Hybrid Dynamic Networks and Higher-Order Statistics

Overview

• Hybrid dynamic networks let you zoom into one region at high resolution (farms) while keeping the rest of the country as regional supernodes. This is a multiscale coarsening strategy: reduce state space without cutting away context.^[5]
• We work with discrete-time temporal graphs (daily snapshots), tracking which connections appear and disappear each day.^[1][2]
• To compare hybrid networks (or models that generate them), we use three families of higher-order statistics: TEA/TNA (edge/node turnover), π-mass (random-walk mass on Farms vs Regions), and magnetic Laplacian spectra (direction-sensitive spectral fingerprints).^[1][10][12]

Interactive hybrid-network simulator

This embedded demo generates a 10-day hybrid network on the Netherlands COROP map (40 regions).^[13] You can pick a focal COROP to keep at farm resolution, tune persistence and birth probabilities, and explore: TEA/TNA (edge/node turnover), π-mass (F vs R), and a magnetic Laplacian spectrum heatmap.

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Why a hybrid representation?

Livestock movement data is naturally a temporal directed network: a shipment from premise u to premise v on day t is a time-stamped directed edge. Temporal ordering matters because disease, information, and material flows must respect time.^[1]

In practice, national-scale premise networks can become very large. But many questions are regional: “What is the import pressure into this region?” “Which local farms are repeatedly connected to outside flows?”. In livestock epidemiology, network-based summaries are routinely used to identify vulnerability, target surveillance, and compare intervention strategies.^[4]

A hybrid network is a compromise: keep the focal region detailed (farm nodes), and contract everything else into regional “supernodes”. This is a form of graph coarsening tuned for interpretation and computation. We have fewer nodes outside, but still explicit cross-region flows.^[5]

Formal definition: hybrid temporal graph

Start from daily shipment records $$r = (t, u, v, w, c_u, c_v),$$ where u and v are premises, w is a shipment weight (e.g., animals moved), and c_u, c_v are COROP codes of origin/destination.^[13] Let $c^*$ be the focal COROP.

Define the hybrid node set as $V' = F \cup R$:

• Farms (F): all premises with COROP = $c^*$.
• Regions (R): one supernode per non-focal COROP $c \neq c^*$.

The contraction map $\varphi$ sends each endpoint to either its farm node if in the focal COROP or its region node otherwise:

$$ \varphi(u) = \begin{cases} u, & \text{if } c_u=c^* \\ c_u, & \text{if } c_u \neq c^* \end{cases} $$

Each day t yields a directed weighted snapshot $G'_t = (V', E'_t, W_t)$ by aggregating shipments with the same hybrid endpoints: $$ W_t(a,b)=\sum_{r:\,\varphi(u)=a,\,\varphi(v)=b} w. $$ Edges can be of four blocks: F→F (within focal), F→R (exports), R→F (imports), and R→R (outside-only superflows).

This “resolution switch” is similar to multilayer and multiscale network thinking: you keep multiple interaction scales in one representation rather than picking a single resolution everywhere.^[3]

Why “higher-order” statistics?

Static summaries of the time-aggregated graph (degree, density, communities) can hide crucial temporal structure: bursts, periodicity, and edge turnover can change reachability and outbreak risk even when the aggregate looks similar.^[1][2] Higher-order statistics here means: summaries that depend on time ordering, persistence, or directional circulation, not just the aggregated adjacency.

Statistic family A — TEA & TNA (temporal turnover)

Let $E_t$ be the set of directed edges present on day t (after aggregation), and let $V_t$ be the set of active nodes (nonzero in- or out-degree). TEA (Temporal Edge Appearance) compares $E_t$ to $E_{t-1}$:

• persist: $E_t \cap E_{t-1}$
• birth: $E_t \setminus E_{t-1}$
• churn: $E_{t-1} \setminus E_t$

Births are further split using a global memory set $E_{\le t-1}$: a birth is new if it has never appeared before, and reactivated if it reappears after absence. TNA (Temporal Node Appearance) repeats the same decomposition on active node sets $V_t$.

Interpretation (livestock trade / epidemiology). Persistent edges often reflect stable trading relationships or logistics “channels”. High new edge rates can indicate exploration of new partners or market-driven rewiring, which increases the chance of connecting previously weakly linked parts of the system—relevant for introduction and spread risk.^[4] Reactivation suggests periodic trade (e.g., production cycles), which can create predictable “windows” of connectivity.

Generic interpretation. TEA/TNA are useful whenever repeated interactions matter: email/contact networks, financial transfers, supply chains, or transportation flows. The “formation vs persistence” viewpoint is standard in statistical models for dynamic networks (e.g., STERGM).^[6]

Statistic family B — π-mass (random-walk mass on Farms vs Regions)

TEA/TNA focus on turnover. π-mass focuses on where flow concentrates. One simple choice is a lazy random walk on the directed graph snapshot:

$$ P_{\mathrm{lazy}} = (1-\alpha)D^{-1}A + \alpha I, $$ where $A$ is the adjacency (unweighted or weighted), $D$ is the out-degree (or out-strength) diagonal, and $\alpha \in (0,1)$ adds a chance of waiting at the current node. Laziness is useful because it removes periodicity within each communicating class, but it does not by itself connect parts of a graph that are unreachable from one another.^[9]

For a node with zero out-strength, replace its undefined transition row with a self-loop in the lazy walk. For PageRank, replace that row with the teleportation distribution $v^\top$. A teleporting PageRank walk is: $$ P_{\mathrm{tele}} = (1-\beta)D^{-1}A + \beta\,\mathbf{1}v^\top, $$ where $v$ is a positive teleportation distribution. This is a broader restart mechanism: instead of sometimes waiting, the walk sometimes jumps to a new node. With positive teleportation weights, every node can still be reached through the restart channel, which gives a well-defined full-graph stationary distribution even when the raw directed snapshot is reducible. In plain language: lazy means “sometimes stay put”, whereas teleporting means “sometimes restart somewhere else”.^[7][14]

Once a walk has been defined, π-mass aggregates stationary probability by node type: $$\pi_F(t)=\sum_{i:\,type(i)=farm} \pi_i,\quad \pi_R(t)=\sum_{i:\,type(i)=region} \pi_i.$$ The subtlety is the state space on which $\pi$ is computed.

A largest strongly connected component (LSCC, or LIC in the simulator labels) is a set of nodes that can all reach one another. But mutual reachability does not imply closure: probability may still leak out to nodes outside that component. A largest closed strongly connected class is both mutually reachable and non-escaping. That difference matters because in a finite Markov chain, stationary mass lives on essential closed classes, not on transient leaky ones.^[9]

The current simulator uses a lazy-walk π-mass computed on the largest strongly connected component because it is fast and easy to teach. That makes it a useful exploratory diagnostic, but it can overstate mass retained in a leaky class. For formal analyses, two safer choices are to use a teleporting PageRank on the full snapshot or to restrict the calculation to a largest closed strongly connected class.^[9][14]

Interpreting the flow split. With the walk and state space fixed, π_F(t) and π_R(t) measure stationary mass on farms and region supernodes. The split depends on edge weights, contraction and connectivity. There is no preferred balance between the two: compare it with the uncontracted network or a model-based reference before judging a representation.

Generic interpretation. Stationary mass is a flow-centrality concept (PageRank is the canonical example).^[7] It is useful for prioritizing nodes for monitoring, intervention, caching, or auditing when you believe the system behaves like repeated movement on edges. For background on random walks, PageRank, and mixing, see Page et al., Lovász, Langville–Meyer, and Levin–Peres–Wilmer.^[8][9][14]

Statistic family C — magnetic Laplacian spectra (direction-sensitive fingerprints)

Many spectral tools (e.g., the normalized Laplacian) are naturally defined for undirected graphs. For directed networks, a magnetic Laplacian encodes direction as a complex phase while producing a Hermitian operator with real eigenvalues. This allows spectral signatures that respond to directed circulation and asymmetry.^[10][12]

The weighted construction used in the simulator starts from a directed adjacency $A$ and defines:

• symmetric magnitude $A_s = \tfrac{1}{2}(A + A^\top)$
• imbalance ratio $R_{uv} = \dfrac{A_{uv} - A_{vu}}{A_{uv} + A_{vu}}$ when $A_{uv}+A_{vu}>0$
• phase $\Theta_{uv} = 2\pi q\,R_{uv}$ (charge parameter $q$)
• magnetic adjacency $H_{uv} = (A_s)_{uv}\,e^{i\Theta_{uv}}$
• magnetic normalized Laplacian $L_q = I - D_s^{-1/2} H D_s^{-1/2}$, where $(D_s)_{uu}=\sum_v (A_s)_{uv}$

The simulator uses the reciprocal-imbalance phase $\Theta_{uv}=2\pi q\,(A_{uv}-A_{vu})/(A_{uv}+A_{vu})$, with zero phase when both weights are zero. Nearly balanced traffic produces a small phase; strongly one-sided traffic produces a larger one. The symmetric weight $A_s$ retains traffic magnitude. Magnetic Laplacians provide the broader framework for encoding direction through complex phases.^[10][12]

In the full-quality setting, the simulator computes the first K eigenvalues of $L_q$ for each day on the full node set and shows them as a heatmap across time. The fast setting uses an active-node approximation and interpolates skipped days for responsiveness. In directed graphs, changes in this spectrum can reflect shifts in “circulation modes” (cycles, one-way backbones, and directional community structure). The heatmap is therefore best read as a direction-sensitive fingerprint. For formal goodness-of-fit, however, a quantitative per-day spectral distance between observed and simulated networks—such as a distance between magnetic spectra or magnetic spectral densities—would be stronger than visual inspection alone.^[11][12][15]

Interpretation (livestock trade / epidemiology). Directionality matters because risk is not symmetric: imports into a region carry different implications than exports. Magnetic spectra are a compact “fingerprint” of directional structure and its evolution, complementing turnover (TEA/TNA) and flow concentration (π-mass).^[4]

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NetSpectra: the Hybrid Simulator project

The simulator you just used is packaged as a standalone, embeddable web app (D3 + Web Workers) for teaching and exploratory analysis. I maintain it alongside the broader HerdLink ecosystem.

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Using these summaries for goodness of fit

For a model check, compare observed statistics with their distribution across networks simulated from the fitted model. Compute the same descriptors and use the same preprocessing for observed and simulated networks.^[16][18]

Compute TEA/TNA, π-mass and magnetic spectral discrepancies for each observed day and its simulated replicates. Plot the observed trajectories against simulation envelopes and summarize discrepancies over time. State the distance measure, replication scheme and treatment of disconnected components so the comparison can be reproduced.^{[12][15][16][18]}

References

1. Holme P, Saramäki J. 2012. Temporal networks. Physics Reports. doi:10.1016/j.physrep.2012.03.001
2. Holme P. 2015. Modern temporal network theory: a colloquium. Eur Phys J B. doi:10.1140/epjb/e2015-60657-4
3. Kivelä M, Arenas A, Barthelemy M, Gleeson JP, Moreno Y, Porter MA. 2014. Multilayer networks. Journal of Complex Networks. doi:10.1093/comnet/cnu016
4. Chaters GL, et al. 2019. Analysing livestock network data for infectious disease control: an argument for routine data collection in emerging economies. Phil Trans R Soc B. doi:10.1098/rstb.2018.0264
5. Chen J, Saad Y, Zhang Z. 2022. Graph coarsening: from scientific computing to machine learning. SeMA Journal. doi:10.1007/s40324-021-00282-x
6. Krivitsky PN, Handcock MS. 2014. A Separable Model for Dynamic Networks. J R Stat Soc Series B. doi:10.1111/rssb.12014
7. Page L, Brin S, Motwani R, Winograd T. 1999. The PageRank Citation Ranking: Bringing Order to the Web. Stanford InfoLab Technical Report. PDF
8. Lovász L. 1993. Random walks on graphs: a survey. In: Combinatorics, Paul Erdős is Eighty (Vol. 2). PDF
9. Levin DA, Peres Y, with contributions by Wilmer EL. 2017. Markov Chains and Mixing Times (2nd ed.). American Mathematical Society. PDF
10. Fanuel M, Alaíz CM, Fernandez A, Suykens JAK. 2018. Magnetic Eigenmaps for the visualization of directed networks. Appl Comput Harmon Anal. doi:10.1016/j.acha.2017.01.004
11. Fanuel M, Alaíz CM, Suykens JAK. 2017. Magnetic eigenmaps for community detection in directed networks. Phys Rev E. doi:10.1103/PhysRevE.95.022302
12. de Resende BMF, Costa LdF. 2020. Characterization and comparison of large directed networks through the spectra of the magnetic Laplacian. Chaos. doi:10.1063/5.0006891
13. Statistics Netherlands (CBS). COROP region (definition and background). cbs.nl definition
14. Langville AN, Meyer CD. 2004. Deeper Inside PageRank. Internet Mathematics. doi:10.1080/15427951.2004.10129091
15. Shore J, Lubin B. 2015. Spectral goodness of fit for network models. Social Networks. doi:10.1016/j.socnet.2015.04.004
16. Vaca-Ramírez F, Peixoto TP. 2022. Systematic assessment of the quality of fit of the stochastic block model for empirical networks. Physical Review E. doi:10.1103/PhysRevE.105.054311
17. Yin M, Zhu L. 2016. Reciprocity in directed networks. Physica A. doi:10.1016/j.physa.2015.12.008
18. Hunter DR, Goodreau SM, Handcock MS. 2008. Goodness of fit of social network models. Journal of the American Statistical Association. doi:10.1198/016214507000000446