Tianjian Qin

TL;DR

• 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). The key question is not just what is connected, but when connections appear and disappear.^[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. Construct a (lazy) random walk on the directed graph snapshot:

$$ P = (1-\alpha)D^{-1}A + \alpha I, $$ where $A$ is the adjacency (unweighted or weighted), $D$ is the out-degree (or out-strength) diagonal, and $\alpha \in (0,1)$ adds “laziness”. A teleporting/lazy step is a standard trick to improve stability and convergence of the stationary distribution on directed graphs.^[7][9]

Because directed graphs may not be strongly connected, we compute the stationary distribution on the largest strongly connected component (LIC), then aggregate stationary mass by node type: $$\pi_F(t)=\sum_{i\in LIC,\,type(i)=farm} \pi_i,\quad \pi_R(t)=\sum_{i\in LIC,\,type(i)=region} \pi_i.$$

Interpretation (hybrid network quality). If contraction is too aggressive, mass can collapse onto region supernodes (π_R≈1), washing out focal structure; if the focal subgraph becomes too isolated, π_F≈1 and the outside context is effectively gone. A “reasonable” hybrid representation tends to keep a meaningful split across time, reflecting both local circulation and external pressure.

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 and mixing, see Lovász and Levin–Peres–Wilmer.^[8][9]

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]

One common construction (used by the simulator) starts from a directed adjacency $A$ and defines:

• symmetric magnitude $A_s = \tfrac{1}{2}(A + A^\top)$
• antisymmetry $\Delta = A - A^\top$
• phase $\Theta_{uv} = 2\pi q\,\mathrm{sign}(\Delta_{uv})$ (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 $(D_s)_{uu}=\sum_v (A_s)_{uv}$

For each day, we compute the first K eigenvalues of $L_q$ and show them as a heatmap across time. In directed graphs, changes in this spectrum can reflect shifts in “circulation modes” (cycles, one-way backbones, and directional community structure).^[11][12]

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.^[4]

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References

Numbers in brackets link to the sources used for the conceptual framing of this post.

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, Wilmer EL. 2009/2017. 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