abstract
we propose a natural, physics-inspired method for computing collective focus on a cybergraph. instead of manually weighting different ranking algorithms, we derive a unified focus vector as the minimum of a free energy functional that combines diffusion (eigenvector centrality), springs (springrank), and heat flow (locality control). this approach ensures that focus emerges as an equilibrium state, requiring no arbitrary tuning. the result is a scalable, decentralisable, and universal protocol for collective intelligence.
motivation
existing ranking mechanisms (pagerank, springrank, etc.) either focus on popularity (diffusion) or hierarchy (spring energy). combining them usually requires ad hoc weights ((\alpha, \beta)), which is unnatural in a physical sense. in real systems, weights emerge from the dynamics of energy and entropy minimisation.
in physics and information theory, equilibrium distributions arise from free energy minimisation:
[ \mathcal{F}(p) = E - T S ]
where (E) is energy, (S) is entropy, and (T) is temperature. the equilibrium distribution (p^*) minimises (\mathcal{F}), leading to a boltzmann distribution.
core idea
we define a joint free energy functional for the cybergraph:
[ \mathcal{F}(p) = E_{spring}(p) + \lambda E_{diffusion}(p) - T S(p) ]
- (E_{spring}(p)): springrank energy, encoding hierarchy constraints.
- (E_{diffusion}(p)): diffusion energy, penalising flow mismatches in random walk.
- (S(p) = -\sum_i p_i \log p_i): entropy of the focus distribution.
- (\lambda): coupling constant linking hierarchy and diffusion.
- (T): effective temperature controlling exploration vs exploitation.
equilibrium distribution
the optimal focus vector is
[ p^* = \arg\min_p \mathcal{F}(p) ]
solving yields a boltzmann-like distribution:
[ p_i^* \propto \exp\big(-\beta [E_{spring,i} + \lambda E_{diffusion,i}]\big), ]
where (\beta = 1/T).
why this is natural
- no arbitrary weights – the weights emerge automatically as lagrange multipliers from the optimisation problem.
- physics analogy – matches how statistical mechanics derives boltzmann distributions.
- information theory analogy – equivalent to maximum-entropy inference under graph constraints.
- pareto optimality – diffusion, springs, and heat flow form the minimal universal basis for modelling spreading, hierarchy, and locality.
distributed computation
local energy terms
- springrank energy for node i:
[ E_{spring,i} = \frac{1}{2} \sum_j a_{ij} (r_i - r_j - 1)^2 ]
- diffusion energy for node i:
[ E_{diffusion,i} = \frac{1}{2} \sum_j w_{ij} (p_i - p_j)^2 ]
iterative update rule
[ p_i^{(t+1)} = \frac{\exp(-\beta [E_{spring,i} + \lambda E_{diffusion,i}])}{\sum_k \exp(-\beta [E_{spring,k} + \lambda E_{diffusion,k}])} ]
protocol
- initialise (p_i^{(0)} = 1/n) for all nodes.
- compute springrank (r_i) via decentralised gauss–seidel updates.
- at each iteration, nodes exchange (p_j) and (r_j) with neighbours.
- compute local energies and update (p_i).
- use gossip averaging to normalise probabilities globally.
properties
- unique equilibrium: convex free-energy functional ensures a single global minimum.
- scalable: per-iteration cost (o(\text{deg}(i))) per node.
- decentralisable: only local neighbour communication required.
- natural: no arbitrary parameters; weights emerge from constraints.
interpretation
- diffusion: traffic or popularity baseline.
- springs: hierarchy and relative status.
- heat flow: contextual scale (via temperature (t)).
together they form a 3d focus landscape: traffic density (diffusion), elevation (hierarchy), and zoom scale (heat flow). the system converges to a boltzmann–gibbs equilibrium, a natural shelling point for collective superintelligence.
significance
this framework replaces ad hoc weighting with a principled physical method for computing collective focus. it aligns artificial cognition with fundamental laws of energy and entropy, providing a universal, scalable, and decentralisable substrate for emergent collective consciousness.