IGMS: Interaction Graph on Minimal Siphons

• IGMS is a directed graph where vertices represent minimal siphons—subsets of species with one-sided production constraints—in chemical reaction networks.
• It connects epidemiologic strains with minimal siphons, enabling acyclic decompositions that yield block-triangular next-generation matrices and classic R0 calculations.
• Its algorithmic implementation leverages Petri-net approaches to efficiently enumerate minimal siphons and detect acyclic partitions despite inherent computational challenges.

Searching arXiv for the specified paper to ground the article and citation. arxiv_search({"^^^^^^^^2query2 OR \2^^^^^^^^"On the Similarity between Epidemiologic Strains, Minimal Self-Replicable Siphons, and autocatalytic cores in (Chemical) Reaction Networks: Towards a Unifying Framework\"","max_results":5,"sort_by":"submittedDate","sort_order":"descending"}) arXiv search results received. Using the identified paper as the primary source. Interaction Graph on Minimal Siphons (IGMS) is a directed-graph construction introduced in a structural program that relates mathematical epidemiology to chemical reaction network theory. In that framework, epidemiologic strains are compared with critical minimal siphons and minimal autocatalytic sets, the disease free equilibrium face is described through unions of minimal siphons or cores, and the Interaction Directed Graph on Minimal Siphons is used to encode how one minimal siphon can produce species in another through the reaction structure. When this graph is acyclic and the infected species decompose as a partition into minimal siphons, the Next-Generation Matrix acquires a block-triangular form, yielding the classical multi-strain formula PRESERVED_PLACEHOLDER_^^^^2query2^^^^ (&&&^^^^2query2^^^^&&&).

^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^. Position within the epidemiology–CRN correspondence

The IGMS construction appears in a review-oriented framework that studies boundary stability and persistence of positive ODEs in mathematical epidemiology by importing structural tools from chemical reaction networks. A central observation is a conceptual correspondence between epidemiologic strains and both critical minimal siphons and minimal autocatalytic sets, or cores, in an underlying CRN; this correspondence is reported to hold in all models examined in the source work (&&&^^^^2query2^^^^&&&).

Within the same framework, the disease free equilibrium face, or infected set, is defined as the union of either all minimal siphons or all cores, and these coincide in the examples discussed. The work also proposes a characterization of mathematical epidemiology models as models with a unique boundary fixed point on the DFE face and an infected-subnetwork Jacobian admitting a regular splitting that permits definition of the next generating matrix. IGMS is introduced precisely at this interface: it is not a species-level interaction graph in the ordinary sense, but a graph whose vertices are minimal siphons, intended to capture directional interdependence among infection-relevant structural modules.

This placement is significant because it ties a graph-theoretic object directly to threshold analysis. A plausible implication is that IGMS serves as a bridge between CRN invariance structure and epidemiological reproduction-number calculations, rather than as a purely combinatorial summary of the reaction network.

2. Minimal siphons as the underlying vertices

Let PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^ be the set of species and $\mathcal{R}$ the set of reactions of a chemical reaction network. Each reaction $\alpha \to \beta$ has reactant complex $\alpha \in \mathbb{N}^{|\mathcal{S}|}$ and product complex $\beta \in \mathbb{N}^{|\mathcal{S}|}$, and induces a stoichiometric column

$\Gamma_{\alpha \to \beta}=\beta-\alpha$

in the stoichiometric matrix $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$ (&&&^^^^2query2^^^^&&&).

A nonempty species set $W \subseteq \mathcal{S}$ is a siphon, or semilocking set, if whenever a reaction produces a species in $W$ it also consumes at least one species in PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2query2^^^^. Equivalently, for every reaction PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2id:(Avram et al., 30 Oct 2025) OR \2^^^^,

PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \22^^^^

A siphon is minimal if it does not strictly contain a smaller siphon. The family of all minimal siphons is written as

PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^3

These minimal siphons provide the vertex set for IGMS. This choice matters structurally: the graph is built on semilocking subsets that already encode one-sided production constraints. Accordingly, IGMS abstracts interactions among these structurally distinguished subsets rather than among arbitrary subnetworks.

A common misunderstanding is to identify minimal siphons directly with individual strains or individual infected species. The source material does not make that identification as an equality. Instead, it reports a conceptual correspondence between epidemiologic strains and critical minimal siphons, and separately uses minimal siphons as the units of the graph construction.

3. Definition of the Interaction Graph on Minimal Siphons

The formal definition is:

“The Interaction Directed Graph on Minimal Siphons (IGMS) is the directed graph PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^4 with * vertex set PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^5, * a directed edge PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^6 PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^7 if and only if there exists at least one reaction PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^8 such that PRESERVED_PLACEHOLDER_^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^9 and $\mathcal{R}$^^^^2query2^^^^. (In words: some species in $\mathcal{R}$^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^ appear as reactants in a reaction that produces at least one species in $\mathcal{R}$2.)” (&&&^^^^2query2^^^^&&&)

If $\mathcal{R}$3 is the $\mathcal{R}$4-th reaction, then

$\mathcal{R}$5

The direction of an edge therefore records a production relation mediated by a reaction: species in $\mathcal{R}$6 occur on the reactant side of a reaction whose product side contains a species in $\mathcal{R}$7. This is narrower than generic coexistence in a reaction and broader than direct one-to-one species conversion. The graph is also explicitly defined only for distinct siphons, since the edge condition is stated for $\mathcal{R}$8.

The resulting object can be read as an inter-siphon dependency graph. In particular, it tracks when the reaction network allows species belonging to one minimal siphon to participate in the generation of species in another minimal siphon. This suggests a structural notion of upstream and downstream infection modules that later becomes formalized through topological ordering.

4. Acyclic Minimal Siphon Decomposition

Acyclic Minimal Siphon Decomposition, abbreviated AMSD, is defined only under a partition condition. Given a CRN whose infected-species set $\mathcal{R}$9 admits a partition into minimal siphons,

$\alpha \to \beta$^^^^2query2^^^^

one forms IGMS on these $\alpha \to \beta$^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^. If IGMS has no directed cycle, the partition is called an Acyclic Minimal Siphon Decomposition (&&&^^^^2query2^^^^&&&).

Two points are structurally decisive. First, the infected set must admit a partition into minimal siphons; the definition is not stated merely for an arbitrary family of overlapping siphons. Second, acyclicity is imposed on the IGMS obtained from that partition. The combination of partition and acyclicity is what drives the later matrix decomposition.

This guards against a possible misconception that acyclicity of IGMS alone is sufficient for the block-triangular threshold result. The theorem reported in the source assumes both that $\alpha \to \beta$2 partition the infected set and that the associated IGMS is acyclic. Without the partition hypothesis, the theorem as stated does not apply.

The notion of AMSD provides a structural criterion for when the infected subsystem can be ordered without feedback loops among minimal siphons. A plausible implication is that AMSD identifies a class of multi-strain or multi-module epidemiologic models whose transmission architecture is hierarchically organized rather than cyclically coupled.

5. Consequences for the Next-Generation Matrix and $\alpha \to \beta$3

Under AMSD, the source gives the following theorem on block triangularity. Let $\alpha \to \beta$4 partition the infected set $\alpha \to \beta$5, form IGMS on these $\alpha \to \beta$6, and assume it is acyclic. Let $\alpha \to \beta$7 be any topological ordering of the siphons in IGMS. Then, by listing all species in $\alpha \to \beta$8 first, then those in $\alpha \to \beta$9, and so on, one obtains:

• $\alpha \in \mathbb{N}^{|\mathcal{S}|}$^^^^2query2^^^^ block-diagonal with $\alpha \in \mathbb{N}^{|\mathcal{S}|}$^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^ blocks;
• $\alpha \in \mathbb{N}^{|\mathcal{S}|}$2 block-lower-triangular with the same block partition;

so that

$\alpha \in \mathbb{N}^{|\mathcal{S}|}$3

is itself block-lower-triangular with $\alpha \in \mathbb{N}^{|\mathcal{S}|}$4 diagonal blocks $\alpha \in \mathbb{N}^{|\mathcal{S}|}$5. In particular,

$\alpha \in \mathbb{N}^{|\mathcal{S}|}$6

These statements are given as the theorem “Block triangularity under acyclic siphon partition” (&&&^^^^2query2^^^^&&&).

The proof sketch is structural. Because no reaction can carry mass “backwards” against the topological order, the Jacobian of new-infection terms $\alpha \in \mathbb{N}^{|\mathcal{S}|}$7 has no off-block entries from higher to lower siphon in that order, so $\alpha \in \mathbb{N}^{|\mathcal{S}|}$8 is block-diagonal. Transitions and removals $\alpha \in \mathbb{N}^{|\mathcal{S}|}$9 link each siphon only “upstream” in the topological order, making $\beta \in \mathbb{N}^{|\mathcal{S}|}$^^^^2query2^^^^ block-lower-triangular. The inverse of a block-lower-triangular Metzler $\beta \in \mathbb{N}^{|\mathcal{S}|}$^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^ is again block-lower-triangular, hence $\beta \in \mathbb{N}^{|\mathcal{S}|}$2 inherits the same triangular form. Since a triangular matrix has spectrum equal to the union of its diagonal-block spectra, the basic reproduction number decomposes as the maximum over the block-wise reproduction numbers.

The significance of this result lies in its recovery of the “classical max structure” of the reproduction number for multi-strain models. In this formulation, the max structure is not introduced axiomatically; it is derived from a reaction-network decomposition of the infected species into acyclically interacting minimal siphons.

6. Algorithmic construction and implementation in Epid-CRN

The source gives a high-level pseudocode algorithm, ComputeIGMSandAMSD(RN, rates), whose outputs are:

• mSi = list of minimal siphons T₁,…,Tₘ
• Edges = list of directed edges (i→j) defining IGMS
• IsAcyclic = Boolean(IGMS acyclic)
• π = topological ordering of siphons if acyclic, else Null
• Perm = permutation of infected-species indices according to π (&&&^^^^2query2^^^^&&&)

The procedure begins by extracting the stoichiometric matrix $\beta \in \mathbb{N}^{|\mathcal{S}|}$3 from the reaction network. It then computes all minimal siphons via structural analysis by forming the Petri-net incidence representation, using the input/output matrices $\beta \in \mathbb{N}^{|\mathcal{S}|}$4, and solving the minimal hitting-set problem: finding all nonempty $\beta \in \mathbb{N}^{|\mathcal{S}|}$5 such that no reaction produces into $\beta \in \mathbb{N}^{|\mathcal{S}|}$6 from outside $\beta \in \mathbb{N}^{|\mathcal{S}|}$7, with $\beta \in \mathbb{N}^{|\mathcal{S}|}$8 minimal. The source notes that this step may use SAT/SMT or specialized Petri-net siphon enumeration and has worst-case exponential complexity in $\beta \in \mathbb{N}^{|\mathcal{S}|}$9.

Once the minimal siphons are known, the algorithm initializes an empty edge set and checks each ordered pair $\Gamma_{\alpha \to \beta}=\beta-\alpha$^^^^2query2^^^^ with $\Gamma_{\alpha \to \beta}=\beta-\alpha$^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^ against each reaction $\Gamma_{\alpha \to \beta}=\beta-\alpha$2. If

$\Gamma_{\alpha \to \beta}=\beta-\alpha$3

the edge $\Gamma_{\alpha \to \beta}=\beta-\alpha$4 is appended. Duplicate edges are then removed. A directed graph on vertices $\Gamma_{\alpha \to \beta}=\beta-\alpha$5 is formed, acyclicity is tested via depth-first search in $\Gamma_{\alpha \to \beta}=\beta-\alpha$6, and, if acyclic, a topological sort $\Gamma_{\alpha \to \beta}=\beta-\alpha$7 is computed. The permutation Perm is obtained by concatenating the ordered lists of species in each $\Gamma_{\alpha \to \beta}=\beta-\alpha$8.

After Perm is known, the global species-index vector $\Gamma_{\alpha \to \beta}=\beta-\alpha$9 is permuted so that infection variables appear in block order. The package then calls its NGM routine to build $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$^^^^2query2^^^^ and $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$^^^^2id:(Avram et al., 30 Oct 2025) OR \2^^^^ in that order and confirms that $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$2 is block-triangular.

The implementation is provided in the Epid-CRN Mathematica package. The source gives the example calls $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$9 where edg=Edges, cyc∈{True,False}=IsAcyclic, and graph is the GraphObject. A subsequent call $W \subseteq \mathcal{S}$^^^^2query2^^^^ produces the permutation Perm to be used by the NGM routine (&&&^^^^2query2^^^^&&&).

7. Complexity, interpretation, and scope

The source identifies minimal siphon enumeration as the computational bottleneck. Its worst-case cost is stated as $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$3, although network sparsity and Petri-net tools are said to keep the problem manageable up to dozens of species. By contrast, edge construction in IGMS is $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$4; cycle detection and topological sorting are linear in graph size; and permutation of variables together with re-indexing of $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$5, $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$6, and $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$7 is $$\Gamma \in \mathbb{Z}^{|\mathcal{S}|\times |\mathcal{R}|}$$8. Once minimal siphons have been computed, IGMS and AMSD detection are described as almost instantaneous (&&&^^^^2query2^^^^&&&).

These complexity statements clarify the role of IGMS in practice. The graph itself is not the expensive component; the cost is concentrated upstream in identifying the minimal siphons on which the graph is defined. This suggests that the usefulness of IGMS depends materially on the tractability of structural siphon analysis in the underlying network class.

In scope, IGMS is designed as part of a unifying framework connecting epidemiologic strains, minimal self-replicable siphons, and autocatalytic cores. Its immediate function is to detect whether the infected subsystem admits an acyclic decomposition compatible with block-triangular NGM analysis. It is therefore best understood as a structural diagnostic for threshold decomposition, rather than as a universal replacement for dynamical analysis. The source confines its strongest conclusions to AMSD models whose minimal siphons partition the infection species; beyond that regime, broader implications would be interpretive rather than established.