[Paper Review] Transitivity Correlation: Measuring Network Transitivity as Comparative Quantity

David Dekker, David Krackhardt+1 more|arXiv (Cornell University)|Aug 2, 2017

Complex Network Analysis Techniques27 references3 citations

TL;DR

This paper introduces two new network transitivity measures—Transitivity Phi (TPhi) and Transitivity Correlation (TC)—that quantify the comparative propensity of ties given two-paths, addressing limitations of traditional clustering coefficients. TC, which correlates tie existence with the number of two-paths across all intermediates, better captures transitivity in highly structured networks like windmills, where it approaches 1, unlike TPhi which tends to 0.

ABSTRACT

This paper proposes that common measures for network transitivity, based on the enumeration of transitive triples, do not reflect the theoretical statements about transitivity they aim to describe. These statements are often formulated as comparative conditional probabilities, but these are not directly reflected by simple functions of enumerations. We think that a better approach is obtained by considering the linear regression coefficient of ties $i\, ightarrow\,j$ on the number of two-paths $i\, ightarrow\,k ightarrow\,j$ for the $(n-2)$ possible intermediate nodes $k$. Two measures of transitivity based on correlation coefficients between the existence of a tie and the existence, or the number, of two-paths are developed, and called "Transitivity Phi" and "Transitivity Correlation". Some desirable properties for these measures are studied and compared to existing clustering coefficients, in both random (Erdös-Renyi) and in stylized networks (windmills). Furthermore, it is shown that under the condition of zero Transitivity Correlation, the total number of transitive triples is determined by four underlying features of any directed graph: size, density, reciprocity, and the covariance between indegrees and outdegrees. Also, it is demonstrated that plotting conditional probability of ties, given the number of two-paths, provides valuable insights into empirical regularities and irregularities of transitivity patterns.

Motivation & Objective

• To address the disconnect between theoretical formulations of transitivity—based on comparative conditional probabilities—and existing measures that rely on simple enumeration of transitive triples.
• To develop measures that reflect the increased propensity of direct ties given indirect paths, aligning with the conceptual definition of transitivity in social network theory.
• To provide a comparative, rather than absolute, measure of transitivity that enables valid cross-network comparisons across different sizes and densities.
• To demonstrate that transitivity is not solely a function of triple counts but is determined by four structural features: size, density, reciprocity, and degree covariance.
• To show that plotting conditional tie probabilities given two-path counts reveals empirical patterns and anomalies in transitivity behavior.

Proposed method

• Define Transitivity Phi (TPhi) as the correlation between the existence of a direct tie (i,j) and the existence of a single random two-path (i→k→j) via a randomly selected intermediate k.
• Define Transitivity Correlation (TC) as the correlation between the existence of a direct tie (i,j) and the total number of two-paths (i→k→j) across all possible intermediates k.
• Use linear regression of tie indicators on two-path counts to derive the correlation-based measures, treating each node pair as a random sample.
• Analyze the measures in stylized networks (e.g., windmill graphs) and random Erdős-Rényi networks to compare behavior and interpretability.
• Derive a theoretical condition under which TC = 0, showing it is equivalent to a specific combination of network features: size, density, reciprocity, and covariance between in- and out-degrees.
• Use ego-networks (induced subgraphs of a node and its out-neighbors) as the structural basis for computing both measures.

Experimental results

Research questions

• RQ1 How can transitivity be measured in a way that reflects its theoretical definition as an increased propensity of direct ties given indirect paths?
• RQ2 Why do traditional clustering coefficients fail to capture the comparative nature of transitivity, and how can this be corrected?
• RQ3 What structural features determine the value of Transitivity Correlation, and can a zero TC value be characterized by a closed-form condition?
• RQ4 How do TPhi and TC differ in their behavior across network types, particularly in highly transitive structures like windmills?
• RQ5 Can visualizing conditional tie probabilities given two-path counts reveal meaningful empirical patterns in real-world networks?

Key findings

• Transitivity Correlation (TC) tends to 1 in windmill graphs with many wings, indicating strong transitive structure, while Transitivity Phi (TPhi) tends to 0, showing TC better captures transitivity in such cases.
• TC is zero if and only if the network’s size, density, reciprocity, and covariance between in- and out-degrees satisfy a specific joint condition, revealing a deep structural constraint.
• The clustering coefficient can take any value in (0,1) under the Erdős-Rényi model regardless of transitivity, highlighting its inability to reflect comparative propensity.
• Plotting conditional tie probabilities given the number of two-paths reveals empirical regularities and irregularities in transitivity, such as deviations from linearity suggesting hidden mechanisms.
• Both TC and TPhi reach 1 in networks composed of disconnected complete subgraphs of equal size, indicating that maximum transitivity is not uniquely captured by correlation measures.
• The proposed measures are invariant to network size and density, enabling valid comparisons across diverse networks, unlike traditional clustering coefficients.