on
population size or reproductive output for population
level analysis, or quantifying conservation value of the
habitat for community level analysis) (Schick and
Lindley 2007). Er}os et al. (2011) used a habitat
availability index, called integral index of connectiv-
ity (Pascual-Hortal and Saura 2006; Saura and Rubio
2010; Bodin and Saura 2010) for selecting and ranking
(b)(a)
(d)(c)
Dam
Model type I Model type II
Model type III Model type IV
Link resolutionHomogeneous Heterogeneous
Nod
e re
solu
tion H
omog
eneo
usH
eter
ogen
eous
Fig. 4 Graph-based representation of a hypothetical stream
network at the segment level and at different levels of network
resolution following the typology of Proulx et al. (2005). Opencircles indicate nodes, whereas lines (or arrows) indicate links.
For example, in a the presence–absence based connectivity
relationships among the segments are shown yielding an
unweighted graph. b Depicting the strength of migration
pathways for a fish population leads to a directed graph, where
the emphasis is on links. c Weighting nodes leads to another type
of graph where for example the size of the populations is
indicated with the area of the circles. d Weighting of both nodes
and edges leads to the most realistic models, where for example
both population size and the strength of migration among the
different patches are indicated
310 Landscape Ecol (2012) 27:303–317
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potential stream segments (nodes) for conservation
based on habitat area, the conservation value of fish
assemblages and connectivity relationships of the
segments in a graph-based model. The authors found
that the integral index ranked segments differently
than the ranking scheme that considered simply the
conservation value of the segments. By considering
connectivity with the help of network based
approaches, Er}os et al. (2011) could better visualize
alternative ways of prioritizing habitats for conserva-
tion. This in turn may be more useful for the long term
persistence of fish assemblage integrity than evalua-
tions considering conservation value only.
Finally, a fourth approach includes weighting of
both nodes and links (Barrat et al. 2004; Drake et al.
2010) and is perhaps the most realistic representation
of natural systems (Fig. 4d; Model type IV). Such
weighted graphs are commonly used in metapopula-
tion studies. Unfortunately, obtaining the input infor-
mation for such models is difficult, especially at large
spatial extent. Schick and Lindley (2007) applied
graph theory to characterize the historical and present
day spatial structure of an endangered anadromous
fish species, to estimate the degree of connectivity
of current populations, and to provide a plausible
mechanism for ecologically successful restoration
(Fig. 6). The nodes in this example represented entire
sub-populations of Chinook salmon (Oncorhynchus
tshawytscha), and were weighted based on the histor-
ical documented extent of habitat. The link adjoining
two nodes was based on the size of each population,
the watercourse distance between them, and the
likelihood fish would stray between them. The authors
showed that despite the inherent treelike structure of
the system, network analysis of running waters (i.e.,
streams and rivers) can reveal surprisingly interesting
information that can inform species conservation
efforts. Note, that although we detail the work of
Schick and Lindley (2007) here, due to its relevance,
their study used a slightly different graph from the
ones mentioned previously. The graph was a quasi-
spatial graph in that the distances between nodes were
based on actual as-the-fish-swims distance between
nodes. However the links depicted in the graph
(Fig. 6) were based on a combination of inter-node
distance, node size, and likelihood of fish from one
node straying to another node. That is, the authors did
not model the topological structure of the dendritic
stream landscape, per se, but rather the dispersal
possibilities between subpopulations (potential
0 20 km
(a) (b)
Fig. 5 a Schematic map of the Zagyva basin, Hungary.
Reservoirs, built on the main channel, which prevent natural
flow of ecological processes are indicated with a half-moon
shape symbol. b Graph-based representation of the Zagyva
riverscape using betweenness centrality. The area of the circles
is proportional to index values. Nodes (segments) of the biggest
component are indicated with open circles, whereas grey circlesindicate other components with more than one element. Filledblack circles show single-node components, which do not have
any connection to other elements of the riverscape. See Er}os
et al. (2011) for further details
Landscape Ecol (2012) 27:303–317 311
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functional connectivity). Indeed given the inherent
structure of dendritic stream landscape, the potential
functional connectivity based on the a spatial graph
captures both the passive (due to the directional flow
of the water) and active dispersal between subpopu-
lations. Using a spatial-graph approach to model the
least-cost path between lakes based on road network,
Drake et al. (2010) used a gravity model to analyse
both the weight of the nodes and links between them.
Such a methodology could be easily expended to
stream network studies to stress both the weights of
the nodes and the links.
Our intent in this review is to argue for the general
applicability of graph based analyses in freshwater
ecology and conservation. Further studies should
address how different phases of graph construction
[e.g., choosing node and link geometry and represen-
tation, weights, graph assembling; see Galpern et al.
(2011) for details] influence modelling structural and/
or functional connectivity. In fact, recent studies on
Fig. 6 Schematic that
shows the progressive
fragmentation of the
network of Chinook salmon
populations in the
Sacramento River,
California, USA. The four
panels depict the addition of
dams to certain rivers in
specific years, and the
accompanying change in the
graph. a Englebright Dam
on the Yuba River (node 18,
1941), b Shasta Dam on the
Sacramento River (nodes
5–7, 1945), c Nimbus Dam
on the American River
(node 19–21, 1955) and
d Oroville Dam on the
Feather River (nodes 14–17,
1968). Nodes are sized
according to the estimated
amount of historical habitat
in the watershed. Links are
weighted based on the
strength of connection
between populations. As the
graph fragments through
time, grey nodes depict
extinct populations
(reproduced, with
permission, from Schick and
Lindley 2007)
312 Landscape Ecol (2012) 27:303–317
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stream fish populations clearly show that potential
functional connectivity can be successfully modelled
with graph-based techniques to terrestrial examples
(Schick and Lindley 2007; Fullerton et al. 2011), even
if the topological structure of stream networks differ
substantially from terrestrial habitats. The most
important is that the movement of organisms in stream
networks is largely constrained by this same dendritic
structure (see also Grant et al. 2007), at least for those
organisms that are confined to the stream for move-
ment (e.g., fish). In dendritic stream networks the
actual route of movement is linear, without alternative
ways for moving from one point to the other, which
makes these systems exceptionally vulnerable to
fragmentation effects (Fig. 2 and the argument in
Model type I). Quantification of connectivity relation-
ships should be therefore one of the key issues in
stream ecology and conservation, at least for migrating
fish populations (Fullerton et al. 2010).
Further, the direction of flow can largely determine
movement patterns in stream networks. Although wind
generated movement occurs regularly for terrestrial
insects, and can constrain migration possibilities,
water current is always uni-directional. Some aquatic
organisms can move against the flow without major
difficulties (e.g., fish, although movement is taxon
specific even within this group), while others must
follow flow direction (e.g., passively moving phyto-
plankton). This feature of stream systems influences
the modelling of habitat availability (reachability) and
connectedness, especially for more detailed graphs,
where links are directed and weighted. Therefore,
results from directed graphs may give quite different
results than graphs that are based on simple undirected
links.
Finally, although stream ecosystems can be charac-
terized from a hierarchical patch-based view, patch
boundaries can be obscured in some cases and
hydrological features (e.g., water level, flow) also can
influence the delineation of habitat patches or (patchy)
populations. If the environment is perceived to be
rather homogenous, continuous populations may be
observed, which may lead to difficulties in modelling
connectivity relationships. Nevertheless, organisms
are usually sampled using ‘‘point-based’’ sampling
data and with the careful consideration of scale these
data can be used in a patch-based context for spatial
network analyses (Dunn and Majer 2007). Grid or
raster based occupancy data are also increasingly used
in terrestrial landscape ecology for graph-based anal-
yses (Pascual-Hortal and Saura 2008; Baranyi et al.
2011), and their application for aquatic organisms
should deserve attention in the future, especially for
fish where detailed data are available and used for
conservation purposes (see Strecker et al. 2011).
A special case: stream–lake networks
Lakes and streams are usually well separated habitats
in the landscape and both stream ecologists and
limnologists have largely ignored the relationship
between these landscape elements (Jones 2010).
However, lakes can be often interconnected with
streams or more precisely, embedded in the stream
network, especially in boreal regions. Therefore, one
has to make distinction between seepage (no lake
outlet) and drainage lakes (connected via stream
network). As we mentioned earlier, measurement of
structural connectivity using graphs proved to be
fruitful to seepage lakes as a proxy for potential
functional connectivity (Fortuna et al. 2006; Riberio
et al. 2011). Here, in the same way one assembles a
terrestrial patch based graph, information about move-
ment habits and dispersal mechanisms, coupled with
between lake distances and other lake features, can be
applied to model potential functional connectivity. In
stream–lake networks of drainage lakes the graph-
based depiction of lake–stream relationships can
provide the habitat template for analysing ecological
flows. If the focus is on lakes the depiction of lakes as
nodes and interconnected streams as links can provide
the base for graph analyses (Olden et al. 2001; Jones
2010). Olden et al. (2001) examined the association
between fish community composition and isolation-
and environment-related factors in 52 drainage lakes in
Ontario, Canada. They provided a way of calculating
link strength based on passive downstream movement
(downstream distance only used in pairwise water-
course distances between lakes) and both active and
passive movement (total watercourse distance). In
addition, elevational change (i.e., channel slope) was
also incorporated in their model to reflect greater
resistance to movement due to rapids and greater
energetic expenditures for organisms. Their analyses
revealed high concordance between patterns in fish
community composition (node attribute) and lake
isolation and environmental conditions.
Landscape Ecol (2012) 27:303–317 313
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Other applications
Although not strongly related to graph-theoretical
analyses, some very promising habitat availability
indices have also been developed very recently for
quantifying connectivity in dendritic networks. Such a
simple and elegant index is the ‘dendritic connectivity
index’. The index is a global index that quantifies the
overall connectivity (permeability) of the stream
network based on the expected probability of an
organism being able to move between two randomly
selected points in the network (Cote et al. 2009). If the
whole system is traversable (there is not any barrier)
the index reaches its maximum value of 100. The
index has two forms, one developed for potadromous
(those that migrate within the freshwater system) and
the other for diadromous species (which migrate
between marine and freshwater habitats). It considers
the stream network as a one-dimensional chain of
habitat patches, where there is only one path to travel
from one place to another [see above and also Labonne
et al. (2008)]. In such a system, it is obvious that the
first few barriers cause the largest changes in land-
scape permeability and the index shows a curvilinear
decline with the number of barriers. However, this is in
sharp contrast with lattice type (terrestrial) networks,
where alternative routes may maintain connectivity
among the patches, providing higher resilience of
these networks to multiple fragmentation events.
Another approach uses transition probability matrices
to model habitat availability between every pair of
reaches (segments) within stream networks (Padgham
and Webb 2010). An advantage of the model is that
changes in both the quality and connectivity attributes
of the reaches can be modified and their effects on the
availability of any other reaches in the system can be
quantified similarly to what has been done in terrestrial
studies (Pascual-Hortal and Saura 2006; Saura and
Pascual-Hortal 2007; Bodin and Saura 2010).
Conclusion
Our review highlights the exciting opportunity for the
application of graph-based approaches to freshwater
ecology and conservation. We have argued that the
graph-based modelling of structural and potential
functional connectivity can follow the same patch-
based scheme in both terrestrial and aquatic
ecosystems. In light of the significant progress
integrating graph-based approaches into freshwater
ecology, we stress the need for additional research in
five key areas to: (1) improve delineation of patches at
different levels of the riverscape hierarchy, (2) model
patchiness and fragmentation effects in relation to
hydrologic changes, (3) sample detailed data on the
movement patterns of different taxa and individual
species in relation to spatio-temporal habitat hetero-
geneity and fragmentation, (4) investigate the effects
of alternative graph constructions on modelling out-
comes, and (5) examine of the performance of
different graph-based and habitat availability indices.
We are aware that most of these points require
extensive field surveys (see e.g., Fausch et al. 2002;
Torgersen et al. 2008), but we also think that this
approach will readily facilitate a detailed understand-
ing of the structure and function of riverscapes at large
spatial extents.
Acknowledgments The work of TE was supported by the
Janos Bolyai Research Scholarship of the Hungarian Academy
of sciences and by the OTKA PD 77684 research fund. JDO
acknowledges funding support from the USGS Status and
Trends Program, the USGS National Gap Analysis Program and
the Environmental Protection Agency Science’s To Achieve
Results (STAR) Program (Grant No. 833834). MJF acknowl-
edges funding support from NSERC Discovery grant. We are
grateful to the anonymous referees for their very constructive
reviews of the manuscript.
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