Characterizing connectivity relationships in freshwaters ...

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]

304 Landscape Ecol (2012) 27:303–317

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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

Landscape Ecol (2012) 27:303–317 305

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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

306 Landscape Ecol (2012) 27:303–317

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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

1

2

3

4

56

78

9

1011

1

5

2

49

1011

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

Landscape Ecol (2012) 27:303–317 307

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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)

308 Landscape Ecol (2012) 27:303–317

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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)

Landscape Ecol (2012) 27:303–317 309

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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

123


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

123


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

123


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

123


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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