One Equation to Rule Them All

The equation that quietly describes an astonishing range of interacting systems, yet not as popular as it deserves to be. Is it because it is deceptively simple and appears in different fields under different names? It's called the generic networked dynamical system equation, the coupled dynamical system equation or interacting dynamical system equation.

Konjugate, at it's core, could be considered a mere user interface for this gem. The mathematical pattern appears in very different domains:

Field	xi	bij	si
Social networks	Opinion of person i	Influence from others	Personal bias or media
Epidemiology	Infection state	Disease transmission	Vaccination or treatment
Robotics	Robot state	Communication with neighbors	Local controller
Power grids	Generator state	Electrical coupling	Mechanical input
Ecology	Population	Predator/prey interactions	Birth/death processes
Neuroscience	Neuron activity	Synaptic inputs	External stimulus
Heat Transfer	Temperature of node i	Thermal coupling (conduction, convection, radiation)	Internal heat generation or external heating/cooling

When a mathematician come across this sort of an equation for the first time, or if you ask a seasoned engineer to explain about a system along this direction, it is more than likely that they would be thinking about or making a whiteboard drawing of a graph with a few nodes and edges, with the state variable (x_i) at the center of the node, interaction between the nodes (b_ij) along the edges and the external input, a.k.a. the source term, (s_i) as an arrow into the node.

And yet, modern simulation tools following model-based design, at least the popular ones, are not designed to be as intuitive as this graph. It is obvious that the amount of "blocks" and "connections" required to model a system would get drastically reduced if a tool is built that is literally centered around this graph. And that, my friends, is the reason why Konjugate in being built.

Konjugate is being built in public as an open-source tool licensed under the Mozilla Public License 2.0. If you are excited to see the idea of graph-first modeling evolving, you're invited to follow its development on GitHub.

Whether your interest lies in mathematics, engineering, physics, biology, social sciences or software development, the goal is the same: to make modeling complex interconnected systems as intuitive as the graph we naturally draw on a whiteboard.