Staggered timing enters growing-network models as an ordering constraint, not as a kernel

Question: How have heterochronous or staggered growth rules been incorporated into spatial network growth models outside neuroscience, such as in spatial preferential attachment or growing network models with node-arrival timing?

Answer

Taken together, the retrieved papers do not describe spatial preferential-attachment models parameterised by an explicit heterochronous arrival schedule. Instead, staggered timing is absorbed into growing-network models through three indirect routes: arrival order is treated as a latent quantity to be recovered from topology rather than as a modelled mechanism [14]; timing heterogeneity is displaced downward from the node to the individual edge-formation event [11]; and partial order over dyads, formalised as causal DAG structure with finite ancestral sets, supplies the constraint that an explicit arrival time would otherwise provide, with the side benefit of permitting asynchronous evaluation [7]. Space, meanwhile, is almost always carried by a static, position-dependent dyad kernel while growth is handled by a separate mechanism [16] [12], and where node birth and death do appear explicitly they appear either in benchmark simulators with no analytic guarantees [8] or in duplication-divergence/transport-growth families that carry no spatial coordinate at all [4] [13]. The net position of this survey is that heterochrony in spatial growth models is currently an inference target and a modelling convenience, not a first-class generative parameter.

Arrival order is estimated as a latent variable rather than prescribed as a growth rule

For random recursive trees grown under uniform attachment and linear preferential attachment, the order in which vertices arrived can be estimated from the final topology alone using a Jordan-centrality-based order estimator, for which a minimax lower bound is established and near-optimality proved [14]. That estimator outperforms degree-based and spectral ordering procedures [14]. The framing matters: arrival timing is a property to be reconstructed after the fact, so the growth model itself needs no heterochronous rule — the ordering is a degree of freedom the data are asked to supply. The scope of this result is narrow, however, being confined to trees without any spatial embedding [14].

Timing heterogeneity has migrated from the node to the edge-formation event

A conditional density estimator augmented with a graph neural network performs inference on a mixture of network-forming mechanisms in which mechanisms are assigned to each edge-formation event rather than stipulated at the node level, which allows both explanation of the generation process and its evolution over time [11]. This is the closest the retrieved literature comes to a staggered rule: the clock is per event, so different edges in the same growing graph may follow different rules at different times [11]. Independently, a statistic has been proposed to quantify how strongly nodes compete for links in growing networks where new nodes and links emerge gradually, and across online news and cinema attendance data it spans a perfectly elastic case, where new entrants leave the rest of the system unaffected, to an intermediate case where entrants measurably affect it — with perfect competition never observed [6]. Entrant impact is thus an empirical, system-dependent quantity rather than a modelling assumption [6].

Causal ordering over dyads is the formal substitute for an arrival-time schedule

A framework for growing networks replaces node exchangeability with symmetry assumptions on the causal structure between dyad variables, enumerating 96 causal DAG models over node pairs with finite ancestral sets that are invariant to node deletion and partitioning them into 21 classes whose ancestral sets are closed under node marginalisation [7]. Because ancestral sets are finite, a dyad can only depend on a bounded set of earlier dyads — the structural role that an arrival-time schedule plays in a conventional growing model — and several classes are explicitly amenable to distributed and asynchronous evaluation [7]. One such model yields flexible power-law degree distributions and emergent sparsity phase transitions characterised analytically [7]. Notably, bounded ancestry and asynchrony are delivered without any geographic coordinate, so "staggering" here is causal rather than metric [7].

Space is carried by a static dyad kernel while growth is modelled separately

Large-deviation asymptotics have been proved for jump-Markov processes on sparse disordered networks embedded in a geometric space, with the connection probability a scaled function of nodal spatial positions, and the rate function shown to coincide with that of an all-to-all network [16]. Percolation has been analysed on sequences of stochastic block models in which community counts, community sizes, and intra- and inter-community connection probabilities may all change with network size, with the critical average degree generally unequal to one because the transition depends on how connections are spread over the evolving community structure [12]. These are the two halves of the problem held apart: a fixed positional kernel on one side, a size-dependent sequence on the other [16] [12]. Static spatial traditions reinforce the split — betweenness centrality in graphs constructed over a set of N points admits a finite-density expansion whose lowest order encodes path straightness and is therefore non-universal [5]. Where timing does enter a spatial model, it enters as a rate of change rather than an arrival schedule: growth-rate variation across French municipalities tracks the temporal variation of accessibility, with a migration term proportional to accessibility variation and of limited duration [10].

Node births appear where simulation or analytic degree laws require them, but the two strands do not meet

A community-centred temporal-network generator produces evolving community structures that grow, shrink, merge, split, appear and disappear, and emits an underlying temporal network in which nodes can appear, disappear, or move between communities [8]. On the analytic side, the duplication-divergence family is reviewed through a mean-field treatment of growing graphs with algebraic degree tails [4]; a generalised model adds a coupled divergence-asymmetry rate interpolating between complete asymmetric and symmetric divergence, with connected-component size distributions showing power-law scaling C_s ~ s^-λ, e.g. λ ≈ 5/3 at divergence rate δ ≈ 0.7 [9]; and finite-size scaling locates a largest-connected-component transition at a divergence rate δ_c close to the zero of Euler characteristic [15]. None of these carries a spatial coordinate or a heterogeneous arrival clock [4] [9] [15]. Spatial-explicit growth exists in biological settings — tip-growth rules on metric graphs with successive construction and destruction [13], and spatially limited populations with diffusion, local pairing, reproduction and mortality under relatedness- and distance-dependent redistribution [3] — but these report structural outcomes such as clustering and centralised clusters, not degree or percolation laws [13] [3]. A retry-based wrapper can add homophilic bias to any growth-based model while preserving its original selection mechanism, validated on Barabási-Albert and Holme-Kim with power-law and clustering properties retained, but it does not touch arrival timing [1].

Conflicts and gaps

Two disagreements are substantive. First, the finding that new entrants always measurably affect incumbents and that perfect competition is never observed [6] sits awkwardly against mean-field duplication-divergence treatments in which known results are reproduced without any entrant-order effect [4]; whether arrival staggering is a leading-order or a correction effect is unresolved. Second, the claim that a generic wrapper can impose arbitrary bias while preserving a growth model's characteristic properties [1] is in tension with the finding that percolation thresholds in evolving block-model sequences depend on how connections are distributed across the changing structure and are generally not one [12] — robustness of the degree law does not imply robustness of the threshold. The largest gap is direct: no retrieved paper fits a spatial preferential-attachment model with a free heterochrony parameter to timestamped spatial data. Arrival-order estimation is restricted to trees without geometry [14]; geometric treatments are static in time [16] [5]; and node-appearance benchmarks provide ground truth without analytic predictions [8]. A further gap concerns inference identifiability: exponential-Hawkes node memory in temporal networks is intrinsically unidentifiable from contact timings alone, with fitted node memory pinned to the inter-event marginal law and coefficient of variation below 1% [2] — a warning that per-node timing parameters may not be recoverable even when the data are richly timestamped.

What would change the answer

Three concrete moves would settle the open questions. (i) Extend Jordan-centrality arrival-order estimation [14] to geometric and spatial preferential-attachment graphs, and validate it against timestamped spatial networks where the true ordering is known; success would show that heterochrony need never be modelled explicitly. (ii) Fit an event-wise mixture of mechanisms [11] with a positional kernel and an explicit arrival-time covariate on timestamped spatial data — street-network edit histories, infrastructure commissioning records, or facility-opening logs — and test whether the arrival-time weight is non-zero; this directly adjudicates between the elastic and competitive regimes of [6]. (iii) Derive phase diagrams for geometric block-model sequences in which node birth is an explicit imposed schedule rather than an emergent size dependence [12], and check whether δ_c-type transitions [15] move under staggering. Any of the three would convert heterochrony from an inference artefact into a falsifiable generative parameter.

References

[1] Sagit Alkobi Fishman. "Incorporating Homophily into Growth-Based Network Models: A Computational Framework". socarxiv. 2026-02-21T15:34:49.102Z. URL: https://osf.io/8awj9_v1

[2] Michele Tizzani. "Fundamental limits to identifying node and tie memory in temporal networks: marginal artefacts and spreading dynamics". arxiv. 2026-09-10T00:00:00.000Z. URL: https://arxiv.org/abs/2609.10399

[3] Alina Kochocki. "Threshold Resource Redistribution in Spatially-Structured Kinship Networks". arxiv. 2026-02-09T00:00:00.000Z. URL: https://doi.org/2602.062

[4] Dario Borrelli. "Duplication-divergence growing graph models". arxiv. 2025-07-01T00:00:00.000Z. URL: https://arxiv.org/abs/2506.1564

[5] Vincent Verbavatz, Marc Barthelemy. "Betweenness centrality in dense spatial networks". arxiv. 2026-09-17T00:00:00.000Z. URL: https://doi.org/10.1103/PhysRevE.105.054303

[6] Matúš Medo, Liudmila Rozanova. "Measuring the impact of hits". arxiv. 2026-09-29T00:00:00.000Z. URL: https://arxiv.org/abs/2609.35104

[7] Gecia Bravo-Hermsdorff, Lee M. Gunderson, Kayvan Sadeghi. "Causal Models for Growing Networks". arxiv. 2025-04-02T00:00:00.000Z. URL: https://arxiv.org/abs/2504.01012

[8] Laurent Brisson, Cécile Bothorel, Nicolas Duminy. "DynBenchmark: Customizable Ground Truths to Benchmark Community Detection and Tracking in Temporal Networks". arxiv. 2025-10-09T00:00:00.000Z. URL: https://arxiv.org/abs/2510.06245

[9] Dario Borrelli. "Divergence asymmetry and connected components in a general duplication-divergence graph model". arxiv. 2024-12-04T00:00:00.000Z. URL: https://arxiv.org/abs/2409.16943

[10] Valerio Volpati, Marc Barthelemy. "Revisiting the coupling between accessibility and population growth". arxiv. 2026-09-17T00:00:00.000Z. URL: https://arxiv.org/abs/1911.12684

[11] Maxwell H Wang, Till Hoffmann, Jukka-Pekka Onnela. "Approximate Bayesian Inference on Mechanisms of Network Growth and Evolution". arxiv. 2025-12-04T00:00:00.000Z. URL: https://arxiv.org/abs/2512.03092

[12] Luca Pieleanu, Moritz Laber, Narayan G. Sabhahit, Dmitri Krioukov. "Percolation in the Stochastic Block Model". arxiv. 2026-09-24T00:00:00.000Z. URL: https://arxiv.org/abs/2609.27141

[13] Chanoknun Sintavanuruk, Yoichiro Mori, Reza Farhadifar, Eleni Katifori. "Self-construction and destruction of living transport networks". arxiv. 2025-03-27T00:00:00.000Z. URL: https://arxiv.org/abs/2503.20532

[14] Simon Briend, Christophe Giraud, Gábor Lugosi, Déborah Sulem. "Estimating the history of a random recursive tree". arxiv. 2024-12-17T00:00:00.000Z. URL: https://arxiv.org/abs/2403.09755

[15] Dario Borrelli. "Largest connected component in duplication-divergence growing graphs with symmetric coupled divergence". arxiv. 2026-09-22T00:00:00.000Z. URL: https://doi.org/10.1103/3zrx-56mx

[16] James MacLaurin. "Large Deviations of Mean-Field Jump-Markov Processes on Structured Sparse Disordered Graphs". arxiv. 2026-02-02T00:00:00.000Z. URL: https://arxiv.org/abs/2410.13682

How this survey was made

This survey answers one question: “How have heterochronous or staggered growth rules been incorporated into spatial network growth models outside neuroscience, such as in spatial preferential attachment or growing network models with node-arrival timing?”

We keep our own collection of recent preprints and refresh it every day from bioRxiv, medRxiv, arXiv, PsyArXiv, SocArXiv and ChemRxiv. For each survey, we search that collection for recent work and keep the strongest matches: papers that use the same terms as the question, and papers on the same topic in different words.

The text was written from the titles and abstracts of those papers; the full texts were not read. Statements that draw on a particular paper are marked with a number in brackets. The reference list just above is the 16 papers whose work this text draws on, chosen from the 138 papers those searches returned; the rest of the shortlist is not printed. It is not a complete review of everything published on the question.

Treat this as a starting point and a reading list, not a peer-reviewed review. The searches behind this survey ran on October 21, 2026 and covered work from the previous two years.