Pattern Formation over Multigraphs

Andras Gyorgy, Murat Arcak

Research output: Contribution to journalArticlepeer-review


Two of the most common pattern formation mechanisms are Turing-patterning in reaction-diffusion systems and lateral inhibition of neighboring cells. In this paper, we introduce a broad dynamical model of interconnected modules to study the emergence of patterns, with the above mentioned two mechanisms as special cases. Our results do not restrict the number of modules or their complexity, allow multiple layers of communication channels with possibly different interconnection structure, and do not assume symmetric connections between two connected modules. Leveraging only the static input/output properties of the subsystems and the spectral properties of the interconnection matrices, we characterize the stability of the homogeneous fixed points as well as sufficient conditions for the emergence of spatially non-homogeneous patterns. To obtain these results, we rely on properties of the graphs together with tools from monotone systems theory. As application examples, we consider patterning in neural networks, in reaction-diffusion systems, and in contagion processes over random graphs.

Original languageEnglish (US)
Pages (from-to)55-64
Number of pages10
JournalIEEE Transactions on Network Science and Engineering
Issue number1
StatePublished - Jan 1 2018


  • Nonlinear dynamics
  • large-scale systems
  • multigraphs
  • networks
  • pattern formation

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Computer Networks and Communications


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