Abstract
Compression, filtering, and cryptography, as well as the sampling of complex systems, can be seen as processing information. A large initial configuration or input space is nontrivially mapped to a smaller set of output or final states. We explored the statistics of filtering of simple patterns on a number of deterministic and random graphs as a tractable example of such information processing in complex systems. In this problem, multiple inputs map to the same output, and the statistics of filtering is represented by the distribution of this degeneracy. For a few simple filter patterns on a ring, we obtained an exact solution of the problem and numerically described more difficult filter setups. For each of the filter patterns and networks, we found three key numbers that essentially describe the statistics of filtering and compared them for different networks. Our results for networks with diverse architectures are essentially determined by two factors: whether the graphs structure is deterministic or random and the vertex degree. We find that filtering in random graphs produces much richer statistics than in deterministic graphs, reflecting the greater complexity of such graphs. Increasing the graph’s degree reduces this statistical richness, while being at its maximum at the smallest degree not equal to two. A filter pattern with a strong dependence on the neighbourhood of a node is much more sensitive to these effects.
Highlights
Many systems of great interest from different domains, from the brain to ecosystems to social systems and technological systems, share the characteristic of complex behaviours that emerge from the interactions between their numerous elements
In a previous work [14], we showed that a simple filtering problem produces analogous behaviour of the Entropy 2020, 22, 1149; doi:10.3390/e22101149
We demonstrate this process by calculating the full degeneracy distributions for various degree regular graphs with 30 or more nodes, while using two example filter patterns
Summary
Many systems of great interest from different domains, from the brain to ecosystems to social systems and technological systems, share the characteristic of complex behaviours that emerge from the interactions between their numerous elements. The filter outputs a 1 for every instance of a particular pattern of states on a node and its immediate neighbours, and a 0 when the pattern is absent. This generalises the filtering problem that is examined in Ref. We studied this problem on a variety of degree-regular graphs. Just as in our previous study on rings, we show that the principal characteristics of the degeneracy distribution are asymptotically described by three key numbers These numbers may be obtained exactly by simple arguments. Our results for regular graphs of diverse architectures essentially depend only on a vertex degree, as Figure 6 demonstrates
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
