Referential Calculus Across Domains

Referential Calculus for Multiscale Network Dynamics Daniel Spargo Abstract This note presents a compact operator formalism for multiscale modeling on graphs that preserves node identity during aggregation. A microstate is treated as an indexed referential family together with a relation object (graph/kernel). A shell map defines identity-preserving “shell objects” and a coarse operator producing macro representatives (e. g. , shell means). For linear flows on graphs, we define an evolution–coarse-grain defect (RG defect) and a residue functional that measures how well coarse-graining commutes with dynamics over a time window. The residue provides a falsifiable criterion for when a shell/frame can be treated as a macro node. We prove a synchronization-based closure bound and give a canonical refinement rule that splits low-coherence shells using spectral (Fiedler) cuts. A minimal worked example illustrates how incorrect shell choices produce large RG residue. The framework is intended as an implementation-ready calculus for identity-aware coarse-graining and adaptive multiscale structure discovery in networked dynamical systems. 1. Introduction Problem: coarse-graining graph/network dynamics without losing identity. Motivation: partitions are often ad hoc; need a diagnostic and refinement principle. Contribution: identity-preserving shell objects + RG defect/residue + synchronization closure + spectral refinement. 2. Core objects and notation 2. 1 Referential family (identity-preserving state) Let I=\1, , N\ index nodes and let xᵢᵐ. Define the referential family X=R₈ ₈ xᵢ, ᵢ (X) =xᵢ. 2. 2 Decorated system Let R be a relation object (weighted adjacency/kernel). Define the decorated system Y=R (X, R) = (X, R). 2. 3 Dynamics (linear baseline) Assume a generator =G (R) (e. g. , Laplacian-based). For a scalar channel N, =, (t) =e^t₀. 3. Shell construction and coarse-graining 3. 1 Shell map and shell object Let f: I K define a partition into M shells, K=\1, , M\. Define the identity-preserving pushforward XK=f\*X, (XK) ₖ=R₈ ₅^-₁ (k) xᵢ. 3. 2 Representative extractor and coarse operator Let G map each shell fiber to a representative (default mean). Define T₅, ₆=G f\*, X=T₅, ₆ (X). 3. 3 Matrix form for scalar channels Define T^M N by (T) ₖ=1|f^{-1 (k) |}₈ ₅^-₁ (k) ᵢ. Define the synchronized lift S^N M by (S) ᵢ=₅ (₈), so TS=IM and ST=f. 4. RG defect and residue (the diagnostic) 4. 1 Macro generator (closure-aligned default) K: = T S. 4. 2 Defect operator ₑ₆ (t): =Te^t-e^tKT. 4. 3 Residue on a probe initial condition S^RG_ (t) =\|ₑ₆ (t) ₀\|², J (f) =₉=₁ᵖ wⱼ\, S^RG_ (tⱼ). Interpretation: J (f) measures how well “evolve → coarse” matches “coarse → evolve” for the chosen shells. 5. Synchronization closure (when “frame = node” is valid) Proposition 1 (Synchronization-based closure, informal statement) Decompose =₈₍+₎ₔₓ into within-shell and cross-shell parts. If internal mixing contracts within-shell disagreement at rate ₈₍, then the macro evolution tracks shell means with an error bounded by a term that decays like e^-₈₍t and a forcing term scaling like \|₎ₔₓ\|/₈₍. Insert the bound you liked: \| X (t) - X (t) \| C e^-₈₍t (0) +C₀ᵗ e^-₈₍ (t-s) \|₎ₔₓ\|\|X (s) \|ds. 6. Spectral refinement (how shells adapt) 6. 1 Refinement trigger Split shell k if its internal spectral gap ₂ (L^ (k) ) is small or if J (f) fails to decrease. 6. 2 Canonical split (Fiedler cut) Compute the Fiedler vector of the shell’s normalized Laplacian and split by sign to obtain f f'. Iterate until J (f) saturates. 7. Minimal worked example (3 nodes) Define a 3-node path graph 1\!-\!2\!-\!3. Compare two partitions: \1, 2\|\3\ vs \1, 3\|\2\. Show ₈₍>0 for the first and ₈₍=0 for the second. Compute/describe that J (f) is much larger for the disconnected-shell partition.