Tight Practical Bounds for Subgraph Densities in Ego-centric Networks

TL;DR

The paper addresses whether local network structure in ego-centric graphs is shaped by social factors or by intrinsic mathematical constraints. It combines plain flag algebra-based bounds with motif counting and topological data analysis to define the subgraph spread ratio, a metric that quantifies how much of the feasible region of localized subgraph densities is realized. Key contributions include substantially tighter feasible regions (up to about threefold improvement over prior bounds), the introduction of the subgraph spread ratio, and empirical validation across 11 real networks showing social networks have smaller spreads than linkage graphs. The work provides a practical, scalable tool for network comparison and offers a framework for interpreting local graph structure in terms of domain-driven versus mathematically-determined factors.

Abstract

Subgraph densities play a crucial role in network analysis, especially for the identification and interpretation of meaningful substructures in complex graphs. Localized subgraph densities, in particular, can provide valuable insights into graph structures. Distinguishing between mathematically-determined and domain-driven subgraph density features, however, poses challenges. For instance, the lack or presence of certain structures can be explained by graph density or degree distribution. These differences are especially meaningful in applied contexts as they allow us to identify instances where the data induces specific network structures, such as friendships in social networks. The goal of this paper is to measure these differences across various types of graphs, conducting social media analysis from a network perspective. To this end, we first provide tighter bounds on subgraph densities. We then introduce the subgraph spread ratio to quantify the realized subgraph densities of specific networks relative to the feasible bounds. Our novel approach combines techniques from flag algebras, motif-counting, and topological data analysis. Crucially, effective adoption of the state-of-the-art in the plain flag algebra method yields feasible regions up to three times tighter than prior best-known results, thereby enabling more accurate and direct comparisons across graphs. We additionally perform an empirical analysis of 11 real-world networks. We observe that social networks consistently have smaller subgraph spread ratios than other types of networks, such as linkage-mapping networks for Wikipedia pages. This aligns with our intuition about social relationships: such networks have meaningful structure that makes them distinct. The subgraph spread ratio enables the quantification of intuitive understandings of network structures and provides a metric for comparing types of networks.

Figures (7)

• Figure 1: Localized point cloud with respect to the graph on 3 vertices with a single edge ($\overline{P_3}$) of some subset of Facebook pages MUSAE. The red points are outliers, consisting of $5\%$ of the data, determined by our pruning algorithm. The gray shaded area is the feasible region of values given by the plain flag algebra method. The subgraph spread ratio is $0.393$, meaning that roughly $40\%$ of the feasible region is covered by the localized point cloud.
• Figure 2: Localized point cloud with respect to the path graph on four vertices ($P_4$) of some subset of Facebook pages MUSAE. The red points are outliers, determined by our pruning algorithm. The darker shaded area is the feasible region of values given by the plain flag algebra method, whereas the lighter green region is the feasible region given in main. Our new feasible region is less than a third of the size. Note that these feasible regions only hold asymptotically in the size of the ego-centric networks, which accounts for the data-point that falls outside of our bound.
• Figure 3: All graphs on three and four vertices
• Figure 4: An Example Graph
• Figure 5: Plot of all localized point clouds of $H$ for $\lvert V(H)\rvert \le 4$ of Facebook data MUSAE. The gray region is the feasible region and red data points are outliers. A list of subgraph spread ratios can be found in Tables \ref{['myTable1']} and \ref{['myTable2']} under FB, the row is highlighted. The y axes follow the same layout as in Figure \ref{['fig::graphs']}.

Theorems & Definitions (1)

• Definition 1