More precisely, by what scientific criterion can we say that an element of a physical network is (im)material?

Since the scientific method works by studying effects to infer causes, how can one distinguish at the scientific level a material cause from an immaterial cause?

One of my answers, in view of the current scientific mentality: it's not possible. Science doesn't deal with "material" causes, but with causes. To those who say that science proves that only "material" exists, I respond that this is not the case: science tries to deduce causes from their presence, to produce a predictive model. If there's a still unknown cause, no one will consider it immaterial: one of the positions of the scientific method will be to make predictions about what can be understood from it.

Edward Feser, one of the Thomist philosophers I like to follow, makes this critique of "mechanistic philosophy", very present at the beginning of modern sciences (The Last Superstition, 2008, p. 179):

The original idea was to posit the interactions between particles as being as "mechanical" as were the parts of a clock; and to reduce the whole system to a sequence of things pushing others. Such a paradigm didn't last long, because it's simply impossible to explain everything that happens in the material world with such a simplistic model, and as Newton's theory of gravitation, Maxwell's theory of electromagnetism, and quantum physics show, physical science has only moved further and further away from the original idea of "mechanism". [...] Of the original idea of mechanistic philosophy, all that now remains is the simple denial of Aristotelian final cause.

To say that science shows that only the material exists is a misunderstanding of what science does: science excludes the immaterial not by scientific advances, but by principle. As soon as something that wasn't materially conceivable at the origin enters the scientific domain, it's included by a modeling of the problem.

I extend this critique to those who say that the supernatural is impossible, since only the natural exists; or that everything is physical, through Hempel's dilemma. A German philosopher, Carl Gustav Hempel, formulates the dilemma as follows: if we say that everything reduces to physical laws, then either what we say is useless, or grossly false. Indeed, what physical laws are we talking about? Either we're talking about "physical laws" in the current state of discoveries, but this is false, because although we have a general idea of what can be considered physical, there are phenomena that we still don't know how to correctly explain (I'm thinking of the explanation of the mind/body problem, or dark matter, for example); or we're talking about an "ideal" physics, which will correspond to what we may have one day, but in this case, we know nothing about what we're talking about... and the justification becomes circular: everything is physical, because ideally defined physics allows ideally defining what exists.