Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Temporal Triadic Closure: Finding Dense Structures in Social Networks That Evolve

Tom Davot, Jessica Enright, Jayakrishnan Madathil, Kitty Meeks

TL;DR

A definition of temporal c-closed graphs, in which if two vertices u and v have at least c common neighbors during a short interval of time, then u and v are adjacent to each other around that time, which shows that several real-world temporal networks are c-closed for rather small values of c.

Abstract

A graph G is c-closed if every two vertices with at least c common neighbors are adjacent to each other. Introduced by Fox, Roughgarden, Seshadhri, Wei and Wein [ICALP 2018, SICOMP 2020], this definition is an abstraction of the triadic closure property exhibited by many real-world social networks, namely, friends of friends tend to be friends themselves. Social networks, however, are often temporal rather than static -- the connections change over a period of time. And hence temporal graphs, rather than static graphs, are often better suited to model social networks. Motivated by this, we introduce a definition of temporal c-closed graphs, in which if two vertices u and v have at least c common neighbors during a short interval of time, then u and v are adjacent to each other around that time. Our pilot experiments show that several real-world temporal networks are c-closed for rather small values of c. We also study the computational problems of enumerating maximal cliques and similar dense subgraphs in temporal c-closed graphs; a clique in a temporal graph is a subgraph that lasts for a certain period of time, during which every possible edge in the subgraph becomes active often enough, and other dense subgraphs are defined similarly. We bound the number of such maximal dense subgraphs in a temporal c-closed graph that evolves slowly, and thus show that the corresponding enumeration problems admit efficient algorithms; by slow evolution, we mean that between consecutive time-steps, the local change in adjacencies remains small. Our work also adds to a growing body of literature on defining suitable structural parameters for temporal graphs that can be leveraged to design efficient algorithms.

Temporal Triadic Closure: Finding Dense Structures in Social Networks That Evolve

TL;DR

A definition of temporal c-closed graphs, in which if two vertices u and v have at least c common neighbors during a short interval of time, then u and v are adjacent to each other around that time, which shows that several real-world temporal networks are c-closed for rather small values of c.

Abstract

A graph G is c-closed if every two vertices with at least c common neighbors are adjacent to each other. Introduced by Fox, Roughgarden, Seshadhri, Wei and Wein [ICALP 2018, SICOMP 2020], this definition is an abstraction of the triadic closure property exhibited by many real-world social networks, namely, friends of friends tend to be friends themselves. Social networks, however, are often temporal rather than static -- the connections change over a period of time. And hence temporal graphs, rather than static graphs, are often better suited to model social networks. Motivated by this, we introduce a definition of temporal c-closed graphs, in which if two vertices u and v have at least c common neighbors during a short interval of time, then u and v are adjacent to each other around that time. Our pilot experiments show that several real-world temporal networks are c-closed for rather small values of c. We also study the computational problems of enumerating maximal cliques and similar dense subgraphs in temporal c-closed graphs; a clique in a temporal graph is a subgraph that lasts for a certain period of time, during which every possible edge in the subgraph becomes active often enough, and other dense subgraphs are defined similarly. We bound the number of such maximal dense subgraphs in a temporal c-closed graph that evolves slowly, and thus show that the corresponding enumeration problems admit efficient algorithms; by slow evolution, we mean that between consecutive time-steps, the local change in adjacencies remains small. Our work also adds to a growing body of literature on defining suitable structural parameters for temporal graphs that can be leveraged to design efficient algorithms.

Paper Structure

This paper contains 7 sections, 5 theorems, 5 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Temporal $c$-Closed Graphs
  3. Empirical Results.
  4. The Need for Neighborhood Stability
  5. Bound for Maximal Cliques
  6. Bounds for Other Dense Subgraphs
  7. Concluding Remarks

Key Result

lemma 8

Let $\mathcal{G}$ be a locally $\eta$-unstable, $(\Delta_0, \Delta_1, \Delta_2, c)$-closed temporal graph. For any two distinct vertices $u, v \in V(G)$ such that $uv \notin E(G_{[a, b]})$ for some time-interval $[a, b]$ where $b - a \geq \Delta_0 + \Delta_1 + \Delta_2$, it holds that ${\vert {CN_{[

Figures (1)

  • Figure 1: Cumulative closure rate of two real-world temporal networks. Each color corresponds to one choice of $(\Delta_0, \Delta_1, \Delta_2)$. For each $x$-value, the corresponding $y$-value is the cumulative closure rate, i.e., the fraction of tuples $([a,a+\Delta_1],u,v)$ such that $[a,a+\Delta_1]\subseteq [1+\Delta_0,\Lambda - \Delta_2]$ and $u$ and $v$ are distinct vertices that have $x$ common neighbors in $[a, a+\Delta_1]$ and are adjacent to each other during $[a - \Delta_0, a+\Delta_1 + \Delta_2]$.

Theorems & Definitions (19)

  • Definition 1: $(\Delta_0,\Delta_1, \Delta_2, c)$-closed graphs
  • Definition 2: closure number of a vertex
  • Definition 3: weakly $(\Delta_0, \Delta_1, \Delta_2, \gamma)$-closed graphs
  • Definition 4: $\Delta$-clique DBLP:journals/tcs/ViardLM16
  • Example 5
  • Definition 6: locally $\eta$-unstable graphs
  • Definition 7: pairwise $\eta$-unstable graphs
  • lemma 8
  • proof
  • Claim 9
  • ...and 9 more