Symmetry-based coarse-graining of evolved dynamical networks

Abstract

-Networks with a prescribed power-law scaling in the spectrum of the graph Laplacian can be generated by evolutionary optimization. The Laplacian spectrum encodes the dynamical behavior of many important processes. Here, the networks are evolved to exhibit subdiffusive dynamics. Under the additional constraint of degree-regularity, the evolved networks display an abundance of symmetric motifs arranged into loops and long linear segments. Exploiting results from algebraic graph theory on symmetric networks, we find the underlying backbone structures and how they contribute to the spectrum. The resulting coarse-grained networks provide an intuitive view of how the anomalous diffusive properties can be realized in the evolved structures.Introduction. -Networks have become a principal tool for the modeling of complex systems in a broad range of scientific fields [1][2][3]. In a dynamical network the topology describes the couplings between individual units of a dynamical process [4]. The general underlying question of research on dynamical networks is how the network structure shapes the global dynamical behavior.An important class of processes on networks are Laplacian dynamics in which the graph Laplacian is the operator of the (linearized) time evolution. The Laplacian spectrum and eigenvectors then describe the overall dynamical behavior. This class comprises many physical processes such as dynamics of Gaussian spring polymers [5], transport processes [6], random walks [7], and synchronization of oscillators [8][9][10].Another dynamical aspect is the evolution of network structure. Many systems change their connectivity structure in the course of time which affects the global dynamical behavior. The time scales of these two processes are often well separated with fast dynamics and slowly responding structural evolution. As an example, think of changes in neuronal activity in a brain on the one hand and the formation of synaptic connections on the other hand. In the opposite case, when dynamics and evolution happen on similar time scales one speaks of coevolutionary or adaptive networks [11]. Since the functionality of a system is often closely associated with dynamical behavior evolutionary forces will drive an evolving dynamical network towards some optimized structure.The method of network evolution adopts this strategy

Exploiting symmetry in network analysis

“…To facilitate dissemination, we provide full implementations of all the algorithms described in this article 26 . Our results supersede 11,13 and help to understand other network symmetry results thereafter 12,[27][28][29][30][31] . We focus on structural and spectral properties, and symmetries commonly found in real-world networks: For a more general study of arbitrary symmetry in (networks of) dynamical systems, see refs.…”

Reconstruction of evolved dynamic networks from degree correlations

“…Finally, we remark that at this point we cannot answer the question why the observed core-periphery structures generate the power-law Laplacian spectra and, thus, anomalous diffusion behavior. In a different settingwhere evolving networks were kept homogeneous-it was observed that networks with loops and dangling ends of different lengths are also able to generate such behavior [35]. A distribution of paths of different lengths through the network might as well be generated by the core-periphery structures found here.…”

“…In both cases the integrated spectral densities display several kink-like features which reflect accumulations of eigenvalues. A possible mechanism underlying such spectral degeneracies are network symmetries [35]. These features are absent from the spectra obtained using the Co and Co-Cl algorithms, which both resemble the target power law closely and almost equally well.…”

The Renormalization Group and Its Applications to Generating Coarse-Grained Models of Large Biological Molecular Systems

“…While this is not by itself a deterrent to the application of Decimateas it only needs to be used once on a systemthere is definitely room for improvement. For symmetric systems such as viral capsids, their symmetric organization may provide simplification . More generic systems require aggressive optimizations of the current implementation of Decimate.…”

“…For symmetric systems such as viral capsids, their symmetric organization may provide simplification. 76 More generic systems require aggressive optimizations of the current implementation of Decimate. We reserve those improvements for future studies.…”