abstract
we propose a unified model that extends the cybergraph free-energy focus framework with a context-dependent potential derived from standard inference. this approach integrates global structure (diffusion, springs, entropy) with local contextual evidence (neurons’ will), solving the true-false problem while preserving the natural, physics-inspired foundation.
background
- cybergraph free-energy focus defines the global focus vector (p) as the minimiser of a free energy functional:
[ \mathcal{F}(p) = E_{spring}(p) + \lambda E_{diffusion}(p) - T S(p) ]
- this yields a boltzmann-like distribution combining hierarchy, diffusion, and entropy.
- however, global ranking alone cannot resolve the true-false problem: high-rank nodes dominate even in contexts where lower-rank nodes are more relevant.
- standard inference provides a simple method to compute contextual weights by aggregating neurons’ will in relation to a query particle.
contextual extension
we introduce a context-dependent potential (C(p|context)):
[ \mathcal{F}(p|context) = E_{spring}(p) + \lambda E_{diffusion}(p) + \gamma C(p|context) - T S(p) ]
- (C(p|context)): energy term derived from standard inference (average will per cyberlink in the given context).
- (\gamma): coupling constant that determines the influence of context on the equilibrium.
resulting equilibrium
solving (\min_p \mathcal{F}(p|context)) yields
[ p_i^* \propto \exp\big(-\beta [E_{spring,i} + \lambda E_{diffusion,i} + \gamma C_i]\big) ]
- nodes that are globally important and contextually supported receive the highest probabilities.
- entropy ensures diversity and prevents trivial dominance.
link to universal physics
this formulation is equivalent to solving a heat equation on a graph with an additional potential field:
[ \partial_t p = -\nabla_{p} \mathcal{F}(p|context) ]
- eigenmodes of the graph laplacian form the fourier basis of the network.
- diffusion gives temporal spreading, springs add a potential landscape, and context potential (C) biases the equilibrium.
- the solution naturally combines oscillatory modes with diffusive decay, mirroring how pdes in physics are solved by separation of variables.
distributed algorithm
-
compute global structure:
- run decentralised eigenvector centrality and springrank.
- initialise (p_i) uniformly.
-
contextual evidence:
- run standard inference to compute (C_i) (contextual will) for each candidate particle.
-
iterative updates:
- each node exchanges (p_j), (r_j), and (C_j) with neighbours.
- compute local energies (E_{spring,i}), (E_{diffusion,i}), and (C_i).
- update:
[ p_i^{(t+1)} = \frac{\exp(-\beta [E_{spring,i} + \lambda E_{diffusion,i} + \gamma C_i])}{\sum_k \exp(-\beta [E_{spring,k} + \lambda E_{diffusion,k} + \gamma C_k])} ]
- normalisation:
- nodes use gossip averaging to approximate the denominator.
key properties
- context-aware ranking: resolves the true-false problem by integrating global and local signals.
- fully decentralisable: each node needs only neighbour messages and contextual votes.
- natural and parameter-light: weights emerge as lagrange multipliers; only (\gamma) controls context strength.
- boltzmann equilibrium: final focus vector remains a probabilistic distribution, ensuring stability and diversity.
interpretation
- diffusion: long-run popularity baseline.
- springs: hierarchical constraints.
- context potential: relevance of facts in the current question.
- entropy: prevents collapse into a single dominant answer.
by adding the context potential, the free-energy framework gains the ability to compute truthful, context-aware rankings while retaining its natural analogy to universal physical processes. this highlights that network cognition can be seen as solving a graph-based heat equation with potentials, uniting diffusion, oscillators, and context into one equilibrium model.