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Temporal Networks

Petter Holme, Jari Saramäki

arXiv:1108.1780v2nlin.AOcs.SIphysics.data-anphysics.soc-ph

TL;DR

Static network models miss when interactions occur, even though timing can shape contagion and information diffusion. This review defines temporal-network representations, surveys analytical methods and models, and shows that temporal heterogeneity can produce slower spreading than Poissonian assumptions.

  • Problem

    Static network models separate interaction timing from network structure, limiting analysis of systems whose dynamics depend on when edges activate.

  • Method

    The review categorizes contact-sequence and interval representations, surveys methods for temporal structure, and discusses models linking temporal networks to dynamical behavior.

  • Results

    Broad inter-event-time distributions produce slower late-stage epidemic spreading than Poissonian timing, matching empirical email data and computer-worm dynamics.

  • Takeaways & Limitations

    Temporal interaction patterns should be represented in the network because burstiness and heterogeneous timing can substantially alter information and disease-spreading dynamics.

  • Takeaways & Limitations

    A suitable reference or null model for temporal motifs remains unresolved, particularly when sparse event times make random reshuffling misleading.

Abstract

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A great variety of systems in nature, society and technology -- from the web of sexual contacts to the Internet, from the nervous system to power grids -- can be modeled as graphs of vertices coupled by edges. The network structure, describing how the graph is wired, helps us understand, predict and optimize the behavior of dynamical systems. In many cases, however, the edges are not continuously active. As an example, in networks of communication via email, text messages, or phone calls, edges represent sequences of instantaneous or practically instantaneous contacts. In some cases, edges are active for non-negligible periods of time: e.g., the proximity patterns of inpatients at hospitals can be represented by a graph where an edge between two individuals is on throughout the time they are at the same ward. Like network topology, the temporal structure of edge activations can affect dynamics of systems interacting through the network, from disease contagion on the network of patients to information diffusion over an e-mail network. In this review, we present the emergent field of temporal networks, and discuss methods for analyzing topological and temporal structure and models for elucidating their relation to the behavior of dynamical systems. In the light of traditional network theory, one can see this framework as moving the information of when things happen from the dynamical system on the network, to the network itself. Since fundamental properties, such as the transitivity of edges, do not necessarily hold in temporal networks, many of these methods need to be quite different from those for static networks.

I. INTRODUCTION · II. TYPES OF TEMPORAL NETWORKS · A. Person-to-person communication

Temporal networks represent pairwise interactions together with when they occur, enabling analysis of dynamical systems whose behavior depends on temporal contact patterns. The review surveys temporal-network systems and methods while emphasizing that projecting temporal structure into static graphs can lose important information.

  • I. INTRODUCTION: Graphs represent complex systems as vertices denoting system units and edges denoting interacting pairs.Examples span the Internet, metabolism, proteomes, and sexual-contact systems.
  • I. INTRODUCTION: Temporal ordering can alter reachability and spreading, so temporal networks need not preserve static-network properties such as edge transitivity.The figure illustrates disease spreading through a contact sequence at successive times.
  • I. INTRODUCTION: Temporal-network approaches move information about when interactions happen from the dynamical system onto the network structure.This contrasts with traditional modeling, which separates a static underlying network from dynamics occurring on it.
  • I. INTRODUCTION: Temporal-network analysis suits systems of pairwise agents whose interactions combine randomness and regularity without being excessively random or regular.The framework also requires temporal structure to matter rather than being safely discarded.
  • I. INTRODUCTION: Projecting a temporal network onto a static graph always loses information, although that loss may be insignificant when contact times are sufficiently uncorrelated or evenly distributed.The added analytical complexity is therefore justified mainly when timing materially affects the dynamics.
  • I. INTRODUCTION: A temporal-network model is unnecessary when the dynamical system evolves too rapidly relative to changes in contacts or edge activity.The Internet is given as an example because data packets travel faster than its topology changes.
  • I. INTRODUCTION: The review covers interdisciplinary temporal-network terminology, real-world temporal graphs, structural measurements, static representations, and dynamics on temporal networks.It aims to help readers across disciplines understand related research rather than impose one unified theory.
  • A. Person-to-person communication: Electronic one-to-one communication records are well suited to temporal-network analysis of information or electronic-virus spreading.Such data include timestamped person-to-person messages and dialogues occurring within time intervals.

B. One-to many information dissemination … H. Ecological networks

The paper applies temporal-network analysis across information dissemination, human and animal proximity, biological systems, computing, infrastructure, brain connectivity, and ecological interactions. These applications show that changing contacts, activity, and connectivity can shape spreading, computation, and biological dynamics, while some slowly changing networks may not require temporal modeling.

  • B. One-to many information dissemination: One-to-many information dissemination, including blogs, microblogs, and chain-letter emails, is framed as a domain that could benefit from temporal-network analysis.Chain-letter email represents an intermediate form between one-to-one and one-to-many communication.
  • C. Physical proximity: Human proximity networks record who is close to whom at what time, supporting studies of airborne pathogens and word-of-mouth information spreading.Electronic devices have made large-scale proximity data cheaper to collect than earlier confined-space fieldwork.
  • D. Cell biology: Temporal-network models can represent biological systems including molecular interactions, gene regulation, and metabolism, whose active components and relationships change over time.Metabolic networks contain a larger reaction system, but only part is active at a given time and subcellular location.
  • E. Distributed computing: Distributed computing systems provide early theoretical settings for temporal networks because parallel computation depends on information spreading between relatively independent units.The units may need to operate with information of different ages.
  • F. Infrastructural networks: Many infrastructural networks change too slowly relative to their operating dynamics for temporal-network modeling to be useful.For the Internet, data flows operate on a seconds scale, whereas major topology changes occur much more slowly.
  • G. Neural and brain networks: Neural networks may benefit from temporal-network analysis across scales ranging from individual-neuron spiking to coarse-grained connectivity between brain areas.Experimental modalities for functional connectivity involve tradeoffs between spatial and temporal resolution.
  • G. Neural and brain networks: A temporal framework can enrich functional brain-network studies by treating motifs as temporal subnetworks rather than only analyzing static correlations.The passage contrasts static EEG-correlation networks with approaches that directly incorporate the time domain.
  • H. Ecological networks: Ecological and animal proximity or mobility networks are dynamic systems in which seasonal interactions, disease, and information spreading can be studied with temporal networks.Ecological networks may be trophic or mutualistic, while animal movement networks represent premises as vertices and cattle movements as edges.

I. Other systems · III. PRELIMINARIES

The review identifies further potential applications of temporal-network modeling and introduces two overlapping representations: contact sequences for negligible-duration interactions and interval graphs for interval-based edge activity. It also notes extensions to weighted, traversable, or duration-dependent interactions and alternative representations that reduce or transform temporal information.

  • I. Other systems: Temporal-network modeling may apply to many systems beyond those already discussed, including systems identified by examining whether complex networks have sufficient temporal structure.The authors suggest using the complex-network literature to find additional candidate applications.
  • I. Other systems: The review’s two fundamental temporal-network representations are contact sequences and interval graphs.Figure 4 distinguishes contacts occurring at points in time from contacts represented over active intervals.
  • I. Other systems: Supply networks in manufacturing were an early application, while economic systems and citation-network growth may also benefit from temporal-network analysis.Citation networks are usually treated as strictly growing, but their growth may contain temporal effects.
  • III. PRELIMINARIES: Contact sequences represent N vertices interacting at negligible-duration times as C triples (i, j, t), where t records each contact time.This is the first of two rough and overlapping temporal-network classes.
  • III. PRELIMINARIES: Interval graphs represent edges as sets of active intervals, with unprimed times marking interval beginnings and primed times marking their ends.The static graph containing an edge whenever two vertices have any contact is the time-aggregated graph; proximity networks are a natural example.
  • III. PRELIMINARIES: The framework can be extended to weighted temporal networks, nonzero traversal times, and contacts completed only after duration δt.Such duration-dependent contacts can be represented as quadruples (i, j, t, δt).
  • III. PRELIMINARIES: Other representations either map temporal graphs to static graphs, as discussed later, or model them with logic programming, an approach rarely followed in the literature.The latter refers to Harary and Gupta’s suggestion.

IV. MEASURES OF TEMPORAL-TOPOLOGICAL STRUCTURE … D. Connectivity and components

Temporal networks require revising static topological measures because time constrains paths, reachability, waiting times, and connectivity. These temporal constraints affect applications such as disease spreading and produce system-dependent reachability patterns.

  • A. Introduction: Temporal networks require rethinking static measures based on neighboring nodes, node sets, paths, diameters, and centralities.Including time adds a degree of freedom to the network description, so many static measures must be revised.
  • B. Time-respecting paths and reachability: Time-respecting paths must follow temporally ordered link activations, rather than merely forming adjacent edge sequences as in static graphs.This definition makes paths relevant to dynamical processes on temporal networks.
  • B. Time-respecting paths and reachability: Time-respecting paths are not transitive: paths from i to j and j to k do not ensure a path from i to k.The concatenated path exists only when the first j−k contact occurs after the last i−j contact.
  • B. Time-respecting paths and reachability: Within an observation window, the set of vertices reachable from i defines its influence set, which can represent vertices eventually infected from i.The corresponding source set contains vertices that could have reached i, and its size is i’s source count.
  • C. Time-respecting paths with limits on waiting times: Waiting times along time-respecting paths may be bounded below or above, depending on transport or spreading dynamics.Lower bounds require waiting before the next contact, whereas upper bounds can require transmission before infectious nodes recover.
  • C. Time-respecting paths with limits on waiting times: The reachability ratio increased sharply around about two days in a mobile-phone network and 30 minutes in an airline network.These characteristic times reflect long inter-call intervals in bursty mobile-phone sequences and distinct temporal structure in airline connections.
  • D. Connectivity and components: Temporal connectivity partitions vertices according to whether time-respecting paths connect them, constraining dynamics on the network.Strongly connected components in temporal graphs can be mapped to maximal cliques in affine graphs.
  • D. Connectivity and components: Temporal connectivity properties depend on the measurement time window and include strong, weak, and transitive connectivity.Thus, connectivity classifications are generally valid only within a specified observation window.

E. Distances, latencies, and fastest paths · F. Average latency · G. Diameter, network efficiency

Temporal networks replace static path-length notions with time-respecting durations and latencies, while making averages, diameters, and efficiency sensitive to finite observation windows, reachability, and infinite latencies. These measures characterize how quickly information can propagate and how compact a temporal network is.

  • E. Distances, latencies, and fastest paths: Temporal distances extend static geodesic ideas by measuring how quickly vertices can transmit information through contact sequences.In static networks, shortest-path lengths influence propagation speed and determine average and distributional compactness.
  • E. Distances, latencies, and fastest paths: Time-respecting paths have durations, while latency is the shortest time in which one vertex can reach another.Duration is measured from the first to the last contact on a path; latency is also called temporal distance.
  • E. Distances, latencies, and fastest paths: Information latency λi,t(j) = t − φi,t(j) measures how old information from j is at i at time t.The corresponding vector of latest reachable information times is i’s vector clock.
  • F. Average latency: Averaging path durations alone ignores path frequency and waiting time before the first contact.One path and ten paths with the same one-unit duration produce the same duration average under this measure.
  • F. Average latency: Average latency is difficult to define because it varies over time, becomes infinite near the observation-window boundary, and depends on how finite values are averaged.Its saw-tooth pattern changes when new fastest paths begin, while incomplete paths near the end create infinite latency.
  • F. Average latency: Long observation periods introduce ambiguity because vertex entry and exit dynamics cannot generally be distinguished from edge activation dynamics.Observing only telephone-call sequences, for example, does not establish whether a person joined before the first call or left after the last.
  • G. Diameter, network efficiency: Temporal diameter can be defined as the longest average latency, but infinite latencies require a convention for disconnected vertex pairs.In static networks, diameter is the largest pairwise distance, while efficiency averages inverse path lengths.
  • G. Diameter, network efficiency: Network efficiency for temporal networks can be defined as the harmonic average of latency, combining average latency with reachability ratio by a harmonic mean.The two components represent how many vertex pairs are connected by time-respecting paths and how quickly information travels between them.

H. Minimum spanning tree · I. Centrality measures

The section extends path-based concepts from static graphs to temporal networks, including a temporal analogue of the minimum spanning tree and time-aware centrality measures. Temporal centralities must account for time-respecting paths, latency, temporal reachability restrictions, contact recency, and possible update failures.

  • H. Minimum spanning tree: A temporal analogue of the minimum spanning tree is introduced as a related concept for connecting vertices through time-dependent paths.In static weighted graphs, a minimum spanning tree is a minimum-total-weight tree connecting all vertices and represents the cheapest way to reach them when edge costs are weights.
  • H. Minimum spanning tree: Temporal-network reachability can also model navigation through cuboid atmospheric volumes when changing winds determine whether a balloon can move between them.This is presented as an unusual application of reachability analysis, while noting that more complex topologies may normally be needed.
  • I. Centrality measures: Temporal closeness centrality measures how quickly a vertex can on average reach other vertices, replacing static distance with time-dependent latency.Static closeness is based on inverse total distance to other vertices and is high for vertices close to all others.
  • I. Centrality measures: Temporal closeness is undefined unless the required time-respecting paths exist and arrive by the specified time, making late observation times restrictive.The restriction is especially difficult when t is relatively late in the observation period.
  • I. Centrality measures: Latency averaging can remove time dependence, but it may discard information from intervals with infinite latencies, motivating alternative centrality metrics.One option integrates the latency function over an interval where all pairwise latencies are finite; another uses a boundary-condition method.
  • I. Centrality measures: Temporal betweenness centrality generalizes static betweenness by counting the fraction of shortest or fastest time-respecting paths passing through a focal vertex.Shortest paths minimize the number of contacts, whereas fastest paths prioritize travel speed.
  • I. Centrality measures: A contact-driven temporal eigenvector centrality becomes time-aggregated, with ζ controlling transmitted centrality and larger ζ emphasizing recent contacts.A sensible range for ζ is (0, 1/2]. However, the algorithm can get stuck when centrality is not updated after a vertex’s last contact, requiring a remedy analogous to PageRank or Katz centrality.

J. Persistent patterns · K. Motifs

The review distinguishes persistent subgraphs, whose recurrence across time can reveal temporal structure, from motifs that identify statistically overrepresented subgraphs. Temporal motif definitions increasingly preserve event order, but suitable temporal null models remain unresolved.

  • J. Persistent patterns: Persistent subgraphs are identified through a support set S(G′), defined as the timesteps when subgraph G′ is contained in the active graph G_t.The framework extends temporal-network analysis beyond paths to recurring temporal patterns and subgraphs.
  • J. Persistent patterns: Static structure in time-sliced proximity networks depends strongly on the sampling window ∆, while abrupt changes in scaling reveal characteristic behavioral timescales.The analyzed quantities include average degree k, clustering coefficient C, and adjacency correlation γ averaged over vertices and times.
  • K. Motifs: Network motifs are equivalence classes of subgraphs overrepresented relative to a null model, commonly a degree-preserving configuration model.Motif analysis compares observed subgraph counts with randomized reference systems.
  • K. Motifs: Snapshot or aggregated-edge motif analyses can find overrepresented dense four-vertex motifs, but they discard the order of communication events.Braha and Bar-Yam’s email study used daily aggregated contacts and an edge-rewiring null model.
  • K. Motifs: Temporal communication motifs link events sharing vertices and occurring within a time limit ∆t, while other approaches encode gene-state transitions or temporal subgraphs.These definitions incorporate temporal constraints in different ways, including event succession and relative transition timing.
  • K. Motifs: Kovanen et al. define motifs from temporal subgraphs whose equivalence depends on contact order, with ∆t-adjacency and maximal ∆t-connected event sets.Temporal isomorphism preserves event-order similarity but not exact timings; temporal subgraphs can be mapped to static directed graphs for counting.
  • K. Motifs: The appropriate reference ensemble for temporal motifs remains unclear because static null models remove structural correlations, whereas temporal motifs must also account for temporal organization.Temporal occurrence counts and their time dependence may provide additional information in communication and functional networks.

L. Measuring inter-contact times and burstiness · M. Entropies and other information-theoretic measures

The section focuses on contact-sequence timing, especially inter-contact-time correlations and burstiness measures, and introduces entropy-based approaches for analyzing temporal uncertainties. It also frames information-theoretic methods as a potential direction for temporal-network research.

  • L. Measuring inter-contact times and burstiness: Inter-contact-time correlations influence the duration of time-respecting paths and the latency between vertices.Temporal-network analysis can therefore examine vertices, edges, and their associated contact sequences as basic building blocks.
  • L. Measuring inter-contact times and burstiness: Burstiness can be measured using the coefficient of variation, στ/mτ, defined as the standard deviation of inter-contact times divided by their mean.For a Poissonian contact sequence, στ/mτ = 1.
  • L. Measuring inter-contact times and burstiness: For finite contact sequences, the burstiness measure B lies in (−1, 1).The supplied passage states that finite sequences always have finite variance.
  • L. Measuring inter-contact times and burstiness: B = 1 indicates a most bursty sequence, B = 0 indicates Poissonian inter-contact times, and B = −1 indicates a completely periodic sequence.These values provide distinct reference points for interpreting burstiness.
  • M. Entropies and other information-theoretic measures: Information theory studies the limits of storing, communicating, and compressing data.It has already been applied in network-clustering research and could be extended to temporal networks.
  • M. Entropies and other information-theoretic measures: Reachability graphs can represent temporal contact sequences.The supplied figure caption shows a contact sequence and its corresponding reachability graph.
  • M. Entropies and other information-theoretic measures: An entropy-based method has been proposed to account for temporal uncertainties in communication networks.The approach is not deterministic, unlike most of the review’s temporal-network methods.
  • M. Entropies and other information-theoretic measures: Entropy-based temporal-network analysis is presented as an interesting direction for future research.Related ideas have also been developed in ecosystem analysis, including Ulanowicz’s measure of “ascendency.”

V. REPRESENTING TEMPORAL DATA AS A STATIC GRAPH … 1. Temporal exponential random graphs

The review describes static representations that preserve selected temporal and topological properties, including reachability, line, and transmission graphs. It then introduces temporal exponential random graphs as models for analyzing, comparing, and generating temporal contact structures.

  • V. REPRESENTING TEMPORAL DATA AS A STATIC GRAPH: Aggregating contacts over time produces static graphs that can capture both temporal and topological properties while making temporal networks easier to analyze.This is presented as the most straightforward static representation.
  • A. Reachability graphs: Reachability graphs draw a directed edge from A to B when a time-respecting path allows A to affect B.Their average degree equals the average worst-case outbreak size minus one.
  • A. Reachability graphs: When contacts are only known to occur within intervals, reachability graphs may be non-unique, so maximal and minimal versions can be plotted.This treatment is illustrated by dating data from high-school students.
  • C. Transmission graphs: Transmission graphs extend line graphs by incorporating disease dynamics through δ, which combines incubation time and disease duration.They encode directionality arising from the order of non-concurrent relationships.
  • C. Transmission graphs: Transmission graphs cannot represent cases where a vertex avoids infection, and their paths need not be time respecting.These limitations accompany their treatment of temporally ordered transmission opportunities.
  • 1. Temporal exponential random graphs: Exponential random graphs infer parameters describing the importance and frequency of topological elements, and temporal extensions apply this framework to observed state evolution.The cited examples include structures such as triangles and stars.
  • 1. Temporal exponential random graphs: Temporal exponential random graph models reduce to a time-varying partition function, enabling reference-model and generative uses.They can measure biases in temporal and topological structure or tune contact sequences for dynamical-system simulations.

2. Models of social group dynamics · B. Contact network models

The reviewed models represent social and contact networks as evolving structures, capturing group dynamics, partnership turnover, disease transmission, assortativity, and temporal overlap. They range from rate-equation frameworks to stochastic pair formation, rewiring, and dynamic random graphs with memory.

  • 2. Models of social group dynamics: Rate-equation frameworks model ephemeral social ties and expected changes in the number of people belonging to groups of different sizes.They capture observations about how interaction duration and isolation affect an agent’s likelihood of leaving or joining groups.
  • 2. Models of social group dynamics: Jo et al.’s model combines contact timings with network evolution, incorporating focal and cyclic closure, tie-strength reinforcement, triad interactions, and priority-queue task execution.The model produces networks with communities of strong ties.
  • B. Contact network models: Volz and Meyers extend static graphs through neighbor turnover by rewiring pairs of edges probabilistically at each time step for disease-contact simulations.Because rewiring preserves an existing network, accumulated-network topology and cross-edge contact patterns require separate models.
  • B. Contact network models: Kretzschmar et al.’s synthetic sexual-contact model generates interval graphs while tuning assortativity through a mixing function and parameter ξ.The model can represent assortative or disassortative mixing, with kmax limiting degree; a modern approach would draw degrees from a distribution before applying φ.
  • B. Contact network models: The same framework can impose serial monogamy by setting φ = 1 when k_i = k_j = 0 and φ = 0 otherwise, enabling control of partnership concurrency.Concurrency is defined as the temporal overlap of partnerships.
  • B. Contact network models: Later work developed related evolving-graph models, including Turova’s dynamic random graphs with memory and more structured evolving interval graphs.Turova’s framework models similar time-evolving graphs while remaining more tractable for analytic calculations.

C. Randomized reference models · 1. Randomized edges (RE)

Temporal randomized reference models reshuffle event sequences to remove selected temporal structure and correlations, revealing their roles by observing resulting dynamical changes. The randomized-edges model rewires edges while preserving their contact sequences and aggregated degrees, thereby isolating topology-related effects.

  • C. Randomized reference models: Temporal reference models randomize or reshuffle event sequences to remove chosen time-domain structure and correlations.Comparing dynamics across reference models helps identify which temporal or topological correlations matter.
  • C. Randomized reference models: The review considers temporal null models for contact sequences, with some methods applicable or adaptable to interval graphs.It also summarizes guidelines for choosing reference models.
  • 1. Randomized edges (RE): Randomized edges (RE) mirrors the static configuration model by rewiring edges while carrying each edge’s contact sequence to its new endpoints.The method is defined algorithmically through sequential edge processing and pairwise rewiring.
  • 1. Randomized edges (RE): RE isolates effects of the original wiring diagram while retaining aggregated-network degrees, although each vertex’s contact counts and timings change.Because contact sequences follow rewired edges, temporal correlations and inhomogeneities associated with those sequences are retained.
  • 1. Randomized edges (RE): RE processes edges sequentially, selects another edge for each one, and randomly chooses between two endpoint-swapping alternatives with probability 1/2 each.The alternatives are (i, j′), (i′, j) or (i, i′), (j, j′).
  • 1. Randomized edges (RE): RE rejects and repeats any rewiring that creates a self-edge or multiple edge.This constraint preserves the intended simple-graph structure.
  • 1. Randomized edges (RE): Contact times on each edge remain constant, while the two rewiring alternatives prevent spurious correlations caused by fixed endpoint ordering.Otherwise, the number of times a vertex appears in the first argument could remain conserved and substantially affect empirical results.

2. Randomly permuted times (RP) … 7. Edge randomization (ER)

The section presents temporal-network null models that selectively preserve or destroy structural, temporal, and edge-level correlations. Together, these models isolate effects of timing patterns, contact counts, topology, and inter-contact dynamics.

  • 2. Randomly permuted times (RP): RP randomly permutes contact times while preserving network structure and the number of contacts between every vertex pair.It exchanges or reshuffles timestamps without requiring the edge-rewiring checks used by RE; interval graphs require non-overlap checks.
  • 3. Randomized edges with randomly permuted times (RE + RP): RE + RP first randomizes network structure using RE, then reshuffles all contact timestamps.This destroys structural and temporal correlations except overall contact-rate patterns such as daily or weekly rhythms.
  • 4. Random times (RT): RT assigns each contact on each edge a random time within the observation window while conserving network structure and total contacts per edge.Comparing RP with RT isolates the effects of network-level temporal patterns in the empirical data.
  • 4. Random times (RT): RT destroys burstiness, triggered chains, and circadian or weekly timing patterns that RP retains at the aggregate network level.The RP ensemble preserves the original set of timestamps and aggregate event rate, whereas RT randomizes contact times independently.
  • 5. Randomized contacts (RC): RC keeps graph topology fixed but redistributes contacts randomly among edges, producing binomial rather than broad, right-skewed contact counts per edge.This tests how the distribution of contacts per edge combines with event ordering.
  • 6. Equal-weight edge randomization (EWER): EWER exchanges complete edge contact sequences only among edges with the same number of contacts, preserving single-edge burstiness and inter-contact-time distributions.Its purpose is to remove timing correlations between adjacent-edge contact sequences while retaining edge-associated temporal characteristics.
  • 7. Edge randomization (ER): ER exchanges complete contact sequences among edges with any contact counts, randomizing edge weights and destroying weight-topology correlations.Because sequences are relocated without alteration, their inter-contact-time distributions remain unchanged.

8. Time reversal (TR) … A. Bursty event dynamics and slow spreading in communication networks

Temporal-network analysis uses time-reversal and randomized reference models to identify causal temporal correlations, while spreading dynamics are shaped by heterogeneous event timing and event ordering. In communication networks, bursty heavy-tailed activity produces slower-than-Poissonian spreading and altered long-time infection decay.

  • 8. Time reversal (TR): Time reversal runs an event sequence backward to assess the frequency and importance of causal contact sequences.If temporal correlations alone generate consecutive contacts, comparable sequence counts should appear in forward and reversed time.
  • 9. Summary and guidelines: Comparing randomized reference models reveals which topological and temporal correlations matter most for dynamical processes.RE and RP permute edges and contact times, while RE+RP destroys all correlations except those retained by the model.
  • VII. SPREADING DYNAMICS AND COMPARTMENTAL MODELS ON TEMPORAL GRAPHS: Temporal heterogeneity and event ordering can have important, sometimes drastic effects on spreading dynamics on temporal graphs.These effects have been studied using empirical interaction sequences, simulations, and analytical tools.
  • A. Bursty event dynamics and slow spreading in communication networks: Human communication is almost universally bursty, with inter-event times better described by heavy-tailed or power-law distributions than by uniform or Poissonian statistics.This invalidates the Poissonian approximation commonly assumed in spreading models on static networks.
  • A. Bursty event dynamics and slow spreading in communication networks: Power-law waiting times produce a power-law long-time decay in new infections, rather than the exponential decay expected under Poissonian spreading.The decay exponent is determined by the generation-time distribution, according to theoretical and simulation studies of the SI model.
  • A. Bursty event dynamics and slow spreading in communication networks: 31,183 individuals forwarding email recommendations provided empirical evidence that heterogeneous response times slow information spreading at the collective level.Response times followed a lognormal distribution, and secondary spreaders typically forwarded recommendations simultaneously rather than remaining spreaders longer.
  • A. Bursty event dynamics and slow spreading in communication networks: Broad inter-event time distributions also slow the ordering dynamics of the Voter model.This parallels the slowing observed in compartmental spreading models.

B. Burstiness and other temporal and structural inhomogeneities · C. Utilizing temporal structure for disease control

Temporal heterogeneities and correlations can either accelerate or slow spreading, depending on the dynamics, data, and reference model. Temporal contact structure also enables vaccination strategies that improve targeting while respecting the limited predictive value of past and future contacts.

  • B. Burstiness and other temporal and structural inhomogeneities: The daily call-frequency pattern had little effect on spreading, while removing temporal triggering slightly slowed short-term and slightly accelerated long-term spreading.The comparison used a Poissonian event-generating model and the EWER scheme.
  • B. Burstiness and other temporal and structural inhomogeneities: Temporal order and correlations accelerated epidemic spreading in an Internet-mediated prostitution contact dataset, contrary to other reported observations.The review notes that the origin of this difference remains unclear.
  • B. Burstiness and other temporal and structural inhomogeneities: Group conversations and correlated contact sequences produced larger SIR spreading cascades below the percolation point, but smaller cascades at large transmission probability λ.The comparison was with a time-shuffled reference model whose relay-time distribution approaches Poisson.
  • B. Burstiness and other temporal and structural inhomogeneities: SI spreading speed is linked to temporal path lengths, latency, and reachability, while fastest temporal paths can represent SIR pathways under timing constraints.This relationship is especially noted for deterministic SI, where infection occurs upon contact.
  • B. Burstiness and other temporal and structural inhomogeneities: Temporal inhomogeneities also slowed SI-like information flow among ants relative to a kinetic null model based on random gas-particle-like collisions.The temporal networks came from 30-minute recordings across 6 ant colonies.
  • C. Utilizing temporal structure for disease control: Vaccination can exploit contact structure because outbreaks may be blocked at partial population coverage rather than requiring universal vaccination.The passage introduces vaccination as the most common preventive intervention and denotes coverage by f.
  • C. Utilizing temporal structure for disease control: Neighborhood vaccination uses local contact information by randomly selecting a person, vaccinating a named contact, and repeating until the desired coverage f is reached.In static-network epidemiology, a person’s spreading importance is described as proportional to k2.
  • C. Utilizing temporal structure for disease control: Temporal-network vaccination must account for contact timing, and asking about the most recent or historically most frequent contact improved neighborhood vaccination in some real datasets.The response differed across datasets, including prostitution, email, and hospital-proximity networks.

VIII. FUTURE OUTLOOK

Temporal networks treat contact timings as part of network structure rather than as a separate component of dynamical models. The field remains young, with open challenges spanning generative models, structural measures, mechanisms, inference, mesoscopic structure, visualization, and applications.

  • Temporal network modeling incorporates contact timings into the network itself instead of treating them as a separate spreading-model component.This reframing makes temporal structure integral to the contact structure used to study dynamics.
  • Generative models for temporal networks: Generative models remain scarce, especially parametrized models reproducing skewed inter-contact times, burstiness, and circadian or weekly rhythms.The review identifies constructing and studying such models as a major open issue.
  • Measures for temporal network structure: Existing temporal-structure measures largely generalize static-network measures, leaving substantial room for improvement.The field knows temporal structure can affect dynamics, but not exactly how or why.
  • Understanding the driving mechanisms: A largely unexplored challenge is explaining why contacts occur when they do, including the causes of skewed inter-contact-time distributions.The review suggests that a vertex’s other connections may influence its contact timing.
  • Inference problems: Inference challenges include constructing temporal networks from partial vertex or edge states and inferring spreading chains from incomplete temporal data.Static-network inference problems may differ because properties such as edge transitivity and Menger’s theorem do not generally carry over directly.
  • Further open directions concern temporal mesoscopic structure and visualization, while the paradigm’s real test is application to concrete problems across many scientific and social domains.The review frames these theoretical and methodological developments as preliminary to applications in biology, neuroscience, social science, economics, chemistry, and related fields.
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