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.
Ferenc Jordán
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.
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.
2.1 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 ( D i
) is the sum of its prey (in-degree,
D in,i
) and predators (out-degree, D out,i
):
D i

D in , i

D out , 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 ( K i
) is defined as:
K i

K bu , i

K td , i
 n  c

1
1 d c
( 1

K bc
)
 m  e

1
1
( 1

K te
) , (2) f e where n is the number of predators eating species i , d c
is the number of prey of its c th predator and K bc
is the bottom-up keystone index of the c th predator. And symmetrically, m is the number of prey eaten by species i , f e
is the number of predators of its e th prey and K te
is the top-down keystone index of the e th prey. For node i , the first sum in the equation ( i.e.

1/ d effect ( K bu,i
) while the second sum ( i.e.

1/ f c
(1+ K bc
)) quantifies the bottom-up e
(1+ K te
)) quantifies the top-down effect ( K td,i
). After rearranging the equation, terms including K bc

K bc
/ d c
+

K te
/ f e containing K bc and K te
) refer to indirect effects for node i ( K indir,i
), while terms not and K te
( i.e.

1/ d c
( i.e.
+

1/ f e
) refer to direct ones ( K dir,i
). Both K bu,i
+
K td,i
and K indir,i
+ K dir,i
equals K i
. 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 a n,ij
as the effect of j on i when i can be reached from j in n steps. The simplest mode of calculating a n,ij
is when n =1 ( i.e.
the effect of j on i in
1 step): a
1, ij
= 1/ D i
, where D i
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 ( multiplicative. Similarly, the effect of j on i i.e.
a
1, kj
through h
× a
1, ik
), therefore the term
equals to a
1, hj, 1
× a
1, ih
. To determine the 2-step effect of j on i ( a
2, ij
), we simply sum up those two individual
2-step effects ( i.e.
a
2, ij
= a
1, kj
× a
1, ik
+ a
1, hj
× a
1, 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 , i
 j
N 

1 a n , ij
(3) 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 , i
 j
N 

1 a n , ji
(4) 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  m

1
 m , i m n N 

1 j

1 a m , ji
TI i n  
(5) n n 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 KPP-
1, 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

 i s i
( s i
N ( N

1 )

1 )
(6) ranges from zero to 1 (for detailed explanation, see [20]). In eqn (6), s i
is the number of nodes in the i th 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

N ( N j
1 d ij

1 )
, (7) where d ij
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 ( R n
) reachable within a given n -step distance from a given set of
M nodes. The other is a distance-weighted reachability approach, R D , considering differences between individual paths (not dichotomising them as equal or shorter than n or longer). This measure is:
 j
1 d
Mj
R
D 
(8)
N where d
Mj
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
F D 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.

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
Table 1: Positional importance ranks based on seven different network indices. rank
2
D K dir
K indir
K td
K bu
K TI 10
10 8 4.92 1 26.83 1 31.17 33 7.44 1 31.50 8 2.41
4
20
24
10 3 4.78 2 14.82 2 18.49 19 4.37 2 18.34 2 2.18
9 1 4.33 3 5.34 3 10 7 3.35 3 9.95 1 2.01
9 2 3.78 33 4.94
8 7 2.89 7 4.51
8
7
6
4.01
30
10
3
2.59
7
33
7.46
7.44
22
7
2.01
1.82
7 19 2.78 34 2.68
7 22 2.78 6 2.46
7 33 2.5
6 5 2
8
19
2.18
1.83
34
5
22
15
3.68
2.68
2.41
1.79
32
6
26
9
2.26
2.11
1.72
1.29
8
19
5
6
7.27
4.62
3.81
3.81
3
15
19
14
1.64
1.54
1.53
1.33
6 15 1.83 5 1.79
6 32 1.7 30 1.55
5 14 1.67 10 1.45
5 18 1.5 9 0.83
4 30 1.5 11 0.72
4 27 1.37 12 0.72
4 6 1.33 13 0.68
4 23 1.19 26 0.66
6
14
18
23
1.68
1.59
1.38
0.83
27
17
5
8
1.28
1.19
1.11
1.1
16 0.54 11 1.1
4 0.5
9 0.5
12 1.1
13 1.1
34
22
30
10
3.81
3.46
3.11
2.59
18
23
27
12
1.33
1.32
1.1
1.06
32 2.20 9 0.91
15 2.15 32 0.89
9 1.79 33 0.89
12 0.46 28 0.69 26 1.78 26 0.86
4 17 1.16 32 0.56
3 10 1.14 27 0.25
3 34 1.11 22 0.17
3 26 1.07 23 0.17
3 9 0.95 20 0.16
3 12 0.84 21 0.16
3 11 0.59 24 0.16
2 16 0.58 28 0.16
2 25 0.57 15 0.15
2 29 0.57 16 0.15
2 28 0.53 14 0.1
2 4 0.5 25 0.1
2 13 0.42 29 0.1
2 31 0.42 31 0.09
1 21 0.39 18 0.06
1 20 0.14 17 0.02
1 24 0.14 4 0
27 0.33 25 0.67 27 1.67 11 0.84
19 0.25 29 0.67 18 1.57 16 0.65
21 0.25 22 0.54 12 1.56 17 0.65
31 0.25 23 0.54 14 1.44 25 0.65
11 0.21 20 0.3
10 0 21 0.3
13 0
17 0
24 0.3
23
11
1.37
1.31
29
28
0.65
0.64
17 1.19 30 0.64
31 0.26 13 1.10 13 0.63
20 0
24 0
25 0
26 0
28 0
29 0
30 0
32 0
33 0
14
15
16
18
3
2
34
1
4
0.19
0.19
0.19
0.19
0.12
0.11
0.11
0
0
16
25
29
21
4
28
31
20
24
0.73
0.73
0.73
0.55
0.50
0.45
0.44
0.30
0.30
5
6
34
10
21
31
4
20
24
0.59
0.54
0.53
0.46
0.45
0.44
0.35
0.23
0.23
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, K dir
, indirect, K indir
, top-down, K td
, and bottom-up, K bu
, components) as well as the topological importance index for n =10 ( TI 10 ); 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 K indir
rank is 8). The differences between the ranks of D and TI 10 are caused by only the local versus global nature of information considered. But, since directed, K td
and K bu
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, K bu
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, TI 10 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 TI 10 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 TI 10 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 F D 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.

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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