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Navigating Small-World Networks with Distance Predictions

Ladan Kian, Ming Ming Tan, Dariusz Kowalski

arXiv:2609.10885v1cs.DCcs.DS

TL;DR

Classical decentralized greedy routing assumes exact distance information and takes Θ(log^2 n) expected steps, motivating the study of routing with imperfect predictions. The paper analyzes history-conditioned distance predictions under two levels of coordinate awareness. Graph-distance predictions yield an asymptotic improvement, while lattice-distance predictions still give finite expected delivery when coordinates are hidden.

  • Problem

    Classical analyses assume exact distance information, leaving open how navigability degrades when routing uses imperfect predictions.

    Source: §1 · p. 2

  • Method

    The paper analyzes greedy routing with fresh, history-conditioned predictions of graph distance when coordinates are known and lattice distance when they are hidden.

    Source: §1 · p. 3, §1.1 · p. 4

  • Results

    Graph-distance predictions yield O(log n/(1 − 4kεδ)) delivery time when 4kεδ < 1, while lattice-distance predictions give finite expected delivery with a linear bound.

    Source: §1.1 · p. 3, §5 · p. 17, §4.2 · p. 17

  • Takeaways & Limitations

    Predicted graph distance can improve on Kleinberg’s Θ(log^2 n) bound, and history-conditioned lattice-distance predictions support finite delivery without coordinate disclosure.

    Source: §1.1 · p. 3, §1 · p. 3, §5 · p. 17

    The analysis does not cover how to construct the oracle or the cost of acquiring its information.

    Source: §3.4 · p. 11

Abstract

from arXiv · show

The small-world phenomenon was given an algorithmic foundation by Kleinberg, who showed that in an augmented $k$-dimensional lattice a decentralized greedy algorithm delivers a message in $O(\log^2 n)$ expected steps. We study predicted-greedy routing, in which a mobile agent forwarding the message moves at each step to the neighbor minimizing a noisy $(\varepsilon,\delta)$-prediction of its distance to the target, redrawn at every step from an oracle conditioned on the full routing history. Two cases arise from what this agent can observe. An agent with the coordinate awareness can still compute lattice distance exactly, but not graph distance in the shortcut-augmented network, since that depends on the shortcuts of nodes it has not yet visited; given an $(\varepsilon,\delta)$-prediction of graph distance, information the classical model never supplies, it achieves expected delivery time $O(\log n/(1-4k\varepsilon\delta))$, an asymptotic improvement over $\Theta(\log^2 n)$. An agent with no coordinate awareness at all, the natural model for a privacy-preserving network whose nodes never disclose their coordinates, cannot compute even lattice distance; given an $(\varepsilon,\delta)$-prediction of lattice distance instead, it still reaches the target in $O(n/(1-4k\varepsilon\delta))$ expected steps. Together these results show that a modest amount of predicted information, of the right kind, is enough to accelerate decentralized routing well below Kleinberg's classical bound, and that even when nodes reveal no coordinates at all, reliable delivery remains achievable.

1 Introduction

The paper studies how noisy distance predictions change decentralized routing in Kleinberg’s small-world model, considering both coordinate-aware and coordinate-hidden crawlers. It shows an asymptotic improvement over classical greedy routing with graph-distance predictions and finite expected delivery with lattice-distance predictions.

1 Introduction

  • Classical greedy routing is tight at Θ(log^2 n), despite an expected graph diameter of Θ(log n), and classical analyses assume exact distance information.The paper asks how navigability changes when that information is imperfect.Source: §1 · p. 2
  • The crawler receives a fresh, history-conditioned prediction of each candidate neighbor’s target distance, with ordering error ε and additive error δ.Predictions are regenerated at every step, including after revisits.Source: §1 · p. 2
  • Full coordinate awareness makes lattice distance computable but leaves graph distance unknown, whereas hiding both coordinates makes lattice-distance prediction the useful primitive.The coordinate-hidden case is motivated as a privacy-preserving setting.Source: §1 · p. 3

1.1 Our Contribution

  • O(log n/(1 − 4kεδ)) expected delivery time is achieved with graph-distance predictions when 4kεδ < 1, improving asymptotically on Kleinberg’s Θ(log^2 n) greedy bound.As prediction error tends to zero, the bound approaches the graph’s Θ(log n) expected diameter up to constants.Source: §1.1 · p. 3
  • With lattice-distance predictions and no coordinate awareness, routing terminates in finite expected time, with a linear rather than polylogarithmic delivery-time bound.The paper attributes termination to fresh, history-conditioned predictions at every step.Source: §1.1 · p. 3
  • Both analyses bound one-step excess distance using the oracle’s guarantees and telescope the resulting drift without assuming independence across steps.The graph metric gives a two-sided excess-distance bound; lattice distance gives only a one-sided bound, explaining the weaker rate in Case 2.Source: §1.1 · p. 4

2 Related Work

Related work covers classical navigability in Kleinberg’s model, methods that improve greedy routing with additional exact local information, and routing under imperfect geometric information. The paper positions its contribution within the algorithms-with-predictions paradigm and privacy-motivated hidden-metric models.

Classical small-world navigability

  • Kleinberg’s critical shortcut exponent yields Θ(log^2 n) greedy delivery, while Martel and Nguyen established a smaller Θ(log n) expected graph diameter.The literature also studies navigability across other geometries and shortcut exponents,,, [1].Source: §2 · p. 4

Improving greedy with additional information

  • Other work improves greedy delivery through lookahead, nearby-shortcut awareness, or one-hop lookahead combined with a power-law contact distribution.That line studies benefits of additional exact local knowledge, unlike this paper’s focus on imperfect information.Source: §2 · p. 5

Geographic and geometric routing under imperfect location information

  • Geographic routing studies imperfect location information, but prior work reports empirical failure rates under localization error rather than closed-form expected delivery bounds.The cited setting uses fixed geometric embeddings and deterministic recovery, unlike a repeatedly queried stochastic oracle in a Kleinberg lattice.Source: §2 · p. 5

Hidden-metric and privacy-motivated models

  • Hidden-metric models motivate the coordinate-free case, where positions represent latent attributes; here, the crawler instead receives noisy per-query predictions without seeing coordinates.The related work describes hidden geometry as statistically recovered from the graph rather than supplied by an oracle at query time.Source: §2 · pp. 5–6

Algorithms with predictions

  • Algorithms with predictions studies untrusted, error-parameterized oracles, but cited distributed applications predict static graph properties or outputs rather than query-specific distances at each routing step [6] [3].The paper identifies decentralized navigation in a small-world topology as unaddressed in that literature.Source: §2 · p. 6

3 Model and Preliminaries

The model pairs a Kleinberg grid with predicted-greedy routing under two information settings: coordinates with graph-distance predictions, or hidden coordinates with lattice-distance predictions. The section defines the oracle-history conditions, compares the information each oracle requires, and specifies delivery time as routing hops.

3.1 Network Model

  • The model is a k-dimensional grid with local links and one independently sampled, fixed directed shortcut per node; at Kleinberg’s α=k setting, lattice-greedy delivery is Θ(log^2 n).Source: §3.1 · pp. 6–7
  • Lattice distance ignores shortcuts, while graph distance is shortest-path distance in the network and never exceeds lattice distance.Source: §3.1 · p. 7

3.2 Case 1: Graph-Distance Predictions

  • With coordinates, the crawler can compute lattice distances but not graph distances affected by unseen shortcuts, so it chooses the neighbor with minimum predicted graph distance.The oracle supplies a new prediction vector at each step; its guarantees are conditional on the pre-step history, and successive vectors need not be independent.Source: §3.2 · p. 8

3.3 Case 2: Lattice-Distance Predictions

  • Without coordinates, the crawler sees identifiers and receives lattice-distance predictions; a shortcut can change that distance by an arbitrary amount, unlike local neighbors.Predictions may depend on the complete routing history and are refreshed at every step.Source: §3.3 · pp. 9–10

3.4 Model Comparison and Oracle Information

  • Both cases use the same topology, but graph-distance prediction requires instance-specific information about unobserved shortcut effects; lattice-distance prediction does not.Constructing the oracle and acquiring its information are outside the analysis’s scope.Source: §3.4 · pp. 10–11

3.5 Delivery Time

  • Delivery time is the number of routing hops until the target is reached, with T=∞ if it is never reached.For fixed graph and endpoints, the stated expectation is over oracle outputs alone; any averaging over shortcuts or endpoints is specified separately.Source: §3.5 · p. 11

4 Expected Delivery Time

The section bounds predicted-greedy delivery using graph distance when coordinates are known and lattice distance when they are not. Under 4kεδ < 1, the resulting bounds differ: logarithmic for graph-distance predictions and linear on average for lattice-distance predictions.

4.1 Case 1: Graph-Distance Predictions

  • Under 4kεδ < 1, graph-distance predictions yield O(log n/(1-4kεδ)) expected delivery time on the random Kleinberg graph.The fixed-graph bound scales with the source-target graph distance; averaging uses the expected graph diameter.Source: §4.1 · pp. 13–14

4 Expected Delivery Time

  • The proofs turn one-step bounds into delivery-time bounds by applying drift to a stopped process and then taking the stopping limit.For Case 1, the drift is positive when 4kεδ < 1; the Case 2 proof likewise bounds the stopped delivery time.Source: §4.1 · p. 13, §4.2 · p. 16

4.1 Case 1: Graph-Distance Predictions

  • In Case 1, the drift bound holds after every feasible history, so finite expected delivery time also implies almost-sure arrival despite revisits.The prediction is redrawn at each step; shortcut links are not resampled, and predictions need not be independent across steps.Source: §4.1 · p. 14

4.2 Case 2: Lattice-Distance Predictions

  • Under 4kεδ < 1, lattice-distance predictions give E[T | G, s, t] ≤ r(s, t)/(1-4kεδ).Averaging over uniformly random source and target gives an O(n/(1-4kεδ)) bound because their expected lattice distance is Θ(n).Source: §4.2 · pp. 15–16
  • In Case 2, fresh history-conditioned predictions restore the drift guarantee at revisited nodes, yielding finite expected delivery time and almost-sure termination.No bound on revisit frequency or independence across prediction vectors is required.Source: §4.2 · p. 16
  • The Case 2 guarantee assumes the oracle satisfies its prediction conditions after every feasible history, not merely at the outset.The result is for a sufficiently adaptive oracle, not an arbitrary fixed predictor.Source: §4.2 · p. 17

5 Conclusion

The paper analyzes predicted-greedy routing with two distance-prediction primitives: graph distance when coordinates are known and lattice distance when they are hidden. It establishes an asymptotic improvement for the first case and finite expected delivery for the second, while leaving a stronger bound for the second case open.

5 Conclusion

  • Graph-distance predictions improve asymptotically on Kleinberg’s Θ(log^2 n) bound, while lattice-distance predictions yield finite expected delivery even without coordinate awareness.The lattice-distance result uses a dynamic, history-conditioned oracle and requires no memory of past visits.Source: §5 · p. 17
  • The lattice-distance analysis certifies only a linear rate because its drift uses guaranteed local-neighbor progress, not the shortcut’s power-law placement.The passage does not rule out a sharper analysis of the same algorithm.Source: §5 · p. 17
  • Whether plain greedy routing with lattice-distance predictions can achieve a polylogarithmic bound without auxiliary memory remains open.A proof would need to control shortcut-information independence across revisits without explicit cycle avoidance.Source: §5 · p. 18

A Pseudo-code of the Predicted-Greedy Routing Algorithm

Predicted-Greedy Routing initializes the crawler at the source and repeatedly selects a neighbor using the current prediction vector until it reaches the target.

A Pseudo-code of the Predicted-Greedy Routing Algorithm

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