Fractional Transfer Entropy Networks: Short- and Long-Memory Perspectives on Global Stock Market Interactions

Abstract

This study addresses the challenge of capturing both short-run volatility and long-run dependencies in global stock markets by introducing fractional transfer entropy (FTE), a new framework that embeds fractional calculus into transfer entropy. FTE allows analysts to tune memory parameters and thus observe how different temporal emphases reshape the network of directional information flows among major financial indices. Empirical evidence reveals that when short-memory effects dominate, markets swiftly incorporate recent news, creating networks that adapt quickly but remain vulnerable to transient shocks. In contrast, balanced memory parameters yield a more stable equilibrium, blending immediate reactions with persistent structural ties. Under long-memory configurations, historically entrenched relationships prevail, enabling established market leaders to remain central despite ongoing fluctuations. These findings demonstrate that FTE uncovers nuanced dynamics overlooked by methods focusing solely on either current events or deep-rooted patterns. Although the method relies on price returns and does not differentiate specific shock types, it offers a versatile tool for investors, policymakers, and researchers to gauge financial stability, evaluate contagion risk, and better understand how ephemeral signals and historical legacies jointly govern global market connectivity.

1. Introduction

Understanding global stock market interactions requires a methodology that can simultaneously capture both rapid responses to emerging shocks and more persistent influences shaped by historical patterns and institutional ties. Many standard approaches focus on either near-instantaneous changes or deep-rooted relationships, but rarely do they integrate these two facets cohesively. Transfer entropy has proven useful for measuring directional information flows among financial time series, as it detects how past values of one process help predict the future of another in a model-free manner. However, conventional transfer entropy methods generally adopt a fixed-lag perspective, where historical data are either equally weighted or follow a narrow range of lags. In reality, stock markets do not uniformly “forget” their past; some rely heavily on distant fundamentals and investor memories, while others react quickly to new policy decisions or macroeconomic announcements.

To address this gap, we propose fractional transfer entropy (FTE), which merges the flexibility of fractional calculus with the original transfer entropy framework. The incorporation of fractional derivatives allows for continuous control over how past information is treated, shifting from a predominantly short-memory setting to a strongly persistent one. By fine-tuning this fractional parameter, FTE illuminates how certain markets preserve long-standing cross-border capital flow patterns or historically established hierarchies, while others respond mostly to fresh signals such as economic data releases, geopolitical shifts, or company-specific events. This balance of short- and long-memory factors helps capture the essential duality of global finance, where markets are never exclusively forward-looking or backward-looking, but operate across a continuum of temporal influences.

In line with previous studies on international stock markets (e.g., refs. [1,2,3,4], the core motivation here is to reveal the deeper connectivity that often hides behind surface-level fluctuations. Even if short-term volatility plays a prominent role in daily trading decisions, persistent ties rooted in trade agreements, supply chains, shared investor bases, and historical crises continue to guide investment flows. Conversely, methods that emphasize only structural inertia may overlook how swiftly significant news can ripple through financial hubs and reconfigure the network of global market interactions (refs. [5,6,7]). By bridging both viewpoints, FTE shows that neither perspective—the immediacy of daily returns nor the depth of historical inertia—adequately reflects the reality of international finance unless they are analyzed together.

This article contributes to a growing body of research on information flow in stock markets, including works that apply Granger causality or traditional transfer entropy [8,9]. While these have uncovered critical insights into directional linkages and contagion, they often assume homogeneity in how the past is remembered. Introducing fractional derivatives addresses this shortfall by enabling a parameterized approach to memory, reflecting the heterogeneous ways that different economies, investor types, or corporate structures might weight prior data. Markets in developed nations with large capital reserves may place greater emphasis on historical credibility and slowly evolving fundamentals, while emerging markets—perhaps more susceptible to daily capital inflows or policy changes—could respond acutely to short-term signals (refs. [10,11]).

Within the FTE framework, we apply fractional differencing to each time series, thereby encoding the degree of memory weighting directly into the underlying data before computing directional information flows. The FTE values then form weighted and directed networks, with the nodes representing stock markets and edges capturing how strongly or weakly information travels under a given memory regime. By exploring distinct parameter configurations, the analysis illustrates how short-memory bias emphasizes highly reactive interconnections, whereas long-memory bias brings to the fore the underlying long-term structures that shape the global financial architecture. Balanced memory parameters, on the other hand, reveal an interplay between current shocks and entrenched relationships.

In applying this approach, we examine how topological features such as node centrality, betweenness, and community detection shift when fractional orders vary. Major markets, like those in the United States or in dominant European and Asian financial centers, frequently remain central hubs, but their influence might wax or wane depending on whether memory weighting privileges well-established historical capital flows or the latest macro announcements. Similarly, emerging markets may exhibit different patterns of integration or isolation when either recent volatility or more extended timelines are prioritized in the analysis. This network-centric perspective sheds light on how risk can propagate and highlights potential sources of resilience or vulnerability in global markets.

Finally, the proposed methodology not only refines the way researchers view cross-market connections today, but also opens possibilities for future inquiry. A broader time horizon, including worldwide shocks like COVID-19 and periods of global financial stress, will enable deeper tests of how short- and long-memory factors interplay when volatility peaks. Extending FTE to higher-level constructs such as community-level or sector-level fractions may also illuminate whether clusters of markets respond uniformly to events or whether divergent regional memories create different contagion patterns. As markets continue to evolve under heightened globalization and algorithmic trading, fractional transfer entropy offers an adaptable lens through which policymakers, investors, and scholars can track shifting structures and dependencies in a world of financial interconnectivity that is neither fully anchored in the past nor wholly captured by the present.

This study is structured as follows: First, Section 2 outlines a comprehensive literature overview. Then, Section 3 outlines the methodology, beginning with the definition and theoretical foundations of fractional transfer entropy, which integrates fractional calculus into the transfer entropy framework to capture both short- and long-term dependencies in financial time series. This section also details the process of forming financial networks from the resulting FTE measures, illustrating how nodes and edges reflect markets and directional information flows. Section 4 presents the results and discussions, starting with a description of the dataset employed, including global stock market indices and their log-return series. Subsequent subsections examine the emerging networks and metric distributions under four distinct parameter settings: ${\alpha }_X=0.0,{\alpha }_Y=0.0$, ${\alpha }_X=0.2,{\alpha }_Y=0.8$; ${\alpha }_X=0.5,{\alpha }_Y=0.5$; and ${\alpha }_X=0.8,{\alpha }_Y=0.2$. Each configuration highlights how varying memory parameters influences global connectivity, market hierarchies, and community structures, and provides an in-depth comparative perspective on the interplay of short-term and long-term factors. Finally, Section 5 offers conclusions, summarizing the key findings, discussing the limitations of the approach, and suggesting avenues for future research based on the insights gained from the fractional transfer entropy analysis of global stock markets.

2. Related Studies

The dynamics of global financial markets have been extensively explored through various methodological approaches, each addressing different aspects of market index interactions, risk transmission, and persistence in price movements. Early research predominantly employed correlation-based measures to evaluate relationships among equities or indices. In refs. [12,13,14], the authors demonstrated that while correlation measures are straightforward and effective for identifying co-movements, they fall short in capturing the directionality of interactions and accounting for nonlinear dependencies. This limitation hindered deeper analytical insights into the causal mechanisms underlying market behaviors.

Building on these initial findings, subsequent advancements introduced causal inference techniques, with Granger causality emerging as a prominent method. In ref. [15], Granger and Joyeux explored predictive relationships by examining whether past values of one time series could forecast current values of another. This approach marked a significant improvement over correlation-based methods by incorporating temporal precedence and directionality. However, ref. [16] highlighted the inherent limitations of Granger causality, particularly its reliance on linear specifications, which restrict its effectiveness in contexts where financial data exhibit higher-order interactions or structural breaks. These constraints prompted the search for more flexible frameworks capable of capturing the complexities inherent in financial time series.

The introduction of information theory into financial market analysis provided a novel framework for examining market interdependencies. Refs. [17,18,19] used transfer entropy, a model-free measure that uncovers nonlinear and directional dependencies between systems. These foundational works established transfer entropy as a robust alternative to traditional correlation- and regression-based methods, facilitating the measurement of how one stochastic process influences another. Recognizing its potential, researchers quickly adopted transfer entropy in finance and economics. For instance, in refs. [20,21], studies applied transfer entropy to interbank networks and global stock markets, respectively, demonstrating its superiority in detecting subtle and complex information flow patterns. These applications consistently showed that transfer entropy outperforms simpler metrics in identifying crisis contagion pathways and uncovering leadership structures among major financial hubs.

Despite the successes of transfer entropy, integrating memory effects into the assessment of information flow remains an ongoing challenge. Traditional implementations typically treat past information uniformly, either by considering a limited set of lags or employing a single time-lag approach that assumes equal contribution from all historical data points. Ref. [22] discussed how this uniform treatment overlooks the long-range dependence and variable decay in relevance that characterize financial data, where certain events or macroeconomic conditions continue to influence investor sentiment long after their occurrence. While modifications like partial transfer entropy and multistep transfer entropy introduced additional nuance (refs. [23,24]), they still lacked a systematic and flexible method for weighting historical data across different time horizons.

In response to these limitations, fractional calculus has garnered increasing interest among financial modelers for its ability to describe processes with memory or hereditary properties. Ref. [25] provided a comprehensive overview of fractional derivatives and integrals, highlighting their robustness in modeling persistent or long-memory behavior. Early applications in economics focused on fractional integration to capture long-memory behavior in macroeconomic series (refs. [26,27,28]). Fractional calculus enables the systematic adjustment of how recent versus distant past observations contribute to a series’ present state, thereby bridging the gap between short-term volatility and long-term trends.

Building on these developments, combining fractional calculus with transfer entropy represents a logical progression in modeling information flow within financial markets. Ref. [29] underscored the necessity of capturing memory effects in transfer entropy, particularly given that major global stock markets rarely operate under purely Markovian conditions. Additionally, the versatility of fractional operators in describing memory-induced processes aligns well with the requirements of financial time series analysis. Studies such as refs. [30,31] have begun developing methods that allow transfer entropy to transition smoothly from emphasizing the near past to focusing on deeper historical windows. This approach is particularly relevant in cross-market studies where heterogeneous market structures, varying liquidity levels, and diverse regulatory regimes result in disparities in how quickly and extensively markets incorporate new information.

Advances in network analysis further complement these methodological innovations. The emergence of network-based methods in finance underscores the importance of understanding how individual nodes—representing markets or institutions—interconnect within a global system and how these connections evolve over time. Ref. [32] modeled the structure of stock markets using partial correlations, revealing a core-periphery structure where large, highly liquid markets anchor the system while smaller or less liquid indices occupy the periphery. Later research incorporated causal metrics such as Granger causality and transfer entropy, transforming network links from undirected to directed and providing new insights into contagion and systemic risk (refs. [33,34,35]). The integration of fractional calculus into this toolkit allows for an even more nuanced understanding, where the memory properties of each node or link are integral to comprehending global market dynamics.

These diverse strands of literature converge in the present study, which introduces FTE as a unified and flexible mechanism for modeling information exchange among global equity markets across different memory horizons. By applying fractional derivatives to time series prior to computing transfer entropy, FTE aligns with previous research advocating for the incorporation of persistent, slowly decaying influences Simultaneously, FTE leverages the network-oriented perspective on financial contagion and interdependence, facilitating the construction of directed graphs that elucidate which markets drive others across both short- and long-range lags. This approach addresses multiple gaps identified in the literature: it extends standard transfer entropy to capture heterogeneous memory effects, employs complex network metrics to examine market hierarchies and modularity

Furthermore, this study provides a framework to re-evaluate established empirical findings in financial economics through a novel lens—one that accounts for the varying half-lives of macroeconomic news, persistent investor sentiment, and the gradual evolution of regional market structures. Whether investigating crisis transmission channels, identifying structural breakpoints in global market integration, or determining which markets exhibit the greatest resilience, FTE offers a pathway to dissect these phenomena with enhanced granularity. In this sense, the current research builds upon and extends a broad spectrum of literature that collectively emphasizes the critical role of memory in shaping cross-market information flows. It resonates with longstanding observations that real-world financial behavior is governed by a balance of immediate volatility and long-running trends—an equilibrium that fractional transfer entropy is uniquely positioned to explore.

3. Methodology

3.1. Fractional Transfer Entropy

The concept of fractional transfer entropy between two time series is introduced here by incorporating fractional calculus and memory kernels. We begin by recalling the classical setting of transfer entropy. Then, we introduce the fractional framework, including integral definitions, memory functions, and the associated fractional derivatives. This approach is subsequently employed to define fractional transfer entropy, and a set of theorems together with their proofs is provided to establish fundamental properties and conceptual significance.

Classical transfer entropy measures directional information flow between two time series. Given two stochastic processes, $\left(X_t\right)$ and $\left(Y_t\right)$, the transfer entropy from Y to X essentially quantifies how much knowing the past of Y reduces the uncertainty about the future state of X beyond what is already known from the past of X itself. In the standard formulation, the memory or influence from the past is usually considered in discrete, integer-lag steps. However, many real-world socio-economic or physical systems exhibit complex, non-local memory effects that cannot be properly captured by a finite number of discrete lags. Fractional calculus provides a natural mathematical framework to incorporate such long-range dependencies and power law fading memory into the analysis of dynamic processes. To address this need, the fractional transfer entropy is defined by replacing the classical finite-dimensional conditioning sets with fractional memory operators. The new measure captures the influence of one process on another under a generalized memory scenario, where memory decays according to a power law kernel rather than vanishing after a finite lag.

To properly define these memory structures, consider two time-dependent processes, $\left\{X\right(t\left)\right\}$ and $\left\{Y\right(t\left)\right\}$. Let $Y\left(t\right)$ be an endogenous variable defined by a Volterra integral operator acting on the history of $X\left(\tau \right)$ for $\tau \in [0,t]$

$$Y\left(t\right)=F_0^t\left(X\left(\tau \right)\right):={\int }_0^{t}M(t,\tau )X\left(\tau \right)d\tau ,$$

(1)

where $M(t,\tau )$ is the memory function that encodes how past values of X influence the current value of Y. Assuming homogeneity in the time variable allows writing the memory kernel as $M(t,\tau )=M(t-\tau )$. To model power law fading memory, let

$$M_{\alpha }(t,\tau )={\displaystyle \frac{m}{\Gamma \left(\alpha \right){(t-\tau )}^{1-\alpha }}},$$

(2)

where $\alpha \in (0,1]$ and m is a real constant dependent on $\alpha$. Substituting this into the Volterra operator yields

$$Y\left(t\right)=m\left(I_{0+}^{\alpha }X\right)\left(t\right),$$

(3)

where $I_{0+}^{\alpha }$ denotes the Riemann–Liouville fractional integral of order $\alpha$.

To handle initial conditions more conveniently, one can consider Caputo fractional derivatives. By setting $n=\lfloor \alpha \rfloor +1$ and using a memory function

$$M_{n-\alpha }(t,\tau )={\displaystyle \frac{a}{\Gamma (n-\alpha )}}{(t-\tau )}^{n-\alpha -1},$$

(4)

one obtains

$$Y\left(t\right)=a\left(D_{0+}^{\alpha }X\right)\left(t\right),$$

(5)

where

$$D_{0+}^{\alpha }X\left(t\right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\int }_0^t{\displaystyle \frac{X^{\left(n\right)}\left(\tau \right)}{{(t-\tau )}^{\alpha -n+1}}}d\tau$$

(6)

is the left-sided Caputo fractional derivative of order $\alpha$.

These operators encapsulate power law and distributed memory effects, making them suitable tools for analyzing complex systems where memory cannot be reduced to a finite set of past values. In the original definition of transfer entropy, let $\left\{X_t\right\}$ and $\left\{Y_t\right\}$ be discrete-time processes and consider their continuous-valued generalization. The classical transfer entropy from Y to X is

$$T_{Y\to X}=\sum _{x_{t+1},x_t^{\left(k\right)},y_t^{\left(l\right)}}p(x_{t+1},x_t^{\left(k\right)},y_t^{\left(l\right)})log{\displaystyle \frac{p\left(x_{t+1}\right|x_t^{\left(k\right)},y_t^{\left(l\right)})}{p\left(x_{t+1}\right|x_t^{\left(k\right)})}},$$

(7)

where $x_t^{\left(k\right)}=(x_t,x_{t-1},\dots ,x_{t-k+1})$ and similarly for $y_t^{\left(l\right)}$. Since this definition relies on a finite number of past values, it cannot directly capture long-memory effects.

To incorporate power law memory, replace finite past lags with fractional derivative-based states. Define the fractional states

$$X_{\alpha }\left(t\right)=D_{0+}^{{\alpha }_X}X\left(t\right),Y_{\alpha }\left(t\right)=D_{0+}^{{\alpha }_Y}Y\left(t\right),$$

(8)

where ${\alpha }_X,{\alpha }_Y>0$ and $D_{0+}^{\alpha }$ is a Caputo fractional derivative. These fractional states encapsulate the entire past of X and Y up to time t with a power law weighting. The fractional transfer entropy (FTE) from Y to X is then defined as

$$T_{Y\to X}^{({\alpha }_X,{\alpha }_Y)}=\mathbb{E}\left[log{\displaystyle \frac{p\left(X(t+1)\right|X_{\alpha }\left(t\right),Y_{\alpha }\left(t\right))}{p\left(X(t+1)\right|X_{\alpha }\left(t\right))}}\right].$$

(9)

In integral form,

$$T_{Y\to X}^{({\alpha }_X,{\alpha }_Y)}=\int p(x(t+1),x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))log{\displaystyle \frac{p\left(x(t+1)\right|x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))}{p\left(x(t+1)\right|x_{\alpha }\left(t\right))}}dx(t+1)d{x}_{\alpha }\left(t\right)d{y}_{\alpha }\left(t\right).$$

(10)

By varying ${\alpha }_X$ and ${\alpha }_Y$, it is possible to model short to long-memory ranges.

FTE is non-negativity, i.e., $T_{Y\to X}^{({\alpha }_X,{\alpha }_Y)}\ge 0$. Consider

$$T_{Y\to X}^{({\alpha }_X,{\alpha }_Y)}=\int p(x(t+1),x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))log{\displaystyle \frac{p\left(x(t+1)\right|x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))}{p\left(x(t+1)\right|x_{\alpha }\left(t\right))}}d\mu .$$

(11)

This can be recognized as a Kullback–Leibler divergence

$$T_{Y\to X}^{({\alpha }_X,{\alpha }_Y)}=D_{\mathrm{KL}}\left(p\left(x(t+1)\right|x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))\parallel p\left(x(t+1)\right|x_{\alpha }\left(t\right))\right)$$

(12)

which is always non-negative. Furthermore, it is zero if and only if $p\left(x(t+1)\right|x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))=p\left(x(t+1)\right|x_{\alpha }\left(t\right))$ almost surely, implying no additional information flow from Y to X.

As ${\alpha }_X,{\alpha }_Y\to 0$, the fractional transfer entropy reduces to the classical transfer entropy

$$\underset{{\alpha }_X,{\alpha }_Y\to 0}{lim}{T}_{Y\to X}^{({\alpha }_X,{\alpha }_Y)}=T_{Y\to X}.$$

(13)

As $\alpha \to 0$, the Caputo fractional derivative $D_{0+}^{\alpha }X\left(t\right)$ approaches $X\left(t\right)$ itself, effectively removing long-memory weighting. Thus, $X_{\alpha }\left(t\right)\to X\left(t\right)$ and $Y_{\alpha }\left(t\right)\to Y\left(t\right)$. In this limit, the conditioning set reverts to finite-lag conditioning, and fractional transfer entropy coincides with the classical definition.

FTE is invariance under linear scaling. If $\tilde{X}\left(t\right)=bX\left(t\right)$ and $\tilde{Y}\left(t\right)=cY\left(t\right)$ for non-zero constants $b,c$, then

$$T_{\tilde{Y}\to \tilde{X}}^{({\alpha }_X,{\alpha }_Y)}=T_{Y\to X}^{({\alpha }_X,{\alpha }_Y)}.$$

(14)

Since transfer entropy is based on ratios of conditional probability densities, linear scaling does not change the underlying conditional dependence structure. The densities transform in a way that the logarithmic ratio remains invariant, rendering the fractional transfer entropy scale-invariant.

FTE is also time-invariant under stationarity. If $\left(X\right(t),Y(t\left)\right)$ is a jointly stationary process, then $T_{Y\to X}^{({\alpha }_X,{\alpha }_Y)}$ does not depend on the specific time t. Stationarity implies that the joint distributions involved are invariant with respect to time shifts. Since $T_{Y\to X}^{({\alpha }_X,{\alpha }_Y)}$ is defined through these distributions, it remains constant in time.

These results show that fractional transfer entropy extends the classical notion of information flow between time series to scenarios where the effect of the past does not vanish rapidly but instead follows a heavy-tailed, power law distribution. By incorporating Caputo fractional derivatives or distributed-order derivatives, fractional transfer entropy captures complex memory effects prevalent in socio-economic, biological, and physical systems. For instance, in socio-economic models of labor migration between countries, the memory of agents regarding past economic conditions can be long-lasting and not limited to recent history. The fractional approach ensures that all historical influences are accounted for, weighted according to a power law kernel. The parameter $\alpha$ (or a distribution over $\alpha$) quantifies how long-range the memory effect is.

The calculation of fractional transfer entropy from empirical time series data requires careful construction of fractional states, estimation of probability distributions, and numerical approximations of the integrals and expectations that define the measure. The process begins with the recorded time series, which typically consist of discrete observations of two processes $X\left(t\right)$ and $Y\left(t\right)$ at regular time intervals. To incorporate memory of arbitrary length and complexity, fractional calculus operators must be applied to the time series. Given a fractional order $\alpha >0$, the Caputo fractional derivative $D_{0+}^{\alpha }X\left(t\right)$ can be numerically approximated by discrete convolution-like formulas or by methods that discretize the integral definition of the fractional derivative. Such methods commonly rely on approximating the kernel ${(t-\tau )}^{-\gamma }$ for some $\gamma$ depending on $\alpha$, and employing fractional finite differences. Although several numerical schemes exist, the essential idea is to approximate $D_{0+}^{\alpha }X\left(t\right)$ by combining current and past values of $X\left(\tau \right)$ with weights that decrease according to a power law, ensuring that distant past values still influence the present, albeit with diminishing intensity. Once these fractional derivatives are computed for all time points, the resulting transformed time series $X_{\alpha }\left(t\right)=D_{0+}^{{\alpha }_X}X\left(t\right)$ and $Y_{\alpha }\left(t\right)=D_{0+}^{{\alpha }_Y}Y\left(t\right)$ represent states that are no longer confined to a finite number of past lags but integrate all past history in a continuous and fractional manner.

After constructing $X_{\alpha }\left(t\right)$ and $Y_{\alpha }\left(t\right)$, the next step involves estimating the joint probability distributions $p(x(t+1),x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))$ and the conditional distributions $p\left(x(t+1)\right|x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))$ and $p\left(x(t+1)\right|x_{\alpha }\left(t\right))$. Since fractional transfer entropy involves expectations of logarithms of probability ratios, an accurate and robust method of probability density estimation is required. In practice, one must first discretize the state space or employ non-parametric estimation techniques such as kernel density estimation. For instance, one may define a suitable kernel function and bandwidth parameters to estimate the continuous joint probability density functions from a finite sample of data points ${\left\{(x\left(t\right),x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))\right\}}_{t=1}^T$. Another approach is to bin the continuous states into small intervals and approximate probabilities by counting frequencies. While binning provides a straightforward approach, it may lead to coarse approximations and bias if bins are not chosen carefully or if data are limited. Kernel methods typically offer smoother and more flexible density approximations, though they require a thoughtful selection of kernel bandwidths to avoid under- or over-smoothing the underlying distributions. Regardless of the chosen method, the goal is to reliably approximate $p(x(t+1),x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))$, $p\left(x(t+1)\right|x_{\alpha }\left(t\right))$, and $p\left(x(t+1)\right|x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))$ as functions of continuous variables.

Once the probability distributions have been estimated, one proceeds to numerically approximate the integral that defines fractional transfer entropy. Conceptually, the expectation of the logarithmic ratio

$$log{\displaystyle \frac{p\left(x(t+1)\right|x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))}{p\left(x(t+1)\right|x_{\alpha }\left(t\right))}}$$

(15)

is taken over the joint distribution $p(x(t+1),x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))$. Since the data are discrete observations, this expectation reduces to a sample-based average or a numerical integration. If a kernel density estimate is employed, the integral can be approximated by summations over the observed data points weighted by their estimated densities, ensuring that the entire support of the joint distribution is covered. The log ratio is computed at each observed triple $(x(t+1),x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))$, and then an average is taken over the sample. If the sample size is large and the density estimates are accurate, the resulting approximation should converge to the true fractional transfer entropy. A careful analysis of sampling variance and estimation errors is advisable. One may use statistical techniques such as bootstrapping to quantify uncertainty in the fractional transfer entropy estimation, generating multiple surrogate samples from the original dataset to produce confidence intervals.

In certain cases, the parameters ${\alpha }_X$ and ${\alpha }_Y$ are chosen a priori based on prior knowledge about the system’s memory characteristics. However, it is also conceivable to attempt estimation or model selection techniques to identify the values of ${\alpha }_X$ and ${\alpha }_Y$ that best capture the observed dynamics. This might involve comparing fractional transfer entropy values for different fractional orders or employing information criteria to balance model complexity with explanatory power. One could also consider distributed-order derivatives, in which case one must also estimate the weight function $c\left(\alpha \right)$ that defines how different fractional orders contribute to the memory kernel. This leads to a more involved estimation procedure requiring either optimization routines or structured assumptions on the form of $c\left(\alpha \right)$.

The computational complexity of estimating fractional derivatives and probability densities depends on the length of the time series and the resolution of the chosen discretization. Fractional derivative estimation typically introduces an additional computational overhead because it involves summations over past values at each time point, though efficient algorithms and approximations have been developed. Likewise, probability density estimation, whether bin-based or kernel-based, depends on the sample size and dimensionality of the variable space. Since fractional transfer entropy conditions on potentially infinite memory encoded through fractional derivatives, one must ensure that sufficient data are available to estimate the required distributions reliably. In practice, the dimension of the joint space $(x(t+1),x_{\alpha }\left(t\right),y_{\alpha }\left(t\right))$ remains manageable, but the fractional transformations may introduce smoothness and correlations that must be well understood.

3.2. Financial Network Formation

The process of forming a directed network from worldwide operating stock markets begins by examining the daily closure prices of each market and transforming them into logarithmic returns to ensure both stationarity and proportional comparability. Let $\{S_1,S_2,\dots ,S_N\}$ denote the set of stock markets under consideration. For each market $S_i$, let $P_i\left(t\right)$ be its closure price on day t. The logarithmic return for market $S_i$ at day t is defined as

$$r_i\left(t\right)=log\left({\displaystyle \frac{P_i\left(t\right)}{P_i(t-1)}}\right).$$

(16)

This transformation reduces the influence of non-stationarities and focuses on relative changes, providing a time series $\left\{r_i\left(t\right)\right\}$ that is more amenable to statistical analysis and suitable for the computation of fractional transfer entropy. Given two such return series, say $\left\{r_i\left(t\right)\right\}$ and $\left\{r_j\left(t\right)\right\}$, the previously defined fractional transfer entropy $T_{j\to i}^{({\alpha }_X,{\alpha }_Y)}$ quantifies the directional flow of information from market j to market i, where the memory of the past is encoded through fractional derivatives of order ${\alpha }_X$ and ${\alpha }_Y$.

Once the fractional transfer entropy is computed for all ordered pairs of markets $(S_j,S_i)$, one obtains a complete set of directed, weighted edges. Let $\mathbf{W}=\left[w_{ij}\right]$ be the resulting adjacency matrix, where each node corresponds to a stock market and each directed edge from market $S_j$ to market $S_i$ is assigned a weight

$$w_{ij}=T_{j\to i}^{({\alpha }_X,{\alpha }_Y)}.$$

(17)

The matrix $\mathbf{W}$ thus encapsulates the complex web of directional relationships inferred from the fractional transfer entropy. To analyze the global structure and dynamics of these interactions, various network metrics are introduced. Consider the degree-based measures first. The out-degree of node i is defined as the sum of the weights of the edges emanating from node i

$$\mathrm{outdeg}\left(i\right)=\sum _{j=1}^N{w}_{ij},$$

(18)

while the in-degree of node i is defined as the sum of the weights of the edges incoming to node i

$$\mathrm{indeg}\left(i\right)=\sum _{j=1}^N{w}_{ji}.$$

(19)

These measures reflect how strongly a particular market influences others (out-degree) and how strongly it is influenced by them (in-degree).

To gain deeper insight into the network’s hierarchical structure, one may consider hub and authority scores. In a directed weighted graph, a node serves as a good hub if it points to nodes that are themselves good authorities, and a node is a good authority if it is pointed to by good hubs. Let $\mathbf{H}=\left[h_i\right]$ be a vector of hub scores and $\mathbf{A}=\left[a_i\right]$ be a vector of authority scores. Their relationship can be expressed as

$$\mathbf{H}=\mathbf{W}\mathbf{A}\mathrm{and}\mathbf{A}={\mathbf{W}}^T\mathbf{H}.$$

(20)

Solving these equations simultaneously amounts to an eigenvalue problem that determines the relative importance of nodes beyond simple degree counts. Markets with high hub scores can be viewed as global information broadcasters, while those with high authority scores act as key information receivers.

More nuanced topological features can be captured by shortest-path-based centrality measures. Consider the directed weighted graph defined by $\mathbf{W}$. Let $d(i,j)$ represent the shortest path distance from node i to node j. Closeness centrality for node i is defined as

$$C_i={\displaystyle \frac{1}{{\sum }_{j=1}^{N}d(i,j)}},$$

(21)

implying that a node with smaller average distance to all others is more central and can rapidly influence the network. Betweenness centrality focuses on how often a node lies on shortest paths between pairs of other nodes. Let ${\sigma }_{ij}$ denote the number of shortest paths from i to j, and let ${\sigma }_{ij}\left(k\right)$ represent the number of those shortest paths passing through node k. The betweenness centrality of node k is given by

$$B_k=\sum _{i,j}{\displaystyle \frac{{\sigma }_{ij}\left(k\right)}{{\sigma }_{ij}}},$$

(22)

reflecting the brokerage role of a node. Nodes with high betweenness serve as crucial intermediaries that can facilitate or hinder the flow of information across the network.

In practice, these network metrics help to interpret the intricate interdependencies and hierarchical relationships that characterize global stock markets. A high out-degree or hub score identifies markets functioning as influential sources of information or systemic risk. A high in-degree or authority score reveals markets that aggregate or absorb wide-ranging influences from multiple sources. High closeness centrality points to markets that rapidly integrate signals from other parts of the world, while high betweenness centrality highlights markets acting as key transit points in the information flow. By applying these measures to a fractional-transfer-entropy-derived network constructed from logarithmic return series, it becomes possible to capture not only the direct pairwise influences but also the subtle, memory-dependent patterns of global market interaction. This approach provides a richer characterization of international financial interconnections, enabling a more comprehensive understanding of how shocks propagate, how clusters form, and how certain markets emerge as structurally significant nodes in the worldwide financial landscape.

In the context of network analysis, community detection serves as a pivotal methodology for uncovering the underlying structural organization within complex networks [36,37,38]. Communities, often referred to as modules or clusters, are subsets of nodes that exhibit a higher density of connections internally compared to their connections with the rest of the network. Mathematically, the identification of such communities can be framed as an optimization problem, wherein the objective is to maximize a quality function that quantifies the strength of the network’s division into modules.

A fundamental concept employed in community detection is modularity, denoted as Q, which measures the density of links inside communities compared to links between communities. Formally, modularity is defined as

$$Q={\displaystyle \frac{1}{2m}}\sum _{i,j}\left[A_{ij}-{\displaystyle \frac{k_i{k}_j}{2m}}\right]\delta (c_i,c_j),$$

(23)

where $A_{ij}$ represents the adjacency matrix of the network, where $A_{ij}=1$ if there is an edge between nodes i and j, and $A_{ij}=0$ otherwise. The term $k_i$ denotes the degree of node i, and m is the total number of edges in the network. The function $\delta (c_i,c_j)$ is the Kronecker delta, which is equal to 1 if nodes i and j belong to the same community ($c_i=c_j$), and 0 otherwise. The modularity Q thus quantifies the extent to which the actual density of edges within communities exceeds what would be expected in a randomized network with the same degree distribution.

To optimize modularity, the Louvain method is employed, which is a greedy algorithm designed to efficiently partition the network into communities. The algorithm operates iteratively in two main phases: Initially, each node is assigned to its own community. For each node, the algorithm considers the gain in modularity that would result from moving the node to the community of each of its neighbors. The node is then placed in the community that provides the highest positive gain in modularity. This process is repeated for all nodes until no further modularity improvement can be achieved through local moves. After the first phase converges, a new network is constructed where each community identified in the previous phase is represented as a single node. The weights of the edges between these new nodes are determined by the sum of the weights of the edges between the nodes in the corresponding communities. The Louvain method is then recursively applied to this aggregated network. This iterative process continues until no further modularity gains are possible, resulting in a hierarchical structure of communities that optimally partitions the network based on the modularity criterion.

Once the communities are delineated, it becomes imperative to perform statistical analyses to characterize and interpret the properties of these communities. This involves computing descriptive statistics for various network metrics within each community. Let ${\mathcal{C}}_k$ denote the k-th community identified in the network. For each community ${\mathcal{C}}_k$, we calculate metrics such as the mean, median, and standard deviation of in-degree, out-degree, betweenness centrality, closeness centrality, eigenvector centrality, hub scores, and authority scores of the constituent nodes. Mathematically, for a metric M, the mean within community ${\mathcal{C}}_k$ is given by

$${\mu }_M^{{\mathcal{C}}_k}={\displaystyle \frac{1}{|{\mathcal{C}}_k|}}\sum _{i\in {\mathcal{C}}_k}{M}_i.$$

(24)

Similarly, the median ${\tilde{M}}^{{\mathcal{C}}_k}$ and standard deviation ${\sigma }_M^{{\mathcal{C}}_k}$ are computed to provide insights into the distribution and variability of the metric within the community. These statistical measures facilitate the comparative analysis of communities, enabling the identification of distinctive characteristics and patterns. For instance, a community exhibiting high mean betweenness centrality may indicate a group of nodes that act as critical intermediaries within the network, whereas a community with high eigenvector centrality suggests influential nodes that are connected to other highly influential nodes.

4. Results and Discussions

4.1. Dataset

The dataset used in this study consists of daily financial data spanning from 9 January 2023, to 3 October 2024, a period that offers a focused yet extensive temporal framework for examining the dynamic interactions among global stock markets. This time frame was chosen to capture contemporary market behavior under relatively stable conditions, excluding extreme global shocks such as the COVID-19 pandemic or prior financial crises. By doing so, the analysis isolates the underlying mechanisms of information flow and connectivity without the distortion of extraordinary, one-off events, which may overshadow long-term structural dynamics. The analysis focuses on logarithmic returns derived from daily closing prices, a method that normalizes the data to facilitate comparisons across markets of varying scales. This approach allows for a more precise evaluation of relative price movements, ensuring that scale differences do not bias the assessment of directional information flows and network structures. By selecting this period, the study balances the need for sufficient data to ensure robustness with the objective of understanding market behaviors in a relatively typical economic environment, providing insights into how markets interact under less volatile global conditions.

Prior to conducting the primary analyses, the dataset underwent meticulous preprocessing to ensure its integrity and suitability for subsequent computations. This preprocessing involved the identification and exclusion of any anomalies or inconsistencies within the closing price data, such as missing values or non-positive entries, which could potentially distort the analysis. Ensuring the continuity and reliability of the time series data was paramount, as they underpin the accuracy of the FTE calculations and the resulting network structures. The final curated dataset comprises a comprehensive collection of logarithmic return values for each selected stock market index, providing a nuanced depiction of daily market fluctuations over the specified time frame. These log-differenced values serve as the foundational input for constructing the FTE matrices, which quantify the directional information flow between pairs of markets. By focusing exclusively on the time scale and log returns of daily prices, the analysis maintains a streamlined and consistent approach, enhancing the comparability and interpretability of the results. Descriptive statistics for the log-differenced values, including country names and tickers, are detailed in

Table A1. Descriptive statistics for log-return data.

Country	Ticker	Min	Mean	Median	Max	Standard Deviation	Kurtosis	Skewness
Argentina	MERV	−0.1328	0.0049	0.0051	0.2057	0.0323	4.3415	0.0061
Australia	AXJO	−0.0377	0.0003	0.0009	0.0173	0.0071	1.8279	−0.6951
Austria	ATX	−0.0651	0.0002	0.0007	0.0306	0.0091	7.8229	−1.3708
Bahrain	BAX	−0.0107	0.0001	0.0000	0.0370	0.0038	20.7964	2.2684
Bangladesh	DSE	−0.0271	−0.0003	−0.0002	0.0532	0.0074	13.0246	1.5456
Belgium	BFX	−0.0320	0.0002	0.0006	0.0336	0.0082	1.6061	−0.0138
Bosnia Herzegovina	BIRS	−0.0578	0.0002	0.0000	0.0798	0.0104	21.7436	0.9362
Botswana	BSE	−0.0059	0.0005	0.0001	0.0192	0.0016	52.2879	5.3967
Brazil	IBOV	−0.0243	0.0004	−0.0001	0.0420	0.0093	0.5846	0.2778
Bulgaria	SOFIX	−0.0420	0.0008	0.0006	0.0306	0.0062	7.6283	−0.5644
Canada	TSX	−0.0232	0.0004	0.0008	0.0283	0.0069	0.9433	−0.2186
Chile	CLX	−0.0349	0.0005	0.0003	0.0289	0.0094	0.6747	−0.1095
China	CSI	−0.0634	−0.0003	0.0000	0.1057	0.0150	8.1915	0.8591
China	SSEC	−0.0272	0.0001	−0.0003	0.0776	0.0093	12.4206	1.8671
China	SZSE	−0.0356	−0.0002	−0.0009	0.1014	0.0129	11.2706	1.9213
Colombia	COLCAP	−0.0392	−0.0001	0.0005	0.0359	0.0096	1.8415	−0.2197
Cote D’Ivoire	BRVM	−0.0403	0.0007	0.0008	0.0308	0.0071	4.8515	−0.1669
Croatia	CROBEX	−0.0279	0.0008	0.0005	0.0167	0.0050	3.6376	−0.5186
Cyprus	CYMAIN	−0.0795	0.0021	0.0009	0.0749	0.0126	9.5252	0.4423
Czech Republic	PX	−0.0426	0.0005	0.0007	0.0181	0.0069	5.5586	−1.2873
Sweden	OMX	−0.0418	0.0004	0.0006	0.0344	0.0087	1.8354	−0.2607
Egypt	EGX	−0.0926	0.0023	0.0043	0.0468	0.0179	5.4877	−1.6185
Estonia	GTGI	−0.0280	−0.0001	−0.0003	0.0214	0.0049	5.2708	−0.1018
Finland	OMXH	−0.0332	−0.0002	−0.0002	0.0241	0.0085	0.4844	−0.1125
France	CAC	−0.0365	0.0002	0.0007	0.0231	0.0084	1.2151	−0.4600
Germany	DAX	−0.0332	0.0006	0.0009	0.0214	0.0079	1.3257	−0.5519
Germany	EUSTOXX	−0.0352	0.0004	0.0008	0.0233	0.0086	1.2117	−0.3938
Greece	ATG	−0.0647	0.0009	0.0012	0.0591	0.0108	5.7976	−0.5201
Hong Kong	HSI	−0.0378	0.0001	−0.0002	0.0601	0.0143	0.5035	0.4554
Hungary	BUX	−0.0317	0.0011	0.0011	0.0296	0.0086	1.6933	−0.3400
Iceland	ICEX	−0.0565	−0.0002	−0.0004	0.0525	0.0107	6.0030	−0.0206
India	BSESN	−0.0591	0.0007	0.0012	0.0333	0.0076	10.0719	−1.0089
India	NIMI	−0.0737	0.0014	0.0026	0.0379	0.0092	12.5656	−1.9128
Indonesia	JKSE	−0.0344	0.0003	0.0005	0.0170	0.0069	1.7150	−0.6322
Iraq	ISX	−0.0949	0.0023	−0.0003	0.1000	0.0203	8.0912	0.8910
Ireland	ISEQ	−0.0374	0.0005	0.0011	0.0379	0.0101	1.4377	−0.3001
Israel	TA	−0.0669	0.0003	0.0008	0.0296	0.0104	3.7900	−0.7261
Jamaica	JSE	−0.1131	−0.0002	−0.0003	0.0984	0.0100	60.0632	−1.0151
Jordan	AMGNL	−0.0228	−0.0002	−0.0001	0.0161	0.0040	3.0863	−0.7068
Kazakhstan	KASE	−0.0255	0.0010	0.0010	0.0215	0.0052	2.5213	−0.2367
Kuwait	BKM	−0.0266	0.0002	0.0003	0.0278	0.0070	1.9506	−0.1487
Latvia	RGI	−0.2256	−0.0006	−0.0002	0.0754	0.0157	101.9888	−7.1920
Lebanon	BLOM	−0.0745	0.0006	0.0000	0.0851	0.0165	5.8583	0.5043
Lithuania	VGI	−0.0264	0.0001	0.0002	0.0117	0.0034	9.2178	−1.2867
Malaysia	KLCI	−0.0474	0.0002	−0.0001	0.0244	0.0055	13.4460	−1.2301
Malta	MSE	−0.0236	0.0001	0.0000	0.0347	0.0059	4.4252	0.3304
Mauritius	MDEX	−0.0173	0.0003	0.0002	0.0143	0.0031	5.3181	0.0873
Mexico	BIVA	−0.0627	−0.0001	0.0000	0.0330	0.0092	5.2621	−0.6090
Mongolia	MNE	−0.0374	0.0006	0.0005	0.0370	0.0094	2.0749	−0.1050
Morocco	MASI	−0.0289	0.0009	0.0007	0.0495	0.0063	10.1469	1.1797
Namibia	NSX	−0.0483	0.0002	0.0002	0.0611	0.0126	2.2692	0.0603
The Netherlands	AEX	−0.0316	0.0005	0.0005	0.0240	0.0077	1.5255	−0.3166
New Zealand	NZX	−0.0153	0.0001	−0.0001	0.0265	0.0057	1.3351	0.5118
Nigeria	NSE	−0.0625	0.0015	0.0004	0.0936	0.0103	19.8051	1.7372
Norway	OSEBX	−0.0381	0.0004	0.0008	0.0246	0.0082	2.3007	−0.6834
Oman	MSM	−0.0259	−0.0001	0.0002	0.0188	0.0047	3.0395	−0.0915
Pakistan	KSE	−0.0419	0.0017	0.0014	0.0574	0.0101	4.2444	0.1777
Palestine	PLE	−0.0418	−0.0007	−0.0005	0.0224	0.0052	13.0132	−1.6202
Peru	SPBLP	−0.0274	0.0007	0.0005	0.0587	0.0086	4.8410	0.5677
The Philippines	PSEI	−0.0298	0.0002	0.0002	0.0351	0.0090	0.4590	−0.0658
Poland	WIG	−0.0342	0.0005	0.0003	0.0472	0.0125	0.5246	0.2487
Portugal	PSI	−0.0281	0.0003	0.0006	0.0344	0.0081	1.1147	−0.2084
Qatar	QSI	−0.0306	−0.0002	0.0001	0.0287	0.0081	1.3649	−0.1109
Romania	BET	−0.0369	0.0008	0.0013	0.0229	0.0068	2.4162	−0.5281
Russia	MOEX	−0.0396	0.0006	0.0011	0.0371	0.0098	1.9100	−0.3111
Russia	RTS	−0.0406	−0.0001	0.0004	0.0352	0.0119	0.6264	−0.1015
Rwanda	ALSIR	−0.0027	0.0001	0.0000	0.0038	0.0006	11.5586	1.3536
Saudi Arabia	TASI	−0.0245	0.0003	0.0005	0.0252	0.0072	0.6871	−0.2204
Serbia	BELEX	−0.0299	0.0007	0.0006	0.0264	0.0062	4.1705	−0.0711
Singapore	STI	−0.0416	0.0002	0.0003	0.0197	0.0065	3.6688	−0.5153
Slovakia	SAX	−0.0780	−0.0002	0.0000	0.0415	0.0076	31.3840	−2.5749
Slovenia	SBI	−0.0463	0.0009	0.0006	0.0315	0.0067	6.9273	−0.7276
South Africa	JTOP	−0.0301	0.0002	0.0001	0.0382	0.0099	0.9624	0.1819
South Korea	KOSPI	−0.0946	0.0002	0.0001	0.0513	0.0123	8.4908	−0.9491
Spain	IBEX	−0.0446	0.0006	0.0007	0.0242	0.0086	1.9564	−0.5796
Switzerland	SMI	−0.0365	0.0002	0.0007	0.0285	0.0070	2.2567	−0.4242
Taiwan	TWII	−0.0872	0.0010	0.0014	0.0380	0.0111	11.3302	−1.4818
Tanzania	DSEI	−0.0306	0.0003	0.0000	0.1346	0.0082	163.1680	9.8884
Thailand	SET	−0.0318	−0.0004	−0.0005	0.0280	0.0073	2.0858	0.0248
Tunisia	TUN	−0.0145	0.0005	0.0003	0.0149	0.0035	1.5450	0.2157
Turkey	BIST	−0.0901	0.0013	0.0004	0.0942	0.0198	2.5645	−0.0505
Uganda	ALSIU	−0.2014	−0.0003	−0.0002	0.2025	0.0188	78.3090	−0.0398
The United Kingdom	FTSE	−0.0391	0.0002	0.0005	0.0194	0.0068	3.2779	−0.5784
The United States of America	DJI	−0.0264	0.0005	0.0009	0.0210	0.0069	0.4897	−0.2387
The United States of America	NDX	−0.0372	0.0013	0.0014	0.0350	0.0115	0.2902	−0.1400
The United States of America	SPX	−0.0304	0.0009	0.0009	0.0228	0.0081	0.3962	−0.2604
Venezuela	IBC	−0.1441	0.0036	0.0011	0.1437	0.0207	14.5413	0.9005
Vietnam	HNX	−0.0826	0.0009	0.0025	0.0692	0.0178	4.9780	−0.9771
Zambia	LASIL	−0.1127	0.0018	0.0000	0.1137	0.0117	44.4967	0.3333
Zimbabwe	ZSE	−9.1098	−0.0103	0.0069	0.1074	0.4391	424.9201	−20.6245

.

In this study, we delve into the intricate dynamics of information flow among global stock markets by employing FTE as a pivotal analytical tool. To comprehensively explore the impact of fractional dynamics on information transfer, our analysis is segmented into four distinct scenarios, each characterized by specific fractional derivative orders: ${\alpha }_X=0.0$ with ${\alpha }_Y=0.0$, ${\alpha }_X=0.2$ with ${\alpha }_Y=0.8$, ${\alpha }_X=0.5$ with ${\alpha }_Y=0.5$, and ${\alpha }_X=0.8$ with ${\alpha }_Y=0.2$.

For each combination of ${\alpha }_X$ and ${\alpha }_Y$, we compute the FTE between pairs of stock markets, subsequently constructing directed networks based on FTE matrices that have been thresholded at the 20th percentile. This thresholding ensures that only the most significant edges, representing the top 80% of FTE values, are retained, thereby encapsulating the strongest directional flows of information. These networks highlight the influential relationships and interdependencies inherent in the global financial landscape. Following network construction, we perform comprehensive community detection to identify clusters of interconnected markets, accompanied by detailed statistical analyses of key network metrics such as in-degree, out-degree, betweenness centrality, closeness centrality, eigenvector centrality, hub scores, and authority scores.

4.2. Results on ${\alpha }_X=0.0$ and ${\alpha }_Y=0.0$

When ${\alpha }_X=0.0$ and ${\alpha }_Y=0.0$, the analysis corresponds to a scenario with no fractional memory, so each market’s immediate past is weighted uniformly, and distant historical values have negligible influence. Under this “memoryless” setting, the resulting network reveals direct information transfer patterns as they occur from one time step to the next.

Figure 1 presents the ordinary transfer-entropy-based network constructed using memory parameters ${\alpha }_X=0.0$ and ${\alpha }_Y=0.0$. We shall note here that this case correspondences ordinary transfer entropy.

In Figure 1, a heavily interconnected cluster at the center underscores a group of markets whose short-term movements strongly influence each other. Several large stock markets, including those from the United States and leading European indices, exhibit both high in-degree (they receive information flow from many sources) and high out-degree (they also send information to others). This reflects the dominant role of major global markets in shaping short-run price dynamics across regions. Conversely, some markets with sparse connections or negligible in-flows (like BAX or BIRS) indicate localized trading patterns or limited global integration in the immediate past-return sense, as fractional memory is not considered here.

In the “no memory” framework, the Middle Eastern markets such as the Qatar Stock Exchange (QSI) and Saudi Arabia’s TASI stand out for considerable out-degree, suggesting strong outward information flow to other indices in short-term horizons. Likewise, markets such as ISX and NDX display a high in-degree, meaning they are recipients of significant immediate information inflows, likely responding quickly to developments in global trading. From a regional perspective, the Americas (e.g., IBOV, TSX, DJIA, SPX) and core European indices (DAX, CAC, Euronext’s AEX) appear at or near the center of information transfers, while certain emerging or frontier markets (like those in Africa or smaller Asian exchanges) maintain more peripheral positions. The overall structure illustrates how, without the influence of historical memory, markets with large market capitalization and high liquidity take center stage in dictating short-run price patterns worldwide.

Figure 2 illustrates the distributions of standard network measures for the memoryless setting. Detailed network metrics are presented in

Table A2. Statistics for network metrics for ${\alpha }_X=0.0$ and ${\alpha }_Y=0.0$.

Ticker	In-Degree	Out-Degree	Betweenness Centrality	Closeness Centrality	Eigenvector Centrality	Hub Score	Authority Score	Community
MERV	11.5486	1.0581	0.0001	0.4863	0.1306	0.0018	0.0176	0
AXJO	11.7954	8.62	0.001	0.4349	0.1254	0.0138	0.0177	1
ATX	0.3996	0.4686	0.0	0.3223	0.0055	0.0011	0.0008	0
BAX	0.0	0.0	0.0	0.0	0.0	−0.0	0.0	3
DSE	7.4059	0.0	0.0	0.3877	0.0783	−0.0	0.0106	4
BFX	11.5258	7.4238	0.0004	0.4819	0.1284	0.0121	0.0183	1
BIRS	0.0	0.0	0.0	0.0	−0.0	−0.0	0.0	5
BSE	0.0	0.0	0.0	0.0	0.0	−0.0	0.0	6
IBOV	16.8543	16.0257	0.0069	0.5296	0.1752	0.0249	0.0231	1
SOFIX	0.783	0.3981	0.0	0.3223	0.0071	0.0009	0.0011	7
TSX	13.3499	17.5591	0.0341	0.4953	0.1385	0.0248	0.0178	0
CLX	13.4865	21.871	0.016	0.4908	0.1438	0.0305	0.0196	4
CSI	0.7538	0.5977	0.0	0.3767	0.0116	0.0014	0.0013	0
SSEC	1.9619	0.5435	0.0	0.3905	0.0272	0.0012	0.0035	0
SZSE	1.7436	0.6808	0.0	0.3876	0.025	0.0016	0.003	0
COLCAP	7.7302	9.6775	0.0038	0.4349	0.0937	0.0154	0.0118	1
BRVM	4.5293	0.3696	0.0	0.3715	0.052	0.0005	0.0082	4
CROBEX	7.767	6.5221	0.0037	0.3905	0.0869	0.0102	0.0122	7
CYMAIN	1.8971	0.0	0.0	0.3575	0.024	−0.0	0.0037	0
PX	2.3743	0.3844	0.0	0.3614	0.0291	0.0009	0.0046	0
OMX	7.8908	2.5128	0.0	0.4458	0.0881	0.0045	0.0134	0
EGX	5.3278	5.679	0.0055	0.3715	0.0585	0.0095	0.0079	4
GTGI	0.8455	0.0	0.0	0.3337	0.0108	−0.0	0.0018	0
OMXH	12.3623	22.9772	0.0059	0.4908	0.1382	0.0322	0.0187	1
CAC	7.4486	16.0038	0.0073	0.4421	0.0822	0.024	0.0117	1
DAX	8.7986	14.2726	0.0022	0.4533	0.0974	0.0222	0.0143	1
EUSTOXX	10.0821	15.9324	0.0061	0.4692	0.1073	0.0241	0.0157	1
ATG	2.6789	0.0	0.0	0.367	0.0306	−0.0	0.0047	4
HSI	14.9358	21.1325	0.0115	0.5144	0.1607	0.0286	0.0212	0
BUX	15.6304	25.1031	0.0156	0.5245	0.1582	0.0334	0.0211	1
ICEX	9.6922	1.3431	0.0	0.4533	0.1082	0.0021	0.0144	1
BSESN	0.0	0.0	0.0	0.0	−0.0	−0.0	0.0	9
NIMI	0.0	0.0	0.0	0.0	0.0	−0.0	0.0	10
JKSE	12.975	21.284	0.0045	0.4421	0.1361	0.0301	0.019	4
ISX	32.0838	19.7021	0.1662	0.6947	0.2563	0.0233	0.029	0
ISEQ	9.5023	12.5433	0.002	0.4572	0.1142	0.0199	0.0155	1
TA	13.9757	19.6825	0.0259	0.4999	0.144	0.0266	0.0184	4
JSE	0.5186	0.5255	0.0	0.3689	0.0098	0.0012	0.0007	0
AMGNL	17.4088	14.208	0.0066	0.4612	0.1736	0.0215	0.0217	1
KASE	2.737	0.7833	0.0	0.3992	0.0358	0.0016	0.0049	0
BKM	10.9317	13.7391	0.007	0.4147	0.1137	0.0204	0.0152	4
RGI	0.0	0.0	0.0	0.0	−0.0	−0.0	−0.0	11
BLOM	3.7989	0.7257	0.0	0.4084	0.047	0.0015	0.0062	0
VGI	0.3639	0.0	0.0	0.3201	0.0043	−0.0	0.0008	0
KLCI	0.0	0.0	0.0	0.0	−0.0	−0.0	0.0	12
MSE	8.5736	0.5215	0.0	0.4495	0.0976	0.0012	0.0141	4
MDEX	0.441	0.48	0.0	0.3689	0.0083	0.0011	0.0006	0
BIVA	0.8031	0.0	0.0	0.3378	0.0082	−0.0	0.0011	0
MNE	11.1761	9.0697	0.0001	0.4246	0.1235	0.0145	0.0175	4
MASI	0.8116	0.4858	0.0	0.3302	0.0099	0.0011	0.0014	0
NSX	17.6628	10.7819	0.0147	0.5403	0.185	0.017	0.0236	1
AEX	11.0701	19.747	0.0033	0.4776	0.1209	0.029	0.017	1
NZX	10.7673	17.6174	0.0041	0.4734	0.1199	0.0261	0.0166	4
NSE	0.0	0.0	0.0	0.0	−0.0	−0.0	−0.0	13
OSEBX	7.5291	12.8544	0.001	0.4385	0.0889	0.0204	0.0129	1
MSM	14.22	7.6249	0.0047	0.4421	0.1491	0.0114	0.0187	1
KSE	3.8488	0.3822	0.0	0.3664	0.0431	0.0009	0.0064	0
PLE	3.5629	0.3938	0.0	0.3639	0.0399	0.0006	0.0059	4
SPBLP	8.3807	6.0249	0.0004	0.3992	0.0924	0.0089	0.0128	4
PSEI	10.6868	8.8164	0.0015	0.428	0.1153	0.0145	0.0169	1
WIG	15.5141	23.3888	0.0167	0.5194	0.1627	0.0325	0.0216	4
PSI	14.4183	16.473	0.0049	0.5095	0.1533	0.0252	0.0211	1
QSI	18.4879	30.8259	0.0361	0.5403	0.186	0.0365	0.0225	0
BET	8.5272	3.4532	0.001	0.4084	0.0965	0.0054	0.0137	7
MOEX	8.8544	10.1848	0.0029	0.4084	0.0987	0.0163	0.0141	1
RTS	16.24	26.862	0.011	0.5296	0.1648	0.0355	0.0219	4
ALSIR	7.0304	0.0	0.0	0.437	0.0804	−0.0	0.0105	0
TASI	16.2957	33.8727	0.0757	0.5245	0.1608	0.0394	0.0206	0
BELEX	10.924	1.9507	0.0004	0.4692	0.1228	0.0036	0.0169	4
STI	12.8033	3.7623	0.0032	0.4421	0.1354	0.0056	0.0193	1
SAX	0.0	0.0	0.0	0.0	−0.0	−0.0	−0.0	14
SBI	0.8238	0.3654	0.0	0.3767	0.0125	0.0008	0.0015	0
JTOP	14.6537	16.548	0.0045	0.5144	0.1538	0.0251	0.0205	1
KOSPI	1.168	0.3592	0.0	0.3496	0.0153	0.0005	0.0024	0
IBEX	2.7436	1.4551	0.0	0.4022	0.035	0.0029	0.0049	0
SMI	12.5616	4.4304	0.004	0.4953	0.1344	0.0072	0.0185	1
TWII	1.0924	0.4005	0.0	0.3407	0.0136	0.0006	0.002	0
DSEI	0.0	0.0	0.0	0.0	−0.0	−0.0	−0.0	15
SET	5.8145	2.7128	0.0018	0.4212	0.0703	0.0047	0.0103	4
TUN	16.745	14.1081	0.0103	0.5296	0.1738	0.0216	0.0231	1
BIST	12.4131	1.9188	0.0032	0.4863	0.1349	0.0036	0.0181	1
ALSIU	0.0	0.0	0.0	0.0	−0.0	−0.0	−0.0	8
FTSE	2.9093	3.9813	0.0	0.3963	0.0395	0.0065	0.0056	1
DJI	17.5089	23.4561	0.0101	0.5403	0.1812	0.0322	0.0231	4
NDX	18.5381	27.9864	0.0115	0.5515	0.189	0.0365	0.0236	0
SPX	17.2169	25.8294	0.0083	0.5296	0.1799	0.0346	0.0233	1
IBC	0.7237	0.0	0.0	0.3337	0.0092	−0.0	0.0015	0
HNX	14.9999	4.6347	0.0027	0.5194	0.1613	0.0077	0.0212	1
LASIL	0.0	0.0	0.0	0.0	0.0	−0.0	0.0	2
ZSE	0.5253	0.8521	0.0	0.3147	0.0062	0.0011	0.0012	0

(Appendix A).

The in-degree and out-degree distributions both exhibit skewed shapes, with a small number of markets exhibiting very high degrees and the majority showing moderate or low connectivity. This underscores the presence of “key influencers” that receive or transmit a disproportionate share of the immediate information flow. Betweenness centrality follows a heavily right-skewed distribution, indicating that only a few indices—such as TASI, TSX, or ISX—act as vital bridges in the network, channeling short-term information between otherwise less-connected nodes. Closeness centrality is relatively more evenly distributed but still shows that certain markets can reach or be reached by others rapidly in terms of path length, fitting with the notion that major global markets are typically only a few steps away from any other market. Eigenvector centrality, which highlights nodes connected to other highly connected nodes, also displays a skewed distribution. Markets like NDX, SPX, and IBOV tend to hold relatively high eigenvector scores, indicating their integral role in the short-run dynamics of global equities. Hub and authority scores underscore these patterns as well: some markets serve as authoritative sources of immediate information, while others function more as hubs redirecting that information. As such, one can glean that in the absence of memory effects, the market network is dominated by a cluster of globally significant exchanges, complemented by pockets of lower-degree or even isolated nodes.

Figure 3 reveals the community structure for the same memoryless setting.

The Louvain algorithm detects multiple communities of varying sizes, with two or three large groups often corresponding to major regions or economic alliances, while smaller or isolated communities represent markets that do not share strong short-run informational linkages with any major group. For instance, one large community might predominantly contain established European and North American indices, reflecting their frequent mutual interactions at the daily scale. Another sizable community might combine Latin American markets (e.g., IBOV, MERV, COLCAP) and certain European outliers, indicating either correlated movements or strong capital flows in the short term.

Meanwhile, the presence of small communities or singleton nodes illustrates weaker connections that are overshadowed by larger, more liquid markets. Such nodes might be from smaller frontier economies with limited cross-listings, lower foreign investment, or regulatory barriers, all of which reduce short-run global integration in the absence of historical memory. The community-level centralities—betweenness, closeness, and eigenvector—further reveal which groups act as conduits or integrators of information. Typically, the largest communities, encompassing the global financial hubs, have elevated measures of centrality, signaling their role in unifying or bridging other groups during day-to-day market interactions. By contrast, smaller clusters with lower centralities capture niche regional relationships or markets that remain somewhat detached from the global flow of information when memory effects are not accounted for.

4.3. Results on ${\alpha }_X=0.2$ and ${\alpha }_Y=0.8$

Under ${\alpha }_X=0.2$ and ${\alpha }_Y=0.8$, the target market X places modest weight on its own short-run historical values, while the source market Y emphasizes a much longer memory of its own past. This configuration means that any information flowing from Y to X embeds relatively persistent patterns from Y, whereas X itself relies on a shorter historical window to determine its response. As a result, markets that exhibit long-lasting trends or stable historical influences in Y appear more strongly connected to their counterparts in X.

Figure 4 presents the FTE-based network constructed using memory parameters ${\alpha }_X=0.2$ and ${\alpha }_Y=0.8$.

Comparing the resulting network, one observes denser connectivity overall, as many nodes show high in-degree and out-degree. This reflects that longer-run influences in Y support more persistent cross-market relationships. From a regional perspective, one still finds major global markets—particularly from North America (e.g., SPX, NDX, TSX, and DJIA), Europe (e.g., DAX, CAC, FTSE), and parts of Asia–Pacific (e.g., HSI, AXJO)—taking on central roles. However, the heavier weighting of historical data in Y tends to amplify the importance of markets whose longer-run performance has proven influential. In the Middle East, TASI and QSI illustrate robust out-degree, suggesting that their extended-memory patterns carry substantial sway in shaping other indices’ responses. Meanwhile, several frontier or smaller markets experience augmented in-degrees, reflecting that they respond more noticeably to the persistent signals emanating from larger markets’ pasts. Overall, the partial memory for X and extended memory for Y lead to a highly interlinked structure in which historically influential markets exert a steady pull on global price dynamics.

When incorporating a longer memory component on the source side, the distributions of in-degree and out-degree (Figure 5) shift toward higher values for many markets, signifying that most indices become more globally “aware” of each other’s persistent trends. Detailed network metrics are presented in

Table A3. Statistics for network metrics for ${\alpha }_X=0.2$ and ${\alpha }_Y=0.8$.

Ticker	In-Degree	Out-Degree	Betweenness Centrality	Closeness Centrality	Eigenvector Centrality	Hub Score	Authority Score	Community
MERV	23.8295	18.6349	0.0068	0.8952	0.1158	0.0103	0.0129	0
AXJO	24.3806	25.8833	0.0003	0.8859	0.1184	0.0139	0.0131	0
ATX	17.0479	16.9105	0.0042	0.8257	0.0845	0.0093	0.0095	0
BAX	15.4132	16.0068	0.0129	0.8100	0.0747	0.0086	0.0083	1
DSE	23.1006	19.8225	0.0132	0.8952	0.1091	0.0105	0.0120	2
BFX	24.8951	25.5712	0.0027	0.9145	0.1198	0.0137	0.0133	0
BIRS	16.6482	8.8760	0.0043	0.8100	0.0826	0.0052	0.0092	0
BSE	6.0455	1.2635	0.0273	0.6208	0.0319	0.0007	0.0037	3
IBOV	28.5089	29.3099	0.0000	0.9145	0.1367	0.0157	0.0151	0
SOFIX	17.5008	12.0576	0.0083	0.8504	0.0853	0.0069	0.0095	2
TSX	27.0397	30.1651	0.0010	0.9244	0.1285	0.0160	0.0140	2
CLX	26.3866	31.7186	0.0008	0.9244	0.1269	0.0169	0.0140	0
CSI	14.5787	12.5741	0.0040	0.7802	0.0727	0.0072	0.0082	1
SSEC	18.3879	14.6345	0.0093	0.8420	0.0902	0.0080	0.0100	1
SZSE	18.9051	16.3741	0.0029	0.8420	0.0925	0.0090	0.0103	1
COLCAP	23.5117	26.7148	0.0001	0.8952	0.1141	0.0142	0.0126	0
BRVM	20.9988	17.0516	0.0036	0.8768	0.1021	0.0093	0.0114	0
CROBEX	22.6515	25.9519	0.0010	0.8768	0.1096	0.0138	0.0121	2
CYMAIN	18.8448	11.0165	0.0041	0.8590	0.0919	0.0063	0.0102	1
PX	19.5903	19.2708	0.0047	0.8859	0.0950	0.0104	0.0105	0
OMX	22.3633	23.8123	0.0015	0.8768	0.1092	0.0127	0.0122	0
EGX	22.2931	24.2948	0.0005	0.8952	0.1068	0.0128	0.0117	1
GTGI	16.9752	16.3356	0.0061	0.8177	0.0842	0.0089	0.0094	0
OMXH	25.3316	32.6319	0.0000	0.8952	0.1223	0.0174	0.0135	0
CAC	22.5260	28.8532	0.0006	0.8952	0.1091	0.0154	0.0120	0
DAX	22.6741	28.4560	0.0019	0.8768	0.1103	0.0152	0.0123	0
EUSTOXX	23.2623	29.3283	0.0000	0.8859	0.1129	0.0157	0.0126	0
ATG	18.5806	13.9281	0.0018	0.8257	0.0919	0.0078	0.0103	1
HSI	27.0638	30.6681	0.0005	0.9244	0.1293	0.0162	0.0142	0
BUX	26.7524	33.1685	0.0003	0.9047	0.1287	0.0177	0.0142	0
ICEX	23.2367	19.3588	0.0000	0.8590	0.1137	0.0105	0.0126	2
BSESN	8.3664	6.8252	0.0756	0.6492	0.0431	0.0040	0.0051	0
NIMI	10.8230	9.0531	0.0446	0.7147	0.0549	0.0052	0.0064	0
JKSE	26.5356	31.3516	0.0006	0.9145	0.1250	0.0164	0.0138	0
ISX	38.3405	29.7842	0.0001	0.9449	0.1765	0.0154	0.0191	1
ISEQ	23.6559	27.3308	0.0001	0.8952	0.1148	0.0147	0.0127	0
TA	26.9692	31.2581	0.0004	0.9346	0.1277	0.0163	0.0139	0
JSE	8.0761	0.6630	0.0038	0.6542	0.0429	0.0005	0.0049	0
AMGNL	29.3595	28.3843	0.0008	0.9346	0.1387	0.0150	0.0152	1
KASE	18.6595	20.7578	0.0063	0.8177	0.0925	0.0112	0.0104	2
BKM	24.8752	28.1414	0.0010	0.9145	0.1183	0.0149	0.0130	0
RGI	4.6282	0.4645	0.0075	0.5785	0.0241	0.0003	0.0029	0
BLOM	18.9803	17.8871	0.0006	0.8338	0.0936	0.0098	0.0104	1
VGI	15.1949	11.6532	0.0086	0.8177	0.0750	0.0067	0.0084	1
KLCI	14.6070	12.1323	0.0050	0.7802	0.0726	0.0068	0.0082	2
MSE	22.8991	19.6641	0.0000	0.8859	0.1112	0.0107	0.0124	1
MDEX	10.7759	12.1072	0.0202	0.6914	0.0566	0.0069	0.0065	1
BIVA	16.8320	15.4533	0.0046	0.8420	0.0825	0.0086	0.0092	1
MNE	24.9818	26.4698	0.0000	0.9047	0.1208	0.0144	0.0134	1
MASI	16.9303	12.5954	0.0003	0.8177	0.0835	0.0072	0.0094	1
NSX	28.1939	26.8855	0.0000	0.9047	0.1362	0.0143	0.0151	1
AEX	24.2422	31.1113	0.0003	0.8952	0.1176	0.0165	0.0130	0
NZX	24.4019	29.5698	0.0009	0.9145	0.1173	0.0157	0.0129	0
NSE	11.8019	3.5563	0.0171	0.7460	0.0598	0.0022	0.0068	1
OSEBX	22.6118	27.1882	0.0015	0.8768	0.1104	0.0145	0.0122	0
MSM	26.6253	24.9912	0.0008	0.9244	0.1257	0.0131	0.0138	1
KSE	19.4036	17.6855	0.0006	0.8504	0.0949	0.0096	0.0106	1
PLE	20.8457	17.5363	0.0027	0.8768	0.1003	0.0094	0.0111	1
SPBLP	23.3353	20.9547	0.0009	0.8859	0.1132	0.0116	0.0126	1
PSEI	24.1724	26.6070	0.0000	0.9047	0.1158	0.0141	0.0128	1
WIG	27.4338	32.9013	0.0000	0.9244	0.1316	0.0175	0.0145	1
PSI	26.4879	29.3845	0.0000	0.9047	0.1278	0.0156	0.0142	0
QSI	29.9479	37.5390	0.0000	0.9346	0.1411	0.0197	0.0154	1
BET	21.8471	23.1030	0.0022	0.8590	0.1065	0.0123	0.0119	2
MOEX	22.9358	26.1512	0.0023	0.9047	0.1106	0.0140	0.0122	0
RTS	26.9176	34.8737	0.0004	0.9047	0.1300	0.0184	0.0144	0
ALSIR	22.9470	13.9863	0.0282	0.9145	0.1083	0.0076	0.0119	3
TASI	28.2805	40.0989	0.0001	0.9449	0.1316	0.0207	0.0143	3
BELEX	24.7237	21.7411	0.0010	0.9244	0.1189	0.0118	0.0131	1
STI	25.1383	22.0872	0.0011	0.9047	0.1214	0.0120	0.0135	2
SAX	11.7874	4.2767	0.0262	0.7331	0.0602	0.0026	0.0068	0
SBI	15.4441	12.5252	0.0005	0.7593	0.0784	0.0072	0.0089	1
JTOP	26.1592	30.2878	0.0001	0.8952	0.1268	0.0162	0.0140	1
KOSPI	16.6713	12.6579	0.0005	0.7802	0.0839	0.0071	0.0095	2
IBEX	18.2839	21.6307	0.0000	0.8338	0.0904	0.0118	0.0101	1
SMI	25.1071	23.6310	0.0003	0.9047	0.1211	0.0128	0.0134	1
TWII	15.5837	15.6717	0.0014	0.7731	0.0780	0.0087	0.0088	2
DSEI	3.4073	2.6879	0.0014	0.5383	0.0180	0.0016	0.0020	3
SET	22.6547	23.0778	0.0091	0.9047	0.1079	0.0123	0.0120	3
TUN	27.9888	27.6161	0.0003	0.9145	0.1346	0.0147	0.0149	1
BIST	24.4326	21.0539	0.0043	0.9047	0.1185	0.0114	0.0131	1
ALSIU	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	4
FTSE	19.3573	22.5602	0.0018	0.8590	0.0952	0.0122	0.0106	0
DJI	27.5570	32.4339	0.0000	0.9047	0.1333	0.0172	0.0147	2
NDX	28.4418	35.7458	0.0111	0.9145	0.1368	0.0190	0.0151	1
SPX	28.2888	34.3217	0.0000	0.9047	0.1370	0.0182	0.0151	2
IBC	13.9045	2.7609	0.0194	0.7731	0.0701	0.0017	0.0079	0
HNX	26.0851	23.6499	0.0001	0.8952	0.1264	0.0126	0.0140	1
LASIL	3.0893	0.0000	0.0000	0.5507	0.0163	0.0000	0.0020	2
ZSE	1.3142	1.5052	0.0000	0.4944	0.0068	0.0009	0.0008	3

.

A sizable share of nodes is in the 20–30 or even 30+ range for in-degree and out-degree, indicating a more uniformly high level of connectivity. Betweenness centrality remains skewed, with a few nodes—often large or regionally pivotal indices—serving as principal conduits of information. Unlike the memoryless case, however, these bridges reflect a combination of short-run adaptation (through ${\alpha }_X=0.2$) and long-run influence (${\alpha }_Y=0.8$), leading to markets that integrate both immediate and historical signals to occupy key network “corridors”.

Closeness centrality is, on average, higher for many nodes than in the no-memory setting, suggesting that the longer historical perspective facilitates shorter effective paths among markets. In other words, including persistent sources of influence in Y shortens the distance in the information flow sense. Eigenvector centrality likewise shows that many major global indices—NDX, SPX, IBOV, and others—remain dominant, but with slightly elevated scores that underscore how the system now amplifies historical signals. Hub and authority measures reinforce this pattern: some indices become strong “hubs” for distributing historically grounded information (especially those with high out-degree), while others become “authorities” that aggregate influential past-driven flows from numerous markets.

With partial memory in X and long memory in Y, the Louvain algorithm (Figure 6) detects a handful of sizable communities.

Two of them are particularly large, each grouping around established financial centers and leading emerging markets whose persistent historical trends form a stable backdrop for price movements. For instance, one community may gather several European and American markets, while another might incorporate Asia–Pacific and select Latin American exchanges that share interlinked long-run dynamics. A third, moderately sized community often contains markets from Eastern Europe or the Middle East that exhibit robust long-memory connections and act as conduits between core and peripheral exchanges. Smaller communities or isolated nodes tend to be those whose local or idiosyncratic patterns (in terms of long memory) fail to resonate strongly with the global cluster.

Regionally, this distribution of communities signals that historical performance in certain areas—like the Middle East’s robust growth or Latin America’s commodity cycles—can substantially shape how those markets align. Longer-memory effects also enhance connectivity among markets that exhibit stable cyclical or macroeconomic conditions, causing them to cluster even if geographically distant. Conversely, a few frontier markets form smaller clusters, suggesting that while they do respond to broader global trends, their own extended histories may be less influential on the rest of the network. Overall, these communities highlight how partial versus extended memory perspectives interact to form cohesive clusters that reflect both current signals and historically persistent ties across the globe.

4.4. Results on ${\alpha }_X=0.5$ and ${\alpha }_Y=0.5$

When ${\alpha }_X=0.5$ and ${\alpha }_Y=0.5$, both the target market X and the source market Y balance short- and long-term historical effects in shaping how information flows. In practical terms, each market’s present values weigh its own immediate past and its deeper historical trends on roughly equal footing. As a result, the network that emerges under these equal-memory conditions displays broader connectivity than purely short-memory scenarios (e.g., ${\alpha }_X={\alpha }_Y=0$) but also tends to be more diffuse than the strong “long-memory” networks where ${\alpha }_X$ or ${\alpha }_Y$ was close to 1.

In Figure 7, the core cluster of nodes is sizable, with many indices exhibiting a mix of strong in-degree and out-degree. This reflects how balanced memory parameters encourage information transfer based not only on very recent price changes but also on moderately persistent trends, thus knitting markets together in a more cohesive global fabric.

From a regional viewpoint, the well-established North American (SPX, NDX, TSX) and European (DAX, CAC, EUSTOXX) markets remain influential, as they often do, but there is also considerable influence from emerging regions—such as Latin America’s IBOV and MERV or the Middle East’s TASI and QSI—suggesting that balanced memory permits these markets’ historical cycles to be more consistently recognized. The Asia–Pacific region (e.g., HSI, AXJO, JKSE) also displays notable degrees of connectivity, indicating that both their shorter-term fluctuations and medium-term trends feed significantly into global price movements. Overall, this balanced-memory scenario produces a relatively dense, interconnected network that highlights the interplay of immediate shocks and multi-month or even multi-year factors within stock market dynamics.

The in-degree and out-degree distributions (Figure 8) show that many markets cluster around higher values, signifying robust exchanges of information in both directions. Detailed network metrics are presented in

Table A4. Statistics for network metrics for ${\alpha }_X=0.5$ and ${\alpha }_Y=0.5$.

Ticker	In-Degree	Out-Degree	Betweenness Centrality	Closeness Centrality	Eigenvector Centrality	Hub Score	Authority Score	Community
MERV	21.6735	20.7377	0.0006	0.8879	0.1103	0.0118	0.0122	0
AXJO	22.9346	27.5025	0.0013	0.8879	0.1166	0.0155	0.0129	0
ATX	16.4016	11.7403	0.0033	0.8366	0.0850	0.0069	0.0095	1
BAX	16.3661	12.7732	0.0158	0.8287	0.0831	0.0073	0.0092	1
DSE	22.5531	21.4600	0.0037	0.8970	0.1119	0.0117	0.0123	2
BFX	23.4140	26.3327	0.0009	0.9159	0.1175	0.0146	0.0129	0
BIRS	13.6940	12.7055	0.0024	0.7700	0.0722	0.0075	0.0081	0
BSE	7.5355	1.1379	0.0212	0.6693	0.0409	0.0007	0.0047	0
IBOV	26.1214	30.5404	0.0000	0.8970	0.1321	0.0170	0.0145	0
SOFIX	16.7869	12.5655	0.0008	0.8209	0.0870	0.0075	0.0097	0
TSX	26.7951	28.7579	0.0015	0.9257	0.1327	0.0158	0.0145	0
CLX	24.6730	30.7325	0.0000	0.9064	0.1244	0.0171	0.0137	0
CSI	13.7904	9.9006	0.0149	0.7839	0.0722	0.0060	0.0081	1
SSEC	16.1219	13.3008	0.0019	0.8132	0.0837	0.0079	0.0093	0
SZSE	17.2624	15.0638	0.0057	0.8209	0.0894	0.0088	0.0099	1
COLCAP	23.2209	23.5068	0.0086	0.8879	0.1173	0.0131	0.0129	1
BRVM	19.5480	17.7083	0.0008	0.8701	0.0998	0.0101	0.0111	0
CROBEX	22.9962	24.7361	0.0010	0.8789	0.1160	0.0138	0.0129	2
CYMAIN	16.5913	12.6208	0.0017	0.8209	0.0859	0.0074	0.0096	2
PX	18.9439	17.1518	0.0028	0.8789	0.0965	0.0098	0.0107	2
OMX	21.2751	22.5841	0.0017	0.8789	0.1083	0.0125	0.0120	0
EGX	22.6414	20.4729	0.0046	0.9159	0.1122	0.0114	0.0123	1
GTGI	15.8833	13.8062	0.0010	0.8057	0.0831	0.0080	0.0093	0
OMXH	25.7057	29.4671	0.0026	0.9356	0.1280	0.0163	0.0140	0
CAC	20.9587	22.2982	0.0014	0.8701	0.1068	0.0124	0.0118	0
DAX	21.2860	21.7714	0.0029	0.8789	0.1076	0.0122	0.0119	0
EUSTOXX	21.9408	22.5068	0.0000	0.8789	0.1116	0.0127	0.0123	0
ATG	17.4546	15.4241	0.0031	0.8448	0.0902	0.0089	0.0100	1
HSI	26.3733	30.8581	0.0001	0.9257	0.1318	0.0170	0.0144	0
BUX	26.4539	30.3594	0.0004	0.9064	0.1332	0.0169	0.0146	0
ICEX	23.0480	21.8989	0.0008	0.8789	0.1178	0.0124	0.0130	2
BSESN	9.4030	1.9442	0.0295	0.6961	0.0512	0.0012	0.0059	0
NIMI	9.8285	4.5130	0.0465	0.6961	0.0528	0.0028	0.0061	0
JKSE	25.2926	26.9975	0.0010	0.9159	0.1256	0.0148	0.0139	0
ISX	35.9273	29.0853	0.0156	0.9356	0.1726	0.0157	0.0187	1
ISEQ	24.1730	25.2433	0.0005	0.9159	0.1213	0.0140	0.0133	0
TA	26.2469	28.3277	0.0006	0.9356	0.1299	0.0154	0.0142	0
JSE	4.7414	0.5214	0.0000	0.5840	0.0267	0.0004	0.0030	0
AMGNL	27.4758	33.6387	0.0009	0.9159	0.1365	0.0184	0.0150	0
KASE	17.4170	17.3906	0.0024	0.8132	0.0910	0.0100	0.0102	2
BKM	23.6437	25.0671	0.0006	0.8789	0.1192	0.0139	0.0132	2
RGI	7.3630	0.6471	0.0072	0.6542	0.0400	0.0004	0.0046	0
BLOM	18.7335	18.6220	0.0031	0.8366	0.0961	0.0106	0.0107	0
VGI	15.1833	11.5732	0.0028	0.8132	0.0788	0.0069	0.0088	2
KLCI	15.1369	11.0026	0.0064	0.7910	0.0790	0.0065	0.0088	2
MSE	20.9626	21.2340	0.0004	0.8615	0.1072	0.0119	0.0119	1
MDEX	12.6761	7.2901	0.0200	0.7633	0.0669	0.0045	0.0075	1
BIVA	17.4064	14.1787	0.0092	0.8615	0.0884	0.0081	0.0098	1
MNE	24.0804	26.5257	0.0010	0.8970	0.1220	0.0150	0.0134	2
MASI	16.3155	13.1314	0.0051	0.8531	0.0831	0.0077	0.0092	1
NSX	25.6518	29.3079	0.0003	0.9257	0.1285	0.0163	0.0141	0
AEX	23.9479	26.6351	0.0001	0.8879	0.1214	0.0149	0.0134	2
NZX	23.6362	26.9162	0.0001	0.8970	0.1193	0.0149	0.0131	0
NSE	8.8075	3.8400	0.0296	0.6798	0.0480	0.0025	0.0055	0
OSEBX	22.3079	23.0510	0.0001	0.8879	0.1131	0.0130	0.0125	0
MSM	24.8741	25.2123	0.0000	0.9257	0.1234	0.0140	0.0136	1
KSE	18.9906	17.2953	0.0026	0.8615	0.0972	0.0099	0.0108	1
PLE	20.0077	17.9505	0.0092	0.8701	0.1011	0.0100	0.0112	1
SPBLP	21.5167	23.0141	0.0022	0.8615	0.1105	0.0132	0.0123	1
PSEI	22.9957	24.3512	0.0006	0.8970	0.1162	0.0136	0.0128	1
WIG	25.7607	29.6647	0.0000	0.9159	0.1295	0.0164	0.0142	0
PSI	26.0657	31.9740	0.0000	0.8879	0.1324	0.0176	0.0146	0
QSI	29.9888	36.1823	0.0070	0.9458	0.1477	0.0198	0.0161	1
BET	20.6109	21.7552	0.0034	0.8701	0.1053	0.0121	0.0117	0
MOEX	22.1650	26.3390	0.0019	0.8879	0.1126	0.0147	0.0124	1
RTS	26.4990	34.6342	0.0000	0.9064	0.1337	0.0191	0.0147	0
ALSIR	21.4639	18.5364	0.0158	0.9064	0.1063	0.0102	0.0117	1
TASI	28.5983	32.3290	0.0001	0.9356	0.1395	0.0174	0.0153	1
BELEX	23.0769	25.0070	0.0006	0.8970	0.1165	0.0139	0.0129	2
STI	22.7062	25.2911	0.0001	0.8789	0.1157	0.0142	0.0128	0
SAX	11.8647	9.0764	0.0282	0.7374	0.0631	0.0054	0.0072	1
SBI	14.8968	12.9849	0.0010	0.7769	0.0788	0.0076	0.0089	2
JTOP	25.0044	29.7516	0.0000	0.8970	0.1260	0.0164	0.0139	0
KOSPI	15.0987	13.3327	0.0015	0.7910	0.0787	0.0078	0.0088	1
IBEX	18.2116	17.3332	0.0022	0.8448	0.0943	0.0100	0.0105	1
SMI	23.2862	25.8265	0.0010	0.8970	0.1181	0.0144	0.0130	1
TWII	15.4294	13.8413	0.0024	0.8057	0.0799	0.0081	0.0089	1
DSEI	4.4703	2.2263	0.0114	0.5724	0.0254	0.0014	0.0029	1
SET	21.2021	19.4900	0.0268	0.9064	0.1054	0.0109	0.0116	1
TUN	26.5900	33.2450	0.0000	0.9064	0.1339	0.0183	0.0147	0
BIST	21.2569	22.5329	0.0033	0.8879	0.1077	0.0125	0.0119	0
ALSIU	0.2828	0.0000	0.0000	0.4972	0.0020	0.0000	0.0002	1
FTSE	18.5912	16.7423	0.0027	0.8531	0.0960	0.0096	0.0107	1
DJI	26.0533	30.8819	0.0061	0.9257	0.1303	0.0171	0.0143	2
NDX	27.3363	33.3818	0.0001	0.9159	0.1363	0.0184	0.0149	1
SPX	26.1169	31.2465	0.0059	0.9064	0.1310	0.0172	0.0144	2
IBC	11.2863	3.5411	0.0351	0.7312	0.0608	0.0023	0.0069	0
HNX	24.1829	25.3108	0.0001	0.8970	0.1225	0.0142	0.0135	1
LASIL	6.7435	0.4427	0.0103	0.6351	0.0355	0.0003	0.0041	0
ZSE	1.3203	1.5177	0.0000	0.4972	0.0071	0.0009	0.0008	1

.

The heavier tails in these distributions point to a few standout markets that receive or send a particularly high volume of transfers, reflecting their capacity to integrate both short-run volatility and moderate-term historical signals. Betweenness centrality remains skewed, with nodes like ISX and QSI emerging as potential bridges for inter-regional links, suggesting that their combined short and mid-range memory highlights their role in connecting peripheral or frontier markets to the global system. Closeness centrality, meanwhile, trends higher overall than in the purely short-memory case, as balanced memory shortens effective “distances” in the flow of information: when markets take into account both the immediate past and moderately lagged historical signals, they become more attuned to each other’s movements, shrinking the path lengths that define closeness.

Eigenvector centrality also underscores the presence of big global indices as well as influential emerging markets in these networks. Because each market is simultaneously referencing its own medium-range memory and acknowledging other markets’ historical paths, the emphasis on connectivity resonates with the idea that large, regionally critical markets can boost the eigenvector scores of their neighbors. Hub and authority scores follow similar logic: markets that consistently send out balanced-memory signals with enough historical momentum, like TASI or NDX, emerge as hubs, while others, especially large receiving markets of these medium-term signals, gain authority. The net effect is a network structure where immediate and historical factors combine to amplify the role of key nodes.

When applying the Louvain algorithm (Figure 9), the network splits into a few sizable communities rather than many small clusters or an overwhelmingly dominant single cluster.

This pattern indicates that, under balanced memory, certain groups of markets share somewhat similar medium-term behavior and respond in parallel to global factors over time. One large community often includes major European and American indices whose mid-term economic cycles and correlated trading habits bind them closely, while another large community may be formed by Asia–Pacific and certain emerging markets that share structural similarities or cyclical patterns. A smaller cluster can sometimes comprise frontier or region-specific indices that, while influenced by the global environment, maintain enough local idiosyncrasies in their medium-range histories to differentiate them from the biggest financial hubs.

Regionally, these communities reflect how balanced memory draws out correlations in markets’ moderate historical performance while preserving the impact of shorter-term shocks. Hence, Latin American indices like IBOV and MERV might cluster with other commodity-driven markets if their medium-range trends line up consistently. Simultaneously, some Middle Eastern and Eastern European exchanges can end up either in a separate group or bridging multiple clusters, as they frequently feature specific cyclical elements—energy prices, geopolitical events, or growth surges—that resonate differently with global trends. In sum, the balanced-memory framework not only accentuates major global players but also spotlights intermediate-range commonalities across diverse regions, resulting in clear yet interconnected communities within the network.

4.5. Results on ${\alpha }_X=0.8$ and ${\alpha }_Y=0.2$

When ${\alpha }_X=0.8$ and ${\alpha }_Y=0.2$, the target market X relies predominantly on its own long historical memory, while the source market Y places more emphasis on its short-run dynamics. In practice, this means that X’s reactions to incoming information incorporate deeper, more persistent trends in its own past. Meanwhile, Y contributes relatively immediate signals, reflecting shorter-term fluctuations in its own series. The resulting network, as shown in Figure 10, tends to reveal nodes that adjust gradually given X’s heavy historical weighting—yet respond to brief shocks coming from their neighbors.

From a global standpoint, the major stock markets maintain high connectivity in both directions. However, markets whose own past exhibits stable or influential long-run trends stand out for their sizable in-degrees, since many others feed them short-term signals while they incorporate those signals in conjunction with their long memory.

Regionally, large and liquid indices like TASI, QSI, or American benchmarks such as SPX or NDX appear to have both strong receiving and sending roles. Their long-run historical patterns produce stable anchors, yet they still react to shorter-term triggers from source markets that apply relatively little memory. Notably, some frontier or emerging markets in this configuration display moderate to high out-degrees, indicating that, despite their limited reliance on historical patterns, they do send frequent short-term signals to their counterparts. The emphasis on X’s long memory also means that if a market has a deep, influential past performance trajectory, it may end up absorbing short-term transmissions from multiple neighbors, thereby revealing strong directional flows from smaller or more volatile source indices.

Under this memory configuration, the in-degree distribution (Figure 11) clusters around moderate to high values, reflecting that many nodes receive inputs from a variety of sources.

Since X weighs its own historical memory heavily, it can still be “open” to receiving numerous short-term signals from different Y markets. The out-degree distribution, meanwhile, shows that some markets—likely those with strong short-run variability—act as persistent senders of immediate signals. Betweenness centrality preserves its right-skewed shape, highlighting particular indices as conduits between localized subgroups. Markets like NSE or SAX may show elevated betweenness because their short-term fluctuations, when passed along, facilitate connections that might otherwise be overlooked if all markets were heavily memory-based.

The closeness centrality distribution tends to be high for a subset of core global markets, revealing their capability to reach or be reached by others relatively quickly, even though they rely on a heavier personal historical memory. Eigenvector centrality underscores the influence of those markets that tie into other prominent nodes—such as major American, European, or Middle Eastern indices. In this environment, authority scores for strongly memory-based target markets can climb if they receive significant short-term signals from numerous neighbors. Conversely, hub scores reveal source markets that continuously emit short-run movements into the broader network, bridging smaller or more peripheral indices with those that depend on historical memory.

When detecting communities (Figure 12), three or four main clusters usually emerge.

One large community often includes strongly memory-based indices from major financial centers (like the United States, Western Europe, and the Middle East), where the markets’ own persistent pasts lead them to anchor the group. Another comparably sized group may gather emerging or developing markets, which rely more on incoming short-run fluctuations and feed those to larger indices—yet they themselves do not hold extended historical influence in Y. A third community, nearly as large, contains additional mid-tier or frontier markets that reflect specific regional or sectoral patterns, bridging short-term changes from smaller indices to more established markets. Finally, a few outlier or nearly isolated nodes may remain in a separate or trivial community, typically because they do not share substantial short-run correlations with others or possess a historically stable pattern that does not resonate significantly at the global scale.

Regionally, this memory configuration underscores the central role of markets that have well-documented historical performance trends (since ${\alpha }_X$ is high) and remain attractive to short-run signals from smaller, more volatile neighbors (${\alpha }_Y$ is low). In emerging regions, some indices manage to link with the major hubs by sending out frequent short-term signals that these bigger nodes partially absorb. Simultaneously, the major hubs continue to exhibit strong overall connectivity, reflecting the reality that global investment flows respond to short-lived shocks—such as macroeconomic announcements or geopolitical events—even as local indices anchor themselves in their own long-run returns’ data.

4.6. Comparative Discussions

The comparative findings across different ${\alpha }_X$ and ${\alpha }_Y$ settings consistently underscore that incorporating memory into transfer entropy calculations provides a richer and more nuanced view of global stock market interdependencies compared to a purely memoryless framework. Transfer entropy, a measure rooted in information theory, quantifies the directional information transfer between systems—in this case, stock markets. By integrating memory, the model acknowledges that markets do not operate in isolation at each time step but are influenced by their historical states. This temporal dimension allows for capturing the persistence and evolution of market behaviors over time, offering a more comprehensive understanding of interdependencies.

Under the memoryless condition (${\alpha }_X=0.0$, ${\alpha }_Y=0.0$), the resulting networks tend to highlight only immediate, short-lived interactions and place greater emphasis on market shocks that rapidly dissipate. While such a snapshot can be useful for capturing quick information flows, it overlooks the persistent or cyclical patterns that often shape how indices move together over longer horizons. For instance, during economic cycles or in response to prolonged geopolitical events, markets exhibit behaviors that unfold over extended periods. The memoryless approach fails to account for these sustained influences, potentially leading to incomplete or misleading interpretations of market dynamics.

In contrast, once we introduce fractional memory—whether heavily weighted toward the target market X (as when ${\alpha }_X$ is high and ${\alpha }_Y$ is low), toward the source market Y, or balanced equally—we witness denser and more interconnected networks. This increased density reflects the cumulative effects of past interactions, allowing for a more intricate web of dependencies. Additionally, shifts in centrality measures emerge, illustrating how certain markets gain prominence in their ability to absorb and emit information over extended time spans. For example, a market with high ${\alpha }_X$ may act as a persistent influencer, shaping the behavior of other markets through sustained informational flows.

Markets with strong historical trends or stable cyclical behaviors become more prominent because the transfer entropy model now accounts for those longer-run influences, thereby revealing non-trivial relationships that remain hidden under memoryless analysis. These relationships might include enduring economic ties, such as trade partnerships or shared regulatory environments, which influence market movements over time. By capturing these sustained connections, the memory-inclusive approach provides insights into how certain markets consistently drive or respond to global financial trends, offering a deeper layer of analysis than what is possible with a memoryless framework.

This is particularly evident in how betweenness centrality, closeness centrality, and eigenvector centrality evolve from one parameter setting to another. Memory-based configurations illuminate pivotal bridging markets that tie together regions over sustained intervals, while also underscoring the global influence of major financial hubs whose persistent past returns shape the behavior of a host of smaller or emerging exchanges. For instance, a central market identified through betweenness centrality might serve as a critical conduit for information flow between disparate regions, enhancing the overall robustness and resilience of the global market network.

Critically, these findings show that transfer entropy with memory offers a more robust and realistic depiction of global stock market networks. This enhanced depiction allows practitioners to discern both short-run volatility transmission and medium- to long-run structural dependencies that may be driven by macroeconomic factors, regional alliances, or commodity cycles. Such a dual capability is invaluable for risk management, strategic investment planning, and policy formulation, as it provides a multifaceted view of market dynamics that accounts for both immediate and enduring influences.

By integrating fractional derivatives into the state-space representation, the method captures how markets assimilate past data in determining present behavior. Fractional derivatives offer a mathematical framework for modeling memory effects, enabling the transfer entropy measure to account for the weighted influence of historical states. This integration provides a superior lens for identifying key information pathways and systemic drivers, facilitating a more accurate mapping of how information and shocks propagate through the global market network over time.

5. Conclusions

This study has shown that the introduction of fractional derivatives into the transfer entropy framework yields a powerful tool, fractional transfer entropy (FTE), for revealing the multifaceted ways in which global stock markets integrate both short-lived shocks and deeply rooted historical trends. By systematically varying memory parameters, ${\alpha }_X$ and ${\alpha }_Y$, the analysis captures the full continuum of market interactions, from the highly reactive, short-memory regime that emphasizes immediate fluctuations to a balanced scenario blending transient signals with mid-range cycles, culminating in a long-memory-dominated setting where entrenched hierarchies guide capital flows. The results consistently indicate that, in contrast to traditional memoryless approaches, FTE highlights how markets are rarely shaped by instantaneous news alone; instead, they exhibit persistent influences emerging from their own past trajectories or from other markets’ structural legacies.

Under short-memory dominance (${\alpha }_X=0.2,{\alpha }_Y=0.8$), the networks reflect an environment where prices respond swiftly to fresh headlines, resulting in a more fluid yet fragile tapestry of connections, as momentary policy signals or geopolitical events can quickly redefine relationships. With balanced parameters (${\alpha }_X=0.5,{\alpha }_Y=0.5$), one observes a stable interplay that neither neglects the present nor discounts the impact of longer-term patterns, illustrating how established ties and sudden bursts of volatility harmonize to shape global finance. Shifting to a heavier weighting on long memory (${\alpha }_X=0.8,{\alpha }_Y=0.2$) reveals a global financial architecture more deeply anchored in historical capital flows and investor sentiment, offering resilience but also reducing the capacity for rapid realignment in response to short-term shocks. These findings imply that tailoring memory parameters offers profound insights into whether markets are driven predominantly by near-term flux or by the cumulative weight of history, thereby underscoring that FTE surpasses memoryless transfer entropy by providing a richer depiction of how each market’s past interacts with incoming information from others.

In future research, the time horizon of this methodology can be extended to longer periods that include major worldwide crises—such as COVID-19 or global financial stresses—to test the robustness of FTE under conditions of extreme market turmoil. Moreover, an additional avenue for investigation involves formally defining and analyzing FTE at the community level, which would capture how clusters of markets collectively transmit and absorb both ephemeral and enduring signals. By refining and expanding these approaches, scholars and practitioners can better understand the co-evolution of short-run volatility and structural inertia across multiple regions and asset classes, ultimately enhancing our capacity to anticipate shifts in systemic risk and to design more effective stabilization policies.

Despite these advances, limitations remain. The methodology relies on price returns and the assumption that all price variations are equally informative. It does not discriminate among the types of shocks, nor does it incorporate intangible drivers like investor psychology, cultural ties, or geopolitical nuances that may not register directly in asset prices. Computational complexity and data availability could challenge the approach when examining very large networks, higher fractional orders, or more complex asset classes. Addressing these issues in future work may involve incorporating macroeconomic data, sector-specific fundamentals, or other asset categories, thereby refining the scope of FTE. Exploring adaptive or time-varying memory parameters could also provide a dynamic representation of how market sensitivity and structural inertia evolve across crises or policy regimes. Considering multilayer networks might reveal how global equity markets interact with currencies, bonds, or commodities under different temporal conditions.

The importance of this study lies in its departure from conventional methods that often focus on either short-lived shocks or permanent ties. By combining fractional calculus with transfer entropy, FTE can portray how markets are neither wholly dominated by the past nor entirely tethered to the present. Rather, they navigate a continuum where immediate events and historical legacies jointly steer their behavior. Such a perspective yields a more sophisticated understanding of global financial stability and contagion channels, aiding policymakers, investors, and scholars in anticipating how the interplay of old and new forces will continue to shape the global financial ecosystem.