Discrete Harmonic Morphisms: Theory & Applications

• Discrete harmonic morphisms are maps that preserve harmonicity in graphs by enforcing locally constant horizontal multiplicities and balancing conditions.
• They bridge divisor theory, chip-firing, and tropical geometry, effectively pulling back harmonic functions and preserving Laplacian properties in various settings.
• These morphisms extend classical Riemann–Hurwitz identities to arithmetical, metric, and probabilistic frameworks, offering new insights into network coarse-graining.

Discrete harmonic morphisms are structure-preserving maps in discrete and tropical geometry whose defining feature is compatibility with harmonicity in a graph-theoretic, metric, or polyhedral sense. In the classical setting of finite graphs, they are graph morphisms with locally constant horizontal multiplicity above each edge adjacent to a target vertex; equivalently, for surjective maps between graphs, they are precisely the maps whose pullback sends harmonic functions on the target to harmonic functions on the source. The notion sits at the intersection of divisor theory, chip-firing, tropical geometry, metrized complexes, and nonarchimedean skeleta, and it has recently also been used as a criterion for diffusion-preserving network coarse-graining (Archer et al., 11 Apr 2025, Waeterschoot, 18 Mar 2025).

1. Classical graph-theoretic definition

In the standard discrete formulation, one considers connected graphs $\Gamma_2$ and $\Gamma_1$ together with a graph morphism $\phi:\Gamma_2\to\Gamma_1$. For an edge $e=(v,w)\in E(\Gamma_2)$, either $\phi(v)=\phi(w)$ and $e$ is vertical, or $\phi(v)$ and $\phi(w)$ are adjacent and $e$ is horizontal. If $f$ is an edge incident to $\Gamma_1$0, the local horizontal multiplicity $\Gamma_1$1 is the number of edges incident to $\Gamma_1$2 that map to $\Gamma_1$3. The morphism is harmonic at $\Gamma_1$4 when $\Gamma_1$5 is independent of the choice of $\Gamma_1$6, and the common value is the horizontal multiplicity $\Gamma_1$7. In the loopless multigraph notation used in divisor theory, one also writes the vertical multiplicity $\Gamma_1$8, and the local identity is

$\Gamma_1$9

For non-constant harmonic morphisms, the degree is independent of the chosen target edge, and fiberwise sums of horizontal multiplicities recover that degree (Archer et al., 11 Apr 2025, Cao et al., 2022).

A complementary formulation uses harmonic functions directly. For graphs $\phi:\Gamma_2\to\Gamma_1$0 and $\phi:\Gamma_2\to\Gamma_1$1 with their simple random walk Laplacians, a surjective map $\phi:\Gamma_2\to\Gamma_1$2 is a harmonic morphism if, whenever $\phi:\Gamma_2\to\Gamma_1$3 is harmonic at $\phi:\Gamma_2\to\Gamma_1$4, the pullback $\phi:\Gamma_2\to\Gamma_1$5 is harmonic at every $\phi:\Gamma_2\to\Gamma_1$6. Urakawa’s theorem, as used in recent network work, identifies this with horizontal conformality: adjacent fine vertices may only map to adjacent or equal coarse vertices, and for fixed $\phi:\Gamma_2\to\Gamma_1$7 the counts

$\phi:\Gamma_2\to\Gamma_1$8

must be constant for all $\phi:\Gamma_2\to\Gamma_1$9. This equivalence makes precise that harmonic morphisms are discrete analogues of horizontally conformal maps rather than arbitrary graph homomorphisms (Guadagnuolo et al., 9 Apr 2026).

Two points of terminology are essential. First, a harmonic morphism is stricter than a plain graph morphism because it imposes a balancing condition on local edge counts. Second, it is also different from a generic discrete harmonic map into Euclidean space: in the classical theory the target is itself a graph or a closely related combinatorial object, and local multiplicity data play the role of discrete branching.

2. Divisors, ramification, and Riemann–Hurwitz theory

The divisor-theoretic formulation, originating in the Baker–Norine framework and developed further in later work, makes harmonic morphisms behave like branched coverings of curves. For a finite graph $e=(v,w)\in E(\Gamma_2)$0, the canonical divisor is

$e=(v,w)\in E(\Gamma_2)$1

and a harmonic morphism $e=(v,w)\in E(\Gamma_2)$2 satisfies a discrete Riemann–Hurwitz identity

$e=(v,w)\in E(\Gamma_2)$3

where the ramification divisor is

$e=(v,w)\in E(\Gamma_2)$4

Taking degrees yields

$e=(v,w)\in E(\Gamma_2)$5

so non-constant harmonic morphisms satisfy $e=(v,w)\in E(\Gamma_2)$6. In this formulation, the local coefficient $e=(v,w)\in E(\Gamma_2)$7 is the discrete ramification term measuring the discrepancy between the source canonical divisor and the pullback of the target canonical divisor (Cao et al., 2022).

This divisor theory extends systematically beyond finite graphs. For vertex-weighted graphs, one replaces ordinary morphisms by indexed morphisms with edge indices $e=(v,w)\in E(\Gamma_2)$8, and the canonical divisor becomes

$e=(v,w)\in E(\Gamma_2)$9

For metric graphs, horizontal edges carry integer slopes

$\phi(v)=\phi(w)$0

and the ramification term replaces vertical multiplicity by slope defects. For vertex-weighted metric graphs and metrized complexes of algebraic curves, analogous formulas persist, with additional contributions from vertex weights and from finite morphisms on the curve components. In every case the characteristic identity has the same form: canonical divisor on the source equals pullback of canonical divisor on the target plus an explicit ramification divisor (Cao et al., 2022).

A further development is a family of second main theorems in the sense of Nevanlinna theory. In the finite graph case, if $\phi(v)=\phi(w)$1 are distinct target vertices and $\phi(v)=\phi(w)$2, then

$\phi(v)=\phi(w)$3

This turns ramification data into a counting inequality for inverse images of selected target vertices. Weighted, metric, and metrized-complex analogues replace the vertical-edge correction by the corresponding weight and slope defects, thereby placing discrete harmonic morphisms in a value-distribution framework parallel to the classical theory of maps between algebraic curves (Cao et al., 2022).

3. Arithmetical structures and generalized Laplacians

A recent extension studies harmonic morphisms together with arithmetical structures on graphs. An arithmetical structure on a connected graph $\phi(v)=\phi(w)$4 is a pair $\phi(v)=\phi(w)$5 of positive integer functions with $\phi(v)=\phi(w)$6 and

$\phi(v)=\phi(w)$7

The associated arithmetical Laplacian is

$\phi(v)=\phi(w)$8

and the defining condition is equivalent to $\phi(v)=\phi(w)$9. The natural structure is $e$0 and $e$1, which recovers the ordinary graph Laplacian (Archer et al., 11 Apr 2025).

If $e$2 is a non-constant harmonic morphism and $e$3 is an arithmetical structure on $e$4, then the pullback structure on $e$5 is given explicitly by

$e$6

With $e$7 the vertex-map matrix and $e$8 the diagonal matrices of horizontal and vertical multiplicities, the key compatibility is

$e$9

This identity shows that harmonicity is exactly the condition making generalized Laplacians functorial under pullback (Archer et al., 11 Apr 2025).

The same paper extends Baker–Norine-style functoriality to arithmetical critical groups. For a divisor $\phi(v)$0, the degree is weighted by $\phi(v)$1,

$\phi(v)$2

principal divisors are those of the form $\phi(v)$3, and the arithmetical critical group is

$\phi(v)$4

A harmonic morphism induces a surjective pushforward

$\phi(v)$5

and an injective pullback

$\phi(v)$6

The paper also proves an arithmetical Riemann–Hurwitz formula. Writing the arithmetical canonical divisor as $\phi(v)$7 and the ramification divisor as

$\phi(v)$8

one has

$\phi(v)$9

The associated arithmetical genus satisfies a degree formula formally parallel to the classical and Baker–Norine Riemann–Hurwitz theorems (Archer et al., 11 Apr 2025).

4. Metric, tropical, and skeletal generalizations

One line of generalization replaces graphs by higher-dimensional $\phi(w)$0-PL spaces arising as dual complexes of toroidal schemes over a DVR. In that setting, a finite cover $\phi(w)$1 is balanced at a ridge $\phi(w)$2 over $\phi(w)$3 when the local degree

$\phi(w)$4

is independent of the adjacent facet $\phi(w)$5. The central theorem states that a balanced finite cover preserves the tropical Laplacian under pullback,

$\phi(w)$6

For dual complexes of finite toroidal morphisms this yields harmonic morphisms of skeleta, and in dimension one the identification is explicit: ridges become vertices, facets become edges, the local degree at a ridge becomes the classical local degree at a vertex, and the balancing condition is exactly the graph-theoretic harmonicity condition. A further consequence is a skeletal Riemann–Hurwitz formula in any dimension, expressed as

$\phi(w)$7

where $\phi(w)$8 is the different function and $\phi(w)$9 is the tropical relative canonical divisor (Waeterschoot, 18 Mar 2025).

A second, closely related but conceptually distinct line comes from tropical geometry. There a tropical curve $e$0 is a metric graph with infinite leaves, and a harmonic tropical morphism

$e$1

is a proper continuous map that is affine linear on each edge or leaf and satisfies the balancing condition

$e$2

at every vertex. Every such morphism arises uniquely from a residue matrix $e$3 via integration of exact tropical $e$4-forms, and ordinary tropical morphisms are the special case in which all edge slopes are integral. The target, however, is $e$5, not another graph or polyhedral complex. For that reason, these objects are best viewed as harmonic maps from metric graphs to affine space, or as vector-valued harmonic functions with residues at the ends, rather than as harmonic morphisms between graphs in the Baker–Norine sense. The paper’s analytic significance lies in showing that these balanced affine-linear maps arise as scaling limits of harmonic amoeba maps on degenerating punctured Riemann surfaces (Lang, 2015).

These two directions clarify a major conceptual divide. In the graph-covering tradition, discrete harmonic morphisms are maps between discrete or polyhedral objects that preserve Laplace equations under pullback. In tropical Euclidean-target theory, harmonicity is instead encoded by edgewise linearity and Kirchhoff balancing. The local conservation law is common to both, but the target geometry and the role of local degree are different.

5. Random walks and network renormalization

A recent probabilistic reinterpretation treats discrete harmonic morphisms as the exact symmetry class for dynamically faithful network coarse-graining. In that setting, a coarse-graining is a surjection $e$6 from a fine graph $e$7 to a coarse graph $e$8, both equipped with simple random walk. The random walk Laplacian is

$e$9

and $f$0 is a harmonic morphism precisely when pullback preserves local harmonicity. The main theorem states that this is equivalent to exact preservation of first-exit random-walk transition structure: for each macro-node $f$1, each $f$2, and each neighbor $f$3,

$f$4

where $f$5 is the probability that a random walk started at $f$6 first exits the macro-set $f$7 into $f$8. Thus discrete harmonic morphisms are the minimal condition under which the macro-jump chain is exactly the simple random walk on the coarse graph, up to a random time change coming from the residence time inside macro-sets (Guadagnuolo et al., 9 Apr 2026).

The same work introduces diagnostics for approximate harmonicity. The mean harmonic degree, modified harmonic degree, and harmonic deviation quantify how closely a coarse-graining approximates horizontal conformality. An exact harmonic morphism has $f$9 and $\Gamma_1$00. Empirically, geometric, Laplacian, and GNN-based renormalization schemes produce different “dynamical fingerprints,” and Laplacian renormalization was observed to yield exact harmonic morphisms at specific scales in several real networks, including Facebook, Web-edu, CS Collab, and Yeast. The paper also contrasts this with entropic susceptibility, which detects diffusion scales but does not detect exact first-exit preservation (Guadagnuolo et al., 9 Apr 2026).

This probabilistic reformulation does not replace the divisor-theoretic theory; it reinterprets it. Horizontal conformality becomes a condition on balanced macro-boundary wiring, and local degree becomes the combinatorial mechanism ensuring that exit probabilities from each macro-set agree with the coarse random walk kernel.

6. Adjacent theories and recurring distinctions

A recurring source of confusion is the use of “harmonic” for several nearby but non-equivalent constructions. One adjacent tradition studies discrete harmonic maps and harmonic conjugates on planar triangulations. In a multiply connected planar domain with conductances on the $\Gamma_1$01-skeleton, Hersonsky constructs a discrete Dirichlet solution $\Gamma_1$02, a harmonic conjugate $\Gamma_1$03 defined through a discrete Dirichlet–Neumann problem, and a rectangular combinatorial net whose cells are quadrilaterals bounded by level curves of $\Gamma_1$04 and $\Gamma_1$05. The resulting map to a Euclidean annulus is built from

$\Gamma_1$06

and it preserves the associated quadrilateral measure. This theory is closely tied to discrete uniformization and convergence to conformal maps, but it does not formulate or prove the pullback-of-harmonic-functions property that characterizes harmonic morphisms between graphs (Hersonsky, 2012).

Another adjacent direction studies discrete-harmonic piecewise-linear embeddings of triangulations into planar polygons. For a triangulation $\Gamma_1$07 with positive weights $\Gamma_1$08, the discrete Laplace operator is

$\Gamma_1$09

and interior vertices satisfy $\Gamma_1$10. In that setting, invertibility of the induced PL map into a non-convex polygon is characterized by a cone condition at reflex boundary vertices,

$\Gamma_1$11

or equivalently by positivity of $\Gamma_1$12 on all boundary triangles. These results generalize Tutte-type convex-boundary embedding theorems, but they concern harmonic embeddings into Euclidean domains rather than graph-to-graph harmonic morphisms with local degree data (Kovalsky et al., 2020).

The comparison with tropical Euclidean-target maps points in the same direction. Harmonic tropical morphisms and planar discrete-harmonic embeddings share edgewise linearity and vertex balancing, but the classical discrete harmonic morphism theory is distinguished by three structural features: a combinatorial target, local multiplicity or local degree, and functoriality for Laplacians, divisors, or harmonic functions under pullback. Within that narrower sense, the modern theory now spans finite graphs, weighted and metric graphs, metrized complexes, $\Gamma_1$13-PL skeleta, and random-walk-preserving network quotients.