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.
In 2014 we proposed a solution to May’s paradox: the key structural property of ecosystems was a food-web feature called “trophic coherence”. Ecologists classify species by trophic level in the following way. Plants (nodes with no in-coming edges) have level one, herbivores (species which only have in-coming 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 left-hand 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 right-hand 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.
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 Erdős-Rényi 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 self-regulation 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.
This work leaves open many new questions. Why are some networks significantly coherent, and others incoherent? We can guess at the mechanism behind food-web 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.