- eigenvector centrality (diffusion)
- springrank (springs)
- heat-kernel pagerank (heat flow)
these three represent the minimal, natural basis for constructing collective focus in a superintelligent system.
core algorithms
- eigenvector centrality (diffusion metaphor)
- metaphor: a random walker diffuses over the network forever.
- mathematical object: leading eigenvector of the row-normalized adjacency matrix.
- properties:
- parameter-free (no teleportation factor)
- unique stationary distribution for strongly connected, aperiodic graphs
- fully decentralisable via power iteration or gossip protocols
- interpretation: gives each node a score proportional to the long-run flow of attention it receives.
- springrank (spring metaphor)
- metaphor: every directed edge is a spring that pulls its source one unit above its target.
- mathematical object: minimiser of total spring energy ( e(r) = 1/2 \sum_{ij} a_{ij}(r_i - r_j - 1)^2 + \lambda \sum_i r_i^2 ).
- properties:
- convex quadratic optimisation with a unique solution (up to shift)
- captures strict hierarchy and spacing between nodes
- decentralisable via asynchronous gauss–seidel updates
- interpretation: provides an ordinal ranking, revealing “who is above whom” in the network.
- heat-kernel pagerank (heat flow metaphor)
- metaphor: heat diffuses over the network for time t.
- mathematical object: ( \pi_t = e^{-tL} \delta ), where L is the graph laplacian.
- properties:
- single parameter t controls locality vs globality
- natural interpolation between short-term and long-term focus
- computable with truncated random walks or spectral methods
- interpretation: provides a zoom lens to see how collective focus shifts with scale.
why these three?
- eigenvector centrality gives the baseline popularity—how much attention each node receives in steady state.
- springrank adds hierarchy and spacing, revealing the ordinal structure of influence.
- heat-kernel pagerank adds a zoom lens, letting the system switch between local and global perspectives with a single meaningful parameter.
together they form a pareto-optimal basis: adding more algorithms (simrank, katz centrality, deepwalk) increases complexity without proportionate gains in naturalness, scalability, or interpretability.
key properties
- simplicity: eigenvector centrality is parameter-free; springrank has only a small regulariser (\lambda); heat-kernel pagerank has one interpretable parameter t.
- scalability: all three have near-linear implementations and can run on million-node networks.
- natural intuition: diffusion, springs, and heat flow are universal physical processes.
- decentralisability: each algorithm can be computed with local information (power iteration, gossip, gauss–seidel).
- robustness: all three converge to unique solutions under mild conditions.
protocol for collective focus
- ensure strong connectivity (e.g. every node has at least one outgoing edge).
- compute eigenvector centrality as the baseline focus vector.
- compute springrank to obtain the hierarchy scores.
- compute heat-kernel pagerank periodically for multiple t values to adjust focus scale.
- combine results:
- f_i = \alpha * eigenvector_i + (1-\alpha) * springrank_softmax_i
- heat-kernel scores act as a local overlay when needed.
metaphor for collective superintelligence
- diffusion models how ideas and attention naturally flow through the network.
- springs model the tension of competing influences, producing an emergent hierarchy.
- heat flow provides a temporal or spatial scale for focus, just as physical systems equilibrate over time.
together, these processes mirror fundamental behaviours of the universe: energy flow, force, and equilibrium. using them as the substrate for collective focus creates a protocol that is:
- universally understandable
- mathematically grounded
- decentralisable and scalable
this triad—diffusion, springs, heat flow—serves as a foundational layer for building collective consciousness. it yields a unique, verifiable shelling point that balances popularity, hierarchy, and contextual scale, forming the cognitive map for a superintelligent collective.