Notes on: Universal resilience patterns in complex networks, Gao et al. 2016

Resiliency Overview

Resiliency is a system’s ability to maintain its overall structure and function when disturbances act upon its network.
Some examples include environmental systems, technological systems, and biological systems that are subject to environmental changes, errors, or failures. More resilient systems will be stable despite such perturbations.
Loss of resiliency occurs when some bifurcation converts a stable fixed point into an unstable fixed point. Bifurcations occur when a changing parameter results in one or more fixed points undergoing stability transitions.
Below is the equation of a 1D dynamical system (a flow on the line) with an adjustable parameter β.

Improved Model of Resiliency

Gao et al. developed a higher-dimensional mathematical model to allow for better prediction of network resilience than past models.
The main equation of this model is given below. A_ij represents a weighted adjacency matrix, N is the number of nodes in the network, and x = (x₁,…,x_N)^T is a vector representing the activities at each node. The function F(x_i) describes the activity of each given node x_i independent of the influences of other nodes. The function G(x_i,x_j) is a rule describing the influence of node x_j on node x_i.

Disturbances to networks described by this equation might include gaining nodes or edges, losing nodes or edges, changes in edge weights, or changes in the values of parameters that are part of the functions F and G.

Resiliency Model Applied to Ecological Networks

Below, this resiliency model describes abundance of species. The function x_i(t) is the abundance of a species i. The parameter B_i represents the incoming migration rate of each species. The second term considers logistic growth with carrying capacities K_i as well as the Allee effect. According to the Allee effect, small populations, where x_i<C_i will decline (negative growth). Finally, the third term describes mutualistic interactions between species x_i and x_j (i.e. plant-pollinator relationships). At high abundances of x_i and x_j, this term tends to plateau. The weighted adjacency matrix A_ij represents the strengths of these mutualistic interactions. D_i, E_i, and H_j are further adjustable parameters.

The authors tested the resiliency of experimentally mapped ecological networks using this model. First, a fraction f_n of the plant nodes was removed, simulating plant extinctions. Then, a fraction f_I of the pollinator nodes was removed, simulating pollinator extinctions. Finally, the weights of the matrix A_ij were randomly rescaled such that the average weight decreased to a fraction f_w of the original average weight.
Initially, the ecological networks each only possessed a single stable fixed point, x^H. These fixed points described high average abundances among the species. Upon applying perturbations, the networks maintained stability until thresholds were reached. Each threshold was dependent on its network and the type of perturbation applied (plant node removal, pollinator node removal, or changes to edge weights). After passing the thresholds, each network gained an additional stable fixed point, x^L. These fixed points described low average abundances among the species (an undesirable state).
For example, network1 lost resilience after 35% of its pollinators were removed, while Network5 lost resilience only after 80% of its pollinators were removed.

Resiliency Model Applied to Gene Regulatory Networks

Below, the resiliency model describes the activity of genes in coli and S. cerevisiae using Michaelis-Menten kinetics. The function x_i(t) is the activity of a gene i. The first term represents the degradation rate (f=1) of a protein (from gene i). Alternatively, if the x_i in the first term is squared (f=2), this term can represent dimerization. The second term describes gene expression levels, where the Hill coefficient h represents the level of cooperativity in the regulation of gene i by protein j. The weighted adjacency matrix A_ij represents regulatory connections among the genes.

By computing the average activity of all genes in a cell, the resiliency model provided a method of predicting cell viability. If a cell’s , then the model predicts that the cell will die. If a cell’s , then the model predicts that the cell will be viable.
Once again, depending on the gene regulatory network and the types of perturbations, resilience loss occurred after thresholds (each specific to the network and the perturbation types) were crossed. The types of perturbations for this application included gene deletions, changes to the weights of regulatory interactions, and global environmental changes.