The Local Closure Coefficient

Abstract

The phenomenon of edge clustering in real-world networks is a fundamental property underlying many ideas and techniques in network science. Clustering is typically quantified by the clustering coefficient, which measures the fraction of pairs of neighbors of a given center node that are connected. However, many common explanations of edge clustering attribute the triadic closure to a "head" node instead of the center node of a length-2 path-for example, "a friend of my friend is also my friend. " While such explanations are common in network analysis, there is no measurement for edge clustering that can be attributed to the head node. Here we develop local closure coefficients as a metric quantifying head-node-based edge clustering. We define the local closure coefficient as the fraction of length-2 paths emanating from the head node that induce a triangle. This subtle difference in definition leads to remarkably different properties from traditional clustering coefficients. We analyze correlations with node degree, connect the closure coefficient to community detection, and show that closure coefficients as a feature can improve link prediction.

The Power of Pivoting for Exact Clique Counting

“…The per-vertex and per-edge counts are sometimes called local counts. In clustering applications, the local counts are used as vertex or edge weights, and are therefore even more useful than global counts [4,18,24,30,31,36]. Fig.…”

“…The simplest case of clique counting is triangle counting, which has received much attention from the data mining and algorithms communities. Recent work has shown the relevance of counts of large subgraphs (4, 5 vertex patterns) [5,23,27,32,35]. Local clique counts have played a significant role in a flurry of work on faster and better algorithms for dense subgraph discovery and community detection [4,24,30,31].…”

“…The latter results define the "motif conductance", where cuts are measured by the number of subgraphs (not just edges) cut. This has been related to higher order clustering coefficients [35,36]. These quantities are computed using local clique counts, underscoring the importance of these numbers.…”

“…General clique counting has received much attention in recent times [1,10,13,16,17,19,21]. There is a line of recent work on exploiting clique counts for community detection and dense subgraph discovery [4,18,24,30,31,36].…”

Closure coefficients in scale-free complex networks

“…While the clustering coefficient turns out to be useful in some important network applications, other statistics that measure triadic closure may be more useful in other network applications. One promising such network statistic is the recently introduced the closure coefficient [37]. Where the clustering coefficient measures the fraction of times a vertex of degree k serves as the center of a triangle, the closure coefficient measures the fraction of times a vertex of degree k serves as the head of a triangle.…”