Complex (adj.): 1.
Consisting of many different and connected parts. ‘A complex network of water
channels’.
Oxford English Dictionary
‘Complex
systems’ – like cells, the brain or human society – are often defined as those
whose interesting behaviour emerges from the interaction of many connected
elements. A simple but particularly useful representation of almost any complex
system is therefore as a network (aka a graph). When the connections (edges)
between elements (nodes) have a direction, this takes the form of a directed
network. For example, to describe interactions in an ecosystem, ecologists use directed
networks called food webs, in which each species is a node and directed edges
(usually drawn as arrows) go from prey to their predators. The last two decades
have witnessed a lot of research into the properties of networks, and how their
structure is related to aspects of complex systems, such as their dynamics or
robustness. In the case of ecosystems, it has long been thought that their
remarkable stability – in the sense that they don’t tend to succumb easily to
destructive avalanches of extinctions – must have something to do with their
underlying architecture, especially given May’s paradox: mathematical models
predict that ecosystems should become more unstable with increasing size and
complexity, but this doesn’t seem to happen to, say, rainforests or coral
reefs.
Trophic coherence
In 2014 we proposed a solution to
May’s paradox: the key structural property of ecosystems was a foodweb feature
called “trophic coherence”. Ecologists classify species by trophic level in
the following way. Plants (nodes with no incoming edges) have level one,
herbivores (species which only have incoming edges from plants) are at level
two, and, in general, the level of any species is defined as the average level
of its prey, plus one. Thus, if the network in the top lefthand corner of the
figure below represented a food web, the nodes at the bottom would be plants
(level 1) the next ones up would be herbivores (level 2), the next, primary
carnivores (level 3) and so on. In reality, though, food webs are never quite
so neatly organised, and many species prey on various levels, making food webs
a bit more like the network in the top righthand corner. Here, most species
have a fractional trophic level. In order to measure this degree of order,
which we called trophic coherence, we attributed to each directed edge a
“trophic difference”, the difference between the levels of the predator and the
prey, and looked at the statistical distribution of differences over all the
edges in the whole network. We called the standard deviation of this
distribution an “incoherence parameter”, q, because a perfectly coherent
network like the one on the left has q=0, while a more incoherent one
like that on the right has q>0 – in this case, q=0.7.
It turns out that the trophic
coherence of food webs is key to their stability, and when we simulated (i.e.
generated in the computer) networks with varying levels of coherence, we found
that, for sufficiently coherent ones, the relationship between size and stability
is inverted. Although there are plenty of caveats to this result – not least
the question how one should measure stability – this suggests a solution to
May’s paradox. Since then, further research has shown that trophic coherence
affects other structural and dynamical properties of networks – for instance,
whether a cascade of activity will propagate through a neural network (example papers here, here and here!). But all these results were somewhat anecdotal, since we didn’t have a
mathematical theory relating trophic coherence to other network features. This
is what we set out to do in our most recent paper.

Figure. Four directed networks, plotted so that the height of each node on the vertical axis is proportional in each case to its trophic level. The top two are synthetic networks, generated in a computer with the ‘preferential preying model’, which allows the user to tune trophic coherence [1,3]. Thus, they both have the same numbers of nodes and edges, but the one on the left is perfectly coherent (q=0) while the one on the right is more incoherent (q=0.7). The bottom two are empirically derived: the one on the left is the Ythan Estuary food web, which is significantly coherent (it has q=0.42, which is about 15% of its expected q) and belongs to the loopless regime; the one on the right is a representation of the Chlamydia pneumoniae metabolic network, which is singificantly incoherent (q=8.98, or about 162% of the random expectation) and sits in the loopful regime. The top two networks are reproduced from the SI of Johnson et al, PNAS, 2014 [1], while the bottom two are from the SI of Johnson & Jones, PNAS, 2017 [5]. 
Looplessness
In statistical physics one thinks
about systems in terms of ensembles – the sets of all possible systems which
satisfy certain constraints – and this method has also been used in graph
theory. For example, the ErdősRényi ensemble comprises all possible networks with
given numbers of nodes N and edges L, while the configuration
ensemble also specifies the degree sequence (the degree of a node being its
number of neighbours). We defined the “coherence ensemble” as the set of all
possible directed networks which not only have given N, L and
degree sequences (each node has two degrees in directed networks, one in and
one out) but also specified trophic coherence. This allows us to derive
equations for the expected values of various network properties as a function
of trophic coherence; in other words, these are the values we should expect to
measure in a network given its trophic coherence (and other specified
constraints) if we had no other knowledge about its structure.
Many network properties are
heavily influenced by cycles – that is, paths through a network which begin and
end at the same node. For example, in a food web you might find that eagles eat
snakes, which eats squirrels, which eat eagles (probably in egg form), thus
forming a cycle of length three. These cycles (properly called ‘directed
cycles’ in directed networks), or loops, are related to various structural and
dynamical features of complex systems. For example, feedback loops can
destabilise ecosystems, mediate selfregulation of genes, or maintain neural
activity in the brain. Furthermore, it had been reported that certain kinds of
network – in particular, food webs and gene regulatory networks – often had
either no cycles at all, or only a small number of quite short cycles. This was
surprising, because in (arbitrarily large) random networks the number of cycles of length l grows exponentially with l, so it was assumed that there must be some
evolutionary reason for this “looplessness”. We were able to use our coherence
ensemble approach to derive the probability with which a randomly chosen path
would be a cycle, as a function of q. From there we could obtain
expected values for the number of cycles of length l, and for other
quantities related to stability (in particular, for the adjacency matrix
eigenspectrum, which captures the total extent of feedback in a system). It
turns out that the number of cycles does indeed depend on length exponentially,
but via a factor τ which is a function of trophic coherence. For sufficiently
coherent networks, τ is negative, and hence the expected number of
cycles of length l falls rapidly to zero. In fact, such networks have a
high chance of being completely acyclic. Thus, our theory predicts that
networks can belong to either of two regimes, depending on the “loop exponent” τ:
a loopful one with lots of feedback, or a loopless one in which networks are
either acyclic or have just a few short cycles. A comparison with a large set
of networks from the real world – including networks of species, genes,
metabolites, neurons, trading nations and English words – shows that this is indeed so, and almost all
of them are very close to our expectations given their trophic coherence.
Our theory can also be used to
see how close quantities such as trophic coherence, or mean trophic level, are
to what would be our random expectations, given just N, L and the
degree sequences, for any real directed network. We found, for example, that in
our dataset the food webs tended to be very coherent, while networks derived
from metabolic reactions were significantly incoherent (see the bottom two
networks in the figure: the one on the left is a food web and the one on the
right is a metabolic network). Our gene regulatory networks are interesting in
that, while often quite coherent in absolute terms, they are in fact very close
to their random expectation.
Open questions
This work leaves open many new
questions. Why are some networks significantly coherent, and others incoherent?
We can guess at the mechanism behind foodweb coherence: the adaptations which
allow a given predator, say a wolf, to hunt deer are also useful for catching
prey like goats or elk, which have similar characteristics because they, in
turn, have similar diets – i.e. trophic levels. This correlation between
trophic levels and node function might be more general. For example, we have
shown that in a network of words which are concatenated in a text, trophic
level serves to identify syntactic function, and something similar may occur in
networks of genes or metabolites. If edges tend to form primarily between nodes
with certain functions, this might induce coherence or incoherence. Some
networks, like the artificial neural networks used for “deep learning”, are
deliberately coherent, which suggests another question: how does coherence affect
the performance of different kinds of system? Might there be an optimal level
of trophic coherence for neural networks? And how might it affect financial,
trade, or social networks, which can, in some sense, be considered human
ecosystems? We hope topics such as these will attract the curiosity of other
researchers who can make further inroads. You can read our paper “Looplessness in networks is linked to trophic coherence” for free here and also in the journal PNAS. Sam and Nick.