[Paper Review] Stream Graphs and Link Streams for the Modeling of Interactions over Time

Matthieu Latapy, Tiphaine Viard+1 more|arXiv (Cornell University)|Oct 11, 2017

Complex Network Analysis Techniques114 references184 citations

TL;DR

The paper introduces stream graphs and link streams to directly model interactions over time, extending graph concepts to capture both temporal and structural dimensions in a self-consistent framework that generalizes graphs.

ABSTRACT

Graph theory provides a language for studying the structure of relations, and it is often used to study interactions over time too. However, it poorly captures the both temporal and structural nature of interactions, that calls for a dedicated formalism. In this paper, we generalize graph concepts in order to cope with both aspects in a consistent way. We start with elementary concepts like density, clusters, or paths, and derive from them more advanced concepts like cliques, degrees, clustering coefficients, or connected components. We obtain a language to directly deal with interactions over time, similar to the language provided by graphs to deal with relations. This formalism is self-consistent: usual relations between different concepts are preserved. It is also consistent with graph theory: graph concepts are special cases of the ones we introduce. This makes it easy to generalize higher-level objects such as quotient graphs, line graphs, k-cores, and centralities. This paper also considers discrete versus continuous time assumptions, instantaneous links, and extensions to more complex cases.

Motivation & Objective

• Motivate the need for a formalism that captures both temporal and structural aspects of interactions.
• Generalize classical graph concepts (density, paths, cliques, etc.) to stream graphs and link streams.
• Ensure consistency with graph theory: graph concepts are special cases of the new framework.
• Provide a self-contained, intuitive development with discrete and continuous time considerations.
• Show that higher-level graph constructs (quotients, line graphs, k-cores, centralities) extend naturally to streams.

Proposed method

• Define stream graphs S=(T,V,W,E) and link streams L=(T,V,E) with precise presence functions.
• Systematically redefine elementary graph concepts (density, size, uniformity, compactness) for streams.
• Develop correspondences between stream concepts and graph concepts when the stream is graph-equivalent.
• Extend to substreams, clusters, and cliques, including their properties and induced substreams.
• Discuss discrete vs continuous time, instantaneous links, and extensions to bipartite and other generalizations.

Experimental results

Research questions

• RQ1 How can classical graph concepts be reformulated to jointly capture temporal and structural aspects of interactions?
• RQ2 In what sense do stream graphs and link streams generalize graphs, and how do standard graph relations carry over?
• RQ3 How do time dynamics (discrete vs continuous, instantaneous links) influence definitions like density, degree, and cliques?
• RQ4 What is the relationship between properties of streams and their graph-equivalent counterparts?

Key findings

• A coherent formalism that generalizes graph theory to model interactions over time, preserving relationships between concepts.
• Definitions of size, duration, uniformity, compactness, and density adapted to stream graphs and link streams.
• A mapping where graph concepts are recovered as special cases when the stream is graph-equivalent.
• Extensions to substreams, clusters, and cliques that align with standard graph notions in the appropriate limits.
• Framework accommodates both discrete and continuous time and enables extensions to more complex structures like line streams and k-cores.