Jordán, F. 2005. Topological key players in communities: the network
perspective. In: Tiezzi, E., Brebbia, C. A., Jörgensen, S. and Almorza Gomar, D.
(eds.), Ecosystems and Sustainable Development V, WIT Press, Southampton,
pp. 87-96.
Topological key players in communities: the
network perspective
Ferenc Jordán
Collegium Budapest, Institute for Advanced Study, Budapest, Hungary
Abstract
For both a general understanding of ecosystem functioning and setting
conservation priorities it would be very important to have quantitative methods
for quantifying the relative importance of species. We present an approach from
the network perspective on ecology. Topological keystone species are those being
in critically important positions within the interaction network of an ecological
community. Graph theory makes the quantification of positional importance of
nodes within networks possible. There are many ways how to measure this and,
in particular, we use some network indices recently introduced in social network
analysis. These are suitable for the characterisation of both direct and indirect
interactions. The key players of a hypothetical ecological community will be
identified and the results of different methods will be compared. We emphasise
that the quantitative, comparative approach to species importance is urgent and
very important to outline for effective conservation efforts.
Keywords: keystone species, network perspective, graph theory, trophic field,
indirect effects.
1 Introduction
The two main traditional strategies of conservation practice are to protect habitats
and to save endangered species. Both have theoretical and practical difficulties
and a possible third way could be to protect ecosystem functioning. This is easy
to say but more abstract to imagine in everyday practice, or harder even to define.
The recent claim for more objective and more quantitative studies in identifying
keystone species [1] is a perspective offering a more functional approach to
conservation biology. The characterisation and understanding of keystone,
flagship, umbrella and other kinds of “important” species may be a bridge
between habitat- and species-centric conservation plans [2].
However, the relative importance of species is not easy to define or quantify.
Many ecologists intuitively agree that the importance of a species probably
reflects the secondary effects of its local extinction (or removal). Through direct
effects and indirect pathways of interspecific interactions, local extinctions can
trigger secondary extinctions or, at least, may seriously influence other coexisting
populations. If we accept the view that importance is a measure of triggering
cascades through interspecific interactions, we might quantify important species
by their “centrality” or positional importance within a community interaction
network.
We present conceptual and methodological developments in outlining this
quantitative view on species importance. Graph theory, as the basis of ecological
network analysis helps in mapping the topology of a hypothetical interaction
network. Species in critically important positions will be called topological
keystone species [3] and their suitably defined topological interaction strength
will be called their trophic field [4]. We suggest that developing the field theory
of biology [5] could be useful in this area since understanding the relationship
between species and ecosystems is a very topological problem, where the strength
and the effective range [6] of spreading interactions is a cornerstone. We illustrate
the application of some old and some new network indices in the quest for
keystones in ecological networks.
2 Methods
2.1 The problem to be solved
We are interested in quantifying topological keystone species in the interaction
network of a hypothetical community. The graph is shown in Figures 1 and 2.
There are 34 “species” in this community and our concern is to rank them
according to the relative importance of their network position. A couple of
indices will be presented, the ranks for seven indices will be shown (see Table 1)
and it will be shortly discussed under what circumstances they can be used
suitably.
2.2 Degree (D) – the number of neighbours
The index that is most local but least informative about the topology of a network
is the degree of a node (D). This is the number of other nodes connected directly
to it. In a food web, the degree of a node i (Di) is the sum of its prey (in-degree,
Din,i) and predators (out-degree, Dout,i):
Di  Din,i  Dout,i .
(1)
We note that the degree of a node is called its degree centrality in the sociological
literature [7]. In ecology, several analyses on topological key species have been
focusing on the number of neighbours in food webs, i.e. the degree of nodes in
trophic networks [8, 9, 10, 11]. Since there are indirect effects spreading beyond
the neighbours, degree gives only local information about a graph node. Social
and ecological interaction networks are similar in many aspects and there are
common technical roots partly presented in the followings.
2.3 Keystone index (K) – vertical interaction structure
Degree considers only the links directly connected to a node. We also consider
network indices reflecting also short indirect effects, i.e. the neighbours of
neighbours. We call these indices mesoscale indices, in contrast to the local
nature of degree, and to the global nature of some indices to be presented later
(defined on and characterising the whole network). The keystone index (K, see
[3]) is derived predominantly from the application and modification of the ”net
status” index introduced in sociometry [12] but used also in ecology [13]. The
keystone index of species i (Ki) is defined as:
n
m
1
1
(1  K bc )   (1  K te ) ,
c 1 d c
e 1 f e
K i  K bu,i  K td ,i  
(2)
where n is the number of predators eating species i, dc is the number of prey of its
cth predator and Kbc is the bottom-up keystone index of the cth predator. And
symmetrically, m is the number of prey eaten by species i, fe is the number of
predators of its eth prey and Kte is the top-down keystone index of the eth prey. For
node i, the first sum in the equation (i.e. 1/dc(1+Kbc)) quantifies the bottom-up
effect (Kbu,i) while the second sum (i.e. 1/fe(1+Kte)) quantifies the top-down
effect (Ktd,i). After rearranging the equation, terms including Kbc and Kte (i.e.
Kbc/dc + Kte/fe) refer to indirect effects for node i (Kindir,i), while terms not
containing Kbc and Kte (i.e. 1/dc + 1/fe) refer to direct ones (Kdir,i). Both Kbu,i +
Ktd,i and Kindir,i + Kdir,i equals Ki. The keystone index emphasises vertical over
horizontal interactions (e.g. trophic cascades as opposed to apparent competition).
This index “measures” the trophic field of a species [4] in vertical direction and
has been applied several times in network analysis [6]. Its important feature is the
sensitivity to both distance and degree: it quantifies position at an intermediate
scale rather than giving very local or very global information [14]. We calculated
the keystone indices of trophic groups by the FLKS 1.1 programme (available on
request).
2.4 Topological importance index (TI) – long, undirected effects
An index not biased for vertical interactions, i.e. taking into account also
exploitative and apparent competition, is called topological importance index
(TI). We use it for characterising long indirect effects [cf. 15] and the index itself
is the extension of an earlier one proposed for the analysis of two-steps long,
horizontal, apparent competition interactions in weighted host-parasitoid
networks [16]. This index does not consider directed and signed interactions.
Various indirect effects do spread in both bottom-up and top-down directions
through trophic links and, as a result, horizontally, too. If non-trophic interactions
are also considered, the network typically has direct horizontal links, too. In this
case, trophically mediated indirect chain effects [17] as well as chemically and
behaviourally mediated indirect effects are considered [18, 19]. In an unweighted
network, we define an,ij as the effect of j on i when i can be reached from j in n
steps. The simplest mode of calculating an,ij is when n=1 (i.e. the effect of j on i in
1 step): a1,ij = 1/Di, where Di is the degree of node i (i.e. the number of its direct
neighbours including both prey or predatory species). We assume that indirect
chain effects are multiplicative and additive. For instance, we wish to determine
the effect of j on i in 2 steps, and there are two such 2-step pathways from j to i:
one is through k and the other is through h. The effects of j on i through k is
defined as the product of two direct effects (i.e. a1,kj×a1,ik), therefore the term
multiplicative. Similarly, the effect of j on i through h equals to a1,hj,1×a1,ih. To
determine the 2-step effect of j on i (a2,ij), we simply sum up those two individual
2-step effects (i.e. a2,ij= a1,kj×a1,ik+ a1,hj×a1,ih) and therefore the term additive.
When the effect of step n is considered, we define the effect received by species i
from all species in the same network as:
N
 n,i   an,ij
(3)
j 1
which is equal to 1 (i.e. each species is affected by the same unit effect).
Furthermore, we define the n-step effect originated from a species i as:
N
 n,i   a n, ji
(4)
j 1
what may vary among different species (i.e. effects originated from different
species may be different). Here, we define the topological importance of species i
when effects “up to” n steps are considered as:
n
n
TI 
n
i
  m ,i
m 1
n

N
 a
m 1 j 1
n
m , ji
(5)
which is simply the sum of effects originated from species i up to n steps (one
plus two plus three…up to n) averaged over by the maximum number of steps
considered (i.e., n).
2.5 KeyPlayer indices – position of groups of nodes
Recently, new network indices, search algorithms and a software (KeyPlayer
1.44) were developed for solving the “key player” problem [20]. This problem is
composed of two related but distinct questions [21, 22]. The first, known as KPP1, is: which k nodes have to be deleted in order to maximally disconnect the
network, i.e. either increases the number of components or the average distance
between remaining nodes. The second question, KPP-2, is: if we spread out an
effect from k nodes, which k nodes have to be chosen in order to reach the others
in the fastest way in the intact network. An optimal choice of k nodes is a KP-set.
The first index for the KPP-1 problem quantifies network fragmentation after
node deletion:
F 1
 s (s
i
i
 1)
i
N ( N  1)
(6)
ranges from zero to 1 (for detailed explanation, see [20]). In eqn (6), si is the
number of nodes in the ith component (i.e. disconnected subgraph). This index
characterises network fragmentation or the reduction in “communication”
between nodes, therefore large F values indicate more fragmented networks. If
node deletion does not increase the number of components (leads to no
fragmentation), it is still of interest how the average distance between nodes i and
j changes. This is expressed as
F D 1
2
i j
1
d ij
N ( N  1)
,
(7)
where dij is the distance between nodes i and j. The R reciprocal distance matrix
is used in order to escape the problem of infinity appearing in the distance matrix
D should disconnection yield isolated nodes [23, 24].
There are two possible approaches to the KPP-2 problem. One is simply the
number of nodes (Rn) reachable within a given n-step distance from a given set of
M nodes. The other is a distance-weighted reachability approach, RD, considering
differences between individual paths (not dichotomising them as equal or shorter
than n or longer). This measure is:
1
RD 
d
j
N
Mj
(8)
where dMj is the distance of any node j from a set of M nodes. When this measure
reaches its largest value, our algorithm has found the optimal set of M nodes from
which effects can most easily reach the remaining nodes. Here we only calculated
FD by using the optimisation algorithms of the KeyPlayer 1.44 software [20]. We
looked for KP-sets of key-player nodes in order to examine the concept of
“keystone species complexes” [25]. It is quite possible that the identification of
single nodes as “keystones” might not tell the whole story on community
organisation. It might even be misleading and difficult to interpret. These three
indices are unique in the sense that the positional importance of groups of nodes
can be quantified. Considering the claims for multispecies conservation biology,
and the need for putting species into a multispecies context in order to understand
their ecology, the future application of the KP indices in ecology seems to be
probably fruitful.
Table 1: Positional importance ranks based on seven different network indices.
rank
2
8
1
22
7
3
15
19
14
18
23
12
27
9
11
26
32
33
13
16
25
28
29
30
5
6
10
17
21
31
34
4
20
24
D
10
10
9
9
8
7
7
7
6
6
6
5
5
4
4
4
4
4
3
3
3
3
3
3
2
2
2
2
2
2
2
1
1
1
8
3
1
2
7
19
22
33
5
15
32
14
18
30
27
6
23
17
10
34
26
9
12
11
16
25
29
28
4
13
31
21
20
24
Kdir
4.92
4.78
4.33
3.78
2.89
2.78
2.78
2.5
2
1.83
1.7
1.67
1.5
1.5
1.37
1.33
1.19
1.16
1.14
1.11
1.07
0.95
0.84
0.59
0.58
0.57
0.57
0.53
0.5
0.42
0.42
0.39
0.14
0.14
1
2
3
33
7
34
6
8
19
5
30
10
9
11
12
13
26
32
27
22
23
20
21
24
28
15
16
14
25
29
31
18
17
4
Kindir
26.83
14.82
5.34
4.94
4.51
2.68
2.46
2.18
1.83
1.79
1.55
1.45
0.83
0.72
0.72
0.68
0.66
0.56
0.25
0.17
0.17
0.16
0.16
0.16
0.16
0.15
0.15
0.1
0.1
0.1
0.09
0.06
0.02
0
1
2
3
8
7
34
5
22
15
6
14
18
23
16
4
9
12
27
19
21
31
11
10
13
17
20
24
25
26
28
29
30
32
33
Ktd
31.17
18.49
10
6
4.01
3.68
2.68
2.41
1.79
1.68
1.59
1.38
0.83
0.54
0.5
0.5
0.46
0.33
0.25
0.25
0.25
0.21
0
0
0
0
0
0
0
0
0
0
0
0
33
19
7
30
10
32
6
26
9
27
17
5
8
11
12
13
28
25
29
22
23
20
21
24
31
14
15
16
18
3
2
34
1
4
Kbu
7.44
4.37
3.35
3
2.59
2.26
2.11
1.72
1.29
1.28
1.19
1.11
1.1
1.1
1.1
1.1
0.69
0.67
0.67
0.54
0.54
0.3
0.3
0.3
0.26
0.19
0.19
0.19
0.19
0.12
0.11
0.11
0
0
1
2
3
7
33
8
19
5
6
34
22
30
10
32
15
9
26
27
18
12
14
23
11
17
13
16
25
29
21
4
28
31
20
24
K
31.50
18.34
9.95
7.46
7.44
7.27
4.62
3.81
3.81
3.81
3.46
3.11
2.59
2.20
2.15
1.79
1.78
1.67
1.57
1.56
1.44
1.37
1.31
1.19
1.10
0.73
0.73
0.73
0.55
0.50
0.45
0.44
0.30
0.30
8
2
1
22
7
3
15
19
14
18
23
27
12
9
32
33
26
11
16
17
25
29
28
30
13
5
6
34
10
21
31
4
20
24
TI10
2.41
2.18
2.01
2.01
1.82
1.64
1.54
1.53
1.33
1.33
1.32
1.1
1.06
0.91
0.89
0.89
0.86
0.84
0.65
0.65
0.65
0.65
0.64
0.64
0.63
0.59
0.54
0.53
0.46
0.45
0.44
0.35
0.23
0.23
4 Results and conclusions
Different indices give different importance ranks for the species of our
hypothetical network. Table 1 shows the degree (D), keystone index (K and its
direct, Kdir, indirect, Kindir, top-down, Ktd, and bottom-up, Kbu, components) as
well as the topological importance index for n=10 (TI10); bold numbers
correspond to species codes, while normal numbers are network index values.
The top row presents the actual index for ranking. Considering only the number
of neighbours of the nodes (degree), species #2 and #8 are in the most critical
positions but more developed indices reflecting also indirect effects (i.e. less local
information) refine this view: species #8 has a weak indirect interaction structure
(its Kindir rank is 8). The differences between the ranks of D and TI10 are caused
by only the local versus global nature of information considered. But, since
directed, Ktd and Kbu are not expected to give similar results; these network
indices reflect different aspects of network structure. For example, if a poison is
being accumulated in a food web, Kbu can be the relevant characterisation of
graph nodes. Thinking on long-term processes, we may more emphasise long
indirect pathways being otherwise probably weak: in this case, TI10 is the
adequate measure of positional importance. If, for any reason, trophic cascades
are of primarily interest, K or its components are informative.
Apart of gaining importance ranks for species, it is also possible to map more
deeply the interaction structure of the community. We present a more detailed
study based on TI10 values. Figure 1 shows what is the relative strength of effects
other species have on the arbitrarily chosen species #33 (a medium interactor
based on its TI10 rank, Table 1). Figure 2 shows the relative strength of effects
species #33 has on all other species.
Figure 1: The relative strength of interactions different species have on species
#33 marked by a dark star (drawn by the UCINET IV programme
[26]). Radius of circles is proportional to topological interaction
strength.
Figure 2: The relative strength of effects species #33 (dark star) has on all
other species (drawn like Figure 1).
It can be noted that species #21 is evidently heavily dependent on our chosen
species (Figure 2) since that is its single prey. However it is not affecting very
strongly species #33, even species #8, a second neighbour can have a larger effect
on #33 (Figure 1). Here we note that our analysis does not predict the dynamics
of the system but calculates the topological constraints on system dynamics.
If our concern is to look for key groups of species, we use the KP indices not
shown on Table 1. For example, if we calculate the FD index for k=3, we have
species #3, #8 and #33 in the KeyPlayer set. Looking at Table 1 we have to
conclude that there is no network index suggesting a top three rank like this. We
emphasise that thinking of group of species and developing techniques for
quantifying and comparing their importance can be of high importance in future
conservation biology, especially as the multispecies context is more an more
actual and urgent to develop. The network perspective in ecology [27] and novel
methodological developments [28, 29] are welcome, especially if bridging and
cross-fertilising different scientific fields [3, 13, 30]. A lot of techniques
developed in social network analysis seem to be useful and relevant in network
ecology, and even some key questions are shared (e.g. how long indirect effects
are still significant). The future task is to try to use these topological positional
analyses in setting conservation priorities.
Acknowledgements
I thank S. Borgatti, W-C. Liu, I. Scheuring, A. J. Davis, T. Wyatt and V. Vasas
for help and comments. K. Csankovszki and H. Erdős are kindly acknowledged
for technical support. My research was funded by the Society in Science: The
Branco Weiss Fellowship at ETH Zürich.
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