Which way for simplifying Nonlinear Dynamics at the “Edge” ?
Thursday, October 27th, 2011 by Antonio ManzaliniYesterday I’ve made a talk at ITU Telecom World 2011 on challenges and opportunities of future networks: in particular, I’ve focused the discussion on the “edge”. Technology advances and the related cost reductions are paving the way to a wider and wider embedding of communications, storage and processing powers inside any network nodes, machines, smart things and any Consumers’ electronics devices.
Then we can argue that future networks, more specifically at the edge, will interconnect a huge number of real and virtual entities providing the Users with any services by using local processing and storage resources.
The thesis I’ve put forward during the talk has been that this future growing complexity of edge networks will require looking at them with the mathematical instruments of nonlinear dynamics. Traditional approaches will be no longer applicable.
Nonlinear dynamics is the theory of nonlinear systems and processes, those where the “result” is not proportional to the “cause”. It includes theory of deterministic chaos (which doesn’t mean random disorder). Chaotic systems behave like there were stochastic but in fact they are deterministic: they show predictability in a short-time-scale but non-predictability in a long-time scale due to extremely high sensitivity to initial conditions and to system’s parameters.
So, I’ve argued that the myriad of real-virtual entities interacting with each other at the edge will behave like a chaotic system, a sort of dynamical game where rules will change dynamically. The challenge is modelling the related dynamics (beyond Nash equilibria) and mastering said complexity to extract value.
Overall the vision has been well received and it has created an interesting debate; one of the most interesting questions I’ve got has been: how can we really apply nonlinear dynamics (which has per se a rather complicated mathematics) to such a complex environment as future edge networks, where there will be millions of nodes? I’ve replied that, obviously, I don’t have an answer today, but I a relatively strong feeling, which follows.
In physics state of a system in a given moment of time is characterized by values of state variables (i.e. data). The minimum number of independent state variables that are necessary to characterize the system’s state is called the number of degrees of freedom: if a system has n degrees of freedom then any state of the system may be characterized by a point in an n-dimensional space (with appropriately defined coordinates) called the system’s phase space. Attractor is a subset of the system’s phase space that attracts trajectories (i.e. the system tends towards the states that belong to some attractor).
Why don't model states attractors rather than single neurons ?
The behaviour of a local network (or recursively a node), like a nonlinear systems that change with time, is dominated by a relatively small number of “attractors”, which correspond to activity patterns (i.e. eventually sets of data). So let’s abstract said behaviour of a local network with the attractors. Then, let’s suppose that these local networks communicate with each others by means of multiple connections, that is, by activity patterns (i.e. sets of data). Let’s abstract also this: the degree of influence that the state of one local network would have on the state of other ones would be given by a “multi-dimensional matrix” coupling attractor states.
Turn nonlinear complexity into interactions of attractors and you’ll get a simpler picture. What if we push this approach even up to getting User’s behaviours in terms of attractors ? I think that for a Telecom Operator it’s better knowing the anonymous Users’ attractors rather than the termination attached to them.
We are staying in contact with the guys that I met at the conference to make a toy model and some simulations.
I’ll keep you posted.
