The Existence of Implicit Knowledge Graphs Conditional Construction and Analysis Based on Internal Transformer Representations

Implicit knowledge graphs describe knowledge organization within large models as structures of semantic relations that emerge in response to an input. This theoretical account requires a rigorous existence proof. Using Transformer hidden states and multi-head attention, we provide a constructive existence proof under explicit assumptions. Nodes are selected by numerical salience and semantic-probe conditions. Edge weights are aggregated within specified groups of attention heads to construct an input-conditioned, multi-relational, directed weighted graph. Under assumptions of local regularity, semantic separability and positive threshold margins, the graph is well-defined, and its nodes satisfy classification conditions for specified semantic tasks. The node set and each typed edge set remain unchanged when input perturbations stay within the neighborhood where the assumptions hold and satisfy the corresponding margin bounds. We also give sufficient conditions for identical discrete structures under extractors that use common indices and the same thresholds. Aggregated graph edges are traced to single-head carrier edges, yielding a directional-derivative criterion for functional effects. If a non-degenerate carrier edge satisfies this criterion, sufficiently small weakening perturbations along the prescribed path decrease the target output’s unnormalized score, or logit. We further characterize the restoring gradient contribution of an entropy regularizer to mean attention entropy. These results establish the conditional existence of implicit knowledge graphs as a family of input-conditioned internal representations. They provide a mathematical basis for studying local stability and connections to model outputs.