Stream Graphs and Link Streams for the Modeling of Interactions over Time (1710.04073v1)

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.

• The paper introduces stream graphs and link streams as a formalism extending traditional graph theory to model time-varying interactions.
• It redefines fundamental graph concepts like density, degree, clustering, and connected components to intrinsically handle temporal dynamics without data loss.
• This framework facilitates more accurate analysis, simulation, and prediction of interactions over time compared to traditional static or sliced methods.

Overview of Stream Graphs and Link Streams for Modeling Interactions Over Time

The paper Stream Graphs and Link Streams for the Modeling of Interactions over Time introduces a formalism designed to address the inherent limitations of using traditional graph theory in modeling time-varying interactions. By generalizing graph concepts to accommodate both temporal and structural aspects of interactions, the authors establish a cohesive and consistent framework that allows for the direct paper of temporal dynamics akin to graph-theoretic studies for static relational data.

Temporal Graph Modeling

Traditional graph theory typically struggles to capture the full temporal complexity of interactions, whether it be human communications, social relationships, or network traffic, due to its static nature. By defining stream graphs and link streams, Latapy et al. offer a methodological enhancement that incorporates dynamic changes over time. Stream graphs accommodate dynamics in node presence and linkage, allowing for concepts such as density, degree, and clustering coefficients to be extended to the temporal domain.

Key Concepts and Definitions

• Stream Graphs and Link Streams: Stream graphs formalize dynamic interactions by considering nodes and edges over intervals of time, whereas link streams consider interactions where node dynamics are non-existent.
• Density and Degree: Density is redefined in this context as the probability that links exist when node presence allows for them. Degree metrics account for temporal node and edge contributions.
• Clustering and Cliques: The paper extends clustering and clique definitions to account for temporally maximal overlapping interactions, a crucial aspect for real-world dynamic networks.
• Connected Components and Paths: It introduces notions of reachability and connectedness that retain temporal specificity, crucial for evaluating network robustness and efficiency in dynamic settings.

Practical and Theoretical Implications

The paper's formalism facilitates analysis without necessitating arbitrary time slices or augmentations that often erase critical timing information. This intrinsic handling of time affords far greater fidelity in capturing the dynamics of interactions, which is pivotal for simulations, predictions, and real-time monitoring in complex systems.

The theoretical implications extend into various realms, providing a groundwork for future developments in temporal network analytics, modeling frameworks, and spontaneous network phenomena. Practically, this can influence how social dynamics are understood, aiding in constructing better communication protocols and security measures in networks subject to temporal influences.

Future Directions

Future research should address algorithmic challenges associated with processing large and complex stream graphs to fully leverage the proposed framework’s benefits. Additionally, developing models that accurately predict interaction patterns using stream graph formalism could revolutionize fields reliant on temporal predictions, such as epidemiology and network security.

In conclusion, this paper paves the way for advanced studies in temporally-aware network science, broadening the applicability of graph theoretical tools in analyzing dynamic phenomena across diverse domains. It fosters enriched understanding and manipulation of networks where time plays a critical role in shaping their structure and function.