Memory effects on link formation in temporal networks: A fractional calculus approach

F. Abbasi; F. Rabbani; G.R. Jafari; T. Khraisha

REVIEW 4 major objections 6 minor 1 cited by

Memory effects on link formation in temporal networks: A fractional calculus approach

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Memory in temporal networks makes node activity peak and then decline with age.

desk verdict The memoryless part is clean and correct, but the fractional-calculus derivation is broken, so the aging claim rests on air. read the letter →

arxiv 1908.01999 v2 pith:DAJLGEYC submitted 2019-08-06 physics.soc-ph

F. Rabbani , T. Khraisha , F. Abbasi , G.R. Jafari This is my paper · ORCID

classification physics.soc-ph

keywords memoryeffectstemporalnetworkslinkformationfractionalcalculusCaputoderivativepreferentialattachmentnodeactivityaging

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that long-term memory in temporal networks changes who receives new links: if nodes remember past interactions and favor them, a node's activity no longer grows without bound but rises to a peak and then declines. The authors build a preferential attachment model in which node activity follows an ordinary rate equation, then replace the instantaneous derivative with a fractional, memory-carrying derivative of order α<1. In the memoryless case α=1, activity grows as sqrt(t) early and t later, with a characteristic crossover time. With memory, numerical solution shows effective activity declining with node age, and degree distributions broadening so late nodes can also become hubs. The claimed consequence is that memory redistributes hub potential and makes network evolution depend on memory length.

What carries the argument

The central object is a Caputo-style fractional differential equation and its discrete predictor-corrector solution. The paper's memory mechanism is a power-law kernel κ(t−t')=(t−t')^{α−2}/Γ(α−1), inserted into the rate equation so that current node activity is a weighted convolution of all past activities, with the fractional order α controlling how far back memory reaches. The predictor-corrector scheme with coefficients b_n=((n+1)^α−n^α)/Γ(α+1) supplies the aging weights that make older events contribute less, and the numerical solution of that scheme produces the reported rise-then-fall activity curves.

What would settle it

Check whether the kernel (t−t')^{α−2}/Γ(α−1) is integrable on [t0,t] for α<1, or solve the same preferential-attachment equation using the standard Caputo derivative (with the first derivative of node activity inside the integrand) for α=0.5 and compare the resulting activity curve to the paper's Fig. 3.

Watch

Extended reading notes

Core claim

The central discovery is an aging effect: when link formation retains a power-law memory of past activity, the effective node activity k̄_i(t) reaches a maximum and then declines, so older nodes lose their attractiveness and receive fewer new connections. This is presented as a general property of memory in temporal networks, controlled by fractional order α: α=1 reproduces the ordinary preferential attachment growth, while smaller α (longer memory) lowers and delays the peak. Consequently, in memoryful networks high-degree nodes need not be early ones; the degree distribution becomes broader, and link weights and assortativity differ from the memoryless limit.

Load-bearing premise

The entire aging prediction depends on the claim that inserting a power-law memory kernel into the activity equation yields a valid fractional differential equation of the Caputo form; if that step is not mathematically well-defined, the rise-and-fall of node activity is not a consequence of the model.

Editorial extensions

If this is right

In the memoryless limit α=1, node activity grows as $\sqrt{t}$ at early times and as $t$ later, with a crossover at $t^* = c^2 N^2 / (4m^2)$.
For memory orders α<1, effective activity rises to a peak and then declines, so a node's probability of receiving new links falls as the node ages.
Longer memory (smaller α) reduces and delays the peak activity, slowing the growth of active nodes relative to the memoryless case.
Memory broadens the degree distribution away from the Gaussian form of the memoryless model, so nodes that join later can also become hubs.
Dense temporal networks exhibit the characteristic crossover and memory effects more clearly than sparse ones.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same fractional-memory attachment rule, applied to citation or recommender networks, would suppress rich-get-richer concentration and increase turnover among top nodes; the paper does not draw this implication.
One could fit α from real interaction logs by matching the empirically observed peak time of node activity to the discrete b_n solution, yielding a direct estimate of network memory length.
Because the memoryless model already has a crossover time, dense event streams should reveal aging effects earlier and more sharply than sparse streams, a testable prediction for empirical temporal networks.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know. First, the memoryless preferential-attachment model in Eqs. (1)-(3) is solved correctly, and the simulation matches the analytic curve in Fig. 2. That part is genuinely fine. Second, the fractional-calculus section that is supposed to add memory is mathematically flawed in ways that invalidate the central claim that memory causes node-activity aging.

What is actually new: the idea of coupling node-activity dynamics to a power-law memory kernel via fractional calculus is a reasonable thing to try, and the paper does make the memoryless limit alpha=1 reduce to the BA-like behavior. The qualitative observation that stronger memory (smaller alpha) leads to a peak-and-decline in effective node activity is interesting, if it were supported. But the derivation does not support it.

The soft spots are real and load-bearing. In Eq. (4), the kernel κ(t-t′) = (t-t′)^{α-2}/Γ(α-1) with 0 ≤ α < 1 is not integrable: the integral diverges at t′=t for every α in that range. So calling the right side a fractional integral of order α−1 is wrong; the order is negative, and the expression is undefined as written. Equation (7) for the Caputo derivative omits the derivative of y; the standard Caputo definition for 0<α<1 has y′(s) in the integrand, not y(s). Therefore the operator used to go from Eq. (4) to Eq. (6) is not the Caputo derivative, and the identity claimed does not hold. These are not typos. The entire aging prediction is justified by converting the memory-kernel equation into a Caputo differential equation, and that conversion is invalid.

One nuance: the predictor-corrector scheme in Eq. (9) is the standard discretization of a well-posed Caputo problem of the form ^C D^α k = f. If one simply posits that Caputo equation as an ad hoc model, the numerical solution might indeed show the reported peak-and-decline. But the paper does not derive that equation from the memory-kernel model, and the simulations are presented as validating a derivation that does not hold. There is also no empirical data, no code, and no parameter fitting; the degree-distribution analysis is qualitative. Citation to prior work, including the same group's Safdari et al., makes clear that the aging phenomenon is not new even if this specific fractional formulation is.

This paper is not ready for serious peer review as written. It has a nice memoryless starting point and a plausible qualitative idea, but the core mathematical step is broken and no independent evidence is offered. My recommendation: reject. If the authors reframe the model as an ad hoc fractional differential equation, validate it against real temporal-network data, and provide code, it could be resubmitted as a different paper.

Referee Report

4 major / 6 minor

Summary. The paper proposes a temporal-network model with preferential attachment and memory. In the memoryless case it derives an analytical solution for node activity, identifies a crossover time, and validates the result by simulation. Memory is introduced by replacing the integer-order derivative with an integral equation containing a power-law kernel, which the authors then convert into a Caputo-type fractional differential equation of order α. The fractional equation is solved numerically with a predictor-corrector scheme, and the paper reports that the effective node activity reaches a peak and then declines, interpreting this as an aging effect. It also reports changes in the degree and link-weight distributions for different fractional orders. The central claim is that memory causes decay of node activity and reduces the chance of older nodes to receive new connections.

Significance. If the derivation were mathematically sound, the paper would offer a compact fractional-calculus framework for introducing long-term memory into temporal-network models and a concrete, testable prediction of aging. Credit is due for the correct memoryless analytical solution, the identified crossover time, the straightforward simulation of the ordinary differential model, and the use of a standard predictor-corrector algorithm. However, the central derivation from the memory-kernel equation to the fractional differential equation contains fundamental mathematical errors. As written, the aging claim is not established as a consequence of the proposed memory mechanism, and the numerical experiments solve a fractional initial-value problem whose connection to the original network model is not demonstrated.

major comments (4)

[Sec. 2, Eq. (7)] Equation (7) defines the Caputo derivative of order α as 1/Γ(1−α) times the integral of (t−s)^{−α} y(s) ds. This is not the standard Caputo derivative for 0<α<1, which instead has y′(s) in the integrand. With the definition as written, the fractional derivative of a constant is nonzero (it equals (t−t0)^{1−α}/Γ(2−α)), contradicting the property of Caputo derivatives on which the paper relies. Because Eq. (6) is obtained by applying this operator, the conversion from Eq. (4) to Eq. (6) is invalid.
[Sec. 2, Eq. (4)] The memory kernel κ(t−t′)=(t−t′)^{α−2}/Γ(α−1) is non-integrable for all 0≤α<1: the exponent α−2 is less than −1, so the integral of u^{α−2} diverges at u=0. Consequently the integral in Eq. (4) is not a well-defined fractional integral of positive order for continuous integrands, and the notation cD^{−(α−1)} in Eq. (5) is not justified. The paper's premise that this kernel yields a fractional integral equation is load-bearing and false as stated.
[Sec. 2, Eqs. (5)–(6)] Even setting aside the kernel singularity, the operator manipulation from Eq. (5) to Eq. (6) is inconsistent. Equation (5) contains D^{−(α−1)} acting on the bracket after dki/dt, while Eq. (6) states D^α ki equals the bracket. The required identities from fractional calculus (such as the composition of fractional integral and derivative operators) are not provided, and with the definitions used in the paper they do not hold. Thus Eq. (6), the equation actually solved numerically, is not derived from the memory model in Eq. (4).
[Sec. 2, Fig. 3 and Eq. (9)] The numerical scheme in Eq. (9) is the standard predictor-corrector discretization of a Caputo-type initial-value problem, and the peak-and-decline in Fig. 3 may be a genuine property of that fractional equation. However, the paper does not establish that this initial-value problem is equivalent to the memory mechanism defined by Eq. (4). The abstract and Fig. 3 caption attribute the decline to 'the aging process' and memory, but the theoretical basis for that attribution is missing. This is the central claim of the paper, so the error is not a minor presentation issue.

minor comments (6)

[Sec. 2, paragraph before Eq. (4)] There is a typo: 'momory' should be 'memory'.
[Sec. 2, Eq. (4)] The integral notation in Eq. (4) is confusing: the differential dt′ appears after the kernel rather than multiplying the bracket; the intended expression is κ(t−t′)[m/N + mki(t′)/∑kj(t′)]dt′.
[Sec. 2, Eqs. (5)–(6)] The text says 'applying a fractional Caputo derivative of order α−1' but then writes Eq. (6) with order α; the stated and used orders do not match.
[Sec. 2, Fig. 2 and text after Eq. (3)] The characteristic time is written as t∗=c²N²/(4m²) in the text but as t∗=c²N²/(4m) in the Fig. 2 caption; one of these is inconsistent.
[Sec. 2, Fig. 5 discussion] The sentence 'The BA model B predicts that after a transient period the connectivity distribution of all nodes becomes a Gaussian around its mean value' is incorrect: the Barabási–Albert model produces a power-law degree distribution, not a Gaussian. This affects the interpretation of the deviation reported in Fig. 5(a).
[Sec. 2, Fig. 3 caption] The caption says 'with as initial condition m=10 nodes and every new node connecting to earlier nodes,' which is inconsistent with the fixed-N model described in Sec. 1 and with the simulation parameters N=1000 reported elsewhere.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: predictions emerge from solving the stated fractional model with no fitted target; self-citations are contextual.

full rationale

The paper's central prediction—that fractional-order memory yields a peak and decline of effective node activity—is obtained by solving the stated fractional initial-value problem (Eqs. 6-9) for scanned values of alpha, not by fitting alpha or any parameter to the observed peak. The memoryless solution (Eq. 2) and its characteristic time follow analytically from Eq. (1) and are checked against simulation. The self-citations [12,28,29,30] appear in background lists of fractional-calculus memory models and are not load-bearing: the conversion from the memory-kernel formulation to the Caputo form is attributed to Caputo [34] and Podlubny [35], and the numerical method to Diethelm/Ford/Freed [43] and Garrappa [44]. The mathematical inconsistency noted by the skeptic—Eq. (4)'s kernel is non-integrable for alpha<1 and Eq. (7) omits the derivative of y—is a correctness concern, not a circularity: even if the derivation is flawed, the aging claim is not assumed as an input, and the discretized problem in Eq. (9) can be evaluated independently. No prediction reduces by construction to a fitted input or to a self-citation chain.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the choice of fractional derivative and kernel. The fractional order α is the main tunable parameter. No new physical entities are introduced, but the mathematical objects themselves are not well-defined due to the errors in Eq. (4) and Eq. (7).

free parameters (4)

α (fractional order) = 0.2, 0.4, 0.6, 0.8, 1 (chosen)
Controls memory strength; smaller α means stronger memory. Not fitted to data, scanned in simulations.
m (selected nodes per time step) = 10 (in most simulations)
Number of source nodes that create events each step; a model parameter set by hand.
c (integration constant) = depends on initial condition k0, not specified
Appears in the analytical solution Eq. (2) and determines the characteristic time t*. Its value is taken from initial conditions but not given explicitly.
N (number of nodes) = 1000 (Fig 2), 200 (Fig 5)
System size, varied across simulations.

assumptions (4)

domain assumption Node activity evolves according to the mean-field ODE Eq. (1) in the memoryless case.
The model assumes a deterministic mean-field equation for activity, neglecting stochastic fluctuations.
domain assumption Memory can be modeled by replacing the integer-order derivative with a Caputo fractional derivative of order α.
This is a modeling choice borrowed from fractional calculus; it is not derived from data or first principles.
standard math A unique solution to the fractional differential equation exists on [0,t] for given initial conditions.
Stated in Section 2 with reference to the predictor-corrector method; assumes standard existence theory for fractional ODEs.
ad hoc to paper The kernel (t-t')^{α-2}/Γ(α-1) is a valid memory kernel.
This specific power-law kernel is chosen to match a fractional integral representation, but as written it is not integrable for α<1, undermining Eq. (4).

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Cite this review

Pith. "Pith review of Memory effects on link formation in temporal networks: A fractional calculus approach." pith.science (2026). https://pith.science/paper/DAJLGEYC

@misc{pith2026190801999,
  author       = {Pith},
  title        = {Pith review of: Memory effects on link formation in temporal networks: A fractional calculus approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DAJLGEYC}},
  note         = {Machine review of arXiv:1908.01999}
}

read the original abstract

Memory plays a vital role in the temporal evolution of interactions of complex systems. To address the impact of memory on the temporal pattern of networks, we propose a simple preferential connection model, in which nodes have a preferential tendency to establish links with most active nodes. Node activity is measured by the number of links a node observes in a given time interval. Memory is investigated using a time-fractional order derivative equation, which has proven to be a powerful method to understand phenomena with long-term memory. The memoryless case reveals a characteristic time where node activity behaves differently below and above it. We also observe that dense temporal networks (high number of events) show a clearer characteristic time than sparse ones. Interestingly, we also find that memory leads to decay of the node activity; thus, the chances of a node to receive new connections reduce with the node's age. Finally, we discuss the statistical properties of the networks for various memory-length.

Figures

Figures reproduced from arXiv: 1908.01999 by the authors.

**Figure 2.** (a) The analytical solution of Eq.2 for the node activity [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** (a)The numerical solution of Eq. 9 for the e [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Schematic representation of fractional vs homogeneous time axis: in the homogeneous time order, all units of time have the same length, [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: In this figure, we show the probability distribution function of degree (P(k)) and link weight (P(w)) for di [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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A modified preferential-attachment model with a linear link cost predicts that networks stop growing when the cost parameter exceeds the link-creation rate, with a trade-off boundary at α=3m.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Clauset, C

A. Clauset, C. R. Shalizi, M. E. Newman, Power-law distributions in empirical data, SIAM review 51 (4) (2009) 661–703

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