Adaptive Rewiring in Weighted Networks Shows Specificity, Robustness, and Flexibility

Abstract

Brain network connections rewire adaptively in response to neural activity. Adaptive rewiring may be understood as a process which, at its every step, is aimed at optimizing the efficiency of signal diffusion. In evolving model networks, this amounts to creating shortcut connections in regions with high diffusion and pruning where diffusion is low. Adaptive rewiring leads over time to topologies akin to brain anatomy: small worlds with rich club and modular or centralized structures. We continue our investigation of adaptive rewiring by focusing on three desiderata: specificity of evolving model network architectures, robustness of dynamically maintained architectures, and flexibility of network evolution to stochastically deviate from specificity and robustness. Our adaptive rewiring model simulations show that specificity and robustness characterize alternative modes of network operation, controlled by a single parameter, the rewiring interval. Small control parameter shifts across a critical transition zone allow switching between the two modes. Adaptive rewiring exhibits greater flexibility for skewed, lognormal connection weight distributions than for normally distributed ones. The results qualify adaptive rewiring as a key principle of self-organized complexity in network architectures, in particular of those that characterize the variety of functional architectures in the brain.

Keywords: evolving network model; functional connectivity; hebbian plasticity; network diffusion; robustness; specificity; structural plasticity; structure function relation.

Figure 1

Adjacency matrices for two example networks evolving through adaptive rewiring with different values of τ. The smaller rewiring interval (τ = 3; upper row of adjacency matrices), produces a modular network, i.e., with dense connections within communities and sparse connections between them. The larger rewiring interval (τ = 4.5; lower row of adjacency matrices) has a different effect on the final rewired network; it gives rise to a centralized connectivity structure, where a few nodes have a large number of connections and the rest are sparsely connected or unconnected.

Figure 2

Modularity for both normal and lognormal networks grows as τ increases, but then decreases for larger τ values. (A) Q as a function of τ for p_random = 0. (B) Same as (A) for p_random = 0.2. Vertical lines indicate standard deviations from 100 instantiations of the rewiring algorithm.

Figure 3

For larger τ values, rewired networks have a large proportion of degree outliers from the mean with the majority of the nodes being sparsely connected and some heavily connected. (A) Proportion of nodes with outlier degrees as a function of τ for normal and lognormal networks. Vertical lines indicate the standard deviations from 100 instantiations of the rewiring algorithm. (B) The degree distribution for modular normal and lognormal networks (left to right; τ_normal = 3, τ_lognormal = 4.5). Inset plots show the corresponding strength distributions. (C) Same as in (B), but for centralized networks (τ_normal = 5, τ_lognormal = 7). In all cases p_random = 0.2. For (B,C) we took the aggregate of 1,000 rewiring instantiations and normalized them so that the sum of the proportions adds to 1.

Figure 4

Normal networks show a uniform modularity distribution and are more prone to connectivity transitions after slight perturbations to τ_transition compared to lognormal ones which have a modularity distribution with a distinct peak. (A) Normal network. Modularity distributions for τ values at and near the transition point: (τ_transition – 2δτ, τ_transition – δτ, τ_transition, τ_transition + δτ, τ_transition + 2δτ) = (3.95, 4.05, 4.15, 4.25, 4.35). For each τ we obtained 1,000 modularity values each corresponding to a different instantiation of the rewiring algorithm (B). Same as (A) for the lognormal network τ = (5.3, 5.4, 5.5, 5.6, 5.7).

Figure 5

The rewiring process for lognormal networks at τ_centralized shows stability. (A) Scatter plot of the modularity values, Q, of networks after 4,000 rewirings at τ_modular (τ_normal = 3, τ_lognormal = 4.5) against those of their initial random configuration show no correlation. (B) Networks in the transition point (τ_normal = 4.15, τ_lognormal = 5.5). Scatter plots show weak positive correlation. (C) For normal networks in the centralized regime there is still no correlation, however the modularity of the random network for lognormal networks is positively correlated with the final rewired network (τ_normal = 5, τ_lognormal = 7), with a linear fit of slope 1 and intercept 0.

Figure 6

Robustness and specificity of the rewiring process depend on the value of τ_test. The Q values of networks after 4,000 and 8,000 rewirings along with their linear fits are shown. For the first 4,000 rewirings τ_transition (τ_normal = 4.15, τ_lognormal = 5.5) was used, for the subsequent 4,000 rewirings τ_test. We collect data from 1,000 rewiring instantiations of a condition. In all cases p_random = 0.2. (A) τ_test = τ_modular (τ_normal = 3, τ_lognormal = 4.5). (B) τ_test = τ_modular. (C) τ_test = τ_centralized (τ_normal = 5, τ_lognormal = 7).

Figure 7

Lognormal networks show greater flexibility compared to normal networks. Squared Pearson correlation coefficient (R²) between Q₄₀₀₀ and Q₈₀₀₀ for different τ_test values. We used bootstrapping to calculate the mean and standard deviation at each point. More specifically for each point we randomly selected 100 Q pairs (Q₄₀₀₀, Q₈₀₀₀) with replacement from a sample of 200 pairs and estimated R². We repeated this process 1,000 times. We calculated the mean and standard deviation from this 1,000 generated data.