Diagrammatic Rules for Triad Census

TL;DR

The paper revisits Moody's matrix-based triad census and introduces diagrammatic rules that convert the tally of all thirteen fully connected triads into compact closed-form expressions derived from the adjacency matrix $A$. By defining edge-type matrices $X$, $Y$, and $Z$ from $A$ and organizing counts through a unified form $t_\alpha = \frac{1}{s_\alpha}[ \sum(B_\alpha) - \operatorname{tr}(B_\alpha) ]$, it provides explicit matrix expressions for various triads (e.g., $t_1$, $t_2$, $t_3$, $t_6$, $t_8$, $t_{13}$) and demonstrates equivalences such as $t_{13} = \frac{1}{6} \operatorname{tr}(Y^3)$. The approach generalizes to all 13 fully connected triads, clarifies the role of symmetry factors, and offers a pathway to scalable motif counting beyond triads, with potential extensions to larger subgraph structures.

Abstract

In network theory, a triad census is a method designed to categorize and enumerate the various types of subgraphs with three nodes and their connecting edges within a network. Triads serve as fundamental building blocks for comprehending the structure and dynamics of networks, and the triad census offers a systematic approach to their classification. Typically, triad counts are obtained numerically, but lesser-known methods have been developed to precisely evaluate them without the need for sampling. In our study, we build upon Moody's matrix approach, presenting general diagrammatic rules that systematically and intuitively generate closed formulas for the occurrence numbers of triads in a network.

Figures (1)

• Figure 1: There are 16 possible types of triads, each representing a unique arrangement of relationships between three nodes in a network. In this work we focus on the thirteen fully connected ones.