---
title: Dynamic Influence Centrality
url: https://www.emergentmind.com/topics/dynamic-influence-centrality-dic
type: topic
---

# Dynamic Influence Centrality

Searching arXiv for recent 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 [2507.09055] [1006.0526] [1002.4042].

## 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 [1810.09981].

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 [1005.4882].

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 [1602.03780].

## 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 | \(DC_i(\alpha,\gamma,\Delta_{1,n})=\sum_j RC^d_{ij}(\alpha,\gamma,\Delta_{1,n})\) [1006.0526] | Total influence via attenuated time-respecting paths |
| Friedkin–Johnsen opinions | \(\mathbf c=\frac{P^\top \mathbb 1_n}{n}\), \(P=\bigl(I-(I-\beta)W\bigr)^{-1}\beta\) [2412.20112] | Average contribution of each initial opinion to final opinions |
| Dynamical influence | \(\dot x=Mx\), \(\text{DIC}_i=c_i\), \(cM=0\) [1002.4042] | Effect of node \(i\)'s initial state on the dominant collective mode |
| Iterative OSN DIC | \(\text{DIC}_{t+1}(v_i)=\text{DIC}_t(v_i)+\sum_{u\in N_{\mathrm{in}}(v_i)}\text{DIC}_t(u)\), with \(\text{DIC}_0(v_i)=1\) [2507.09055] | Temporal accumulation of structural influence over repeated rounds |
| Competition CON | \(\mathrm{CON}(u,v)=\sum_k \min(A_{uk},A_{vk})\), \(\mathrm{CON}(u)=\sum_v \mathrm{CON}(u,v)\) [1909.06810] | Shared competitive influence through common out-neighbors |
| Minimum-energy control | \(V_c(i)=\left\|x_{fi}-\frac{c}{n}\mathbf 1\right\|_2\), \(x_{fi}=\frac{c}{t_f}\int_0^{t_f} e^{-L\tau}e_i\,d\tau\) [2511.00339] | 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 \(i\) to \(j\) is only valid if its edges appear in the correct temporal order. The memoryless dynamic centrality matrix is
\[
C^d_{t_1\to t_n}(\beta,\alpha)
=
\beta A(t_1)+\beta\alpha A(t_1)A(t_2)+\cdots+\beta\alpha^{n-1}A(t_1)\cdots A(t_n),
\]
and the retained version introduces a memory parameter \(\gamma\) through retained adjacency matrices \(R(t,\gamma)\). The resulting node-level centrality
\[
DC_i(\alpha,\gamma,\Delta_{1,n})=\sum_j RC^d_{ij}(\alpha,\gamma,\Delta_{1,n})
\]
measures expected total information sent by \(i\) 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 [1006.0526].

A second line treats DIC as **influence centrality under changing topology**. In the Friedkin–Johnsen model,
\[
\mathbf x(k+1)=(I-\beta)W\,\mathbf x(k)+\beta\,\mathbf x(0),
\qquad
\mathbf x_f=P\mathbf x(0),
\]
with
\[
P=\bigl(I-(I-\beta)W\bigr)^{-1}\beta,
\qquad
\mathbf c=\frac{P^\top \mathbb 1_n}{n}.
\]
Here \(c_i\) is the average contribution of node \(i\)'s initial opinion to final opinions, and in the two-influencer case the relevant scores satisfy \(c_p+c_q=1\). Edge modifications of type \((a,b,d)\) preserve row-stochasticity by adding \((a,b)\) and decreasing \((d,b)\). Signal flow graph analysis yields two structural results: some modifications are **redundant** and leave \(\mathbf c\) 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 [2412.20112].

A third temporal line makes DIC **community-aware** in polarized networks. Temporal degree and temporal closeness are defined on a time-expanded adjacency \(\mathbb M\), eigenvector-based temporal scores are obtained from a supra-centrality matrix \(\mathbb C(\varepsilon)\), temporal Katz centrality uses
\[
\mathcal Q=(I-\alpha A^{(1)})^{-1}(I-\alpha A^{(2)})^{-1}\cdots(I-\alpha A^{(t)})^{-1},
\]
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 [2507.17177].

## 4. Diffusion, competition, and misinformation

A central DIC interpretation comes from **dynamical influence** in linearized spreading and consensus dynamics. For
\[
\dot x=Mx,
\]
if the dominant eigenvalue is \(0\) and \(cM=0\), then the asymptotic state depends on the initial condition only through \(c\cdot x(0)\), and \(c_i\) quantifies how strongly node \(i\)'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 [1002.4042].

A closely related spreading-specific formulation is **dynamics-sensitive centrality** for SIR and SI models. With
\[
\mathbf H=\beta \mathbf A + (1-\mu)\mathbf I,
\qquad
\mathbf S(t)=\sum_{r=0}^{t-1}\beta \mathbf A \mathbf H^r L,
\]
the \(i\)-th component \(S_i(t)\) approximates the spreading influence of node \(i\) at time \(t\). For SIR with \(\mu=1\),
\[
\mathbf S(t)=\beta\mathbf A L+\beta^2\mathbf A^2L+\cdots+\beta^t\mathbf A^tL.
\]
This interpolates between degree centrality at \(t=1\) and eigenvector centrality when \(\beta\) and \(t\) are large, and empirically outperforms degree, \(k\)-shell, and eigenvector centrality for identifying influential spreaders on several real networks [1504.06672].

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 [1005.4882].

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,
\[
\mathrm{CON}(u,v)=\sum_k \min(A_{uk},A_{vk}),
\qquad
\mathrm{CON}(u)=\sum_v \mathrm{CON}(u,v),
\]
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
\[
A_2=A+A^2,
\]
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 [1909.06810] [2501.19220].

An explicit use of the name **Dynamic Influence Centrality** appears in health misinformation analysis on online social networks. There each node starts with
\[
\mathrm{DIC}_0(v_i)=1,
\]
and influence is iteratively accumulated by
\[
\mathrm{DIC}_{t+1}(v_i)=\mathrm{DIC}_t(v_i)+\sum_{u\in N_{\mathrm{in}}(v_i)} \mathrm{DIC}_t(u).
\]
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% [2507.09055].

## 5. Control, directionality, and signed dynamics

Several DIC formulations are explicitly **control-theoretic**. Katz centrality can be reinterpreted as the steady state of
\[
\dot{\mathbf x}=\mathbf 1+aA\mathbf x-\mathbf x,
\qquad
\mathbf r=(I-aA)^{-1}\mathbf 1.
\]
A node-specific perturbation analysis with decay vector \(\boldsymbol\gamma\) yields an influence matrix
\[
C=MR,
\qquad
M=(I-aA)^{-1},
\qquad
R=\mathrm{diag}(r_1,\ldots,r_n),
\]
and a global impact approximation
\[
\boldsymbol\sigma=\mathbf r\circ \mathbf s,
\qquad
\mathbf s=M^\top \mathbf 1.
\]
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
\[
\dot x(t)=-Lx(t)+Bu(t),
\]
with single-node control \(B=e_i\), the terminal state that achieves aggregate threshold \(c\) at horizon \(t_f\) is
\[
x_{fi}=\frac{c}{t_f}\int_0^{t_f}e^{-L\tau}e_i\,d\tau,
\]
and the centrality score is
\[
V_c(i)=\left\|x_{fi}-\frac{c}{n}\mathbf 1\right\|_2.
\]
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 [2511.00339].

Dynamic influence can also depend on **global directionality**. Trophic analysis assigns trophic levels \(h\) through
\[
\Lambda h=v,
\qquad
\Lambda=\mathrm{diag}(u)-A-A^\top,
\qquad
v_i=k_i^{\mathrm{in}}-k_i^{\mathrm{out}},
\]
and measures global directionality through trophic incoherence
\[
F=\frac{\sum_{i,j}A_{ij}(h_j-h_i-1)^2}{\sum_{i,j}A_{ij}}.
\]
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 [2210.12081].

In **signed networks**, DIC can be defined through the ability to steer the outcome of structural-balance dynamics. For the nonlinear model
\[
\dot X=X^2,
\]
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 \(t=0\). This leads to the **Structural Balance Influence Index**
\[
SBII_{v^*}(i)=\left\|V^{-1}\left(\lambda_1(X_i)I-X_i\right)\hat v^*\right\|,
\]
which measures the magnitude of local perturbation needed for agent \(i\) to realize a desired balanced sign pattern \(v^*\). Smaller \(SBII\) means greater influence over the dynamic emergence of factions [1306.5338].

## 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 \(O(n|E|)\) over \(n\) time steps in a sparse representation [1006.0526]. Influence-based centralities derived from triggering models admit reverse-reachable-set approximations: for SNI,
\[
\psi_u^{\mathrm{SNI}}=n\cdot\mathbb E_R[I\{u\in R\}],
\]
and for Shapley centrality,
\[
\psi_u^{\mathrm{Shapley}}=n\cdot\mathbb E_R\!\left[\frac{I\{u\in R\}}{|R|}\right],
\]
yielding scalable algorithms with near-linear dependence on graph size under standard assumptions [1602.03780]. The broader stochastic sphere-of-influence family also admits efficient approximation through RR-set sampling once \(f\) is additive [1810.09981]. Dynamic CON computation proceeds round by round by building the current adjacency matrix and its square [2501.19220]. The misinformation DIC update has time complexity \(O(Tm)\) for \(T\) iterations and \(m\) edges [2507.09055]. On ultrametric river trees, the CTMC-based dynamic centrality
\[
C_{\mathrm{CTMC}}(i)=\left(\int_0^\infty (H_t(i,i)-1)\,dt\right)^{-1}
\]
admits a fully explicit closed form computable in \(O(n)\) total time from the tree structure alone [2605.16328].

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 [1006.0526]. Community-aware temporal centrality has been used on polarized Twitter data around the Irish abortion referendum [2507.17177]. CON-based dynamic competition centralities have been tested on Survivor, conflict networks, food webs, Chess.com, and Dota 2 [1909.06810] [2501.19220]. Signed-dynamics centrality has been illustrated on UN General Assembly voting data from 1946 to 2008 [1306.5338]. CTMC-based dynamic centrality has been applied to 49 natural river basins across the United States

Source: https://www.emergentmind.com/topics/dynamic-influence-centrality-dic