Temporal Degree Centrality

• Temporal degree centrality is a network measure that counts a node’s time-stamped incident edges to assess its relevance over specific time intervals.
• It encompasses variants such as instantaneous snapshot degree, interval-aggregated degree, and activity-based degree, each capturing different aspects of temporal interactions.
• As a local and computationally efficient metric, it forms the core of more complex analyses like temporal communicability and causal multilayer centrality.

Temporal degree centrality denotes a family of local centrality measures for temporal networks in which the relevance of a node is quantified by the number of incident edges it has as a function of time, over an interval, or in a time-unfolded representation. In the literature considered here, the term covers instantaneous snapshot degree, interval-based counts of distinct neighbors or active adjacency relations, and normalized inbound-plus-outbound degree on temporal-node constructions; in several works it also appears as the short-time or small-parameter limit of broader temporal centralities, or as a degree-like ingredient inside multilayer constructions (Farahi et al., 2024, Pena et al., 23 Jul 2025, Schwengber et al., 2021).

1. Formal definitions and principal variants

A central feature of the subject is that there is no single canonical definition. Instead, temporal degree centrality is specified relative to a temporal graph model and to a choice of aggregation over time. In the discrete-time contact-network setting, the most direct definition is the instantaneous degree

$D_t(i) = \sum_j \mathcal{L}(i,j,t),$

or equivalently

$k_i(t) = \sum_j A^{[t]}_{ij},$

which counts the neighbors of node $i$ at time $t$. Over an interval $[t_s,t_e]$, one can instead count distinct neighbors

$$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$

or total active adjacency relations

$$d_{i,\mathcal{G}(t_s,t_e)} = \sum_{u=t_s}^{t_e} \sum_j \mathcal{L}(i,j,u).$$

These alternatives correspond, respectively, to “how many different people” and “how often” a node is in contact (Farahi et al., 2024).

Variant	Definition	Interpretation
Instantaneous temporal degree	$D_t(i)=\sum_j \mathcal{L}(i,j,t)$ or $k_i(t)=\sum_j A^{[t]}_{ij}$	Neighbors at one time step
Interval aggregated degree	$$\deg_{[t_s,t_e]}(i)=|\Gamma_{i,\mathcal{G}(t_s,t_e)}|$$	Distinct neighbors contacted at least once
Activity-based temporal degree	$k_i(t) = \sum_j A^{[t]}_{ij},$0	Total active adjacency relations over time
Temporal-node total degree	Normalized total inbound plus outbound temporal edges on $k_i(t) = \sum_j A^{[t]}_{ij},$1, excluding self-edges	Causal degree in a time-unfolded graph

A different, but closely related, formulation is used in temporal-node constructions. In the polarised-network setting, a block matrix $k_i(t) = \sum_j A^{[t]}_{ij},$2 is formed with one layer per time slice, and each original directed edge $k_i(t) = \sum_j A^{[t]}_{ij},$3 in slice $k_i(t) = \sum_j A^{[t]}_{ij},$4 becomes a temporal edge $k_i(t) = \sum_j A^{[t]}_{ij},$5. In that representation, temporal degree is defined as the normalized total number of inbound edges to and outbound edges from node $k_i(t) = \sum_j A^{[t]}_{ij},$6 on the time interval $k_i(t) = \sum_j A^{[t]}_{ij},$7, disregarding the self-edges from $k_i(t) = \sum_j A^{[t]}_{ij},$8 to $k_i(t) = \sum_j A^{[t]}_{ij},$9 (Pena et al., 23 Jul 2025). This definition is still local, but it is explicitly causal because all counted edges point forward in time.

A recurrent misconception is that “temporal degree centrality” always refers to a single scalar attached to the whole observation period. The surveyed literature instead treats it as a family of degree-like observables: a time series $i$0, an interval aggregate, a total count of time-stamped contacts, or a causal degree on a multilayer unfolding. This suggests that the term is best understood as a modeling choice rather than as a fixed invariant.

2. Temporal network representations and degree across time

Temporal degree depends strongly on how time is encoded. In temporal contact networks, the standard representation is a sequence of adjacency matrices

$i$1

where $i$2 if a contact is active during interval $i$3. In that setting, snapshot degree, sliding-window degree, binarized aggregated degree, and duration-weighted aggregated degree are all natural local observables, but they differ in whether they retain ordering, duration, or only the existence of contact (Chen et al., 2016).

Time-unfolded and multilayer constructions make the causal structure explicit. In the temporal-node matrix

$i$4

the $i$5 blocks encode persistence of the same node across successive layers, whereas the $i$6 blocks encode actual interactions. Temporal degree is then computed from the non-self interlayer edges only (Pena et al., 23 Jul 2025). In a related, more general supracentrality framework, a temporal network is written as a multiplex with intralayer centrality matrices $i$7 and interlayer coupling $i$8, assembled into

$i$9

That framework is eigenvector-based rather than degree-based, but it shows how trajectories of node importance can be coupled across time by a chosen interlayer topology (Taylor et al., 2019).

A plausible implication is that temporal degree centrality can be read at two levels: as a purely local count on each slice, and as a degree trajectory embedded in a causal multilayer object. The first emphasizes contact volume; the second emphasizes temporal admissibility.

3. Degree as the local endpoint of broader temporal centralities

Several temporal-centrality frameworks recover degree in a mathematically precise limit. In the entropy-based, scale-dependent centrality defined from diffusion

$t$0

the node score

$t$1

satisfies a small-time equivalence: for sufficiently small $t$2, the ranking induced by $t$3 coincides exactly with degree ranking, and

$t$4

Thus, degree appears as the local endpoint of a continuum that later becomes eigenvector-like and then closeness-like (Schwengber et al., 2021).

In dynamic communicability, the temporal Katz-like matrix

$t$5

defines broadcast and receive centralities through row and column sums. Expanding $t$6 for small $t$7 yields

$t$8

so broadcast centrality converges to aggregated degree in the $t$9 limit (Chen et al., 2016). Likewise, in Temporal Walk Centrality, if one fixes $[t_s,t_e]$0, uses $[t_s,t_e]$1, and then restricts attention to walks of length one, the resulting centrality equals out-degree centrality (Oettershagen et al., 2022).

A related result appears in TempoRank. There the snapshot strength

$[t_s,t_e]$2

is the temporal analogue of degree at time $[t_s,t_e]$3, and the long-run random-walk centrality is well approximated not by simple aggregate strength, but by the in-strength of an effective time-respecting directed network: $[t_s,t_e]$4 This places temporal degree-like quantities on a spectrum from local contact count to time-respecting path-based in-strength (Rocha et al., 2014).

Taken together, these constructions indicate that temporal degree centrality often serves as the zeroth- or first-order approximation to richer temporal notions. What changes across models is not the local counting intuition, but the extent to which temporally ordered walks, diffusion, or interlayer persistence are admitted to reshape that count.

4. Degree-like formulations in financial temporal networks

In correlation-based financial networks, the terminology becomes more delicate. In the stock-market temporal-network model, nodes are stocks and each time slice $[t_s,t_e]$5 is a planar maximally filtered graph derived from a rolling correlation matrix $[t_s,t_e]$6. The paper explicitly states that it does not define a “temporal degree centrality” per se; its temporal centrality is eigenvector-based and built on a supra-evolution matrix (Zhao et al., 2017).

The degree-like quantity in that framework is the correlation strength

$[t_s,t_e]$7

which summarizes how strongly stock $[t_s,t_e]$8 is correlated with all other stocks at time $[t_s,t_e]$9. Each $$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$0 is modeled as an ARMA or ARIMA process, and the autoregressive coefficients define diagonal interlayer couplings $$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$1 in a lower block-triangular supra-evolution matrix. Temporal eigenvector centrality is then obtained from

$$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$2

The paper’s own interpretation is that this construction generalizes the idea of degree over time: within-layer PMFG connectivity is combined with temporal persistence in correlation strength. It therefore uses a degree-like scalar $$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$3 as the dynamical ingredient, but the final ranking is not a temporal degree ranking; it is a time-aware global centrality. Empirically, low-centrality, peripheral-stock portfolios have lower risk and better efficient-frontier behavior than central-stock portfolios, so the degree-like core–periphery interpretation becomes financially consequential even though the operative score is eigenvector-based rather than degree-based (Zhao et al., 2017).

5. Applications in epidemic spreading, influence analysis, and critical-node detection

In epidemic applications, the performance of temporal degree centrality is strongly task-dependent. In an empirical emergency-department contact network, binarized aggregated degree $$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$4, duration-weighted aggregated degree $$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$5, broadcast centrality $$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$6, and simple temporal markers such as duration observed $$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$7 and time of first appearance $$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$8 were compared against outbreak statistics from an SI process. The study found that dynamic communicability identifies some “dynamic communicators” that aggregated degree misses, especially for worst-case spreading, but it also found that simple predictors can perform as well or better for average-outcome prediction; in particular, $$\Gamma_{i,\mathcal{G}(t_s,t_e)} = \{ v \in \mathcal{N} : \exists t \in [t_s,t_e] \text{ with } (i,v,t) \in \mathcal{E} \},$$9 combined with $$d_{i,\mathcal{G}(t_s,t_e)} = \sum_{u=t_s}^{t_e} \sum_j \mathcal{L}(i,j,u).$$0, $$d_{i,\mathcal{G}(t_s,t_e)} = \sum_{u=t_s}^{t_e} \sum_j \mathcal{L}(i,j,u).$$1, and staff status provided the largest gain in predictive accuracy in the regression framework (Chen et al., 2016).

In critical-node detection on temporal social networks, temporal degree itself appears mainly through instantaneous degree $$d_{i,\mathcal{G}(t_s,t_e)} = \sum_{u=t_s}^{t_e} \sum_j \mathcal{L}(i,j,u).$$2 and through Temporal Degree Deviation,

$$d_{i,\mathcal{G}(t_s,t_e)} = \sum_{u=t_s}^{t_e} \sum_j \mathcal{L}(i,j,u).$$3

That work concludes that purely degree-based temporal measures are often insufficient for identifying influential nodes under SIR dynamics, and that semi-local or cycle-based refinements such as TSLC, TSLI, and TSCR can perform better in epidemic control and robustness scenarios (Farahi et al., 2024).

A contrasting result arises in polarised social networks. Using temporal-node representations and influence-band analysis, one study reports that a modified temporal independent cascade model and temporal degree centrality perform the best in that setting, because they are able to reliably isolate nodes into their bands. There, temporal degree is the normalized total number of inbound and outbound time-respecting edges over the observation interval, excluding self-edges, and it aligns closely with the cascade-based benchmark on both synthetic BandNet networks and the RT8 Twitter network (Pena et al., 23 Jul 2025).

These findings are not inconsistent. They indicate that temporal degree centrality is effective when the objective is to recover broad influence strata or when contact volume is already strongly aligned with cascade potential, but less effective when influence depends on second-order temporal neighborhoods, cycle participation, or other nonlocal temporal structures.

6. Interpretation, limitations, and adjacent alternatives

Temporal degree centrality is computationally attractive because it is local. Computing $$d_{i,\mathcal{G}(t_s,t_e)} = \sum_{u=t_s}^{t_e} \sum_j \mathcal{L}(i,j,u).$$4 for all nodes and times is essentially a pass over the time-stamped edges, whereas dynamic communicability requires products of resolvents, walk-based methods require counting time-respecting walks, and coverage centralities require reachability structure over temporal vertices (Farahi et al., 2024, Colman et al., 2016). This computational advantage helps explain why degree-like measures remain common baselines.

Its main limitation is equally clear: degree counts incident temporal edges but does not, by itself, encode whether those edges can form effective time-respecting paths. In transportation networks with non-zero travel times, this limitation becomes structural. Trip Centrality was introduced precisely because walk-counting temporal centralities that ignore link durations can count infeasible itineraries; when $$d_{i,\mathcal{G}(t_s,t_e)} = \sum_{u=t_s}^{t_e} \sum_j \mathcal{L}(i,j,u).$$5 is very small, Trip Centrality approaches temporal degree-like behavior, but multi-leg, schedule-respecting connectivity requires a richer object than contact count (Zaoli et al., 2019). Likewise, coverage centralities for temporal vertices assess whether a vertex-time pair lies on fastest temporal paths for many source–target pairs, revealing bottleneck times that a local degree measure would not directly detect (Takaguchi et al., 2015).

A second limitation is interpretive ambiguity. “Temporal degree centrality” may mean degree at a time, degree over an interval, number of distinct temporal neighbors, number of active contacts, or total causal degree in a time-unfolded graph. A practical implication is that comparisons across studies are only meaningful when the temporal graph model and the aggregation rule are explicitly specified.

A third limitation is that local degree can be outperformed by temporal refinements that preserve order, waiting time, duration, or higher-order neighborhoods. Dynamic communicability separates broadcast and receive roles; entropy-based diffusion centrality interpolates between degree, eigenvector, and closeness; semi-local temporal centralities extend degree to time-respecting two-step neighborhoods; and supracentrality frameworks smooth centrality trajectories across time layers (Schwengber et al., 2021, Oettershagen et al., 2022, Taylor et al., 2019). Temporal degree centrality remains the local baseline against which these constructions are often interpreted.

In this sense, temporal degree centrality is best regarded not as a single metric but as the local core of temporal centrality theory: the immediate count of time-stamped adjacency relations from which more global, path-aware, and scale-dependent notions depart.