Local Limits of Preferential Attachment with Deterministic or Random Outdegree

Definitions and mathematical statement

The supplied record concerns two preferential-attachment models: an affine model with fixed outdegree, denoted schematically by $G_n^{\mathrm{fix}}$, and a corresponding Bernoulli preferential-attachment model described as having Poisson outdegree, denoted by $G_n^{\mathrm{Ber}}$. Edges are understood to point from a newly arriving vertex toward an older vertex. The indegree of a vertex counts incoming edges, and its outdegree counts outgoing edges, with multiplicity when applicable. The question requires attachment to depend only on indegree. An affine attachment weight is a function $f(k)=ak+b$ of the indegree $k$; the coefficients, their admissible range, and the rule converting these weights into attachment probabilities are not supplied.

To express the local-limit question schematically, fix a common category of rooted graphs and a neighborhood convention. Let $B_r(G,o)$ denote the rooted neighborhood of radius $r\in{\mathbb{Z}}_{\geq 0}$. Whether directions, multiplicities, birth times, or other marks are retained must be specified as part of that convention. For roots $o_n^{\mathrm{fix}}$ and $o_n^{\mathrm{Ber}}$, suppose that the respective rooted graphs converge locally in distribution: \[ (G_n^{\mathrm{fix}},o_n^{\mathrm{fix}}) \xrightarrow{\mathrm{loc},\,d}(\mathcal G^{\mathrm{fix}},o^{\mathrm{fix}}), \qquad (G_n^{\mathrm{Ber}},o_n^{\mathrm{Ber}}) \xrightarrow{\mathrm{loc},\,d}(\mathcal G^{\mathrm{Ber}},o^{\mathrm{Ber}}). \] Here local convergence means convergence in distribution of each fixed-radius rooted neighborhood in the chosen category.

The supplied clean statement asks whether, for the intended corresponding models with attachment based only on indegree, their local limits agree. Under the preceding conditional notation, this asks whether \[ \mathcal L\!\left(B_r(\mathcal G^{\mathrm{fix}},o^{\mathrm{fix}})\right) = \mathcal L\!\left(B_r(\mathcal G^{\mathrm{Ber}},o^{\mathrm{Ber}})\right) \qquad\text{for every }r\in{\mathbb{Z}}_{\geq 0}. \] The supplied original question is broader: "What is the relation between the local limits of the two models?" Thus equality is the target of the supplied clean formulation, whereas the original wording also permits a different relationship.

Unresolved qualifications. These expressions are a conditional formulation, not a uniquely specified conjecture. The record does not give the two graph construction laws, initial graphs, fixed outdegree, Poisson parameter, affine coefficients, parameter correspondence, root sampling laws, or neighborhood conventions. It also does not say whether Poisson outdegree describes a finite-stage law or an asymptotic law. In particular, the graph information retained in the comparison must be specified before equality has a definite meaning. No numerical parameter range or universal quantifier over model parameters can be recovered from this record.

Short English statement

Compare the rooted local limits of a fixed-outdegree affine preferential-attachment model and its intended Bernoulli counterpart with Poisson outdegree, using indegree alone for attachment. The supplied clean statement asks whether these limits have the same law; the original wording asks more generally how they are related. Construction rules, parameter matching, root selection, and the information retained in local neighborhoods are missing, so this remains a conditional statement requiring clarification and human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005451$, titled Local Limits of Preferential Attachment with Deterministic or Random Outdegree; source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2023.
• Supplied primary-source citation: MATRIX-MFO Tandem Workshop: Stochastic Reinforcement Processes and Graphs (2023), Oberwolfach Reports, pp. 639--679, DOI 10.4171/owr/2023/12. The background describes these reports as documenting mathematical workshops and their open-problem discussions. Supplied source link and literature-source label "original Oberwolfach report": https://doi.org/10.4171/owr/2023/12.
• Supplied literature-source label "universality of the local limit of preferential attachment models": https://arxiv.org/abs/2212.05551.
• Supplied literature-source label "Dereich--Mörters, random networks with sublinear preferential attachment": https://arxiv.org/abs/0807.4904.
• Supplied literature assessment dated exactly 2026-08-22: the registry labels the problem partially solved and reports that local limits are identified for broad preferential-attachment families and for Bernoulli preferential attachment, with coupling and universality results, while the general relation in the extracted record is not a single fully resolved theorem. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; the background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says it was checked against the official report and preserves its hypotheses, notation, and target. These are supplied verification claims, not verification performed for this draft. The record does not include the report passage needed to check those claims or reconstruct the model definitions.