Dynamic Influence Centrality

• Dynamic Influence Centrality is a framework that defines node importance through explicit influence processes over time rather than static summaries.
• It leverages diverse mathematical formulations—including temporal paths, opinion dynamics, and control solutions—to capture influence in dynamic settings.
• The approach is applied to online misinformation control, competitive networks, and scalable influence algorithms for evolving graph structures.

Searching arXiv for papers on dynamic influence centrality and closely related influence-based centrality frameworks. Dynamic Influence Centrality (DIC) denotes a family of centrality constructions in which node importance is defined through an explicit influence process rather than through a purely static graph summary. In the available literature, the exact term is sometimes introduced directly and sometimes used as an interpretive label for related notions such as dynamic centrality, dynamical influence, influence centrality, CON score, or task-aware control centrality. Across these formulations, the common object is a mapping from temporal cascades, equilibrium opinion dynamics, control responses, or topology changes to node-level scores that quantify how strongly a node shapes downstream states, activation patterns, or collective outcomes (Sikosana et al., 11 Jul 2025, Lerman et al., 2010, Klemm et al., 2010).

1. Conceptual foundations

A general influence-based framework models diffusion as an influence instance $I=(V,E,P_I)$, where $P_I$ assigns, for each seed set, a distribution over cascading sequences $(S_0,S_1,\ldots,S_{n-1})$. These sequences are monotonic and $G$-continuous, so they encode activation over time rather than only final reachability. The corresponding influence spread is the expected terminal cascade size, and a broad class of influence-based centralities takes the form

$$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$

where $\vec d$ records activation times in the realized cascade. Within this framework, layered graphs form a basis for the space of influence-cascading-sequence profiles, and for anonymous $f$, the resulting $\psi[f]$ is the unique Bayesian centrality conforming with the corresponding graph-theoretical centrality on BFS instances (Chen et al., 2018).

This dynamic viewpoint is closely related to the distinction between conservative and non-conservative processes. Random walks and Markov flows preserve total mass, whereas information propagation in online social networks duplicates mass across multiple outgoing links. The literature argues that influence models should match the underlying process: conservative models such as PageRank are appropriate for conservative dynamics, while path-summing formulations such as alpha-centrality are more appropriate for broadcast-like information diffusion (Ghosh et al., 2010).

Two influential special cases clarify what DIC means operationally. Single Node Influence (SNI) centrality measures the spread of a singleton seed,

$$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$

and is suited to assessing individual influence in isolation. Shapley centrality instead uses the Shapley value of the influence spread function and measures expected marginal contribution in group influence settings. The comparative study of these two measures shows that DIC can be grounded either in standalone diffusion potential or in coalition-sensitive marginal impact (Chen et al., 2016).

2. Major mathematical formulations

The literature contains several mathematically distinct constructions that are all naturally interpreted as DIC. They differ in what “dynamic” means: temporal paths, changing topology, explicit differential equations, iterative accumulation, or control effort.

Setting	Core expression	Interpretation
Temporal paths	$$DC_i(\alpha,\gamma,\Delta_{1,n})=\sum_j RC^d_{ij}(\alpha,\gamma,\Delta_{1,n})$$ (Lerman et al., 2010)	Total influence via attenuated time-respecting paths
Friedkin–Johnsen opinions	$P_I$0, $P_I$1 (Shrinate et al., 2024)	Average contribution of each initial opinion to final opinions
Dynamical influence	$P_I$2, $P_I$3, $P_I$4 (Klemm et al., 2010)	Effect of node $P_I$5's initial state on the dominant collective mode
Iterative OSN DIC	$P_I$6, with $P_I$7 (Sikosana et al., 11 Jul 2025)	Temporal accumulation of structural influence over repeated rounds
Competition CON	$P_I$8, $P_I$9 (Bonato et al., 2019)	Shared competitive influence through common out-neighbors
Minimum-energy control	$(S_0,S_1,\ldots,S_{n-1})$0, $(S_0,S_1,\ldots,S_{n-1})$1 (Zheng et al., 1 Nov 2025)	Ability of a node to unify states under Laplacian dynamics

These formulations are not interchangeable. Some score expected reach or activation-time profiles, some score equilibrium sensitivity, some score shared influence regions, and some score control efficacy under a prescribed objective. A useful unifying description is that DIC maps a specified network dynamics and an associated performance functional to node-wise influence scores.

3. Temporal networks and topology variation

One major DIC line studies influence on temporal networks through time-respecting paths. In the dynamic centrality framework for evolving graphs, a path from $(S_0,S_1,\ldots,S_{n-1})$2 to $(S_0,S_1,\ldots,S_{n-1})$3 is only valid if its edges appear in the correct temporal order. The memoryless dynamic centrality matrix is

$(S_0,S_1,\ldots,S_{n-1})$4

and the retained version introduces a memory parameter $(S_0,S_1,\ldots,S_{n-1})$5 through retained adjacency matrices $(S_0,S_1,\ldots,S_{n-1})$6. The resulting node-level centrality

$(S_0,S_1,\ldots,S_{n-1})$7

measures expected total information sent by $(S_0,S_1,\ldots,S_{n-1})$8 that reaches all other nodes over a time window. The framework shows that static aggregation can overestimate or underestimate influence because many paths counted in the aggregate graph are not temporally feasible (Lerman et al., 2010).

A second line treats DIC as influence centrality under changing topology. In the Friedkin–Johnsen model,

$(S_0,S_1,\ldots,S_{n-1})$9

with

$G$0

Here $G$1 is the average contribution of node $G$2's initial opinion to final opinions, and in the two-influencer case the relevant scores satisfy $G$3. Edge modifications of type $G$4 preserve row-stochasticity by adding $G$5 and decreasing $G$6. Signal flow graph analysis yields two structural results: some modifications are redundant and leave $G$7 unchanged, whereas others induce a monotone reallocation of influence from one stubborn agent to the other, independent of the exact new weights as long as the admissibility constraints are satisfied (Shrinate et al., 2024).

A third temporal line makes DIC community-aware in polarized networks. Temporal degree and temporal closeness are defined on a time-expanded adjacency $G$8, eigenvector-based temporal scores are obtained from a supra-centrality matrix $G$9, temporal Katz centrality uses

$$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$0

and a modified temporal independent cascade model provides a diffusion benchmark. Nodes are aggregated into high-, mid-, and low-influence bands, and community influence is summarized by Marginal Community Centrality. In that setting, the modified temporal independent cascade model and temporal degree centrality perform the best, because they are able to reliably isolate nodes into their bands (Pena et al., 23 Jul 2025).

4. Diffusion, competition, and misinformation

A central DIC interpretation comes from dynamical influence in linearized spreading and consensus dynamics. For

$$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$1

if the dominant eigenvalue is $$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$2 and $$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$3, then the asymptotic state depends on the initial condition only through $$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$4, and $$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$5 quantifies how strongly node $$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$6's initial state affects the final collective state. At critical SIR or SIS spreading, this reduces to the leading eigenvector of the adjacency matrix; for diffusive processes it weights each node’s contribution to the final consensus; and for oscillator networks it predicts the most effective driving nodes (Klemm et al., 2010).

A closely related spreading-specific formulation is dynamics-sensitive centrality for SIR and SI models. With

$$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$7

the $$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$8-th component $$\psi[f]_v(I) = \mathbb{E}\big[f(\vec d(\{v\},S_1,\ldots,S_{n-1}))\big],$$9 approximates the spreading influence of node $\vec d$0 at time $\vec d$1. For SIR with $\vec d$2,

$\vec d$3

This interpolates between degree centrality at $\vec d$4 and eigenvector centrality when $\vec d$5 and $\vec d$6 are large, and empirically outperforms degree, $\vec d$7-shell, and eigenvector centrality for identifying influential spreaders on several real networks (Lin et al., 2015).

In online information diffusion, the choice of DIC is tied to whether the process is conservative or non-conservative. On Digg, information propagation was modeled as a non-conservative broadcast process, and normalized alpha-centrality provided one of the best predictors of empirically observed influence, outperforming conservative random-walk-based measures in that setting (Ghosh et al., 2010).

In competition networks, influence is adversarial rather than persuasive. The Dynamic Competition Hypothesis states that leaders in dynamic competition networks should have high CON scores, high closeness, high out-degree, and low in-degree. The CON score is based on common out-neighbors,

$\vec d$8

and is interpreted as a signature of leadership because influential actors shape whom others target. A later dynamic analysis defined first-order and second-order CON scores on round-by-round competition graphs, with

$\vec d$9

and used them as features in supervised learning for Survivor, Chess.com, and Dota 2 competitions. In that study, CON consistently outperformed PageRank, closeness, and betweenness centrality in classification tasks, and the dynamic CON score emerged as a powerful predictor of node rankings (Bonato et al., 2019, Bonato et al., 31 Jan 2025).

An explicit use of the name Dynamic Influence Centrality appears in health misinformation analysis on online social networks. There each node starts with

$f$0

and influence is iteratively accumulated by

$f$1

The process is run for a small fixed number of timesteps and then normalized. In experiments on FibVID, traditional metrics identified 29 influential nodes, the new metrics uncovered 24 unique nodes, and the combined set contained 42 nodes, an increase of 44.83%. Baseline interventions reduced health misinformation by 50%, while incorporating the new metrics increased this to 62.5%, an improvement of 25% (Sikosana et al., 11 Jul 2025).

5. Control, directionality, and signed dynamics

Several DIC formulations are explicitly control-theoretic. Katz centrality can be reinterpreted as the steady state of

$f$2

A node-specific perturbation analysis with decay vector $f$3 yields an influence matrix

$f$4

and a global impact approximation

$f$5

This construction quantifies the net impact of a node’s absence from the steady state and privileges nodes that both receive flux from others and pass it on (1711.01891).

A later task-aware extension for Laplacian dynamics defines U-centrality through minimum-energy control of average opinion. For

$f$6

with single-node control $f$7, the terminal state that achieves aggregate threshold $f$8 at horizon $f$9 is

$\psi[f]$0

and the centrality score is

$\psi[f]$1

U-centrality aligns with degree centrality in the short-time horizon and converges, over longer time scales, to a new centrality closely related to current-flow closeness centrality (Zheng et al., 1 Nov 2025).

Dynamic influence can also depend on global directionality. Trophic analysis assigns trophic levels $\psi[f]$2 through

$\psi[f]$3

and measures global directionality through trophic incoherence

$\psi[f]$4

Low-trophic-level nodes in coherent directed networks can reach the most others, strongly shape majority-vote and voter outcomes, influence synchronized frequency in directed Kuramoto dynamics, and determine successful strategies in generalized rock-paper-scissors games. The corresponding notion of influenceability is therefore mediated by global directionality as well as by local hierarchy (Rodgers et al., 2022).

In signed networks, DIC can be defined through the ability to steer the outcome of structural-balance dynamics. For the nonlinear model

$\psi[f]$5

the asymptotic sign pattern is determined by the dominant eigenvector of the initial friendliness matrix. A single agent can force any desired structurally balanced state by perturbing only its own row and column at $\psi[f]$6. This leads to the Structural Balance Influence Index

$\psi[f]$7

which measures the magnitude of local perturbation needed for agent $\psi[f]$8 to realize a desired balanced sign pattern $\psi[f]$9. Smaller $$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$0 means greater influence over the dynamic emergence of factions (Summers et al., 2013).

6. Computation, applications, and limitations

Algorithmically, DIC spans several complexity regimes. Dynamic centrality based on time-respecting paths admits a dynamic programming implementation with total time $$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$1 over $$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$2 time steps in a sparse representation (Lerman et al., 2010). Influence-based centralities derived from triggering models admit reverse-reachable-set approximations: for SNI,

$$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$3

and for Shapley centrality,

$$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$4

yielding scalable algorithms with near-linear dependence on graph size under standard assumptions (Chen et al., 2016). The broader stochastic sphere-of-influence family also admits efficient approximation through RR-set sampling once $$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$5 is additive (Chen et al., 2018). Dynamic CON computation proceeds round by round by building the current adjacency matrix and its square (Bonato et al., 31 Jan 2025). The misinformation DIC update has time complexity $$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$6 for $$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$7 iterations and $$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$8 edges (Sikosana et al., 11 Jul 2025). On ultrametric river trees, the CTMC-based dynamic centrality

$$\psi_v^{\mathrm{SNI}}(\mathcal I)=\sigma_{\mathcal I}(\{v\}),$$9

admits a fully explicit closed form computable in $$DC_i(\alpha,\gamma,\Delta_{1,n})=\sum_j RC^d_{ij}(\alpha,\gamma,\Delta_{1,n})$$0 total time from the tree structure alone (Ledezma, 5 May 2026).

Applications are correspondingly broad. Dynamic centrality has been used on an arXiv high-energy theory citation network to identify influential papers and latent precursors in citation chains (Lerman et al., 2010). Community-aware temporal centrality has been used on polarized Twitter data around the Irish abortion referendum (Pena et al., 23 Jul 2025). CON-based dynamic competition centralities have been tested on Survivor, conflict networks, food webs, Chess.com, and Dota 2 (Bonato et al., 2019, Bonato et al., 31 Jan 2025). Signed-dynamics centrality has been illustrated on UN General Assembly voting data from 1946 to 2008 (Summers et al., 2013). CTMC-based dynamic centrality has been applied to 49 natural river basins across the United States