Structural self-assembly and avalanchelike dynamics in locally adaptive networks

Johannes Gräwer, Carl D. Modes, Marcelo O. Magnasco, and Eleni Katifori

Phys. Rev. E 92, 012801 – Published 2 July 2015

Abstract

Transport networks play a key role across four realms of eukaryotic life: slime molds, fungi, plants, and animals. In addition to the developmental algorithms that build them, many also employ adaptive strategies to respond to stimuli, damage, and other environmental changes. We model these adapting network architectures using a generic dynamical system on weighted graphs and find in simulation that these networks ultimately develop a hierarchical organization of the final weighted architecture accompanied by the formation of a system-spanning backbone. In addition, we find that the long term equilibration dynamics exhibit behavior reminiscent of glassy systems characterized by long periods of slow changes punctuated by bursts of reorganization events.

Received 30 May 2014
Revised 6 April 2015

DOI:https://doi.org/10.1103/PhysRevE.92.012801

Authors & Affiliations

Johannes Gräwer¹, Carl D. Modes², Marcelo O. Magnasco², and Eleni Katifori^1,3,*

¹Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Goettingen, Germany
²Center for Studies in Physics and Biology, The Rockefeller University, New York, New York 10065, USA
³Department of Physics and Astronomy, University of Pennsylvania, 19104, Philadelphia, Pennsylvania 19104, USA

^*katifori@sas.upenn.edu

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 92, Iss. 1 — July 2015

Reuse & Permissions

Access Options

Article Available via CHORUS

Download Accepted Manuscript

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) and (b) Time-evolution of the network conductivities. The conductivities are colored based on the magnitude of the normalized maximum derivative of the conductivity [plotted in (c)] during the rearrangement event around $t = 120$ . After an initial period of transient dynamics ( $t ≃ 0 - 10$ ), the system settles into a pattern of long times of slow or steady dynamics punctuated by rapid avalanchelike rearrangements of the edge conductivities, such as the periods highlighted in yellow (the last four rearrangement events seen for this dataset). Only a subset of the edges in the network typically participate in each of these avalanche events.
Reuse & Permissions
Figure 2
Time dependence of distance between datasets with the same topology but different initial conditions $G (t)$ for $γ = 1.4$ (a) average of ER topologies, (b) average of BA topologies, and (c) average of WS topologies. $G (t)$ does not relax to zero, indicating a dependence on the initial conditions.
Reuse & Permissions
Figure 3
Time dependence of distance between datasets with the same topology but different initial conditions $G (t)$ for $γ = 0.9$ (a) average of ER topologies, (b) average of BA topologies, and (c) average of WS topologies. $G (t)$ decreases to zero in all cases, indicating independence of initial conditions.
Reuse & Permissions
Figure 4
The development of a self-assembled backbone in the network conductivities under the action of the sigmoidal local feedback function. (a) Left panel: The initial state of an ER network with the top 10% conductive edges shown in red. These edges are randomly placed around the network with no additional underlying structure or connectivity. Middle panel: The same network during the initial transient dynamics ( $t = 5$ ). Some rearrangement has occurred among the most conductive edges, but no clear structure has yet emerged. Right panel: The network final state ( $t = 1000$ ) after avalanche events have apparently ceased. The most conductive edges now form a contiguous chain. (b) Initial, intermediate, and final states for a BA network. (c) Initial, intermediate, and final states for a WS network. (d) Normalized average shortest path length $s (t)$ for WS, BA, and ER graphs. $s (t)$ decreases as the backbone is formed. The inset shows $s (t)$ in logarithmic scale in the initial stages of the dynamics.
Reuse & Permissions
Figure 5
The dependence of the evolved “loopiness” of a network on the exponent $γ$ in the sigmoidal feedback function. ER, WS, BA, and four-regular initial underlying network topologies are all considered. In all cases there is a threshold value $γ^{*} \approx 1$ (marked with a gray line) below which loop evolution does not occur. Above $γ^{*}$ all networks rapidly shed loops until $L / max (L)$ approaches a constant value $< 1$ and significantly $< 1$ in the case of the BA topology.
Reuse & Permissions

Physical Review E

covering statistical, nonlinear, biological, and soft matter physics