1. Purpose of the Model

This note presents a toy model for analyzing elite interlinkages using filtered simplicial complexes, ordinary cohomology, and sheaves.

The purpose is not to prove wrongdoing, collusion, corruption, or intent. The purpose is to show how a richer mathematical object can represent structures that a plain graph tends to flatten.

A graph can show that entities are connected. A simplicial complex can show that several entities participate in a jointly observed higher-order relation. A sheaf can show whether local interpretations of evidence can be glued into a globally consistent account.

This matters for OSINT and political-network analysis because elite systems are rarely reducible to pairwise links. They often involve events, institutions, intermediaries, knowledge flows, source contradictions, and ambiguous interpretations.

The toy model below is deliberately small. It contains only a few entities, one higher-order event, and one deliberate source contradiction.

2. Entities and Layers

Suppose we observe a small elite system with three layers:

1. people,
2. corporations or institutions,
3. events or knowledge nodes.

Let the people be:

$$
P={p_1,p_2,p_3}.
$$

Let the corporate or institutional entities be:

$$
C={c_1,c_2}.
$$

Let the events be:

$$
E={e_1,e_2,e_3}.
$$

Let the knowledge or policy nodes be:

$$
Q={k_1,k_2}.
$$

The full vertex set is therefore:

$$
V=P\cup C\cup E\cup Q.
$$

This gives ten vertices:

$$
V={p_1,p_2,p_3,c_1,c_2,e_1,e_2,e_3,k_1,k_2}.
$$

The layers are analytically meaningful. A person is not the same kind of object as a corporation. An event is not the same kind of object as a knowledge node. The model can still represent them inside one complex, but the analyst should not forget that the vertices are typed.

3. Observed Relations

Suppose the following weighted dyadic relations are observed.

$$
(p_1,c_1):0.95
$$

$$
(c_1,e_1):0.90
$$

$$
(p_2,c_1):0.88
$$

$$
(e_2,e_3):0.86
$$

$$
(k_1,p_2):0.85
$$

$$
(p_3,c_2):0.78
$$

$$
(c_2,k_2):0.74
$$

$$
(p_1,e_2):0.70
$$

$$
(c_1,e_2):0.82
$$

$$
(e_3,k_1):0.81
$$

The weights may represent confidence, evidentiary strength, frequency of contact, or some normalized combination of such measures. In a real model, one would need to define the weighting scheme carefully. Here the weights are illustrative.

Now suppose there is also one directly evidenced higher-order event:

$$
[p_2,e_2,k_1].
$$

This means that $p_2$, $e_2$, and $k_1$ were not merely pairwise related. They were jointly observed in a single event or evidentiary unit.

This is the first point at which a graph becomes inadequate. A graph can encode three pairwise edges. It cannot directly encode the fact that the three-way relation was observed as one higher-order event.

4. The Filtered Complex

We now build a filtered simplicial complex from the toy data.

Let $K_\tau$ denote the simplicial complex obtained by including all observed relations with weight at least $\tau$. Vertices are included once they appear in at least one relation.

At threshold $\tau = 0.85$, suppose we include the strongest observed edges:

$$
K_{0.85}^{(1)} =
{(p_1,c_1),(c_1,e_1),(p_2,c_1),(e_2,e_3),(k_1,p_2)}.
$$

This gives a sparse 1-dimensional complex. It contains several strong pairwise relations, but no higher-order event.

At threshold $\tau = 0.80$, we add two further dyadic relations and the directly evidenced higher-order event:

$$
(c_1,e_2):0.82,
\qquad
(e_3,k_1):0.81,
\qquad
[p_2,e_2,k_1].
$$

Because $K_{0.80}$ must be a simplicial complex, adding the 2-simplex also forces the inclusion of all its faces:

$$
(p_2,e_2),\qquad (e_2,k_1),\qquad (k_1,p_2).
$$

Thus:

$$
K_{0.80}=
K_{0.85}^{(1)}
\cup
{(c_1,e_2),(e_3,k_1)}
\cup
{(p_2,e_2),(e_2,k_1),[p_2,e_2,k_1]}.
$$

This distinction matters. The 2-simplex does not merely add three pairwise edges. It records that $p_2$, $e_2$, and $k_1$ were observed together in a single higher-order event.

A graph projection would reduce this event to three edges. The simplicial complex preserves the fact that the relation was jointly witnessed.

5. Ordinary Cohomology

Let $k$ be a coefficient field, for example $k=\mathbb{F}_2$ or $k=\mathbb{R}$.

For a finite simplicial complex $K$, define the $r$-cochains by:

$$
C^r (K;k) =\operatorname{Hom}(C_r (K;k),k).
$$

The coboundary map is:

$$
\delta^r : C^r(K;k) \rightarrow C^{r+1}(K;k).
$$

The coboundary maps satisfy:

$$
\delta^{r+1} \circ \delta^r=0.
$$

The $r$-th cohomology group is:

$$
H^r(K;k)=\ker \delta^r / \operatorname{im}\delta^{r-1}.
$$

In this toy model, $H^1(K;k)$ captures loops in the relational structure that are not filled by 2-simplices.

Consider the loop:

$$
\gamma =
p_2 \rightarrow c_1 \rightarrow e_2 \rightarrow e_3 \rightarrow k_1 \rightarrow p_2.
$$

This loop records a closed chain linking a person, a corporation, two events, and a knowledge node.

The addition of the 2-simplex

$$
[p_2,e_2,k_1]
$$

fills only the local triangle among $p_2$, $e_2$, and $k_1$. It does not fill the larger loop passing through $c_1$ and $e_3$. Therefore the larger cycle may remain visible as a nontrivial 1-dimensional feature.

Analytically, this means the model distinguishes between a local confirmed meeting and a broader unresolved relational circuit. The local meeting is explained by the 2-simplex. The wider loop is not.

6. A Sheaf of Local Interpretations

Ordinary cohomology captures holes in the relational structure. But OSINT analysis also requires another kind of object: local interpretations that may or may not be globally consistent.

To model this, define a sheaf $\mathcal{F}$ over the complex $K$.

For each vertex or simplex $\sigma$, let the stalk $\mathcal{F}(\sigma)$ be a vector space of possible analytic interpretations.

For simplicity, let each stalk be $k^2$. Use the basis vectors:

$$
\mathbf{p}=
\left(
\begin{matrix}
1 \cr
0
\end{matrix}
\right),
\qquad
\mathbf{s}=
\left(
\begin{matrix}
0 \cr
1
\end{matrix}
\right).
$$

Here, $\mathbf{p}$ denotes policy coordination and $\mathbf{s}$ denotes social or non-policy contact.

Restriction maps encode how interpretations must agree when passed from larger contexts to smaller ones.

Now suppose two sources give contradictory interpretations of the role of $e_2$.

Source $A$ interprets $e_2$ as policy coordination:

$$
s_A(e_2)=\mathbf{p}.
$$

Source $B$ interprets $e_2$ as merely social contact:

$$
s_B(e_2)=\mathbf{s}.
$$

Let $U_A$ be the local subcomplex supported by the evidence from Source $A$. Let $U_B$ be the local subcomplex supported by the evidence from Source $B$. Assume their overlap contains $e_2$:

$$
e_2 \in U_A \cap U_B.
$$

For the two local interpretations to glue into a single global section, they must agree on the overlap:

$$
s_A\vert_{U_A \cap U_B}=s_B\vert_{U_A \cap U_B}.
$$

But on $e_2$, we have:

$$
s_A(e_2)=\mathbf{p},
\qquad
s_B(e_2)=\mathbf{s},
\qquad
\mathbf{p}\neq\mathbf{s}.
$$

Therefore:

$$
s_A\vert_{U_A \cap U_B}\neq s_B\vert_{U_A \cap U_B}.
$$

The local interpretations fail to glue.

This is the sheaf-theoretic obstruction. The obstruction is not that one source must be true and the other false. Rather, the obstruction records that the available local interpretations cannot be merged into one globally consistent account without adding further assumptions, discounting one source, or refining the model.

7. The Sheaf Obstruction as an Analytic Signal

The failure of gluing has a direct OSINT interpretation.

A graph can show that $e_2$ is connected to several entities. It can show that $e_2$ sits between $c_1$, $e_3$, $p_2$, and $k_1$. But it cannot distinguish between the following two situations.

First, all sources may support a coherent interpretation of $e_2$.

Second, different sources may assign incompatible meanings to $e_2$.

The sheaf distinguishes these cases.

In this toy example, the contradiction is not represented merely as a disputed edge weight. It is represented as a failure of compatible local sections.

This matters because contradiction is not always noise. In intelligence analysis, contradiction can itself be evidence. It may indicate compartmentalization, deception, source unreliability, strategic ambiguity, or genuinely different roles played by the same event in different contexts.

The sheaf obstruction therefore functions as an analytic warning light:

$$
\text{local consistency does not imply global consistency}.
$$

The analyst should not collapse the structure into a single averaged interpretation too quickly. The obstruction says that something in the evidentiary structure remains unresolved.

8. Why a Plain Graph Misses the Structure

A graph summary would represent the toy model as vertices connected by edges. This is useful, but incomplete.

The graph can show that $p_2$, $e_2$, and $k_1$ are pairwise connected. But it cannot distinguish between three separate dyadic observations:

$$
(p_2,e_2),\qquad (e_2,k_1),\qquad (k_1,p_2)
$$

and one observed higher-order event:

$$
[p_2,e_2,k_1].
$$

That distinction is analytically important. Three pairwise contacts are not the same thing as one meeting involving all three actors.

The graph also struggles to represent the difference between a missing relation and an unresolved interpretation. If two sources disagree about the meaning of $e_2$, a graph usually handles this by changing edge labels, edge colors, or edge weights. But those are annotations on the graph, not part of the underlying mathematical structure.

The sheaf model makes the contradiction structural.

The graph asks:

$$
\text{Who is connected to whom?}
$$

The simplicial complex asks:

$$
\text{Which higher-order groups or events are jointly observed?}
$$

The sheaf asks:

$$
\text{Can the local interpretations be glued into a globally consistent account?}
$$

These are different questions. For elite interlinkage analysis, all three matter.

9. Analytic Interpretation

The toy model produces three distinct analytic outputs.

First, the filtered complex separates strong dyadic relations from higher-order events. This prevents the analyst from confusing pairwise association with jointly observed coordination.

Second, ordinary cohomology identifies persistent relational cycles. A cycle does not prove conspiracy, collusion, or intent. It indicates an unresolved structural loop in the evidence.

Third, the sheaf detects incompatibility among local interpretations. This is especially useful when sources agree on the existence of a contact but disagree about its meaning.

In this example, the model tells us that the following simplex is a locally confirmed higher-order event:

$$
[p_2,e_2,k_1].
$$

But the larger loop through $c_1$ and $e_3$ remains structurally unresolved. Separately, the conflicting interpretations of $e_2$ fail to glue into a single global account.

That is the analytic payoff.

The model does not say:

$$
\text{therefore wrongdoing occurred}.
$$

It says:

$$
\text{there is a structured evidentiary inconsistency requiring further analysis}.
$$

That is a weaker claim, but a more defensible one.

Links: https://www.lminus1.com/proposal-for-understanding-political-structures/