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Graphon Mixtures

Sevvandi Kandanaarachchi, Cheng Soon Ong

TL;DR

The paper addresses the challenge of modeling networks that combine hub-dominated sparsity with dense communities. It introduces graphon mixtures that couple a sparse line-graph component $U$ with a dense graphon $W$ to generate ($U$,$W$)-mixture graphs, with density controlled by the ratio of sparse to dense nodes. The authors establish a theoretical framework linking line-graph limits, mass-partitions, and disjoint clique graphons to enable hub identification and estimation of the sparse graphon $U$, providing consistency guarantees under finite and infinite partition regimes. Empirically, the method yields accurate top-$k$ degree predictions and improved estimation of $U$ on synthetic and real networks, demonstrating practical value for understanding and predicting the role of hubs in mixed sparse-dense graph regimes.

Abstract

Social networks have a small number of large hubs, and a large number of small dense communities. We propose a generative model that captures both hub and dense structures. Based on recent results about graphons on line graphs, our model is a graphon mixture, enabling us to generate sequences of graphs where each graph is a combination of sparse and dense graphs. We propose a new condition on sparse graphs (the max-degree), which enables us to identify hubs. We show theoretically that we can estimate the normalized degree of the hubs, as well as estimate the graphon corresponding to sparse components of graph mixtures. We illustrate our approach on synthetic data, citation graphs, and social networks, showing the benefits of explicitly modeling sparse graphs.

Graphon Mixtures

TL;DR

The paper addresses the challenge of modeling networks that combine hub-dominated sparsity with dense communities. It introduces graphon mixtures that couple a sparse line-graph component with a dense graphon to generate (,)-mixture graphs, with density controlled by the ratio of sparse to dense nodes. The authors establish a theoretical framework linking line-graph limits, mass-partitions, and disjoint clique graphons to enable hub identification and estimation of the sparse graphon , providing consistency guarantees under finite and infinite partition regimes. Empirically, the method yields accurate top- degree predictions and improved estimation of on synthetic and real networks, demonstrating practical value for understanding and predicting the role of hubs in mixed sparse-dense graph regimes.

Abstract

Social networks have a small number of large hubs, and a large number of small dense communities. We propose a generative model that captures both hub and dense structures. Based on recent results about graphons on line graphs, our model is a graphon mixture, enabling us to generate sequences of graphs where each graph is a combination of sparse and dense graphs. We propose a new condition on sparse graphs (the max-degree), which enables us to identify hubs. We show theoretically that we can estimate the normalized degree of the hubs, as well as estimate the graphon corresponding to sparse components of graph mixtures. We illustrate our approach on synthetic data, citation graphs, and social networks, showing the benefits of explicitly modeling sparse graphs.

Paper Structure

This paper contains 40 sections, 35 theorems, 184 equations, 17 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Limits of sparse graphs and line graphs
  3. Sparse graphs are stars in the limit
  4. Related work
  5. Mixtures of sparse and dense graphs
  6. The graph mixture model
  7. Expectations and edge density
  8. The top-$k$ degrees of sparse $(U,W)$ mixture graphs
  9. Estimating $U$ for finite number of partitions
  10. Estimating $U$ for infinite number of partitions
  11. Experiments
  12. Illustration with synthetic data
  13. Real-world networks
  14. Conclusions
  15. Background: Limits of sparse graphs and line graphs
  16. ...and 25 more sections

Key Result

Theorem 2.3

(Janson2016321 Thm 8.3) A graph limit is a line graph limit if and only if it is a disjoint clique graph limit.

Figures (17)

  • Figure 1: (a) A graph $G$ and its line graph $L(G)$. Vertices in $L(G)$ are edges of $G$. Vertices of $L(G)$ are connected if the corresponding edges in $G$ share a vertex. (b) Line graphs of disjoint stars are disjoint cliques. Thus, the inverse line graph of disjoint cliques are disjoint stars.
  • Figure 2: The overview of $(U,W)$-mixture graphs. A disjoint clique graph $H_m$ is sampled from graphon $U$. The inverse line graph $G_s = L^{-1}(H_m)$ is a disjoint set of stars. A graph $G_d$ is sampled from graphon $W$. Then, $G_s$ and $G_d$ are joined according to graph joining rules resulting in the mixture graph $G_n$.
  • Figure 3: Unique log degree values of a $(U,W)$ mixture graph $G_{n_i}$. The line fitted to points $\{(j, \log( \text{deg }_{G_{n_i}} v_{(j)}))\}_{j = 1}^{k_i}$ is shown in red and the line fitted to degrees generated from $W$ is shown in green. The line fitted to points $\{(j, \log \frac{m_{s_i}}{j}) \}_{j = 1}^{N}$ is shown in blue.
  • Figure 4: Graphons $U$ and $W$ used in Example 1.
  • Figure 5: A $(U,W)$-mixture graph with $n_{d_i} = 100$ and $n_{s_i} = 600$ where $U$ and $W$ are shown in Figure \ref{['fig:UandWEx1']}. The mixture graph $G_{n_i}$ resembles a graph from a sparse sequence.
  • ...and 12 more figures

Theorems & Definitions (76)

  • Definition 2.1: Dense and sparse graph sequences
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 66 more