A complexity and networks approach to sustainability

Abstract

Sustainability is a key issue of the 21st century. A complexity and networks approach is taken to investigate what makes socio-economic systems sustainable, or not, and what lessons that can be learned from the sandpile model, an analogy of observed economic activity.

The sandpile model has been studied extensively on regular lattices with coordination number K = 2. Here, the dynamics is generalised to evolve on less rigid networks. The regular lattice structure is relaxed by introducing an inter-layer distribution for the interactions between nodes. This forces the scaling exponent of the avalanche-size probability density function out of the two-dimensional directed sandpile universality class τ = 4/3, into the mean-field universality class τ = 3/2. An apparent drift in the exponents of the avalanches found in the Monte Carlo simulations on a lattice with K > 2 is attributed to the observed holes and side-branches of avalanche clusters, which leads to large corrections to scaling. The K = 3 case is solved analytically, proving that in fact the critical exponents for K > 2 are identical to the K = 2 case.

The most realistic topology that can be used to model economic activity is the production-consumption network between firms in a real economy. A novel avalanche-size exponent τ ≈ 1.87, that falls outside the two known universality classes, emerges when the model evolves on the Japanese inter-firm network. This network has the typical bow-tie structure of a real-world directed network: in and out-layers, and a strongly connected component. The strongly connected component is not layered and displays a large range of network motifs.

Randomly adding a small proportion of links between non-adjacent layers in one of the layered networks in the mean-field universality class, abruptly takes the system out of the mean-field regime with a non-trivial avalanche-size probability density function, which closely reproduces the behaviour observed on the real-world Japanese network.