Pith. sign in

REVIEW 2 major objections 6 minor 1 cited by

Characterising the dynamics of unlabelled temporal networks

T0 review · 2 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper claims that the dynamical fingerprints of a temporal network—periodicity, memory decay, and sensitive dependence on initial conditions—can be partially recovered from a sequence of unlabelled snapshots, using…

desk verdict A solid proof-of-concept that unlabelled temporal networks retain fingerprints of periodicity, memory, and instability; the main fixes are an analytic typo, a missing topology-only control, and a slightly overreaching monotonicity claim. read the letter →

arxiv 2412.14864 v1 pith:XRCM3WRN submitted 2024-12-19 physics.soc-ph nlin.CDphysics.data-an

Annalisa Caligiuri , Tobias Galla , Lucas Lacasa This is my paper · ORCID
classification physics.soc-phnlin.CDphysics.data-an PACS 89.75.Fb
keywords temporalnetworksunlabelledgraphsgraphinvariantspermutation-invariantpseudo-distancesnetworkautocorrelationLyapunovexponentstrajectoriesdynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Temporal networks are usually studied as sequences of adjacency matrices that require nodes to be tracked over time, but there are many settings—animal swarms, privacy-restricted contact data, molecular graphs—where labels are missing or cannot be matched. This paper asks whether the dynamics of the underlying network can still be characterized after the labels have been removed. The answer it argues for is a qualified yes: quantitative fingerprints such as the period of noisy periodic trajectories and the memory decay of autoregressive network processes remain detectable from unlabelled snapshots, and chaotic expansion of nearby network trajectories stays visible qualitatively, although not with a well-defined Lyapunov exponent. The authors make the case by defining three permutation-invariant pseudo-distances and showing, on synthetic models and three empirical datasets, that autocorrelation-like functions built from them recover the same qualitative behaviour as the labelled ground truth.

What carries the argument

The machinery is the permutation-invariant pseudo-distance: a normalized dissimilarity between two unlabelled graphs computed from a graph invariant, so that it does not depend on which labelling of the nodes is chosen. The paper uses three: the $L^1$ distance between sorted degree sequences ($d_{\mathrm{deg}}$), the $L^1$ distance between ordered adjacency spectra ($d_{\mathrm{spec}}$), and the $L^2$ distance between sorted, normalized eigenvector centrality vectors ($d_{\mathrm{eig}}$). These feed an autocorrelation-like function $c_X(\tau)=1-\frac{1}{T-\tau}\sum_{t=1}^{T-\tau} d_X(G_t,G_{t+\tau})/J$, where $J$ is the mean pairwise pseudo-distance over snapshots, and a recurrence-based expansion analysis for chaos. The load-bearing numerical fact is that, in all tests, each $d_X$ is a continuous, monotonically increasing, nonlinear function of the labelled ground-truth distance $d_{\mathrm{lab}}$; that monotonicity is what makes period, memory, and expansion visible in the unlabelled signal.

What would settle it

Construct a labelled temporal trajectory whose snapshots are pairwise non-isomorphic but all share the same sorted degree sequence, the same adjacency spectrum, and the same sorted eigenvector centrality; if every pseudo-distance stays identically zero and $c_X(\tau)$ shows no structure, then the claimed recovery fails for that dynamical class. Regular graphs with rearranged edges are a natural place to search for such a counterexample.

Watch

Extended reading notes

Core claim

The central discovery is that removing node labels does not erase all dynamical information: graph properties that are invariant under relabelling can stand in for the adjacency matrix. More precisely, the paper defines pseudo-distances $d_{\mathrm{deg}}$, $d_{\mathrm{spec}}$, and $d_{\mathrm{eig}}$ from the sorted degree sequence, the ordered spectrum of the adjacency matrix, and the sorted eigenvector centrality vector, and uses them to build unlabelled autocorrelation functions $c_X(\tau)$. These recover the period of noisy periodic network trajectories, reproduce the flat-then-decaying autocorrelation shape of $DARN(\rho)$ processes, and reveal an expansion phase for initially close trajectories in both low- and high-dimensional chaotic network dynamics. The paper states clearly that the expansion measured by the pseudo-distances is not strictly exponential and that, because the relation between pseudo-distance and labelled distance is monotonic but model-dependent, a model-independent Lyapunov exponent for unlabelled networks cannot currently be defined.

Load-bearing premise

The approach only works when changes in the network show up in its degree list, its spectrum, or its centrality scores; a trajectory in which all three stay the same while the network genuinely changes would look perfectly static to every measure.

Editorial extensions

If this is right

  • The period of a noisy periodic network trajectory can be read off as a peak in $c_X(\tau)$, though it remains detectable to higher noise levels in the labelled case than in the unlabelled one.
  • The memory order $\rho$ of a $DARN(\rho)$ process is visible in unlabelled trajectories: the autocorrelation-like functions stay roughly flat for $\tau \le \rho$ and then decay, with slopes close to the labelled version for the degree- and spectrum-based measures.
  • Sensitive dependence on initial conditions produces an expansion phase in all three pseudo-distances for both low-dimensional and high-dimensional chaotic network dynamics, so chaos is qualitatively detectable without labels.
  • No model-independent Lyapunov exponent can be extracted from unlabelled trajectories, because the relation between each pseudo-distance and the labelled distance is nonlinear and varies with the generative model.
  • Empirical temporal networks with labels deliberately removed—US domestic flights and two face-to-face contact datasets—still show their main dynamical features, including the daily periodicity of the flight network.
  • The pseudo-distances are blind to dynamics that leave the chosen invariants unchanged; for example, any trajectory preserving the total number of edges yields a vanishing degree-sequence pseudo-distance.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Combining several graph invariants into a single pseudo-distance should shrink the set of graphs that look identical to the measure, likely improving period and memory detection; this is a direct testable extension of the paper's single-invariant comparison.
  • The non-exponential shape of the chaotic expansion observed in the pseudo-distances could itself be a usable signature of instability, for instance through the time-dependent local slope of $\log d_X(t)$, even without a universal Lyapunov exponent.
  • If the invariance-blindness limitation can be controlled in practice, these methods open a privacy-friendly route: behavioural rhythms and correlation scales could be measured from anonymized contact or proximity data without tracking individual identities.
  • A model-specific calibration of the monotonic function relating each pseudo-distance to the labelled distance could, in principle, turn the qualitative chaos detection into a quantitative exponent estimate for known generative models, although the paper does not pursue this.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper proposes a framework for characterizing the dynamics of temporal networks when node labels are unavailable. It defines three permutation-invariant pseudo-distances based on the degree sequence, the adjacency spectrum, and the eigenvector centrality, and uses them to construct autocorrelation-like functions and expansion measures for unlabelled network trajectories. The approach is validated on synthetic models of noisy periodic dynamics, DARN processes with memory, and low- and high-dimensional chaotic graph dynamics, as well as on three empirical temporal networks with removed labels. The central conclusion is that periodicity, memory, and, qualitatively, sensitivity to initial conditions can be partially recovered from unlabelled trajectories, while a model-independent Lyapunov exponent cannot be defined.

Significance. The problem addressed is timely and relevant: unlabelled temporal networks arise in privacy-constrained or technically difficult tracking settings, and few tools exist for their dynamical analysis. The paper's strengths are its clear formulation of invariant-based pseudo-distances, the use of labelled ground-truth distances as benchmarks, the explicit comparison against shuffled and permuted baselines, and a candid discussion of limitations, including the non-universal and sublinear relation between pseudo-distances and the labelled distance. If the results hold, the paper provides a useful proof of concept for extending network time-series analysis to unlabelled trajectories. However, the analytical inconsistency in the ground-truth derivation and the unproven monotonicity bridge between pseudo-distances and labelled distance prevent the claims from being fully established as stated.

major comments (2)
  1. [Sec. 2.3.1 / Appendix] Equation (5) is inconsistent with the appendix derivation in Eq. (15). Eq. (15) contains a factor 1/2 multiplying the two exponential terms, whereas Eq. (5) does not; for example, with p=0.5 and sigma=1, Eq. (5) predicts a negative value of d_lab, which is impossible. In addition, the appendix states that Z in Eq. (4) is chosen as N(N-1)/2, while the main text sets Z=N(N-1); with the appendix's normalization, the right-hand side of Eq. (14) would need a factor of 2. Since d_lab is the ground-truth distance used to calibrate the pseudo-distances in Fig. 1, the theoretical curve in Fig. 1(a) and the horizontal axis in Fig. 1(b) should be recomputed with the corrected formula before the results in Sec. 2.3 are taken as quantitative.
  2. [Sec. 2.3 and Sec. 3.1] The interpretation of the expansion phases in Figs. 2 and 3 as evidence of sensitive dependence on initial conditions rests on the assumption that the pseudo-distance d_X is a non-decreasing function of the labelled distance d_lab along the sampled trajectory. The paper only provides numerical evidence for monotonicity of ensemble averages on two perturbation families (ER latent noise and edge displacement) and explicitly concedes that no universal quantitative mapping d_X = f_X(d_lab) exists. This is not sufficient for the inference made: pseudo-distances can vanish for non-isomorphic graphs (e.g., d_deg=0 for any two graphs with the same degree sequence), so inversions or plateaus are possible outside the tested families. The authors should either prove monotonicity for a relevant class of latent graph dynamics, or explicitly restrict the chaotic-fingerprint claims to the tested models and state that the expansion phase is not necessarily diagnostic in general.
minor comments (6)
  1. [Sec. 2.3.1] The reference 'Appendix 4' should be updated to the actual appendix label (e.g., 'Appendix A').
  2. [Sec. 3.2.3] The word 'Autorregressive' in the section title should be spelled 'Autoregressive'.
  3. [Appendix] The misspelling 'egdes' should be corrected to 'edges'.
  4. [Throughout] The phrase 'ground true' is used several times; it should be 'ground-truth'.
  5. [Figure 2 caption] The description of the 'permuted' curve in panel (a) is ambiguous; please clarify whether the node-label permutation is applied independently to each snapshot, and why the corresponding curves are not shown in panels (b)-(d).
  6. [Eq. (11)] The definition of J includes the t=t' term where d_X=0; state this explicitly, as it affects the interpretation of c_X(tau) for small tau.

Circularity Check

0 steps flagged · score 0.0 of 10

No reduction-by-construction found; the unlabelled pseudo-distances are parameter-free and validated against external labelled ground truth.

full rationale

The paper's central quantities are defined, not fitted: the three pseudo-distances d_deg, d_spec and d_eig (Eqs. 1-3) are parameter-free permutation invariants, and the autocorrelation-like measure c_X(tau) = 1 - (1/(T-tau)) sum_t d_X(G_t,G_{t+tau})/J (Eq. 10) is a direct similarity construction rather than a regressed predictor. The periodicity peaks and DARN decay slopes are emergent features of the test signals, not consequences of fitting the measures to those signals. For the chaos experiments, the labelled ground-truth trajectories are generated from known chaotic maps whose Lyapunov exponents are externally established (logistic map; globally coupled tent map with lambda = ln[2(1-alpha)], cited to Refs. [51-53]); the unlabelled distances are then compared with this independent benchmark. The paper explicitly concedes that the relation d_X = f_X(d_lab) is not universal and that no model-independent Lyapunov exponent can be constructed, so the central claim is not forced by construction. The reliance on the authors' prior work [22,23] supplies the labelled autocorrelation baseline and the dictionary/graph-dynamics generators, but the unlabelled invariants and c_X are independent constructions relative to those baselines; there is no self-citation chain that substitutes for the empirical validation. The possible monotonicity failure of d_X versus d_lab is a correctness or robustness concern about interpreting expansion phases, not a circularity: the paper does not define d_X in terms of d_lab or fit d_X to d_lab. The apparent algebraic mismatch between Eq. (5) and Eq. (15) also concerns derivation accuracy, not circularity. Accordingly, no step reduces a prediction to an input by definition or by fitted redefinition.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted parameters in the core construction: the pseudo-distance normalizations are chosen to bound the values in [0,1], and the autocorrelation-like functions are defined, not calibrated. The single hand-chosen quantity that affects quantitative claims is the z > 4 detection threshold in Eq. (12). The load-bearing premises are domain assumptions: equal node counts across snapshots, visibility of the dynamics in the three chosen invariants, and the numerically evidenced monotonic relation between pseudo-distances and the labelled ground-truth distance. The paper pulls no new entities from a hat; its axioms are standard spectral results, cited Lyapunov exponents, and the dictionary construction from prior work.

free parameters (1)
  • z-score detectability threshold z* = 4 (hand-chosen)
    Chosen in Eq. (12) as the criterion for a detectable periodicity peak; the quantitative claims that unlabelled autocorrelations detect periodicity up to noise q about 0.4 (labelled: q about 0.9) are calibrated to this threshold, so a different threshold shifts the quantitative conclusion.
assumptions (6)
  • domain assumption All snapshots have the same number of nodes N, and compared graphs must have equal N.
    Stated in Sec. 2.2: 'we require the graphs G and H to have the same number of nodes (relaxing this constraint is left for future work)'. The entire framework assumes fixed network size.
  • domain assumption The chosen invariants (sorted degree sequence, adjacency spectrum, sorted eigenvector centrality) carry enough information about the latent labelled dynamics for the target fingerprints.
    Sec. 2.1 concedes that invariants 'will not, in general, capture the full information of the graph', and Sec. 2.1 notes degree-based distances fail for edge-count-preserving dynamics. Recovery of fingerprints is conditional on the dynamics being visible in these invariants.
  • domain assumption d_X depends monotonically and continuously on the labelled ground-truth distance d_lab.
    Established only numerically in Sec. 2.3.2 (Fig. 1b): 'numerical evidence suggests... monotonic dependence'. The interpretation of d_X growth in the chaos experiments as a dynamical fingerprint rests on this unproven monotonicity.
  • standard math Standard spectral facts: the spectral radius of a simple graph is at most N-1, the smallest eigenvalue is at least 1-N, and the spectrum and sorted eigenvectors are invariant under node relabelling.
    Used in Sec. 2.2 to normalize d_spec and d_eig; these are standard results (Perron-Frobenius and trace bounds) from Refs. 48-50.
  • standard math Reference Lyapunov exponents: ln 2 for the logistic map and ln[2(1-alpha)] for the globally coupled tent map.
    Used in Secs. 3.1.1 and 3.1.2 as ground-truth exponents against which d_lab growth is checked; the tent-map value is cited to Refs. 51-53. These are prior literature results, not derived here.
  • domain assumption The dictionary of graphs H_0,...,H_L encodes the chaotic time series faithfully, with labelled distance proportional to index separation.
    Sec. 2.3.2 and Sec. 3.1.1 construct the dictionary with one edge displaced per step, giving d_lab(H_n, H_m) proportional to |n-m| (Eq. 7). The low-dimensional chaos experiment assumes the network trajectory inherits the chaotic separation of the underlying logistic map.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Characterising the dynamics of unlabelled temporal networks." pith.science (2026). https://pith.science/paper/XRCM3WRN

@misc{pith2026241214864,
  author       = {Pith},
  title        = {Pith review of: Characterising the dynamics of unlabelled temporal networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XRCM3WRN}},
  note         = {Machine review of arXiv:2412.14864}
}
read the original abstract

Networks model the architecture backbone of complex systems. The backbone itself can change over time leading to what is called `temporal networks'. Interpreting temporal networks as trajectories in graph space of a latent graph dynamics has recently enabled the extension of concepts and tools from dynamical systems and time series to networks. Here we address temporal networks with unlabelled nodes, a case that has received relatively little attention so far. Situations in which node labelling cannot be tracked over time often emerge in practice due to technical challenges, or privacy constraints. In unlabelled temporal networks there is no one-to-one matching between a network snapshot and its adjacency matrix. Characterizing the dynamical properties of such unlabelled network trajectories is nontrivial. We here exploit graph invariants to extend to the unlabelled setting network-dynamical quantifiers of linear correlations and dynamical instability. In particular, we focus on autocorrelation functions and the sensitive dependence on initial conditions. We show with synthetic graph dynamics that the measures are capable of recovering and estimating these dynamical fingerprints even when node labels are unavailable. We also validate the methods for some empirical temporal networks with removed node labels.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Scalar embedding of temporal network trajectories

    physics.soc-ph 2024-11 conditional novelty 5.0 of 10

    Pairwise graph distances compressed by PCA or MDS yield scalar time series that inherit periodicity, memory, and chaos from temporal network trajectories.

Reference graph

Works this paper leans on

58 extracted references · 42 canonical work pages · cited by 1 Pith paper

  1. [1]

    Zhao, K., Stehl´ e, J., Bianconi, G., Barrat, A.: Social network dynamics of face-to-face interactions. Phys. Rev. E 83(5), 056109 (2011). https: //doi.org/10.1103/PhysRevE.83.056109

    work page doi:10.1103/physreve.83.056109 2011

  2. [2]

    Understanding Complex Systems, p

    Palla, G., Pollner, P., Barab´ asi, A., Vicsek, T.: Social group dynamics in networks. Understanding Complex Systems, p. 11. Springer, Berlin, Heidelberg. (2009). https://doi.org/10.1007/978-3-642-01284-6 2

    doi:10.1007/978-3-642-01284-6 2009

  3. [3]

    S., Sporns, O.: Network neuroscience

    Bassett, D. S., Sporns, O.: Network neuroscience. Nature Neuroscience 20(3), 353 (2017). https://doi.org/10.1038/nn.4502

    doi:10.1038/nn.4502 2017

  4. [4]

    I., Laskaris, N

    Dimitriadis, S. I., Laskaris, N. A., Tsirka, V., Vourkas, M., Micheloyannis, S., Fotopoulos, S.: Tracking brain dynamics via time-dependent network analysis. Journal of Neuroscience Methods 193(1), 145 (2010). https:// doi.org/10.1016/j.jneumeth.2010.08.027

    work page doi:10.1016/j.jneumeth.2010.08.027 2010

  5. [5]

    Cambridge University Press, Cambridge (2017)

    Latora, V., Nicosia, V., Russo, G.: Complex Networks: Principles, Meth- ods and Applications. Cambridge University Press, Cambridge (2017). https://doi.org/10.1017/9781316216002

    work page doi:10.1017/9781316216002 2017

  6. [6]

    Oxford University Press, Oxford (2018)

    Newman, M.: Networks. Oxford University Press, Oxford (2018). https: //doi.org/10.1093/oso/9780198805090.001.0001

    arXiv 2018

  7. [7]

    Physics Reports519(3), 97 (2012)

    Holme, P., Saram¨ aki, J.: Temporal networks. Physics Reports519(3), 97 (2012). https://doi.org/10.1016/j.physrep.2012.03.001

    doi:10.1016/j.physrep.2012.03.001 2012

  8. [8]

    Series on Complexity Science, vol

    Masuda, N., Lambiotte, R.: A Guide to Temporal Networks. Series on Complexity Science, vol. 4, p. 252. World Scientific, Europe (2016). https: //doi.org/10.1142/q0033

    work page doi:10.1142/q0033 2016

Show all 58 references
  1. [9]

    (eds.) Graph Metrics for Temporal Net- works, p

    Nicosia, V., Tang, J., Mascolo, C., Musolesi, M., Russo, G., Latora, V.: In: Holme, P., Saram¨ aki, J. (eds.) Graph Metrics for Temporal Net- works, p. 15. Springer, Berlin, Heidelberg (2013). https://doi.org/10. 1007/978-3-642-36461-7 2

    2013

  2. [10]

    M.: Temporal networks: Slowing down diffusion by long lasting interactions

    Masuda, N., Klemm, K., Egu ´ ıluz, V. M.: Temporal networks: Slowing down diffusion by long lasting interactions. Phys. Rev. Lett. 111(18), 188701 (2013). https://doi.org/10.1103/PhysRevLett.111.188701

    2013 doi

  3. [11]

    Delvenne, J.-C., Lambiotte, R., Rocha, L. E. C.: Diffusion on networked systems is a question of time or structure. Nature Communications 6(1), 7366 (2015). https://doi.org/10.1038/ncomms8366

    2015 doi

  4. [12]

    Scholtes, I., Wider, N., Pfitzner, R., Garas, A., Tessone, C. J., Schweitzer, Springer Nature 2021 LATEX template 24 Characterising the dynamics of unlabelled temporal networks F.: Causality-driven slow-down and speed-up of diffusion in non- markovian temporal networks. Nature...

    2014 doi

  5. [13]

    Starnini, M., Baronchelli, A., Pastor-Satorras, R.: Modeling human dynamics of face-to-face interaction networks. Phys. Rev. Lett. 110(16), 168701 (2013). https://doi.org/10.1103/PhysRevLett.110.168701

    2013 doi

  6. [14]

    European Journal of Operational Research 281(1), 50 (2020)

    Mazzarisi, P., Barucca, P., Lillo, F., Tantari, D.: A dynamic network model with persistent links and node-specific latent variables, with an application to the interbank market. European Journal of Operational Research 281(1), 50 (2020). https://doi.org/10.1016/j.ejor.2019.07.024

    2020 doi

  7. [15]

    Scientific Reports 8(1), 15321 (2018)

    Hiraoka, T., Jo, H.-H.: Correlated bursts in temporal networks slow down spreading. Scientific Reports 8(1), 15321 (2018). https://doi.org/10.1038/ s41598-018-33700-8

    2018

  8. [16]

    Van Mieghem, P., van de Bovenkamp, R.: Non-Markovian infection spread dramatically alters the susceptible-infected-susceptible epidemic thresh- old in networks. Phys. Rev. Lett. 110(10), 108701 (2013). https://doi. org/10.1103/PhysRevLett.110.108701

    2013 doi

  9. [17]

    H., Brantefors, P., Fransson, P.: From static to temporal network theory: Applications to functional brain connectivity

    Thompson, W. H., Brantefors, P., Fransson, P.: From static to temporal network theory: Applications to functional brain connectivity. Network Neuroscience 1(2), 69 (2017). https://doi.org/10.1162/NETN a 00011

    2017 doi

  10. [18]

    Chaos: An Interdisciplinary Journal of Nonlinear Science 19(2), 023111 (2009)

    Zanin, M., Lacasa, L., Cea, M.: Dynamics in scheduled networks. Chaos: An Interdisciplinary Journal of Nonlinear Science 19(2), 023111 (2009). https://doi.org/10.1063/1.3129785

    2009 doi

  11. [19]

    The European Physical Journal Special Topics 215(1), 5 (2013)

    Zanin, M., Lillo, F.: Modelling the air transport with complex networks: A short review. The European Physical Journal Special Topics 215(1), 5 (2013). https://doi.org/10.1140/epjst/e2013-01711-9

    2013 doi

  12. [20]

    E., Lillo, F., Latora, V.: Effects of memory on spreading processes in non-Markovian temporal networks

    Williams, O. E., Lillo, F., Latora, V.: Effects of memory on spreading processes in non-Markovian temporal networks. New Journal of Physics 21(4), 043028 (2019). https://doi.org/10.1088/1367-2630/ab13fb

    2019 doi

  13. [21]

    E., Lacasa, L., Mill´ an, A

    Williams, O. E., Lacasa, L., Mill´ an, A. P., Latora, V.: The shape of mem- ory in temporal networks. Nature Communications 13(1), 499 (2022). https://doi.org/10.1038/s41467-022-28123-z

    2022 doi

  14. [22]

    P., Eguiluz, V

    Lacasa, L., Rodriguez, J. P., Eguiluz, V. M.: Correlations of network tra- jectories. Phys. Rev. Res. 4(4), 042008 (2022). https://doi.org/10.1103/ PhysRevResearch.4.L042008 Springer Nature 2021 LATEX template Characterising the dynamics of unlabelled temporal networks 25

    2022

  15. [23]

    M., Di Gaetano, L., Galla, T., Lacasa, L.: Lya- punov exponents for temporal networks

    Caligiuri, A., Egu ´ ıluz, V. M., Di Gaetano, L., Galla, T., Lacasa, L.: Lya- punov exponents for temporal networks. Phys. Rev. E 107(4), 044305 (2023). https://doi.org/10.1103/PhysRevE.107.044305

    2023 doi

  16. [24]

    C., Lacasa, L.: Dynamical stability and chaos in artificial neural network trajectories along training

    Danovski, K., Soriano, M. C., Lacasa, L.: Dynamical stability and chaos in artificial neural network trajectories along training. Frontiers in Complex Systems 2 (2024). https://doi.org/10.3389/fcpxs.2024.1367957

    2024

  17. [25]

    IEEE Access 11, 41426 (2023)

    Thongprayoon, C., Livi, L., Masuda, N.: Embedding and trajectories of temporal networks. IEEE Access 11, 41426 (2023). https://doi.org/10. 1109/ACCESS.2023.3268030

    2023

  18. [26]

    Hartle, H., Masuda, N.: Autocorrelation properties of temporal networks governed by dynamic node variables (2024) arXiv:2408.16270

    2024 arXiv

  19. [27]

    J., Masuda, N., Arola-Fern´ andez, L.: Scalar embedding of temporal network trajectories (2024) arXiv:2412.02715

    Lacasa, L., Mar ´ ın-Rodr ´ ıguez, F. J., Masuda, N., Arola-Fern´ andez, L.: Scalar embedding of temporal network trajectories (2024) arXiv:2412.02715

    2024 arXiv

  20. [28]

    IEEE Trans- actions on Pattern Analysis and Machine Intelligence37(12), 2451 (2015)

    Attanasi, A., Cavagna, A., Del Castello, L., Giardina, I., Jeli´ c, A., Melillo, S., Parisi, L., Pellacini, F., Shen, E., Silvestri, E., Viale, M.: Greta-a novel global and recursive tracking algorithm in three dimensions. IEEE Trans- actions on Pattern Analysis and Machine In...

    2015

  21. [29]

    J., Aarestrup, K., Cooke, S

    Lennox, R. J., Aarestrup, K., Cooke, S. J., Cowley, P. D., Deng, Z. D., Fisk, A. T., Harcourt, R. G., Heupel, M., Hinch, S. G., Holland, K. N., Hussey, N. E., Iverson, S. J., Kessel, S. T., Kocik, J. F., Lucas, M. C., Flemming, J. M., Nguyen, V. M., Stokesbury, M. J. W., Vagle...

    2017 doi

  22. [30]

    IEEE Network 24(4), 13 (2010)

    Zhang, C., Sun, J., Zhu, X., Fang, Y.: Privacy and security for online social networks: challenges and opportunities. IEEE Network 24(4), 13 (2010). https://doi.org/10.1109/MNET.2010.5510913

    2010

  23. [31]

    Zheleva, E., Getoor, L.: In: Aggarwal, C. C. (ed.) Privacy in Social Net- works: A Survey, p. 277. Springer, Boston, MA (2011). https://doi.org/ 10.1007/978-1-4419-8462-3 10

    2011 doi

  24. [32]

    Nature Communications12(1), 1655 (2021)

    Cencetti, G., Santin, G., Longa, A., Pigani, E., Barrat, A., Cattuto, C., Lehmann, S., Salath´ e, M., Lepri, B.: Digital proximity tracing on empiri- cal contact networks for pandemic control. Nature Communications12(1), 1655 (2021). https://doi.org/10.1038/s41467-021-21809-w

    2021 doi

  25. [33]

    E., Battaglini, M., Darias, Springer Nature 2021 LATEX template 26 Characterising the dynamics of unlabelled temporal networks F

    Rodr ´ ıguez, P., Gra˜ na, S., Alvarez-Le´ on, E. E., Battaglini, M., Darias, Springer Nature 2021 LATEX template 26 Characterising the dynamics of unlabelled temporal networks F. J., Hern´ an, M. A., L´ opez, R., Llaneza, P., Mart ´ ın, M. C., RadarCovid- Pilot Group, et al.:...

    2021 doi

  26. [34]

    F., Chen, L., L´ opez-L´ opez, E., Ju´ arez-Mercado, K

    L´ opez-P´ erez, K., Avellaneda-Tamayo, J. F., Chen, L., L´ opez-L´ opez, E., Ju´ arez-Mercado, K. E., Medina-Franco, J. L., Miranda-Quintana, R. A.: Molecular similarity: Theory, applications, and perspectives. Artificial Intelligence Chemistry 2(2), 100077 (2024). https://do...

    2024

  27. [35]

    Bondy, A., Murty, U. S. R.: Graph Theory. Graduate Texts in Mathematics. Springer, London (2008). https://link.springer.com/book/ 9781846289699

    2008

  28. [36]

    M.: Random graph isomorphism

    Babai, L., Erd˝ os, P., Selkow, S. M.: Random graph isomorphism. SIAM Journal on Computing 9(3), 628 (1980). https://doi.org/10.1137/ 0209047

    1980

  29. [37]

    In: Proceedings of the 2016 ACM on International Conference on Multimedia Retrieval (ICMR ’16), p

    Yan, J., Yin, X.-C., Lin, W., Deng, C., Zha, H., Yang, X.: A short survey of recent advances in graph matching. In: Proceedings of the 2016 ACM on International Conference on Multimedia Retrieval (ICMR ’16), p. 167. Association for Computing Machinery, New York, NY, USA (2016)...

    2016

  30. [38]

    D., Piperno, A.: Practical graph isomorphism, ii

    McKay, B. D., Piperno, A.: Practical graph isomorphism, ii. Journal of Symbolic Computation 60, 94 (2014). https://doi.org/10.1016/j.jsc.2013. 09.003

    2014 doi

  31. [39]

    M.: Canonical labeling of graphs

    Babai, L., Luks, E. M.: Canonical labeling of graphs. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing. STOC ’83, p. 171. Association for Computing Machinery, New York, NY, USA (1983). https://doi.org/10.1145/800061.808746

    1983

  32. [40]

    G.: Metrics for graph comparison: A practitioner’s guide

    Wills, P., Meyer, F. G.: Metrics for graph comparison: A practitioner’s guide. PLOS ONE 15(2), 1 (2020). https://doi.org/10.1371/journal.pone. 0228728

    2020 doi

  33. [41]

    A., Carpi, L., D ´ ıaz-Guilera, A., Pardalos, P

    Schieber, T. A., Carpi, L., D ´ ıaz-Guilera, A., Pardalos, P. M., Masoller, C., Ravetti, M. G.: Quantification of network structural dissimilarities. Nature Communications 8(1), 13928 (2017). https://doi.org/10.1038/ ncomms13928

    2017

  34. [42]

    M.: The evolution of unlabelled graphs

    Wright, E. M.: The evolution of unlabelled graphs. Journal of the Lon- don Mathematical Society s2-14(3), 554 (1976). https://doi.org/10.1112/ jlms/s2-14.3.554 Springer Nature 2021 LATEX template Characterising the dynamics of unlabelled temporal networks 27

    1976

  35. [43]

    Jour- nal of the London Mathematical Society s2-26(2), 201 (1982)

    Bollob´ as, B.: The asymptotic number of unlabelled regular graphs. Jour- nal of the London Mathematical Society s2-26(2), 201 (1982). https: //doi.org/10.1112/jlms/s2-26.2.201

    1982 doi

  36. [44]

    Walsh, T. R. S., Wright, E. M.: The k-connectedness of unlabelled graphs. Journal of the London Mathematical Society s2-18(3), 397 (1978). https: //doi.org/10.1112/jlms/s2-18.3.397

    1978 doi

  37. [45]

    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476(2243), 20190744 (2020)

    Hartle, H., Klein, B., McCabe, S., Daniels, A., St-Onge, G., Murphy, C., H´ ebert-Dufresne, L.: Network comparison and the within-ensemble graph distance. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476(2243), 20190744 (2020). https://do...

    2020

  38. [46]

    Paton, J., Hartle, H., Stepanyants, H., van der Hoorn, P., Krioukov, D.: Entropy of labeled versus unlabeled networks. Phys. Rev. E 106(5), 054308 (2022). https://doi.org/10.1103/PhysRevE.106.054308

    2022 doi

  39. [47]

    Linear Algebra and its Applications 436(5), 1425 (2012)

    Jovanovi´ c, I., Stani´ c, Z.: Spectral distances of graphs. Linear Algebra and its Applications 436(5), 1425 (2012). https://doi.org/10.1016/j.laa.2011. 08.019

    2012 doi

  40. [48]

    E., Haemers, W

    Brouwer, A. E., Haemers, W. H.: Spectra of Graphs, 1st edn., p. 250. Springer, New York, NY (2012). https://doi.org/10.1007/ 978-1-4614-1939-6

    2012

  41. [49]

    Social Networks 29(4), 555 (2007)

    Bonacich, P.: Some unique properties of eigenvector centrality. Social Networks 29(4), 555 (2007). https://doi.org/10.1016/j.socnet.2007.04. 002

    2007 doi

  42. [50]

    J.: In: Tur´ an, P

    Schoenberg, I. J.: In: Tur´ an, P. (ed.) Publications of Edmund Lan- dau, p. 335. Springer, Boston, MA (1969). https://doi.org/10.1007/ 978-1-4615-4819-5 23

    1969

  43. [51]

    Physica D: Nonlinear Phenomena 55(3), 368 (1992)

    Kaneko, K.: Mean field fluctuation of a network of chaotic elements: Remaining fluctuation and correlation in the large size limit. Physica D: Nonlinear Phenomena 55(3), 368 (1992). https://doi.org/10.1016/ 0167-2789(92)90066-V

    1992

  44. [52]

    Morita, S.: Scaling law for the lyapunov spectra in globally coupled tent maps. Phys. Rev. E 58(4), 4401 (1998). https://doi.org/10.1103/ PhysRevE.58.4401

    1998

  45. [53]

    Physics Letters A 226(3), 172 (1997)

    Morita, S.: Lyapunov analysis of collective behavior in a network of chaotic elements. Physics Letters A 226(3), 172 (1997). https://doi.org/10.1016/ S0375-9601(96)00940-1 Springer Nature 2021 LATEX template 28 Characterising the dynamics of unlabelled temporal networks

    1997

  46. [54]

    Cambridge University Press, Cambridge (2003)

    Kantz, H., Schreiber, T.: Nonlinear Time Series Analysis, 2nd edn. Cambridge University Press, Cambridge (2003). https://doi.org/10.1017/ CBO9780511755798

    2003

  47. [55]

    J., Musolesi, M.: Spatio-temporal networks: reachability, centrality and robustness

    Williams, M. J., Musolesi, M.: Spatio-temporal networks: reachability, centrality and robustness. Royal Society Open Science 3(6), 160196 (2016). https://doi.org/10.1098/rsos.160196

    2016 doi

  48. [56]

    P., Haenni, S., Phuka, J., Leal-Neto, O

    Ozella, L., Paolotti, D., Lichand, G., Rodr ´ ıguez,J. P., Haenni, S., Phuka, J., Leal-Neto, O. B., Cattuto, C.: Using wearable proximity sensors to characterize social contact patterns in a village of rural malawi. EPJ Data Science 10(1), 46 (2021). https://doi.org/10.1140/ e...

    2021

  49. [57]

    https://doi.org/10.1140/ epjds/s13688-018-0140-1

    G´ enois, M., Barrat, A.: Can co-location be used as a proxy for face-to- face contacts? EPJ Data Science7(1), 11 (2018). https://doi.org/10.1140/ epjds/s13688-018-0140-1

    2018

  50. [58]

    Nature Communica- tions 15(1), 4754 (2024)

    Gallo, L., Lacasa, L., Latora, V., Battiston, F.: Higher-order correlations reveal complex memory in temporal hypergraphs. Nature Communica- tions 15(1), 4754 (2024). https://doi.org/10.1038/s41467-024-48578-6

    2024 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.