Quilting Stochastic Kronecker Product Graphs to Generate Multiplicative Attribute Graphs

We describe the first sub-quadratic sampling algorithm for the Multiplicative Attribute Graph Model (MAGM) of Kim and Leskovec (2010). We exploit the close connection between MAGM and the Kronecker Product Graph Model (KPGM) of Leskovec et al. (2010), and show that to sample a graph from a MAGM it suffices to sample small number of KPGM graphs and \emph{quilt} them together. Under a restricted set of technical conditions our algorithm runs in $O((\log_2(n))^3 |E|)$ time, where $n$ is the number of nodes and $|E|$ is the number of edges in the sampled graph. We demonstrate the scalability of our algorithm via extensive empirical evaluation; we can sample a MAGM graph with 8 million nodes and 20 billion edges in under 6 hours.

💡 Research Summary

The paper addresses the long‑standing challenge of efficiently sampling graphs from the Multiplicative Attribute Graph Model (MAGM), a model that extends the Kronecker Product Graph Model (KPGM) by allowing each node to possess an independent binary attribute vector. While KPGM enjoys a sub‑quadratic sampling algorithm that exploits the fractal structure of its edge‑probability matrix, MAGM’s probability matrix lacks an obvious structure, leading to naïve O(n²) sampling methods that are infeasible for large networks.

The authors’ key insight is that, under a natural representation of node attributes as integer “configurations” λ_i, the MAGM probability matrix Q can be expressed as a permuted sub‑matrix of the KPGM matrix P: Q_{ij}=P_{λ_i,λ_j}. This observation enables a “quilting” strategy. First, nodes are partitioned into sets D₁,…,D_B such that no two nodes within the same set share the same attribute configuration. The partition is constructed by grouping nodes according to the multiplicity of each configuration; the resulting number of non‑empty sets B is provably minimal (Theorem 2).

The edge‑probability matrix Q is then decomposed into B² blocks Q^{(k,l)} that contain only entries where the source belongs to D_k and the target to D_l. Each block is permuted (via the mapping i↦λ_i) into a sub‑matrix of the KPGM matrix P. Because KPGM sampling can be performed in expected O(log n·|E|) time using a recursive quadrant‑selection scheme (Algorithm 1), each block can be sampled independently and efficiently. After sampling, the edges are mapped back to the original node indices, and all B² sampled sub‑graphs are “quilted” together to form a single MAGM sample (Algorithm 2).

The theoretical runtime of the overall procedure is O(B²·log n·|E|). The authors prove that, when attribute probabilities are balanced (μ_k = 0.5 for all k) and n = 2^d, B is with high probability O(log n). This follows from a Chernoff‑type bound on the maximum occupancy of attribute configurations, which behaves like the maximum of n independent Poisson(1) variables. Consequently, the expected runtime becomes O((log n)³·|E|), a dramatic improvement over the quadratic baseline. In the case of highly unbalanced attributes (μ ≠ 0.5), B can grow as Θ(n·μ·log n), leading to a slower but still sub‑quadratic bound O((n·log μ + log n)·|E|).

The paper also discusses extensions to the realistic scenario where n is not an exact power of two. By padding or truncating the node set to the nearest power of two and adjusting the attribute vectors accordingly, the same quilting framework remains applicable.

Empirical evaluation validates the theoretical claims. Experiments on synthetic graphs with node counts ranging from 10⁴ to 10⁷ and various attribute distributions show that the observed B is consistently far below the worst‑case bound, often much smaller than log n. The authors successfully sampled a MAGM graph with 8 million nodes and 20 billion edges in under six hours on a modest compute cluster, demonstrating practical scalability.

In summary, the paper introduces a novel sub‑quadratic sampling algorithm for MAGM by leveraging the structural similarity to KPGM, formalizing a partition‑and‑quilt methodology, providing rigorous probabilistic analysis of the partition size, and confirming scalability through extensive experiments. The work bridges a gap between expressive graph generative models and tractable sampling, opening avenues for large‑scale network simulation and hypothesis testing in domains where attribute‑driven connectivity is essential.