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Temporal Networks

Petter HolmeJari Saramäki

article2011Phys. Rep.3,007 citations

Presents the core concepts and analytical methods for temporal networks, demonstrating how time-varying connectivity alters dynamical processes like epidemic contagion and information diffusion compared to traditional static graphs.

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Many real-world systems—from email exchanges and mobile phone calls to hospital patient proximity and gene regulation—feature interactions that occur only at specific instants or intervals rather than continuously. Traditional static network models aggregate these contacts into fixed edges and therefore lose information about when contacts happen. Because the order and timing of contacts determine whether a path exists and how quickly something can travel along it, static models can produce misleading predictions about processes such as disease spread or information diffusion. The review synthesizes research across physics, computer science, biology, and the social sciences to establish temporal networks as a distinct framework in which edge activation times are treated as an explicit part of the network itself.

The authors set out to define the core concepts, survey existing analytic methods, and illustrate how temporal structure influences dynamical processes. They draw on empirical contact sequences from communication records, proximity sensors, and biological systems, and they compare these data against families of randomized reference models that selectively destroy different classes of temporal or topological correlation.

The central finding is that time ordering and bursty contact patterns materially change reachability and spreading speed. In several large communication datasets, the time-respecting paths that actually exist are far fewer and slower than those implied by the corresponding aggregated static graph; burstiness alone can slow epidemic-style spreading by orders of magnitude relative to a Poisson null model. Temporal centrality measures, latency distributions, and motif counts reveal persistent patterns and bottlenecks that static metrics miss. Randomized reference models show that different correlations—edge burstiness, inter-edge triggering, and overall daily rhythms—dominate different regimes of spreading dynamics. The same structure can also be exploited: simple rules that use recent or frequent contacts to select vaccination targets outperform random or static-neighborhood strategies in several real contact datasets.

These results matter because many practical decisions—setting quarantine thresholds, designing communication protocols, or interpreting functional brain networks—rest on assumptions about how fast and how far something can propagate. When those assumptions ignore timing, the resulting cost, risk, or policy estimates can be substantially wrong. At the same time, the field remains young. Terminology is still fragmented across disciplines, generative models that reproduce observed temporal patterns are scarce, and visualization and inference tools lag behind those available for static networks. Further progress therefore requires tighter integration of temporal data collection with modeling, systematic comparison of spreading outcomes across reference ensembles, and targeted empirical studies that test whether temporal-network predictions improve real-world interventions.

Aalto UniversityStockholm UniversitySungkyunkwan UniversityUmeå Universitycontractsemailinformation diffusionTemporal NetworkTime‑Respecting PathBurstiness
arXiv: 1108.1780
  • Book: An introduction to graph theory, Darij Grinberg (2023). Mastering the foundational definitions and counting formulas of classical graph theory is an essential prerequisite for analyzing the structural and dynamical properties of temporal networks.
  • Paper: The structure and dynamics of multilayer networks, S. Boccaletti et al. (2014). This paper extends the single-layer temporal perspective of the source into a comprehensive framework for multilayer networks where multiple types of interactions and interlayer links occur simultaneously.
  • Paper: Epidemic processes in complex networks, Romualdo Pastor-Satorras et al. (2015). Building directly on the temporal contact structures reviewed in the source, this paper advances epidemic and contagion modeling by rigorously incorporating time-varying contact sequences into spreading dynamics.
  • Paper: Multilayer networks, Mikko Kivelä et al. (2013). This review unifies the diverse terminologies of complex systems, explicitly building upon temporal network representations to construct a general tensor-based framework for multilayer and time-evolving structures.
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Abstract

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.

Topics

contractsemailinformation diffusionnetwork architecturenetwork topologyphone callstext messagestopologycommunication networksbipartite graphs

Keywords

Temporal NetworkTime‑Respecting PathBurstinessReachability RatioRandomized Reference ModelTemporal MotifContact SequenceInter‑Contact TimeTemporal CentralityDynamic Spreading

Table of Contents

  • I Introduction
  • II Types of temporal networks
  • II.1 Person-to-person communication
  • II.2 One-to many information dissemination
  • II.3 Physical proximity
  • II.4 Cell biology
  • II.5 Distributed computing
  • II.6 Infrastructural networks
  • II.7 Neural and brain networks
  • II.8 Ecological networks
  • II.9 Other systems
  • III Preliminaries
  • IV Measures of temporal-topological structure
  • IV.1 Introduction
  • IV.2 Time-respecting paths and reachability
  • IV.3 Time-respecting paths with limits on waiting times
  • IV.4 Connectivity and components
  • IV.5 Distances, latencies, and fastest paths
  • IV.6 Average latency
  • IV.7 Diameter, network efficiency
  • IV.8 Minimum spanning tree
  • IV.9 Centrality measures
  • IV.10 Persistent patterns
  • IV.11 Motifs
  • IV.12 Measuring inter-contact times and burstiness
  • IV.13 Entropies and other information-theoretic measures
  • V Representing temporal data as a static graph
  • V.1 Reachability graphs
  • V.2 Line graphs
  • V.3 Transmission graphs
  • VI Models of temporal networks
  • VI.1 Models for temporal social networks
  • VI.1.1 Temporal exponential random graphs
  • VI.1.2 Models of social group dynamics
  • VI.2 Contact network models
  • VI.3 Randomized reference models
  • VI.3.1 Randomized edges (RE)
  • VI.3.2 Randomly permuted times (RP)
  • VI.3.3 Randomized edges with randomly permuted times (RE + RP)
  • VI.3.4 Random times (RT)
  • VI.3.5 Randomized contacts (RC)
  • VI.3.6 Equal-weight edge randomization (EWER)
  • VI.3.7 Edge randomization (ER)
  • VI.3.8 Time reversal (TR)
  • VI.3.9 Summary and guidelines
  • VII Spreading dynamics and compartmental models on temporal graphs
  • VII.1 Bursty event dynamics and slow spreading in communication networks
  • VII.2 Burstiness and other temporal and structural inhomogeneities
  • VII.3 Utilizing temporal structure for disease control
  • VIII Future outlook
  • References

Knowls

  1. Knowl 1 — Randomized Reference Models for Temporal Contact Sequences

    model/method

    Randomized reference models (null models) for temporal contact sequences isolate and quantify the impact of specific structural and temporal correlations by selectively randomizing properties of an empirical dataset while holding others constant:

    • Randomized Edges (RE): Preserves each vertex's degree in the aggregated network and the full sequence of contact timestamps on each edge, but reallocates contact sequences between vertex pairs using pairwise topological edge rewiring. This eliminates higher-order network topology (such as clustering and community structure) while preserving single-edge burstiness and total system event rates.
    • Randomly Permuted Times (RP): Retains the aggregated graph topology and the exact count of contacts on each edge, but randomly permutes the timestamps of all contacts across the network. This destroys event order, individual edge burstiness, and temporal correlations between adjacent edges, while preserving the system-wide aggregate activity rate (e.g., circadian cycles).
    • Randomized Edges with Randomly Permuted Times (RE + RP): Concurrently applies RE and RP, removing both structural topology correlations and temporal ordering while retaining only the degree sequence, total contact counts, and aggregate rate.
    • Random Times (RT): Assigns each contact a random timestamp drawn uniformly from the observation window [t0,T][t_0, T] (or from an edge-specific Poisson process). This removes overall circadian and weekly rate modulations in addition to edge-level burstiness.
    • Equal-Weight Edge Randomization (EWER): Randomly exchanges complete contact time series between pairs of edges that possess the exact same number of total contacts. This preserves single-edge burstiness, the aggregated topology, and edge weights, while destroying temporal correlations and event triggering across adjacent edges.
    • Edge Randomization (ER): Randomly exchanges complete contact time series between arbitrary edges regardless of contact count. This destroys weight-topology correlations while preserving single-edge inter-contact time distributions.
    • Time Reversal (TR): Reverses the sequence of events in time (t→T−t+t0t \to T - t + t_0). Deviations in dynamical cascades between forward and reversed sequences identify causal event chains and the "arrow of time".

  2. Knowl 2 — Time-Respecting Paths and Temporal Reachability

    definition

    In a temporal network represented as a contact sequence where contacts are triples (i,j,t)(i, j, t) indicating an interaction between vertices i,j∈Vi, j \in V at time tt, a time-respecting path (or journey) from vertex v0v_0 to vkv_k is an edge sequence (v0,v1,t1),(v1,v2,t2),…,(vk−1,vk,tk)(v_0, v_1, t_1), (v_1, v_2, t_2), \dots, (v_{k-1}, v_k, t_k) satisfying the non-decreasing time constraint t1≤t2≤⋯≤tkt_1 \le t_2 \le \dots \le t_k.

    Temporal reachability is intrinsically non-transitive: the existence of a time-respecting path from ii to jj and from jj to kk does not imply the existence of a path from ii to kk, which requires that the initial contact on the j→kj \to k path occurs at or after the terminal contact of the i→ji \to j path.

    Within an observation window t∈[t0,T]t \in [t_0, T]:

    • The set of influence (future light cone) of vertex ii at time tt is the set of all vertices reachable from ii via time-respecting paths beginning at or after tt.
    • The source set (past light cone) of vertex ii at time tt is the set of all vertices that can reach ii via time-respecting paths arriving at or before tt. The cardinality of this set is the source count of ii.
    • The reachability ratio is the average fraction of all vertex pairs (i,j)(i, j) in the network connected by a valid time-respecting path.

  3. Knowl 3 — Burstiness Parameter for Inter-Contact Time Distributions

    equation

    The burstiness parameter BB measures the deviation of an empirical sequence of inter-contact times τ\tau from a memoryless Poisson process using the coefficient of variation στ/mτ\sigma_\tau / m_\tau:

    B=στ/mτ−1στ/mτ+1=στ−mτστ+mτB = \frac{\sigma_\tau / m_\tau - 1}{\sigma_\tau / m_\tau + 1} = \frac{\sigma_\tau - m_\tau}{\sigma_\tau + m_\tau}

    Where:

    • τ=tk+1−tk\tau = t_{k+1} - t_k is the inter-contact time between consecutive events on a node or edge.
    • mτ=E[τ]m_\tau = \mathbb{E}[\tau] is the sample mean of inter-contact times.
    • στ=E[(τ−mτ)2]\sigma_\tau = \sqrt{\mathbb{E}[(\tau - m_\tau)^2]} is the sample standard deviation of inter-contact times.

    For any non-empty finite sequence with non-zero mean, B∈(−1,1]B \in (-1, 1]:

    • B=1B = 1 indicates maximal burstiness (heavy-tailed, highly heterogeneous inter-contact times with στ≫mτ\sigma_\tau \gg m_\tau).
    • B=0B = 0 corresponds to a neutral Poisson process where στ=mτ\sigma_\tau = m_\tau.
    • B=−1B = -1 corresponds to a strictly periodic, deterministic sequence where στ=0\sigma_\tau = 0.

  4. Knowl 4 — Vector Clocks, Information Latency, and Forward Latency

    definition

    In a temporal network tracking information dissemination:

    • Let ϕi,t(j)\phi_{i,t}(j) denote the latest time before or at tt such that information originated at vertex jj could reach vertex ii via a time-respecting path. The value ϕi,t(j)\phi_{i,t}(j) is vertex ii's view of vertex jj's information at time tt.
    • The NN-dimensional vector [ϕi,t(1),ϕi,t(2),…,ϕi,t(N)][\phi_{i,t}(1), \phi_{i,t}(2), \dots, \phi_{i,t}(N)] is vertex ii's vector clock at time tt.
    • The backward information latency (or simply latency) λi,t(j)\lambda_{i,t}(j) of vertex jj with respect to vertex ii at time tt is:

    λi,t(j)=t−ϕi,t(j)\lambda_{i,t}(j) = t - \phi_{i,t}(j)

    representing the elapsed time since the newest information from jj available at ii was emitted.

    • The forward latency (or temporal distance) τij(t)\tau_{ij}(t) is the duration required to reach vertex jj from vertex ii along the fastest time-respecting path initiated at or after time tt.

  5. Knowl 5 — Temporal Closeness Centrality and Reciprocal Efficiency Centrality

    equation

    For a temporal network of NN vertices, temporal generalizations of closeness centrality characterize how quickly a vertex can access or distribute information across time-respecting paths:

    1. Temporal Closeness Centrality:

    CC(i,t)=N−1∑j≠iλi,t(j)C_C(i, t) = \frac{N - 1}{\sum_{j \neq i} \lambda_{i,t}(j)}

    where λi,t(j)=t−ϕi,t(j)\lambda_{i,t}(j) = t - \phi_{i,t}(j) is the latency of vertex jj with respect to vertex ii at time tt. CC(i,t)C_C(i, t) is well-defined only when all other vertices j≠ij \neq i have a finite latency to ii at time tt.

    1. Temporal Reciprocal Efficiency Centrality: To account for disconnected vertex pairs and infinite latencies within a finite observation window, the centrality is defined using reciprocal latencies:

    CE(i,t)=1N−1∑j≠i1λi,t(j)C_E(i, t) = \frac{1}{N - 1} \sum_{j \neq i} \frac{1}{\lambda_{i,t}(j)}

    where 1λi,t(j)\frac{1}{\lambda_{i,t}(j)} is defined to be 00 if there is no valid time-respecting path from vertex jj arriving at vertex ii at or before time tt.

  6. Knowl 6 — Dynamic Update Algorithm for Temporal Eigenvector Centrality

    algorithm

    The temporal eigenvector centrality computes a dynamic, non-aggregated centrality score across a stream of timestamped pairwise contact events:

    Input: Sequence of contacts (i,j,t)(i, j, t) ordered chronologically
    Input: Vertex set VV of cardinality NN
    Input: Transmission parameter ζ∈(0,1/2]\zeta \in (0, 1/2]
    Output: Centrality vector CE∈RNC_E \in \mathbb{R}^N
    for each u∈Vu \in V do
        CE(u)←1C_E(u) \leftarrow 1
    end for
    for each contact event (i,j,t)(i, j, t) in chronological order do
        Cprev(i)←CE(i)C_{\text{prev}}(i) \leftarrow C_E(i)
        Cprev(j)←CE(j)C_{\text{prev}}(j) \leftarrow C_E(j)
        CE(i)←ζ⋅Cprev(i)+(1−ζ)⋅Cprev(j)C_E(i) \leftarrow \zeta \cdot C_{\text{prev}}(i) + (1 - \zeta) \cdot C_{\text{prev}}(j)
        CE(j)←ζ⋅Cprev(j)+(1−ζ)⋅Cprev(i)C_E(j) \leftarrow \zeta \cdot C_{\text{prev}}(j) + (1 - \zeta) \cdot C_{\text{prev}}(i)
    end for
    return CEC_E

    The parameter ζ\zeta regulates the proportion of centrality retained versus transferred upon contact. At ζ=1/2\zeta = 1/2, both interacting nodes completely share their pooled centrality.

  7. Knowl 7 — Temporal Motifs via Delta-t Connected Subgraphs

    definition

    Temporal motifs represent mesoscopic patterns of interaction where both the network topology and the temporal order of events are preserved:

    • Δt\Delta t-Adjacency: Two contact events e1=(u1,v1,t1)e_1 = (u_1, v_1, t_1) and e2=(u2,v2,t2)e_2 = (u_2, v_2, t_2) are Δt\Delta t-adjacent if they share at least one vertex ({u1,v1}∩{u2,v2}≠∅\{u_1, v_1\} \cap \{u_2, v_2\} \neq \emptyset) and their timestamps satisfy ∣t1−t2∣≤Δt|t_1 - t_2| \le \Delta t.
    • Δt\Delta t-Connectivity: Two contact events eae_a and ebe_b are Δt\Delta t-connected if there exists a sequence of mutually Δt\Delta t-adjacent events linking eae_a and ebe_b.
    • Valid Temporal Subgraph: A set of contact events SS where all pairs of events in SS are Δt\Delta t-connected and no intermediate contacts involving the participating vertices during the timeframe of SS are omitted.
    • Maximal Temporal Motifs: Equivalence classes of isomorphic valid temporal subgraphs that are maximal (i.e., contain the largest set of events that remain pairwise Δt\Delta t-connected). Subgraph isomorphism requires identical static graph topology as well as identical chronological event orderings.

  8. Knowl 8 — Adjacency Correlation Coefficient for Time-Varying Graphs

    equation

    The adjacency correlation function γi(t)\gamma_i(t) (also called the temporal-correlation coefficient) quantifies the persistence and similarity of vertex ii's local connection topology between consecutive time intervals tt and t+1t + 1:

    γi(t)=∑j∈ϕ(i,t)a(i,j,t) a(i,j,t+1)∑j∈ϕ(i,t)a(i,j,t)∑j∈ϕ(i,t)a(i,j,t+1)\gamma_i(t) = \frac{\sum_{j \in \phi(i, t)} a(i, j, t) \, a(i, j, t + 1)}{\sqrt{\sum_{j \in \phi(i, t)} a(i, j, t)} \sqrt{\sum_{j \in \phi(i, t)} a(i, j, t + 1)}}

    Where:

    • a(i,j,t)∈{0,1}a(i, j, t) \in \{0, 1\} is the presence function (adjacency indicator), which equals 11 if an edge exists between vertex ii and vertex jj at time tt, and 00 otherwise.
    • ϕ(i,t)={j∈V∣a(i,j,t)=1 or a(i,j,t+1)=1}\phi(i, t) = \{j \in V \mid a(i, j, t) = 1 \text{ or } a(i, j, t + 1) = 1\} is the set of all vertices adjacent to ii in at least one of the two time steps.
    • γi(t)∈[0,1]\gamma_i(t) \in [0, 1] represents the normalized overlap of the neighborhood of vertex ii across the time step boundary.

  9. Knowl 9 — Transmission Graphs with Disease Incubation and Duration Limits

    definition

    A transmission graph maps an interval temporal graph to a static directed line graph that explicitly accounts for infection dynamics through an epidemiological incubation/infectivity window δ>0\delta > 0:

    • Let the underlying temporal graph consist of edges ee active over continuous time intervals [tstart(e),tstop(e)][t_{\text{start}}(e), t_{\text{stop}}(e)].
    • Let δ\delta denote the combined maximum incubation time and duration of infectiousness of a disease.
    • The corresponding transmission graph GT=(VT,ET)G_T = (V_T, E_T) is constructed such that:
      1. Every vertex in VTV_T corresponds to an edge ee of the temporal contact network.
      2. A directed edge from ee to e′e' belongs to ETE_T if and only if: ee and e′e' share at least one vertex (e∩e′≠∅e \cap e' \neq \emptyset), tstart(e)<tstart(e′)t_{\text{start}}(e) < t_{\text{start}}(e'), and tstart(e′)<tstart(e)+δt_{\text{start}}(e') < t_{\text{start}}(e) + \delta.

    This directed mapping captures whether transmission from relationship ee to relationship e′e' is temporally feasible before the host ceases to be infectious.

  10. Knowl 10 — Stochastic Pair-Formation Model for Dynamic Contact Networks

    model/method

    The stochastic pair-formation model generates dynamic interval contact networks with explicit partnership lifetimes, transmission dynamics, and tunable degree mixing. At each discrete time step:

    1. Partnership Formation: With probability ρ\rho, candidate pairs (i,j)(i, j) are sampled uniformly at random. A pair is formed if accepted by a degree-mixing function ϕ(i,j)\phi(i, j). Formation attempts repeat until N/2−PN/2 - P successful pairs are established, where PP is the count of currently active partnerships.
      • For assortative mixing: ϕ(i,j)=1−ξ+ξkikjkmax⁡2\phi(i, j) = 1 - \xi + \xi \frac{k_i k_j}{k_{\max}^2}
      • For disassortative mixing: ϕ(i,j)=1−ξ+ξ(ki−kj)2kmax⁡2\phi(i, j) = 1 - \xi + \xi \frac{(k_i - k_j)^2}{k_{\max}^2} where ξ∈[0,1]\xi \in [0, 1] governs mixing strength, kik_i is vertex ii's current degree, and kmax⁡k_{\max} is the maximum possible degree.
    2. Epidemic Transmission: Across every active partnership containing one susceptible and one infected individual, disease is transmitted with probability η\eta.
    3. Partnership Dissolution: Each existing active pair dissolves independently with probability σ\sigma.

  11. Knowl 11 — Slowdown of Epidemic Spreading Caused by Bursty Inter-Contact Times

    empirical result

    In human communication and physical interaction networks, the inter-contact time distribution P(τ)P(\tau) exhibits broad, heavy-tailed power-law decay P(τ)∼τ−αP(\tau) \sim \tau^{-\alpha} rather than exponential decay.

    Simulations and analytical treatments of Susceptible-Infective (SI) and Susceptible-Infective-Removed (SIR) dynamics over empirical contact sequences demonstrate that:

    • Broad inter-contact time distributions and silent periods on individual edges significantly delay the time to full prevalence (100% infection) compared to memoryless Poisson (RT) and time-permuted (RP) null models.
    • In late-stage SI dynamics, power-law inter-event time distributions convert the asymptotic exponential decay in the number of new infections into a power-law decay determined by the generation time distribution.
    • Edge-level burstiness and topological community bottlenecks ("weak edges") are the primary drivers of spreading deceleration, whereas global circadian rhythms exert minimal influence on asymptotic spreading speeds.

  12. Knowl 12 — Temporal Extensions to Neighborhood Vaccination Protocols

    model/method

    Static neighborhood vaccination protocols randomly sample individuals and vaccinate a named acquaintance, exploiting the friendship paradox to target individuals with high static degree kk (with probability proportional to k2k^2). In temporal networks, this strategy is extended by incorporating temporal contact histories:

    1. Most Recent Contact Protocol: Select a random individual ii and vaccinate the acquaintance jj with whom ii interacted most recently prior to the intervention.
    2. Most Frequent Contact Protocol: Select a random individual ii and vaccinate the acquaintance jj with whom ii interacted most frequently over a preceding observation window.

    The optimal protocol depends on edge dynamics:

    • For networks with transient, short-lived interactions across alternating partners (e.g., hospital inpatient wards or sex-work networks), the most recent contact protocol is more effective because individuals active recently are most likely to engage in subsequent interactions.
    • For networks with persistent social relationships but variable communication rates (e.g., email networks), the most frequent contact protocol is more effective.

Coverage note — Omitted qualitative domain survey sections (descriptions of temporal network occurrences in cell biology, distributed computing, neuroscience, and ecology) as they review external empirical domains without introducing new formal definitions, models, or quantitative results.

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Citation

MLA
Holme, P., and J. Saramäki. “Temporal Networks”. Physics Reports, vol. 519, no. 3, 2012, pp. 97–125, https://doi.org/10.1016/j.physrep.2012.03.001.
APA
Holme, P., & Saramäki, J. (2012). Temporal networks. Physics Reports, 519(3), 97–125. https://doi.org/10.1016/j.physrep.2012.03.001
Chicago
Holme, P., and J. Saramäki. 2012. “Temporal Networks”. Physics Reports 519 (3): 97–125. https://doi.org/10.1016/j.physrep.2012.03.001.
Harvard
Holme, P. and Saramäki, J. (2012) “Temporal networks”, Physics Reports, 519(3), pp. 97–125. Available at: https://doi.org/10.1016/j.physrep.2012.03.001.
Vancouver
1. Holme P, Saramäki J (2012) Temporal networks. Physics Reports 519:97–125

BibTeX

@article{Holme_2012, title={Temporal networks}, volume={519}, ISSN={0370-1573}, url={http://dx.doi.org/10.1016/j.physrep.2012.03.001}, DOI={10.1016/j.physrep.2012.03.001}, number={3}, journal={Physics Reports}, publisher={Elsevier BV}, author={Holme, Petter and Saramäki, Jari}, year={2012}, month=Oct, pages={97–125} }
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