Influence Matrix: Concepts & Applications

Updated 5 July 2026

Influence matrix is a matrix-valued operator that summarizes conditional effect structures between entities across varied domains.
It is employed in diverse methodologies, including per-client effects in personalized federated learning, gradient-based sample influence, and temporal tensor representations in quantum dynamics.
The concept bridges raw interaction data to actionable model insights, enabling optimized predictions in network analysis, recommendation systems, and control theory.

Searching arXiv for recent and relevant papers on “influence matrix” and closely related formulations across domains. Search results reviewed. I will ground the article primarily in the supplied arXiv papers and cite them directly, with additional cross-domain context only where the supplied records support it. An influence matrix is a matrix-valued representation of how entities, states, subsystems, or trajectories affect one another, but the term does not denote a single canonical object across disciplines. In personalized federated learning it can be a per-client, class-wise matrix that measures how much one client benefits another on a specific class (Tan et al., 2024); in recommender systems it can be a trust adjacency or its Laplacian acting on latent user factors (Aravkin et al., 2016); in open and many-body quantum dynamics it can be a temporal tensor or matrix-product-state object encoding the effect of an environment on a local subsystem (Lerose et al., 2020); and in reduced Google matrix analysis it can be the compressed operator that concentrates direct and indirect pathways among a selected set of countries while preserving their PageRank probabilities (Zant et al., 2018). The common thread is not a fixed algebraic form but a fixed role: an influence matrix is an operator that summarizes conditional effect structure at the granularity relevant to a model.

1. Conceptual scope and formal variants

Across the cited literature, “influence matrix” spans several mathematically distinct objects. Some formulations are pairwise and static, some are time-indexed, some are per-target rather than global, and some live on a doubled or folded time contour rather than on a graph of agents.

Domain	Matrix object	Encoded influence
Personalized federated learning	$\mathbf{\Lambda}_m \in \mathbb{R}^{M \times C}$	Client $i$ ’s effect on client $m$ for class $c$ (Tan et al., 2024)
Social recommendation	$W_t \in \mathbb{R}^{m \times m}$ , $L_t=D_t-W_t$	Trust-based coupling of user latent factors (Aravkin et al., 2016)
Sample influence in deep learning	$M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$	Pairwise training/test influence surrogate (Yang et al., 2024)
Randomized opinion dynamics	$H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$	Mapping from prejudices to steady-state mean opinions (Ravazzi et al., 2018)
Mutual influence regression	$B_t=\sum_{k=1}^d \lambda_k W_k(t)$	Time-varying mutual influence among actors (Fan et al., 2022)
Many-body Floquet dynamics	temporal IM tensor/MPS	Environment influence on subsystem trajectories (Lerose et al., 2020)
Geopolitical/trade REGOMAX	$G_R$ and $i$ 0	Direct and hidden indirect influence among selected countries (Zant et al., 2018, Coquidé et al., 2019)

This heterogeneity has immediate formal consequences. Symmetry may or may not hold. In FedC $i$ 1I, the influence matrix is defined per client and is direction-specific; symmetry is not imposed (Tan et al., 2024). In Hessian-free sample influence matrices based on gradient dot products, symmetry follows when both entries are computed at the same parameters (Yang et al., 2024). In homophily-based opinion dynamics, the influence matrix is signed and symmetric with entries in $i$ 2 rather than nonnegative and row-stochastic (Disarò et al., 2023). In reduced Google matrix analysis, the relevant operator is column-stochastic and Perron–Frobenius by construction (Zant et al., 2018).

A second distinction concerns what the matrix indexes. Several models remain actor-by-actor, but others are actor-by-time or class-by-client. The participant-invariant framework defines $i$ 3 as a participant–time matrix of normalized influence increments and then aligns rows by time since first appearance to obtain $i$ 4 (Clark et al., 2023). The teamwork context matrix is the transition matrix in a linear dynamical system, with entries specifying how much each individual’s current behavior is attributable to their own versus every other group member’s past behaviors (Lee et al., 10 Sep 2025). In quantum settings, the influence matrix is instead a tensor over time indices in folded Keldysh space and is naturally represented as an MPS rather than as an adjacency-like array (Lerose et al., 2020).

2. Learning-theoretic and data-driven formulations

The most explicit supervised-learning use of the term appears in influence-oriented personalized federated learning. FedC $i$ 5I defines, for the $i$ 6-th client, an influence matrix $i$ 7 with entries

$i$ 8

where $i$ 9 represents the influence brought by the $m$ 0-th client regarding the $m$ 1-th class, estimated through a leave-one-out loss on a random local batch sampled from $m$ 2 (Tan et al., 2024). This matrix is computed locally at each round, complements the client-level influence vector $m$ 3, and personalizes class-wise classifier aggregation while $m$ 4 personalizes feature-representation aggregation. The paper states that “the model aggregation process is moved from the server side to the client side,” and reports average accuracies of $m$ 5 on Digit-5 and $m$ 6 on Office-10, exceeding FedAvg, FedProx, FedRep, FedRoD, and FedProto under the reported non-IID setups (Tan et al., 2024).

A different learning-theoretic construction appears in Hessian-free influence analysis. There the influence matrix is typically pairwise over examples, with the classical form

$m$ 7

and the Hessian-free surrogate

$m$ 8

or its checkpointed TracIn variant (Yang et al., 2024). The paper emphasizes that dot-product matrices are symmetric when computed at the same parameters, are far cheaper than Hessian-based approximations, and can be used for noisy-label detection, sample selection for LLM fine-tuning, fairness, and robustness. In the reported noisy-label experiments, removing only $m$ 9 most detrimental samples yielded average accuracy $c$ 0 for IP Ensemble, above Self-TracIn at $c$ 1, LiSSA at $c$ 2, TracIn at $c$ 3, DataInf at $c$ 4, EKFAC at $c$ 5, and a vanilla cross-entropy baseline at $c$ 6 (Yang et al., 2024).

Temporal graph learning supplies a third data-driven meaning. TempNodeEmb defines a temporal edge influence matrix

$c$ 7

which rescales adjacency and self-loops by an exponential recency factor and, according to the paper, “normalizes” entries to lie in $c$ 8 while avoiding degree-based normalization (Abbas et al., 2020). This matrix drives a three-layer graph convolution at each time step, after which node orientations are aligned across time by a Given’s-angle method and QR decomposition. On the reported benchmarks, TempNodeEmb achieves ROC $c$ 9 and PRAUC $W_t \in \mathbb{R}^{m \times m}$ 0 on PPI, and ROC $W_t \in \mathbb{R}^{m \times m}$ 1 and PRAUC $W_t \in \mathbb{R}^{m \times m}$ 2 on COLLMsg; DeepWalk is slightly higher on MITC with ROC $W_t \in \mathbb{R}^{m \times m}$ 3 and PRAUC $W_t \in \mathbb{R}^{m \times m}$ 4 compared with TempNodeEmb’s ROC $W_t \in \mathbb{R}^{m \times m}$ 5 and PRAUC $W_t \in \mathbb{R}^{m \times m}$ 6 (Abbas et al., 2020).

These examples already show that an influence matrix in machine learning may be local rather than global, benefit-oriented rather than causal, and either loss-based, gradient-based, or recency-weighted. This suggests that the phrase denotes a modeling interface between raw interaction data and downstream optimization, rather than a single statistical estimator.

In networked opinion and recommendation models, the matrix usually indexes actors directly, but its semantics vary sharply. In dynamic matrix factorization with social influence, $W_t \in \mathbb{R}^{m \times m}$ 7 is a sparse trust adjacency, $W_t \in \mathbb{R}^{m \times m}$ 8 is the unnormalized graph Laplacian, and the regularizer

$W_t \in \mathbb{R}^{m \times m}$ 9

encourages socially connected users to have similar latent profiles (Aravkin et al., 2016). The same paper embeds this term in a state-space smoothing objective for time-evolving matrix factorization and reports a best RMSE of $L_t=D_t-W_t$ 0 with $L_t=D_t-W_t$ 1 and $L_t=D_t-W_t$ 2 on Epinions, compared with the best static RMSE $L_t=D_t-W_t$ 3 at $L_t=D_t-W_t$ 4 (Aravkin et al., 2016). Here the influence matrix is not learned as a free parameter; it is constructed from time-stamped trust links and acts through Laplacian smoothing.

Randomized Friedkin–Johnsen gossip dynamics distinguishes between an inter-agent influence matrix and a steady-state influence mapping. The expected dynamics uses

$L_t=D_t-W_t$ 5

while the matrix

$L_t=D_t-W_t$ 6

maps prejudices $L_t=D_t-W_t$ 7 to steady-state mean opinions, $L_t=D_t-W_t$ 8 in expectation (Ravazzi et al., 2018). The same work shows how to estimate the expected transition matrix $L_t=D_t-W_t$ 9 and then recover topology, $M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 0, and $M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 1 from partial observations via Yule–Walker-type identities. In the reported experiments with random networks of size $M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 2 and degree $M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 3, about $M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 4 samples suffice for near-perfect topology recovery under full observations, and the sparse estimator remains resilient for intermittent observation probabilities $M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 5 (Ravazzi et al., 2018).

A homophily-based extension of Friedkin–Johnsen changes the picture further by allowing signed appraisals. Its time-varying influence matrix is

$M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 6

symmetric and valued in $M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 7 (Disarò et al., 2023). The model always asymptotically converges to a constant solution, and in the single-topic case the asymptotic behavior is obtained in closed form. The paper states that $M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 8 for all $M_{ij}^{\mathrm{IP}}=\langle \nabla L_i,\nabla L_j\rangle$ 9, and either $H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 0 has an eigenvalue equal to $H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 1 and the signed graph is structurally balanced, or $H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 2 is Schur stable (Disarò et al., 2023).

Econometric formulations make the matrix itself the object of regression. In the Mutual Influence Regression model of Fan, Lan, Zou and Tsai, actors obey

$H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 3

with

$H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 4

where the $H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 5 are row-normalized similarity matrices derived from observed attributes (Fan et al., 2022). The model generalizes a spatial autoregressive specification by allowing multiple time-varying similarity matrices, establishes QMLE, introduces an EBIC-type criterion for selecting relevant matrices, and proposes an adequacy test for the influence structure. In the mutual-fund application summarized in the data, the adequacy test yields $H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 6-value $H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 7, and Return, Size, Age, and Volatility are significant and positive, while Alpha is positive but not significant (Fan et al., 2022).

Other social formulations relax adjacency semantics altogether. The participant-invariant framework defines an influence matrix $H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 8 whose entries are normalized influence increments and uses rank-1 NMF,

$H=(I-\Lambda \overline W)^{-1}(I-\Lambda)$ 9

to extract a shared temporal pattern $B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 0 (Clark et al., 2023). On 28 temporal networks, the paper reports reconstruction tolerance below $B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 1 under Frobenius norm and near-identical $B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 2 across random participant subsets in many datasets (Clark et al., 2023). Closely related in spirit, the teamwork context matrix is the transition matrix in a linear dynamical system for group behavior, and its dyadic summary features—Relative Influence, Leader Strength, and Leader Switch Rate—differentiate task contexts and predict accuracy in human eye-tracking experiments (Lee et al., 10 Sep 2025).

4. Quantum, open-system, and nonequilibrium influence matrices

In quantum many-body theory, the influence matrix is often the discrete-time analogue of the Feynman–Vernon influence functional. For interacting Floquet spin chains, the influence matrix is a tensor over the time indices of folded Keldysh degrees of freedom and describes the effect of the system on the dynamics of a local subsystem (Lerose et al., 2020). In translationally invariant settings it becomes the right eigenvector of a dual transfer matrix, admits a self-consistency equation, and at perfect dephaser points reduces to

$B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 3

which the paper interprets as a perfectly Markovian bath acting on itself (Lerose et al., 2020). Near those points, temporal entanglement remains low enough that the influence matrix can be represented efficiently by MPS methods.

The Rule 201 Floquet-PXP work develops the same language in a more algorithmic form. There the influence matrix is the leading eigenvector of the spatial transfer matrix in the folded picture, represented as a temporal MPS with alternating local tensors, and exact solvability is enforced by generalized zipper conditions (Yang et al., 17 Jun 2026). The paper also introduces a numerical bootstrap method that reconstructs exact finite-bond-dimension IMs from finite-time light-cone data, reports exact IMs with bond dimensions up to $B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 4 for certain initial states, and uses the resulting object to analyze long-time local dynamics, entanglement growth, and hidden Markov order (Yang et al., 17 Jun 2026). For a global quench with small $B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 5, it obtains the linear growth

$B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 6

for the second Rényi entropy (Yang et al., 17 Jun 2026).

A closely related program compresses the Feynman–Vernon influence functional directly as matrix product states. In IF-MPS DMFT, the discrete-time influence matrix $B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 7 is assembled from the hybridization function and defines a Gaussian IF state

$B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 8

which is then converted to an MPS and contracted with a local impurity MPO (Nayak et al., 4 Mar 2025). The method yields numerically exact descriptions of metallic states without sharp spectral features at moderate numerical cost, while low-temperature Mott insulators and systems with narrow quasiparticle or spin-polaron peaks are substantially more challenging because they require long time contours or high bond dimensions (Nayak et al., 4 Mar 2025).

The TEMPO literature addresses the same compression problem from another angle. The paper on efficient construction of the Feynman–Vernon influence functional as MPS exploits time-translational invariance of the discrete kernel $B_t=\sum_{k=1}^d \lambda_k W_k(t)$ 9, fits the resulting TTI kernels to sums of exponentials by the Prony method, and replaces the original linearly scaling sequence of GMPS multiplications by a construction whose required number of multiplications is almost independent of total evolution time (Guo et al., 2024). In the reported benchmarks, the TTI construction gives absolute errors around $G_R$ 0 in the Toulouse model while runtime scales roughly linearly with total time rather than quadratically (Guo et al., 2024).

A plausible implication is that, in quantum settings, the phrase “influence matrix” most naturally denotes a multitime kernel or transfer object rather than an adjacency-like table. The matrix language survives because all bath effects are condensed into an operator indexed by discrete contour times.

5. Network, ranking, trade, and control-theoretic influence matrices

In complex-network optimization, the influence matrix can be a stability operator. The optimal-percolation formulation defines a modified non-backtracking operator

$G_R$ 1

whose largest eigenvalue governs the stability of the zero-giant-component solution (Morone et al., 2015). Minimizing $G_R$ 2 over node removals yields the optimal influencer set, and the leading-order expansion produces the Collective Influence score

$G_R$ 3

On an ER graph with $G_R$ 4 and $G_R$ 5, the extrapolated optimal threshold is $G_R$ 6, and on the Twitter mention/retweet network with $G_R$ 7 and $G_R$ 8, CI at $G_R$ 9 finds a dismantling set yielding $i$ 00 while HDA, PR, HD, and $i$ 01-core still give $i$ 02– $i$ 03 at the same $i$ 04 (Morone et al., 2015).

Reduced Google matrix theory defines another influential operator by compressing a large directed network to a selected node set. If the global Google matrix is partitioned into selected nodes $i$ 05 and the complement $i$ 06, the reduced Google matrix is

$i$ 07

with the canonical decomposition

$i$ 08

where $i$ 09 collects direct links, $i$ 10 is a PageRank projector term, and $i$ 11 encodes indirect hidden links through the complement (Zant et al., 2018). For the top-40-country set in English Wikipedia, the paper reports weights $i$ 12, $i$ 13, and $i$ 14; for EU-27, $i$ 15, $i$ 16, and $i$ 17 (Zant et al., 2018). The associated sensitivity analysis perturbs a bilateral tie and measures the logarithmic response of PageRank probabilities through

$i$ 18

revealing edition-dependent but geopolitically meaningful influence patterns (Zant et al., 2018).

The same REGOMAX formalism is used for multiproduct world trade. In that setting, the non-diagonal indirect component

$i$ 19

acts as the operative influence matrix because it quantifies hidden pairwise influence through the global trade network (Coquidé et al., 2019). The paper studies EU sensitivity of trade balance to petroleum and gas price increases from Russia, USA, Saudi Arabia, and Norway. For petroleum in 2016, REGOMAX identifies the Netherlands as the most negatively affected EU country for Russian petroleum, whereas conventional Import–Export analysis places Latvia and Lithuania first and treats Western Europe as almost insensitive (Coquidé et al., 2019). The reported maximal negative sensitivities to Russian petroleum over the sampled years are $i$ 20 in 2004, $i$ 21 in 2008, $i$ 22 in 2012, and $i$ 23 in 2016 (Coquidé et al., 2019).

Control theory introduces yet another semantics. In the discrete-time MIMO ultra-local model, the input influence matrix $i$ 24 is a user-designed matrix that injects the control input into the ultra-local model,

$i$ 25

and is compared multiplicatively with the true plant input matrix through

$i$ 26

(Teng et al., 2022). The local stability analysis reduces to a scalar quadratic for each eigenvalue $i$ 27,

$i$ 28

at the origin, and in the scalar scaling case $i$ 29 the paper states that $i$ 30 places both roots strictly inside the unit circle, whereas $i$ 31 makes one root leave the unit circle (Teng et al., 2022). Here the influence matrix is not estimated from data; it is a design parameter whose mismatch with the plant governs the coupled tracking and estimation error dynamics.

6. Recurrent properties, common misconceptions, and limitations

A recurring misconception is that an influence matrix must be a nonnegative, symmetric, actor-by-actor matrix. The surveyed literature does not support that restriction. FedC $i$ 32I uses a per-client, direction-specific matrix $i$ 33 rather than a single global matrix (Tan et al., 2024). Homophily-based Friedkin–Johnsen uses a signed symmetric matrix with entries in $i$ 34 (Disarò et al., 2023). The reduced Google matrix is column-stochastic and primarily meaningful through its decomposition into direct, projector, and hidden-link terms (Zant et al., 2018). Quantum influence matrices are temporal process tensors or MPS objects, not adjacency operators at all (Lerose et al., 2020).

A second misconception is that influence matrices always encode observed pairwise relations directly. Many of the cited constructions are indirect or counterfactual. FedC $i$ 35I uses leave-one-out local losses and does not use class distributions, gradients, prototypes, or confusion statistics (Tan et al., 2024). Hessian-free sample influence uses gradient alignments rather than retraining (Yang et al., 2024). Dynamic matrix factorization with social influence regularizes latent trajectories through a Laplacian instead of learning edge weights (Aravkin et al., 2016). REGOMAX condenses all indirect pathways through a resolvent term $i$ 36 (Zant et al., 2018). Quantum formulations integrate out an entire environment and encode multitime memory in a folded-time kernel (Guo et al., 2024, Nayak et al., 4 Mar 2025).

Limitations are equally domain-specific. FedC $i$ 37I provides no convergence analysis or theoretical guarantees specific to its influence matrix (Tan et al., 2024). Dynamic matrix factorization with social influence fixes $i$ 38 from trust data and does not learn influence strengths (Aravkin et al., 2016). MIR depends on the construction of similarity matrices and on identifiability conditions such as $i$ 39 (Fan et al., 2022). TempNodeEmb uses a fixed exponential recency factor and explicitly omits degree normalization (Abbas et al., 2020). IF-MPS and related quantum methods become costly when long memory or sharp low-energy features require large bond dimensions or long time contours (Nayak et al., 4 Mar 2025, Guo et al., 2024).

Taken together, these formulations support a broad encyclopedia-level conclusion. “Influence matrix” is a family resemblance term for operators that compress how perturbations, interactions, or histories propagate through a modeled system. What unifies the family is the attempt to replace raw heterogeneous interactions by a matrix-structured object that is computable, compositional, and directly usable in inference, optimization, or dynamical prediction. What distinguishes members of the family are their index sets, normalization rules, symmetry properties, and the specific notion of influence each field chooses to formalize.