Why do some systems, from gene networks to air travel, preserve imprints of the past longer than others? For a long time, network science blamed topology: the number of connections, modularity, spectrum. But two networks with the same skeleton can remember and forget in completely different ways. The key is in the weight geometry: how interaction strengths are distributed along directed paths. The binary structure is just a canvas; the memory picture is painted by the weights.

Imagine a cup of hot coffee. Heat escapes unevenly: the shape, wall thickness, and material set the cooling rate. Network memory is like this heat. Each step into the past is a layer that 'cools down' and stops influencing the present. In the new study, scientists constructed a thermodynamic cascade inspired by the ideas of statistical physics and performed large-scale numerical simulations. They introduced a temperature analog, a parameter that suppresses the contribution of long trajectories, and determined the moments of 'thermalization,' when memory of depth n becomes indistinguishable from equilibrium noise. This revealed a picture: real weights, like a thermos, retain the heat of the distant past, while randomness would have cooled everything long ago.

In 32 of the 34 studied networks — from E. coli genes to connectomes — real weights systematically deepened functional memory compared to random shuffling on the same topology. The mechanism is intuitive: strong interactions correlate along directed paths, amplifying the contribution of the distant past. Randomization destroys these corridors, and memory fades. Applying Gaussian mixtures for automatic classification, the authors identified four dynamic regimes: diffuse networks (broad memory spectrum, like in E. coli), reactive (deep memory with concentrated compression, many food webs), stable, and impulsive — where memory collapses at a single scale, like in air travel or Photosystem I. These regimes resemble states of matter: diffuse gas, reactive liquid, stable crystal, and impulsive plasma — perhaps they are just as fundamental for information organization. Quantitatively, the long-memory measure NMR in real systems is on average 12% higher than in 'shuffled' counterparts.

The discovery undermines the exclusive role of binary topology and calls for a revision of models. The developed GFM test serves as a litmus test: if weight randomization does not change memory depth, the weights carry no functional information. In thermodynamic terms, memory suppression is analogous to the growth of entropy, and the thermalization cascade links memory organization to the second law. Directionality of edges modulates sensitivity: in neural networks it suppresses deep memory, in food webs it supports it. The future lies with temporal and multilayer networks, where weights evolve, giving rise to competing regimes. The connection to algorithmic complexity and Bennett's logical depth promises universal indicators of organization.

We are used to thinking of memory as a property of consciousness or a computer tape. But any network that processes information remembers: its memory is written not in neurons, but in the pattern of weights. This work opens the way to designing resilient infrastructures and understanding cognitive processes where integrating history is crucial. Perhaps we will learn to tune the thermodynamics of memory in artificial systems, creating machines that remember the past with human depth and forget the unnecessary, like a cup of coffee giving up its heat without residue.