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arxiv: 2604.23034 · v1 · submitted 2026-04-24 · 💻 cs.SI

The Decay of Impact with Network Distance in Linear Diffusion Processes

Alexander Murray-Watters , Cheng Wang , John R. Hipp , Cynthia Lakon , Carter T. Butts This is my paper

Pith reviewed 2026-05-08 08:55 UTC · model grok-4.3

classification 💻 cs.SI
keywords social networkslinear diffusioneigenvector centralityimpact decaynetwork distanceadjacency spectrumsocial influence
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The pith

Total impact between nodes in linear diffusion decays exponentially with network distance, approximated by eigenvector centrality product and spectral terms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an approximate expression for the total influence one node exerts on another in forced linear diffusion models on networks. This expression shows the impact as primarily a product of the nodes' eigenvector centralities times a factor that decreases exponentially with their distance, derived from the eigenvalues of the adjacency matrix. Sympathetic readers would care because models like social influence or status scoring rely on such impacts, yet calculating exact totals from all paths is complex. Numerical checks on school networks confirm the average exponential decay and the utility of the first-order approximation. The result allows simpler estimation of influences in social processes.

Core claim

Here, we provide an approximate solution for the total impact of one node on another as a function of network distance, showing that the total impact is given to first order by a product of eigenvector centrality scores together with an expression in terms of the graph spectrum (eigenvalues of the adjacency matrix) that falls exponentially with distance. We also show how this solution can be refined using higher-order eigenvectors of the adjacency matrix. A numerical study on interpersonal networks drawn from educational settings verifies an average exponential decline in impact strength under the linear diffusion model, and shows that the first-order eigenvector approximation can often be a

What carries the argument

The first-order term from the leading eigenvector of the adjacency matrix paired with the dominant eigenvalue, which together produce the exponential decay in summed path contributions.

If this is right

  • Total impact falls exponentially with distance on average in linear diffusion processes.
  • The product of eigenvector centrality scores serves as a practical proxy for pairwise total impact.
  • Higher-order eigenvectors of the adjacency matrix can be used to refine the first-order approximation.
  • This yields a simple model for estimating influence or status effects in social networks without full path summation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approximation may enable faster influence calculations on large networks by avoiding repeated full solutions of the diffusion equation.
  • Local interaction models could be justified when distant impacts become negligible due to the exponential term.
  • Similar spectral approximations might extend to directed networks or those with weighted edges if the adjacency structure permits.

Load-bearing premise

That the first-order term involving only the leading eigenvector and dominant spectral contribution provides a good proxy for total impact across graphs and diffusion parameters in social networks.

What would settle it

A network where exact total impacts computed from the diffusion model show no exponential decay with distance or deviate strongly from the predicted product of eigenvector centralities.

Figures

Figures reproduced from arXiv: 2604.23034 by Alexander Murray-Watters, Carter T. Butts, Cheng Wang, Cynthia Lakon, John R. Hipp.

Figure 1
Figure 1. Figure 1: Network diffusion under the FLDM. (A) Impact of node view at source ↗
Figure 2
Figure 2. Figure 2: Mean total impact as a function of distance from ego, by network. Left-hand panel view at source ↗
Figure 3
Figure 3. Figure 3: Level of correlation between the first order approximation and the true geodesic view at source ↗
Figure 4
Figure 4. Figure 4: Level of correlation between the first order approximation and the true geodesic view at source ↗
Figure 5
Figure 5. Figure 5: Correlation for first and second order eigenvectors as decay increases. Note that view at source ↗
read the original abstract

Many processes related to status, power, and influence within social networks have been modeled using forced linear diffusion models; examples include the highly successful Friedkin-Johnsen model of social influence, the status/power scores of Katz and Bonacich, and the widely used network autocorrelation model. While a basic assumption of such models is that the impact of one individual on another through any given path falls exponentially with path length, the total impact of the first individual on the second involves contributions from walks of all lengths; thus, while total impact is expected to decline with network distance, the relationship is not trivial. Here, we provide an approximate solution for the total impact of one node on another as a function of network distance, showing that the total impact is given to first order by a product of eigenvector centrality scores together with an expression in terms of the graph spectrum (eigenvalues of the adjacency matrix) that falls exponentially with distance. We also show how this solution can be refined using higher-order eigenvectors of the adjacency matrix. A numerical study on interpersonal networks drawn from educational settings verifies an average exponential decline in impact strength under the linear diffusion model, and shows that the first-order eigenvector approximation can often be a good proxy for total impact as obtained from the exact solution. This suggests a simple model that can be used to approximate total impact for social influence or status processes in a range of settings.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript derives an approximate closed-form expression for the total impact of one node on another under linear diffusion processes on graphs. To first order, this impact is proportional to the product of the nodes' components in the leading eigenvector of the adjacency matrix (i.e., eigenvector centralities) multiplied by a spectral factor involving the dominant eigenvalue that decays exponentially with shortest-path distance. Higher-order corrections using additional eigenvectors are outlined, and numerical experiments on educational interpersonal networks are presented to show that impact declines exponentially with distance on average and that the first-order approximation often serves as a reasonable proxy for the exact impact obtained from matrix inversion.

Significance. If the approximation is accurate with controllable error, the work supplies a simple, interpretable link between pairwise impact, eigenvector centrality, and the graph spectrum, enabling efficient estimation of influence decay without full linear-system solves. This is relevant for models such as Friedkin-Johnsen, Katz-Bonacich status, and network autocorrelation. The numerical verification on real social networks adds practical value, though the result's scope is tied to the tested network class.

major comments (1)
  1. [§3] §3 (Derivation of the first-order approximation): The truncation of the spectral expansion of the resolvent (I − αA)^−1 after the leading term is presented without an explicit remainder bound in terms of |λ2/λ1| and the diffusion parameter α. This is load-bearing for the central claim that the leading term provides a good proxy, because the error can become non-negligible when the spectral gap is small, as noted in the stress-test concern; the manuscript supplies no quantitative condition under which the truncation error remains small uniformly in distance d.
minor comments (2)
  1. [Abstract] The abstract states that the total impact 'is given to first order by a product of eigenvector centrality scores together with an expression in terms of the graph spectrum' but does not display the explicit leading-order formula; inserting the concrete expression would improve immediate clarity.
  2. [§2] Notation for the impact function and the precise definition of network distance (shortest-path versus walk-based) should be introduced consistently in §2 before the derivation begins.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying a key point that can strengthen the rigor of our first-order approximation. We address the major comment below and will revise the manuscript to incorporate an explicit error bound.

read point-by-point responses

  1. Referee: [§3] §3 (Derivation of the first-order approximation): The truncation of the spectral expansion of the resolvent (I − αA)^−1 after the leading term is presented without an explicit remainder bound in terms of |λ2/λ1| and the diffusion parameter α. This is load-bearing for the central claim that the leading term provides a good proxy, because the error can become non-negligible when the spectral gap is small, as noted in the stress-test concern; the manuscript supplies no quantitative condition under which the truncation error remains small uniformly in distance d.

    Authors: We agree that an explicit remainder bound is desirable to make the conditions for the approximation fully quantitative. In the revision we will add to §3 a derivation of the truncation error for the resolvent expansion, expressed in terms of the spectral gap ratio |λ2/λ1| and α (under the standing assumption that the graph is connected and α is below the reciprocal of the largest eigenvalue). The bound will be shown to be independent of distance d when the leading eigenvector components are bounded away from zero, and we will discuss the regime in which the error remains small uniformly in d. This addition will also clarify the stress-test scenarios. revision: yes

Circularity Check

0 steps flagged

Spectral truncation approximation derived directly from adjacency matrix eigenvalues without circular reduction

full rationale

The paper's core claim is an approximate closed-form expression for total impact obtained by truncating the spectral expansion of the resolvent (I - αA)^{-1} after the dominant eigenvector term. This is a direct application of standard linear algebra to the forced linear diffusion model and does not reduce to a fitted parameter, self-definition, or self-citation chain. The numerical study on educational networks is described as independent verification of the exponential decay pattern rather than the source of the formula. No load-bearing steps invoke prior author work as an unverified uniqueness theorem or rename an empirical pattern as a derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the spectral decomposition of the adjacency matrix for weighted walks and the premise that per-path impact decays exponentially with length; no free parameters or new entities appear in the abstract.

axioms (1)
  • domain assumption Impact of one individual on another through any given path falls exponentially with path length
    Explicitly identified in the abstract as a basic assumption of the forced linear diffusion models.

pith-pipeline@v0.9.0 · 9911 in / 1078 out tokens · 74349 ms · 2026-05-08T08:55:10.656708+00:00 · methodology

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Reference graph

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