AGP-Dynamic: Dynamic Graph Propagation

• AGP-Dynamic is a dynamic algorithm that extends the Approximate Graph Propagation framework to evolving graphs by supporting edge insertions, deletions, and on-the-fly query parameters.
• It enables various propagation semantics, such as PageRank and GNN feature diffusion, by dynamically adjusting sampling and neighborhood structures while preserving approximation guarantees.
• The method significantly improves query and update efficiency over static methods by leveraging bounded geometric sampling and controlled dynamic maintenance to achieve constant amortized update time.

AGP-Dynamic is a dynamic algorithm for Approximate Graph Propagation (AGP), a unified framework for graph propagation tasks expressed as

$$\boldsymbol{\pi}=\sum_{i=0}^{\infty} w_i \cdot (\mathbf{D}^{-a}\mathbf{A}\mathbf{D}^{-b})^i \mathbf{x},$$

where $\mathbf{A}$ is the adjacency matrix of an undirected graph $G=\langle V,E\rangle$, $\mathbf{D}$ is the diagonal degree matrix, $a,b\in[0,1]$ with $a+b\ge 1$, $w_i\ge 0$ and $\sum_i w_i=1$, and $\mathbf{x}$ is a nonnegative $\ell_1$-normalized input vector. Within this formalism, AGP-Dynamic extends AGP from the original static, fixed-parameter regime to dynamic graphs with edge insertions and deletions and to parameterized queries whose propagation semantics are supplied on the fly; it preserves the approximation guarantees of the static framework while achieving $\mathbf{A}$0 amortized time per update (Zhao et al., 12 Sep 2025).

1. Unified propagation model

The AGP framework treats a broad family of propagation operators as instances of the same series expansion. By suitable choices of $\mathbf{A}$1, it captures L-hop transition probability, PageRank, Personalized PageRank (PPR), Single-target PPR, Heat kernel PageRank, Simplified graph convolution / SGCN, APPNP, and Graph diffusion convolution (GDC) (Zhao et al., 12 Sep 2025).

A query is denoted

$\mathbf{A}$2

where $\mathbf{A}$3 is an oracle that returns both $\mathbf{A}$4 and $\mathbf{A}$5, the smallest $\mathbf{A}$6 such that

$\mathbf{A}$7

This oracle-based formulation is what allows the framework to accommodate multiple weight families, including geometric and Poisson weights, without specializing the algorithm to a single propagation model.

A central operational consequence is that the same backend can serve classical random-walk proximity, GNN feature diffusion, and graph-based retrieval. In graph-RAG terms, the underlying propagation semantics can be changed per request: a query about “What causes Alzheimer’s disease?” may invoke a PPR-style source-centered propagation, whereas “Who affects Trump?” may invoke single-target PPR. AGP-Dynamic is designed precisely for this regime, in which both the graph and the propagation parameters vary over time.

2. Dynamic Approximate Graph Propagation and the limits of AGP-Static

The problem addressed by AGP-Dynamic is Dynamic Approximate Graph Propagation (DAGP): the graph evolves through edge insertions and deletions, and the query parameters $\mathbf{A}$8 are specified on the fly rather than fixed in preprocessing (Zhao et al., 12 Sep 2025).

The paper identifies two distinct weaknesses in the predecessor method, AGP-Static. The first is a query-time issue. AGP-Static’s reported bound assumed an optimal subset sampling routine that did not exist at the time. In practice, its implementation incurred an extra

$\mathbf{A}$9

factor: one $G=\langle V,E\rangle$0 from power-of-two bucketing and another $G=\langle V,E\rangle$1 from generating binomial random variates. A recurring misconception is therefore that AGP-Static’s published query complexity was already fully realizable; the revision explicitly states that it depended on a non-existent ideal sampler.

The second weakness is dynamic maintenance. AGP-Static maintains sorted neighborhood structures, so a single edge update may trigger widespread local repairs, producing

$G=\langle V,E\rangle$2

or effectively $G=\langle V,E\rangle$3-type update cost in the worst case, depending on implementation details. This is the principal reason it is unsuitable for frequently changing graphs.

These two observations motivate a two-step revision. AGP-Static++ repairs the static query algorithm and its analysis, while AGP-Dynamic introduces a dynamic maintenance scheme that preserves the improved query behavior under graph updates.

3. Residue–reserve propagation, AGP-Static++, and the dynamic extension

AGP-Static++ retains the AGP residue/reserve decomposition. For truncation depth $G=\langle V,E\rangle$4, it defines

$G=\langle V,E\rangle$5

with

$G=\langle V,E\rangle$6

and outputs

$G=\langle V,E\rangle$7

Within this structure, AGP-Static++ removes the sorted-list requirement, replaces the earlier binomial-based sampler with bounded geometric sampling, and tightens the analysis so that

$G=\langle V,E\rangle$8

suffices, in place of the earlier $G=\langle V,E\rangle$9-style choice (Zhao et al., 12 Sep 2025).

Its neighbor sampling rule is explicit. Each neighbor is sampled independently with probability

$\mathbf{D}$0

and, if sampled, receives propagated mass

$\mathbf{D}$1

AGP-Dynamic inherits this propagation logic but changes the maintenance policy. Its defining data structure is a reference degree $\mathbf{D}$2 for each vertex $\mathbf{D}$3. The value $\mathbf{D}$4 records the degree at the last rebuild of $\mathbf{D}$5’s neighborhood data, and rebuilding is deferred until the true degree drifts outside a factor-two band: $\mathbf{D}$6 On an update to edge $\mathbf{D}$7, the algorithm adjusts local buckets for $\mathbf{D}$8 and $\mathbf{D}$9, checks these degree-drift conditions, and rebuilds only when the threshold is crossed. The method relies on the observation that exact current degrees are unnecessary for valid sampling structures; it is sufficient to maintain a controlled overestimate $a,b\in[0,1]$0 of the sampling probability and correct by rejection/acceptance. Because the degree distortion is bounded by a constant factor before rebuilding, the resulting amortized update cost becomes constant.

4. Approximation theory, truncation, and complexity bounds

AGP-Dynamic returns a $a,b\in[0,1]$1-approximation: for all vertices $a,b\in[0,1]$2 such that $a,b\in[0,1]$3,

$a,b\in[0,1]$4

The same guarantee applies to AGP-Static++ and is inherited by AGP-Dynamic (Zhao et al., 12 Sep 2025).

The general expected query-time theorem for AGP-style methods is

$a,b\in[0,1]$5

with $a,b\in[0,1]$6 for AGP-Static and $a,b\in[0,1]$7 for AGP-Static++. The specialized corollaries stated in the paper are

$a,b\in[0,1]$8

for AGP-Static and

$a,b\in[0,1]$9

for AGP-Static++. The improvement is described as roughly an $a+b\ge 1$0 gain in the regime where the first term dominates. AGP-Dynamic preserves this query complexity while adding dynamic updates.

The truncation depth is controlled by the weight oracle. The paper proves that truncating at

$a+b\ge 1$1

already yields a $a+b\ge 1$2-approximation. This makes the method practical for decaying weight sequences, because only the first $a+b\ge 1$3 propagation layers need to be realized.

The dynamic result is the update theorem: $a+b\ge 1$4 amortized time for each edge insertion or deletion. This is the principal asymptotic distinction between AGP-Dynamic and the earlier AGP-Static family.

5. Unbiased estimation, variance control, and initialization

The probabilistic propagation estimator is analyzed through conditional expectation and variance bounds. For propagated residue, the paper derives

$a+b\ge 1$5

where

$a+b\ge 1$6

This establishes unbiasedness of the propagation step (Zhao et al., 12 Sep 2025).

The conditional variance satisfies

$a+b\ge 1$7

which yields the global bounds

$a+b\ge 1$8

for deterministic initialization and

$a+b\ge 1$9

under randomized initialization. By Chebyshev’s inequality, selecting

$w_i\ge 0$0

produces the $w_i\ge 0$1-approximation with constant probability.

A separate implementation component is randomized initialization for compactly representable $w_i\ge 0$2. Rather than scanning all $w_i\ge 0$3 coordinates, the algorithm directly assigns large entries and samples within groups for small entries, preserving

$w_i\ge 0$4

with expected initialization time $w_i\ge 0$5. This is particularly useful for PageRank-like inputs and is one reason the framework remains practical when parameterized queries arrive online.

The static subset sampler used in AGP-Static++ also has a standalone complexity statement: bounded geometric sampling yields $w_i\ge 0$6 expected neighborhood sampling time under the Word RAM model, where $w_i\ge 0$7 is the expected output size. This detail matters because the query-time improvement is not solely analytical; it depends on a concrete replacement for the older binomial-based routine.

6. Empirical evaluation and practical significance

The experimental study uses nine real-world graphs, with sizes up to 4.85 million vertices and 117 million edges. Across these datasets, AGP-Static++, AGP-Dynamic, and the original AGP-Static are compared on query efficiency, update efficiency, and initialization cost (Zhao et al., 12 Sep 2025).

For query processing, AGP-Static++ improves efficiency by up to about 10× over AGP-Static, and AGP-Dynamic shows similar query-time behavior to Static++ with only a small additional overhead from dynamic maintenance. On the largest graph, LiveJournal, the paper reports that AGP-Static++ reached about 3.1% ARE in 187s versus 565s for AGP-Static. The intended interpretation is not merely faster execution, but tighter correspondence between the theory and the realized implementation.

For graph updates, AGP-Dynamic is reported to be up to 177× faster than the baselines, and even up to 207× or 553× in some update-pattern settings and datasets. These measurements are presented as empirical validation of the amortized $w_i\ge 0$8 update claim. The initialization subroutine also improves over naive full scans, with up to 3.29× speedup for random $w_i\ge 0$9 and up to 11× for PageRank-style initialization.

The broader significance of AGP-Dynamic lies in the conjunction of three properties that are usually treated separately: support for evolving graphs, support for parameterized propagation semantics decided at query time, and preservation of approximation guarantees. Within graph analytics, this makes the method relevant to continuously changing similarity or influence computations; within GNN pipelines, it supports dynamic propagation operators; within graph-RAG, it provides a backend for query-specific proximity semantics without requiring a distinct precomputation for every propagation model. A plausible implication is that AGP-Dynamic is best viewed not as a single-task algorithm, but as a dynamic propagation substrate for workloads in which both graph structure and propagation semantics are first-class runtime variables.