The origin of motif families in food webs

Food webs have been found to exhibit remarkable “motif profiles”, patterns in the relative prevalences of all possible three-species subgraphs, and this has been related to ecosystem properties such as stability and robustness. Analysing 46 food webs of various kinds, we find that most food webs fall into one of two distinct motif families. The separation between the families is well predicted by a global measure of hierarchical order in directed networks—trophic coherence. We find that trophic coherence is also a good predictor for the extent of omnivory, defined as the tendency of species to feed on multiple trophic levels. We compare our results to a network assembly model that admits tunable trophic coherence via a single free parameter. The model is able to generate food webs in either of the two families by varying this parameter, and correctly classifies almost all the food webs in our database. This is in contrast with the two most popular food web models, the generalized cascade and niche models, which can only generate food webs within a single motif family. Our findings suggest the importance of trophic coherence in modelling local preying patterns in food webs

Emergent patterns in complex networks

Complex interacting systems permeate the modern world. Many diverse natural, social and human made systems—ranging from food webs to human contact networks, to the Internet—can be studied in the context of network science. This thesis is a compendium of research in applied network science, investigating structural and dynamical patterns behind the formation of networks and processes supported on them. Trophic food webs—networks of who eats whom in an ecosystem—have fascinated network scientists since data from field observations of the gut content of species first became available. The empirical patterns in food webs reveal a rich hierarchy of feeding patterns. We study how global structure of food webs relates to species immediate diet over a range of 46 different ecosystems. Our finding suggest that food webs fall broadly into two different families based on the extent of species tendency towards omnivory. Drawing inspiration from food webs, we investigate how trophic networks support spreading processes on them. We find that the interplay of dynamics and network structure determines the extent and duration of contagion. We uncover two distinct modes of operation—short-lived outbreaks with high incidence and endemic infections. These results could be important for understanding spreading phenomena such as epidemics, rumours, shocks to ecosystems and neuronal avalanches. Finally, we study the emergence of structural order in random network models. Random networks serve as null models to empirical networks to help uncover significant non-random patterns but are also interesting to study in their own right. We study the effect of triadic ties in delaying the formation of extensive giant components— connected components taking over the majority of the network. Our results show that, depending on the network formation process, order in the form of a giant component can emerge even with a significant number of triadic tie

The origin of motif families in food webs

Food webs have been found to exhibit remarkable “motif profiles”, patterns in the relative prevalences of all possible three-species subgraphs, and this has been related to ecosystem properties such as stability and robustness. Analysing 46 food webs of various kinds, we find that most food webs fall into one of two distinct motif families. The separation between the families is well predicted by a global measure of hierarchical order in directed networks—trophic coherence. We find that trophic coherence is also a good predictor for the extent of omnivory, defined as the tendency of species to feed on multiple trophic levels. We compare our results to a network assembly model that admits tunable trophic coherence via a single free parameter. The model is able to generate food webs in either of the two families by varying this parameter, and correctly classifies almost all the food webs in our database. This is in contrast with the two most popular food web models, the generalized cascade and niche models, which can only generate food webs within a single motif family. Our findings suggest the importance of trophic coherence in modelling local preying patterns in food webs

Relaxation dynamics of maximally clustered networks

We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an unclustered state under two different edge dynamics---the double-edge swap, corresponding to degree-preserving randomization of the configuration model, and single edge replacement, corresponding to full randomization of the Erd\H{o}s--R\'enyi random graph. We derive expressions for the time evolution of the degree distribution, edge multiplicity distribution and clustering coefficient. We show that under both dynamics networks undergo a continuous phase transition in which a giant connected component is formed. We calculate the position of the phase transition analytically using the Erd\H{o}s--R\'enyi phenomenology

From neurons to epidemics: How trophic coherence affects spreading processes

Trophic coherence, a measure of the extent to which the nodes of a directed network are organised in levels, has recently been shown to be closely related to many structural and dynamical aspects of complex systems, including graph eigenspectra, the prevalence or absence of feed-back cycles, and linear stability. Furthermore, non-trivial trophic structures have been observed in networks of neurons, species, genes, metabolites, cellular signalling, concatenated words, P2P users, and world trade. Here we consider two simple yet apparently quite different dynamical models -- one a Susceptible-Infected-Susceptible (SIS) epidemic model adapted to include complex contagion, the other an Amari-Hopfield neural network -- and show that in both cases the related spreading processes are modulated in similar ways by the trophic coherence of the underlying networks. To do this, we propose a network assembly model which can generate structures with tunable trophic coherence, limiting in either perfectly stratified networks or random graphs. We find that trophic coherence can exert a qualitative change in spreading behaviour, determining whether a pulse of activity will percolate through the entire network or remain confined to a subset of nodes, and whether such activity will quickly die out or endure indefinitely. These results could be important for our understanding of phenomena such as epidemics, rumours, shocks to ecosystems, neuronal avalanches, and many other spreading processes