How directed is a directed network?

The Thermodynamics of Mind

To not only survive, but also thrive, the brain must efficiently orchestrate distributed computation across space and time. This requires hierarchical organisation facilitating fast information transfer and processing at the lowest possible metabolic cost. Quantifying brain hierarchy is difficult but can be estimated from the asymmetry of information flow. Thermodynamics has successfully characterised hierarchy in many other complex systems. Here, we propose the 'Thermodynamics of Mind' framework as a natural way to quantify hierarchical brain orchestration and its underlying mechanisms. This has already provided novel insights into the orchestration of hierarchy in brain states including movie watching, where the hierarchy of the brain is flatter than during rest. Overall, this framework holds great promise for revealing the orchestration of cognition.

Land-use intensity influences European tetrapod food webs

Land use intensification favours particular trophic groups which can induce architectural changes in food webs. These changes can impact ecosystem functions, services, stability and resilience. However, the imprint of land management intensity on food-web architecture has rarely been characterized across large spatial extent and various land uses. We investigated the influence of land management intensity on six facets of food-web architecture, namely apex and basal species proportions, connectance, omnivory, trophic chain lengths and compartmentalization, for 67,051 European terrestrial vertebrate communities. We also assessed the dependency of this influence of intensification on land use and climate. In addition to more commonly considered climatic factors, the architecture of food webs was notably influenced by land use and management intensity. Intensification tended to strongly lower the proportion of apex predators consistently across contexts. In general, intensification also tended to lower proportions of basal species, favoured mesopredators, decreased food webs compartmentalization whereas it increased their connectance. However, the response of food webs to intensification was different for some contexts. Intensification sharply decreased connectance in Mediterranean and Alpine settlements, and it increased basal tetrapod proportions and compartmentalization in Mediterranean forest and Atlantic croplands. Besides, intensive urbanization especially favoured longer trophic chains and lower omnivory. By favouring mesopredators in most contexts, intensification could undermine basal tetrapods, the cascading effects of which need to be assessed. Our results support the importance of protecting top predators where possible and raise questions about the long-term stability of food webs in the face of human-induced pressures.

Disruption of international trade and its propagation through firm-level domestic supply chains: A case of Japan

This study simulates how the disruption of imports from various regions affects the total production of the importer economy. We particularly incorporate the propagation of the economic effect through domestic supply chains using data on more than one million firms and four million supply chain ties in Japan. Our findings are summarized as follows. First, the negative effect of the disruption of intermediate imports grows exponentially as its duration and strength increase due to downstream propagation. Second, the propagation of the economic effect is substantially affected by the network topology of importers, such as the number of importers (affected nodes) and their degree of upstreamness in supply chains, whereas the effect of their degree centrality is heterogeneous depending on their degree of upstreamness. Finally, the negative effect of import disruption can be mitigated by the reorganization of domestic supply chains, even when conducted only among network neighbors. Our findings provide important policy and managerial implications for the achievement of more robust and resilient global supply chains.

Influence and influenceability: global directionality in directed complex networks

Knowing which nodes are influential in a complex network and whether the network can be influenced by a small subset of nodes is a key part of network analysis. However, many traditional measures of importance focus on node level information without considering the global network architecture. We use the method of trophic analysis to study directed networks and show that both 'influence' and 'influenceability' in directed networks depend on the hierarchical structure and the global directionality, as measured by the trophic levels and trophic coherence, respectively. We show that in directed networks trophic hierarchy can explain: the nodes that can reach the most others; where the eigenvector centrality localizes; which nodes shape the behaviour in opinion or oscillator dynamics; and which strategies will be successful in generalized rock-paper-scissors games. We show, moreover, that these phenomena are mediated by the global directionality. We also highlight other structural properties of real networks related to influenceability, such as the pseudospectra, which depend on trophic coherence. These results apply to any directed network and the principles highlighted-that node hierarchy is essential for understanding network influence, mediated by global directionality-are applicable to many real-world dynamics.

metanetwork: A R package dedicated to handling and representing trophic metanetworks

Trophic networks describe interactions between species at a given location and time. Due to environmental changes, anthropogenic perturbations or sampling effects, trophic networks may vary in space and time. The collection of network time series or networks in different sites thus constitutes a metanetwork. We present here the R package metanetwork, which will ease the representation, the exploration and analysis of trophic metanetwork data sets that are increasingly available. Our main methodological advance consists in suitable layout algorithm for trophic networks, which is based on trophic levels and dimension reduction in a graph diffusion kernel. In particular, it highlights relevant features of trophic networks (trophic levels, energetic channels). In addition, we developed tools to handle, compare visually and quantitatively and aggregate those networks. Static and dynamic visualisation functions have been developed to represent large networks. We apply our package workflow to several trophic network data sets.

Inferring missing edges in a graph from observed collective patterns

Many real-life networks are incomplete. Dynamical observations can allow estimating missing edges. Such procedures, often summarized under the term 'network inference', typically evaluate the statistical correlations among pairs of nodes to determine connectivity. Here, we offer an alternative approach: completing an incomplete network by observing its collective behavior. We illustrate this approach for the case of patterns emerging in reaction-diffusion systems on graphs, where collective behaviors can be associated with eigenvectors of the network's Laplacian matrix. Our method combines a partial spectral decomposition of the network's Laplacian matrix with eigenvalue assignment by matching the patterns to the eigenvectors of the incomplete graph. We show that knowledge of a few collective patterns can allow the prediction of missing edges and that this result holds across a range of network architectures. We present a numerical case study using activator-inhibitor dynamics and we illustrate that the main requirement for the observed patterns is that they are not confined to subsets of nodes, but involve the whole network.

Network hierarchy and pattern recovery in directed sparse Hopfield networks

Many real-world networks are directed, sparse, and hierarchical, with a mixture of feedforward and feedback connections with respect to the hierarchy. Moreover, a small number of master nodes are often able to drive the whole system. We study the dynamics of pattern presentation and recovery on sparse, directed, Hopfield-like neural networks using trophic analysis to characterize their hierarchical structure. This is a recent method which quantifies the local position of each node in a hierarchy (trophic level) as well as the global directionality of the network (trophic coherence). We show that even in a recurrent network, the state of the system can be controlled by a small subset of neurons which can be identified by their low trophic levels. We also find that performance at the pattern recovery task can be significantly improved by tuning the trophic coherence and other topological properties of the network. This may explain the relatively sparse and coherent structures observed in the animal brain and provide insights for improving the architectures of artificial neural networks. Moreover, we expect that the principles we demonstrate here, through numerical analysis, will be relevant for a broad class of system whose underlying network structure is directed and sparse, such as biological, social, or financial networks.

System-level consequences of synergies and trade-offs between SDGs: quantitative analysis of interlinkage networks at country level

The Sustainable Development Goals (SDGs) present a complex system of 17 goals and 169 individual targets whose interactions can be described in terms of co-benefits and trade-offs between policy actions. We analyse in detail target-by-target interlinkage networks established by the Institute for Global Environmental Strategies (IGES) SDG Interlinkages Tool. We discuss two quantitative measures of network structure; the leading eigenvector of the interlinkage networks ('eigencentrality') and a notion of hierarchy within the network motivated by the concept of trophic levels for species in food webs. We use three interlinkage matrices generated by IGES: the framework matrix which provides a generic network model of the interlinkages at the target level, and two country-specific matrices for Bangladesh and Indonesia that combine SDG indicator data with the generic framework matrix. Our results echo, and are confirmed by, similar work at the level of whole SDGs that has shown that SDGs 1-3 (ending poverty, and providing food security and healthcare) are much more likely to be achieved than the environmentally- related SDGs 13-15 concerned with climate action, life on land and life below water. Our results here provide a refinement in terms of specific targets within each of these SDGs. We find that not all targets within SDGs 1-3 are equally well-supported, and not all targets within SDGs 13-15 are equally at risk of not being achieved. Finally, we point to the recurring issue of data gaps that hinders our quantitative analysis, in particular for SDGs 5 (gender equality) and 13 (climate action) where the huge gaps in indicator data that mean the true nature of the interlinkages and importance of these two SDGs are not fully recognised.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11625-022-01109-y.

Hierarchy Depth in Directed Networks

In this study, we explore the depth measures for flow hierarchy in directed networks. Two simple measures are defined—rooted depth and relative depth—and their properties are discussed. The method of loop collapse is introduced, allowing investigation of networks containing directed cycles. The behavior of the two depth measures is investigated in Erdös-Rényi random graphs, directed Barabási-Albert networks, and in Gnutella p2p share network. A clear distinction in the behavior between non-hierarchical and hierarchical networks is found, with random graphs featuring unimodal distribution of depths dependent on arc density, while for hierarchical systems the distributions are similar for different network densities. Relative depth shows the same behavior as existing trophic level measure for tree-like networks, but is only statistically correlated for more complex topologies, including acyclic directed graphs.

Directed network Laplacians and random graph models

We consider spectral methods that uncover hidden structures in directed networks. We establish and exploit connections between node reordering via (a) minimizing an objective function and (b) maximizing the likelihood of a random graph model. We focus on two existing spectral approaches that build and analyse Laplacian-style matrices via the minimization of frustration and trophic incoherence. These algorithms aim to reveal directed periodic and linear hierarchies, respectively. We show that reordering nodes using the two algorithms, or mapping them onto a specified lattice, is associated with new classes of directed random graph models. Using this random graph setting, we are able to compare the two algorithms on a given network and quantify which structure is more likely to be present. We illustrate the approach on synthetic and real networks, and discuss practical implementation issues.

Graph hierarchy: a novel framework to analyse hierarchical structures in complex networks

Trophic coherence, a measure of a graph's hierarchical organisation, has been shown to be linked to a graph's structural and dynamical aspects such as cyclicity, stability and normality. Trophic levels of vertices can reveal their functional properties, partition and rank the vertices accordingly. Trophic levels and hence trophic coherence can only be defined on graphs with basal vertices, i.e. vertices with zero in-degree. Consequently, trophic analysis of graphs had been restricted until now. In this paper we introduce a hierarchical framework which can be defined on any simple graph. Within this general framework, we develop several metrics: hierarchical levels, a generalisation of the notion of trophic levels, influence centrality, a measure of a vertex's ability to influence dynamics, and democracy coefficient, a measure of overall feedback in the system. We discuss how our generalisation relates to previous attempts and what new insights are illuminated on the topological and dynamical aspects of graphs. Finally, we show how the hierarchical structure of a network relates to the incidence rate in a SIS epidemic model and the economic insights we can gain through it.

Money flow network among firms' accounts in a regional bank of Japan

In this study, we investigate the flow of money among bank accounts possessed by firms in a region by employing an exhaustive list of all the bank transfers in a regional bank in Japan, to clarify how the network of money flow is related to the economic activities of the firms. The network statistics and structures are examined and shown to be similar to those of a nationwide production network. Specifically, the bowtie analysis indicates what we refer to as a "walnut" structure with core and upstream/downstream components. To quantify the location of an individual account in the network, we used the Hodge decomposition method and found that the Hodge potential of the account has a significant correlation to its position in the bowtie structure as well as to its net flow of incoming and outgoing money and links, namely the net demand/supply of individual accounts. In addition, we used non-negative matrix factorization to identify important factors underlying the entire flow of money; it can be interpreted that these factors are associated with regional economic activities. One factor has a feature whereby the remittance source is localized to the largest city in the region, while the destination is scattered. The other factors correspond to the economic activities specific to different local places. This study serves as a basis for further investigation on the relationship between money flow and economic activities of firms.