Time-Varying Graphs and Social Network Analysis: Temporal Indicators and Metrics

• The interplay between social influence and network change is recognized, yet its resulting dynamical patterns remain qualitatively little understood.Source: §1 · p. 1
• Most social-network instruments are designed for static networks and generally fail to capture the evolution of phenomena and their dynamical properties.Source: §1 · p. 1
• Temporal indicators such as distance and connectivity address fine-grain dynamics but do not show how network properties evolve over longer periods.Source: §1 · p. 1
• The proposed approach tracks atemporal indicators on aggregated static-graph sequences and temporal indicators on successive time-windowed subgraphs of the original time-varying graph.The paper expresses its temporal concepts and evolution properties using the time-varying graph formalism and discusses examples of both indicator types.Source: §1 · p. 1

2.1 The TVG Formalism

• The time-varying graph formalism represents entities, relations, and time-dependent edge presence, while this social-network analysis omits latency because propagation time is often unnecessary for its scope.The formalism is interaction-centric and can describe transportation, communication, and social networks.Source: §2.1 · p. 2, §2 · p. 2

2.2 Journeys and related Temporal Concepts

• A journey is a path over time, with successive edges traversed at nondecreasing times; it has both hop count and end-to-end duration.Source: §2.2 · p. 2
• Shortest, foremost, and fastest distances respectively minimize journey hops, arrival time measured from t, and end-to-end journey duration.These are three distinct distance measures for journeys departing at or after t.Source: §2.2 · p. 2
• Temporal connectivity, eccentricity, diameter, betweenness, and closeness characterize fine-grain node interactions, but not how networks evolve over longer periods.Source: §2.2 · p. 2

3.1.1 Methodological approach

• Footprints aggregate interactions within successive time-windows; changing window size adjusts the granularity of atemporal measurements, and windows may overlap.The node set may also vary between footprints, such as by retaining only nodes with an adjacent edge in each interval.Source: §3.1.1 · p. 3

3.1.2 Indicators and Discussions

• The scientific publishing network’s density declined over time, a trend the authors explain by the increasing number of authors and the changing node sets across footprints.The example uses one-year windows and includes papers with more than 150 citations.Source: §3.1.2 · p. 3, Figure 1 · p. 3
• Clustering first showed sparse connectivity, then shifted around 1999 toward denser sub-communities in the scientific publishing network.The authors present average clustering coefficient evolution over the footprint sequence.Source: §3.1.2 · p. 3, Figure 2 · p. 3
• Modularity showed a phase transition around 1999 from a monotone trend to a decreasing one, interpreted as initially separate groups interconnecting into fewer, larger groups.The authors use modularity to characterize community structure and show its average evolution across footprints.Source: §3.1.2 · p. 4, Figure 3 · p. 4
• The decreasing degree power law indicates that interconnection among groups is driven by low-degree nodes acting as hubs within groups.The authors describe the plotted exponent as indicating how closely each footprint’s degree distribution follows a power law.Source: §3.1.2 · p. 4, Figure 4 · p. 4

3.2.1 Methodological approach

• Because journey-based parameters depend on interaction chronology, their evolution requires sequences of temporal subgraphs rather than aggregated static footprints.Each temporal subgraph restricts the original graph to a time interval; these intervals may be disjoint or overlapping.Source: §3.2.1 · pp. 4–5

3.2.2 Indicators and Discussions

• Temporal distances and views can be tracked across temporal subgraphs to describe how close nodes become in time or in hops.The paper distinguishes shortest, foremost, and fastest distances; temporal view captures the latest possible message-emission time consistent with receipt by a given time.Source: §3.2.2 · p. 5
• The paper proposes studying temporal eccentricity and diameter evolution, which it says had not previously been considered, as a possible lens on fairness or inequality among participants.Eccentricity reflects a node’s reachability capacity and potential impact on the network.Source: §3.2.2 · p. 5