Trophic coherence is a property of
directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pa ...
s (or directed
networks
Network, networking and networked may refer to:
Science and technology
* Network theory, the study of graphs as a representation of relations between discrete objects
* Network science, an academic field that studies complex networks
Mathematics
...
).
It is based on the concept of
trophic levels
The trophic level of an organism is the position it occupies in a food web. A food chain is a succession of organisms that eat other organisms and may, in turn, be eaten themselves. The trophic level of an organism is the number of steps it i ...
used mainly in
ecology
Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...
,
but which can be defined for directed networks in general and provides a measure of hierarchical structure among nodes. Trophic coherence is the tendency of nodes to fall into well-defined trophic levels. It has been related to several structural and dynamical properties of directed networks, including the prevalence of
cycles and
network motif
Network motifs are recurrent and statistically significant Subgraph (graph theory), subgraphs or patterns of a larger complex network, graph. All networks, including biological networks, social networks, technological networks (e.g., computer netw ...
s,
ecological stability
In ecology, an ecosystem is said to possess ecological stability (or equilibrium) if it is capable of returning to its equilibrium state after a perturbation (a capacity known as resilience) or does not experience unexpected large changes in its c ...
,
intervality,
and spreading processes like
epidemics
An epidemic (from Greek ἐπί ''epi'' "upon or above" and δῆμος ''demos'' "people") is the rapid spread of disease to a large number of patients among a given population within an area in a short period of time.
Epidemics of infectious d ...
and
neuronal avalanches.
Definition
![Networks with different trophic coherence](https://upload.wikimedia.org/wikipedia/commons/8/88/Networks_with_different_trophic_coherence.png)
Consider a directed network defined by
the
adjacency matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
In the special case of a finite simp ...
.
Each node
can be assigned a
trophic level
The trophic level of an organism is the position it occupies in a food web. A food chain is a succession of organisms that eat other organisms and may, in turn, be eaten themselves. The trophic level of an organism is the number of steps it ...
according to
::
where
is
's in-degree, and nodes
with
(basal nodes) have
by convention.
Each edge has a ''trophic difference'' associated, defined as
.
The ''trophic coherence'' of the network is a measure of how tightly peaked the distribution of trophic distances,
,
is around its mean value, which is always
.
This can be captured by an ''incoherence parameter''
,
equal to the standard deviation of
:
::
where
is the number of edges in the network.
The figure shows two networks which differ in their trophic coherence. The position of the nodes on the vertical axis corresponds to their trophic level. In the network on the left, nodes fall into distinct (integer) trophic levels, so the network is maximally coherent
.
In the one on the right, many of the nodes have fractional trophic levels, and the network is more incoherent
.
Trophic coherence in nature
The degree to which empirical networks are trophically coherent (or incoherent) can be investigated by comparison with a null model.
This is provided by the ''basal ensemble'', which comprises networks in which all non-basal nodes have the same proportion of basal nodes
for in-neighbours.
Expected values in this ensemble converge to those of the widely used ''configuration ensemble''
in the limit
,
(with
and
the numbers of nodes and edges),
and can be shown numerically to be a good approximation for finite random networks. The basal ensemble expectation for the incoherence parameter is
::
where
is the number of edges connected to basal nodes.
The ratio
measured in empirical networks reveals whether they are more or less coherent than the random expectation.
For instance, Johnson and Jones
find in a set of networks that
food webs are significantly coherent
,
metabolic networks are significantly incoherent
, and
gene regulatory networks
A gene (or genetic) regulatory network (GRN) is a collection of molecular regulators that interact with each other and with other substances in the cell to govern the gene expression levels of mRNA and proteins which, in turn, determine the fun ...
are close to the random expectation
.
Trophic levels and node function
There is as yet little understanding of the mechanisms which might lead to particular kinds of networks becoming significantly
coherent or incoherent.
However, in systems which present correlations between trophic level and other features of nodes,
processes which tended to favour the creation of edges between nodes with particular characteristics could induce coherence or incoherence.
In the case of food webs, predators tend to specialise on consuming prey with certain biological properties (such as size, speed or behaviour)
which correlate with their diet, and hence with trophic level.
This has been suggested as the reason for food-web coherence.
However, food-web models based on a ''niche axis'' do not reproduce realistic trophic coherence,
which may mean either that this explanation is insufficient, or that several
niche dimensions need to be considered.
![Green Eggs and Ham word adjacency network](https://upload.wikimedia.org/wikipedia/commons/2/20/Green_Eggs_and_Ham_word_adjacency_network.png)
The relation between trophic level and node function can be seen in networks other than food webs. The figure shows a word adjacency network derived from the book ''
Green Eggs and Ham
''Green Eggs and Ham'' is a children's book by Dr. Seuss, first published on August 12, 1960. As of 2019, the book has sold 8 million copies worldwide.
The story has appeared in several adaptations, starting with 1973's '' Dr. Seuss on the Loos ...
'', by Dr. Seuss.
The height of nodes represents their trophic levels (according here to the edge direction which is the opposite of that suggested by the arrows, which indicate the order in which words are concatenated in sentences). The syntactic function of words is also shown with node colour. There is a clear relationship between syntactic function and trophic level: the mean trophic level of common nouns (blue) is
, whereas that of verbs (red) is
.
This example illustrates how trophic coherence or incoherence might emerge from node function, and also that the trophic structure of networks provides a means of identifying node function in certain systems.
Generating trophically coherent networks
There are various ways of generating directed networks with specified trophic coherence, all based on gradually introducing new edges
to the system in such a way that the probability of each new candidate edge being accepted depends on the expected trophic difference it would have.
The ''preferential preying model'' is an evolving network model similar to the
Barábasi-Albert model of preferential attachment, but inspired on an ecosystem that grows through immigration of new species.
One begins with
basal nodes and proceeds to introduce new nodes up to a total of
.
Each new node
is assigned a first in-neighbour
(a prey species in the food-web context) and a new
edge is placed from
to
. The new node is given a temporary trophic level
.
Then a further
new in-neighbours
are chosen for
from among those in the network according to their trophic levels. Specifically, for a new candidate in-neighbour
, the probability of being chosen is a function of
. Johnson ''et al''
use
::
where
is a parameter which tunes the trophic coherence: for
maximally coherent networks are generated, and
increases monotonically with
for
.
The choice of
is arbitrary. One possibility is to
set to
,
where
is the number of nodes already in the network when
arrives, and
is a random variable drawn from a
Beta distribution
In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...
with parameters
and
::
(
being the desired number of edges).
This way, the ''generalised cascade model''
is recovered in the limit
, and the degree distributions are as in the ''niche model''
and ''generalised niche model''.
This algorithm, as described above, generates networks with no cycles (except for self-cycles, if the new node
is itself considered among its candidate in-neighbours
). In order for cycles of all lengths to be a possible, one can consider new candidate edges in which the new node
is the in-neighbour as well as those in which it would be the out-neighbour. The probability of acceptance of these edges,
, then depends on
.
The ''generalised preferential preying model''
is similar to the one described above, but has certain advantages. In particular, it is more analytically tractable, and one can generate networks with a precise number of edges
.
The network begins with
basal nodes, and then a further
new nodes are added in the following way. When each enters the system, it is assigned a single in-neighbour randomly from among those already there. Every node then has an integer temporary trophic level
. The remaining
edges are introduced as follows. Each pair of nodes
has two temporary trophic distances associated,
and
. Each of these candidate edges is accepted with a probability that depends on this temporary distance. Klaise and Johnson
use
::
because they find the distribution of trophic distances in several kinds of networks to be approximately
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
, and this choice leads to a range of the parameter
in which
. Once all the edges have been introduced, one must recalculate the trophic levels of all nodes, since these will differ from the temporary ones originally assigned unless
.
As with the preferential preying model, the average incoherence parameter
of the resulting networks is a monotonically increasing function of
for
. The figure above shows two networks with different trophic coherence generated with this algorithm.
References
{{Reflist
External links
Why don't large ecosystems just collapse?Trophic Coherence Could Help Solve the Mystery of Coexistence within Complex Ecosystems*
ttps://www.samuel-johnson.org/ Samuel Johnson's websiteNick Jones's website
Ecology