Source-linked AI summary
Temporal Graphs
Vassilis Kostakos
TL;DR
The paper addresses the difficulty of analysing rich temporal datasets without sacrificing either temporal information or the analytical benefits of graph representations. It introduces temporal graphs and associated metrics, applies them to real-world data, and reports distinct temporal measures that identify key nodes in dynamic processes. The approach assumes temporal events have no duration, limiting direct applicability where events overlap.
Problem
Emulations retain temporal data but sacrifice universal graph metrics and make analyses harder to extrapolate and compare, while graph snapshots sacrifice most rich temporal information.
Method
The paper represents rich temporal datasets as temporal graphs, defines temporal, geodesic, and availability metrics, and demonstrates construction under one-way and two-way communication assumptions.
Results
Temporal metrics are distinct from static graph metrics, quantify relationships over time, and quickly identify key nodes in dynamic processes.
Takeaways & Limitations
Temporal graphs provide a graph-based basis for describing and analysing dynamic data while retaining its temporal information and using standard shortest-path algorithms.
Takeaways & Limitations
The approach assumes temporal events have no duration, which may be inappropriate for face-to-face communication and domains where event overlap is crucial.
Abstract
from arXiv · showhide
We introduce the idea of temporal graphs, a representation that encodes temporal data into graphs while fully retaining the temporal information of the original data. This representation lets us explore the dynamic temporal properties of data by using existing graph algorithms (such as shortest-path), with no need for data-driven simulations. We also present a number of metrics that can be used to study and explore temporal graphs. Finally, we use temporal graphs to analyse real-world data and present the results of our analysis.
Introduction
Existing graph analyses often aggregate relationships into static snapshots, while emulations use temporally rich data but lose universal graph metrics and are harder to compare. Temporal graphs address this tension by retaining temporal information in a graph representation and supporting temporal metrics and real-world analysis.
- Background: Static graphs aggregate intermittently instantiated relationships into discrete time intervals, making analysis tractable but discarding temporal detail.This practice was also historically shaped by the limited availability of rich temporal data.
- Emulations: Emulation-based analysis uses observed events to model dynamic processes such as information sharing or disease spread over time.Analysts can vary diffusion parameters and examine alternative real-world scenarios.
- Emulations: Emulations sacrifice universal node metrics, produce messier analyses, and make extrapolation and cross-dataset comparison more difficult.Graph snapshots offer a compromise but sacrifice most of the dataset’s rich temporal information.
- Temporal graphs: Temporal graphs retain rich temporal information while preserving the analytical benefits of static graph analysis.They define temporal proximity, geodesic proximity, and temporal availability to quantify relationships and node roles over time.
- Method and analysis: The paper constructs temporal graphs under one-way and two-way communication assumptions, then applies them to real-world social-interaction datasets.The analysis reports metrics that differ from static graph metrics and identify key nodes in dynamic processes.
Temporal graphs
Temporal graphs retain event timing while enabling graph-based analysis. They represent temporal instances and interactions explicitly, support temporal metrics, and reveal relationships that static aggregation can misrepresent.
- Construction: Temporal graphs model instantaneous events between time-stamped node instances, preserving temporal information that static aggregation discards.The construction assumes point events with no duration; waiting links connect consecutive instances, while transaction links encode interactions.
- Construction: Static graphs create one node per person and aggregate any observed email relationship into a continuously available directed edge.This representation supports conventional centrality measures but loses event frequency and timing between interactions.
- Temporal paths: Temporal graph paths enforce event order, eliminating apparent static paths such as DBC when D communicates with B after B communicates with C.The same temporal constraints also remove paths such as EDBC and ADB from the static representation.
- Metrics: Temporal proximity measures time-respecting distance between node sets under temporal pre- and post-conditions and can be computed with weighted shortest-path algorithms.The paper also defines average temporal proximity and geodesic proximity, with geodesic proximity retaining temporal restrictions while discarding edge weights.
- Metrics: Temporal availability measures the probability that a path exists between two nodes at a given moment, complementing proximity measures that discard unavailable paths.For example, P(B,C) can be small even though only two of B’s four instances can reach C.
- Empirical analysis: The empirical analysis uses directed temporal links for email, bidirectional links for face-to-face encounters, and reports metric distributions and in/out relationships across datasets.Figures and tables examine degree, instance-set size, link weights, metric relationships, and temporal proximity distributions.
Discussion
Temporal graphs and their metrics reveal temporal differences between datasets that static structural similarities would not expose. The analysis contrasts Enron’s irregular reachability and propagation patterns with Cityware’s more consistent temporal behavior.
- Temporal graph metrics: Temporal graphs retain temporal dynamics while supporting metrics such as P, G, and V for comparing node relationships over time.P measures time to reach another node, G measures hops or events, and V measures reachability probability.
- Dataset comparison: Cityware is denser and tighter than Enron: people are reached more quickly, with fewer hops, while Enron has extremely low temporal availability.The low V in Enron indicates that only a small proportion of temporal paths are available on average.
- Structural similarities: Despite similar static degree distributions, both datasets show approximate power-law structure, their instance-set sizes show exponential decay, and their link-weight distributions also decay exponentially.Instance-set size indicates how often people appear and therefore how many opportunities they have to receive or transmit information; link weight reflects activity frequency.
- Within-metrics analysis: Enron is less structured than Cityware, combining extremely fast and slow individuals, whereas Cityware shows smaller variation between people.In Cityware, individuals with more links tend to reach and be reached by the network more quickly.
- Within-metrics analysis: Enron’s temporal availability divides people into groups that are hard or easier to reach, while Cityware shows an apparent inverse relationship between reaching and being reached.In Cityware, low-degree individuals’ positions depend partly on whether they appear early or late in the observation period.
- Between-metrics analysis: In Cityware, high-degree individuals are quickly and easily reached with high availability, but lower connectivity rapidly increases required hops and reach time.For transmitting information, Cityware paths are often at least 15 hops long, whereas Enron’s low-degree individuals remain relatively quickly and easily reached.
Conclusion and ongoing work
Temporal graphs preserve rich temporal dynamics while retaining graph-based benefits for describing, communicating, and analysing dynamic data. The paper also identifies event duration as an unresolved boundary for the current representation.
- Conclusion: Temporal graphs retain rich temporal information while providing graph-based metrics for describing and analysing inherently dynamic data.The representation is intended to preserve temporal dynamics without losing the communicative and analytical benefits of graphs.
- Conclusion: Temporal graph analysis relies on standard shortest-path algorithms, allowing most existing graph software tools to handle it easily.The paper presents an array of metrics and uses them to compare two real-world datasets.
- Ongoing work: The current model assumes temporal events have no duration, which may not fit face-to-face communication and other domains with overlapping events.The authors identify tools for representing event duration as ongoing work.
Code for generating temporal graphs
The generation code reads timestamped sender–recipient records and constructs temporal-graph instances and links. It supports either unidirectional or bidirectional communication data.
- Directionality: The two-way option controls whether communication is represented only from sender to recipient or in both directions.This corresponds to the paper’s unidirectional versus bidirectional construction cases.
- Instance construction: For each device, the code creates a directed link between consecutive appearances, weighted by the elapsed time between them.A link is added only when the device appeared previously on a different date.
- Input processing: The code reads each input line as a date, sender, and recipient.Comma-separated fields are trimmed before graph construction.
- Communication links: Each communication event connects the sender and recipient instances with a zero-weight link.When two-way mode is enabled, the reverse recipient-to-sender link is also emitted.