# Systemic resilience of networked commodities

Roy Cerqueti, Raffaele Mattera, Saverio Storani

► To cite this version:

Roy Cerqueti, Raffaele Mattera, Saverio Storani. Systemic resilience of networked commodities. Energy Economics, 2025, 143, pp.108270. 10.1016/j.eneco.2025.108270. hal-05109120

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Submitted on 12 Jun 2025

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# Systemic resilience of networked commodities

Roy Cerqueti<sup>a,b</sup> <sup>●,*</sup>, Raffaele Mattera<sup>c</sup>, Saverio Storani<sup>b,d</sup>

## ARTICLE INFO

Keywords:

Resilience

Commodity market

Network modeling

## ABSTRACT

This paper develops a class of complex network-based models whose interconnected nodes are commodities. We assume that the considered commodities are linked on the ground of the similarities of risk profiles and correlations of their returns. In this framework, we explore the resilience of the networks — i.e., their ability to absorb exogenous microscopic shocks. To this aim, we assume that high levels of resilience are associated with small variations of the community structure of the network when an exogenous shock occurs — hence, assuming that the stability of the networked commodities is measured through the maintenance of their connection levels. Shocks are conceptualized as impulsive modifications of the links among the considered commodities. The employed methodological instrument is the clustering coefficient, which is a nodal centrality measure describing the way the adjacent of the nodes are mutually connected. The theoretical proposal is empirically tested over a large set of commodities of different nature.

## 1. Introduction

### 1.1. *Outline of the problem*

Commodities represent a key instrument not only in the real economy context, but also in the environment of financial instruments. Indeed, commodity price dynamics is of fundamental importance to global economic activity (see Labys, 2017), with volatility impacting not only on exporting countries but also on consumers' behavior. In this respect, exports have a major impact on macroeconomic performance (see Cashin and McDermott, 2002; Rossen, 2015); moreover, the dynamics of commodities prices – and thus their volatility – is vital for industrial planning and economic forecasting (see Arezki et al., 2014; Rossen, 2015). There is also evidence in the literature on the role of the stock market and its returns on shaping the volatility of commodities (see Creti et al., 2013; Kang et al., 2020). Specifically, one can find that commodity prices have a negative correlation with the stock market performance. This is particularly accentuated during macroeconomic shocks, so that diversifying one's portfolio using commodities is a well-established portfolio strategy among practitioners (Kang et al., 2017), with a prominent role played by energy commodities (Liu and Tu, 2012; Khalfaoui et al., 2019; Wang et al., 2022). In addition, the volatility of some commodities promotes financial instability by encouraging a weakening of the banking system (see Eberhardt and Presbitero, 2021).

From this perspective, commodities and their interconnections are active actors in the financial and economic contexts. This statement is the ground of the present paper, where we interpret commodities as interconnected elements. Interconnections are suitably weighted through the distance among the considered commodities in terms of their financial features — with special attention to risk profiles and dynamic correlation. In so doing, commodities form an overall system that deserves investigation procedures tailored on the exploration of a unified structure composed by linked granular elements.

We here face the crucial aspect of the resilience of the system of networked commodities, i.e., the ability of the overall structure to absorb an exogenous microscopic shock occurring in a subset of its components — even, in one component. We adopt a holistic approach. Indeed, as argued in different studies (Fernandez-Aviles and Montero, 2016; Fülle and Otto, 2023; Mattera and Otto, 2024; Tabak et al., 2010), several network configurations for the financial and commodity markets are possible depending on the adopted concept of similarity.

In this line, we build different models of networked commodities on the basis of different financial/statistical characteristics of the available empirical samples.

### 1.2. *Methodological ground and contribution of the paper*

Complex networks are powerful instruments for describing frameworks with interconnected elements (see, e.g. Albert and Barabási,

<footnote><sup>a</sup> Department of Social and Economic Sciences, Sapienza University of Rome, Italy</footnote>

<footnote><sup>b</sup> GRANEM, University of Angers, France</footnote>

<footnote><sup>c</sup> Department of Mathematics and Physics, University of Campania "Luigi Vanvitelli", Italy</footnote>

<footnote><sup>d</sup> Department of Computer Science, Sapienza University of Rome, Italy</footnote>

<footnote>* Correspondence to: Department of Social and Economic Sciences, Sapienza University of Rome, Piazzale Aldo Moro, 5. 00185, Rome, Italy.</footnote>

<footnote>E-mail addresses: *roy.cerqueti@uniromal.it* (R. Cerqueti), *raffaele.mattera@unicampania.it* (R. Mattera), *saverio.storani@uniromal.it* (S. Storani).</footnote>
2002; Newman, 2003) and are widely used in finance (e.g. see Cerqueti et al., 2022b; Xi and An, 2018; Xi et al., 2021).

The literature underscores the flexibility of similarity-based networks – i.e., weighted networks having weights that increase with the similarity of the connected nodes – in uncovering undirected relationships across various contexts. In Bardoscia et al. (2021), the authors discuss similarity-based networks, such as those constructed from correlation matrices, as methodological tools to capture the degree of similarity between financial entities (e.g., institutions or assets). While these scholars primarily address the theoretical and methodological challenges of such networks, their insights are highly relevant to our study. Other relevant literature contributions include (Cerqueti et al., 2022b), which constructs a proximity matrix based on the similarity among financial indicators (e.g., total assets or leverage), and Xi et al. (2021), where the nodes represent listed mining companies. In the latter, the connections between nodes are based on the structural similarity of four types of financial indicators, and the strength of these connections reflects the degree of similarity in the companies’ financial profiles.

The versatility of similarity-based networks has also been applied to non-financial contexts. For instance, Cerqueti et al. (2021) analyzes a network based on similar behaviors among municipalities in sorting waste. Additionally, similarity-based models are widely used for link prediction in evolving networks. For example, Li et al. (2018) utilizes both current and future common neighbors based on node similarity to predict potential future connections. These networks use proximity or similarity measures to define the structure of connections, thereby highlighting relationships that are not always evident through direct interactions.

In the context of resilience analysis, several methods have been proposed to describe the ability of a network to absorb a shock (e.g. see Cerqueti et al., 2019; Liu et al., 2022).

Fundamentally, resilience measures can be of three types: topological-based, cascade-based, and shock propagation-based.

The topological-based resilience measures family is grounded on the analysis of the structure of the graph as an instrument for the definition of resilience. The idea is that a shock is nodes and/or arcs removal so that it modifies the topological structure of the network and performance measures include diameter (e.g. see Albert et al., 2000), the largest connected component’s size (e.g. see Xiao et al., 2008), the variation of some nodal centrality measures (e.g. see Ghanbari et al., 2018) and the community structure features (e.g. see Cinelli et al., 2017; Dong et al., 2018).

Cascade-based models move from the assumption that the failure of some components of the network might generate the subsequent failure of a wide part of the entire system. Applications of this approach is very popular when dealing with finance or energy systems (e.g. see Ash and Newth, 2007; Zhang et al., 2007).

In the shock propagation-based models, shocks are assumed to flow dynamically over the system moving from a node to another through the related link. This type of context is particularly useful when describing financial contagion and systemic risk (e.g. see Cerqueti et al., 2023).

In this respect, the network definition is inevitably related to the type of resilience one aims at studying. In our case, we deal with resilience to volatility shocks and dependency shocks. Therefore, we introduce volatility-based and correlation-based network structures. This approach aligns with the broader concept of similarity-based networks, where distance or proximity measures are employed to capture nuanced relationships between entities. In particular, we draw on the similarity of GARCH processes proposed in Otranto (2008), which provides a methodological foundation for constructing such proximity matrices. Moreover, we also consider the correlation coefficient to exploit correlation-based similarities in the commodity market. In this specific case, we exploit the Dynamic Conditional Correlation (DCC) model to derive the matrix of conditional correlations and a

related appropriate distance matrix, by establishing a similarity-based representation of the commodities market, akin to what is done for GARCH models. Therefore, considering both GARCH and DCC-based networks, we evaluate the systemic risk through a novel and different perspective than previous studies. In fact, this choice allows us to study the interconnections within the commodity market from two interesting viewpoints from policymakers and investors, that is both volatility and correlations. The so-obtained proximity matrices are then used for constructing the adjacency matrices of complex networks whose nodes are the considered commodities. Interestingly, we can efficiently use the main devices employed for analyzing complex networks to gain insights on the commodities market. The different network models allow a panoramic view of the reaction of the commodities system to external impulsive solicitations. More in detail, the GARCH-based network relies to shocks of a microscopic nature, involving some targeted commodities and/or specific sectors. Differently, the DCC-based network provides convincing outcomes on the resilience of the commodities framework in the presence of more structural macroscopic shocks over the dynamic correlation structure of the system. This dual perspective highlights the flexibility and robustness of similarity-based networks in addressing different layers of systemic shocks.

In our context, we define the exogenous shock as a variation of the parameters defining the proximity matrix. For the GARCH-based network, we define the exogenous shock as a variation of the volatility persistence parameter. For the DCC case, we shock the pairwise correlation coefficients in a given point in time *t* by modifying the persistency of the conditional correlation process. This introduces a new approach to evaluating systemic resilience in the commodity market, as the network is shocked by the statistical process behind its construction. Moreover, by following Cerqueti et al. (2022a), we measure the resilience of the system by observing the network topology before and after the shock, to check if a remarkable variation of the structure of the network witnesses the effect of the shock. To this end, we exploit the informative content of the nodal clustering coefficient for weighted networks (see Onnela et al., 2005), that describes the strength of their community structure. Resilience is then measured on the basis of the variation of the community structure of the network through the clustering coefficient of its nodes. The financial reasoning behind such an approach lies in the evidence that a remarkable weakening or strengthening of the connections among the commodities leads to very different financial settings. Hence, such variations have to be carefully re-evaluated by the economic and financial actors. A paradigmatic instance of this type of occurrences can be found in the behavior of the investors in optimal allocation problems, since the portfolio strategy in the unshocked case can be radically different from that needed for the shocked one. This said, a high resilience of the commodities networks – i.e., a scarce modification of the communities structures as a consequence of a shock – can be interpreted as high stability of an optimal portfolio strategy (see You and Daigler, 2013) and references therein contained.

### 1.3. Organization of the paper

The paper is organized as follows. Section 2 introduces the main ingredients of the methodological framework. In particular, it provides the data-generating processes of the financial features of the commodities, outlines the considered complex network models, and introduces the definition of the resilience measure. Section 3 presents and discusses some empirical experiments based on real data. Last section concludes and gives some lines of future research.

## 2. Methodology

The starting point of the proposed methodology is the network structure definition, which builds on the concept of similarity between commodities. In this regard, we assume that the more in some sense
similar two commodities, the stronger their connection in a weighted network representation.

For constructing the network we first need to define the main assumption on the Data Generating Processes (DGPs, hereafter) of financial quantities of interest for commodities. Then, relevant financial features are used to define similarities across commodities.

Formally, we consider N commodities whose returns are observed for T consecutive time units. The following matrix collects the commodity returns' time series:

$$ \mathbf{R} = \begin{bmatrix} r_{1,1} & \cdots & r_{1,t} & \cdots & r_{1,T} \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ r_{N,1} & \cdots & r_{N,t} & \cdots & r_{N,T} \end{bmatrix}, \quad (1) $$

where R has dimension N × T and $r_{i,t}$ represents the return of ith commodity at time t.

## 2.1. GARCH-based volatility and distance

To measure the resilience of the commodity market to volatility shocks, we define a volatility-based network. We define $\epsilon_{i,t} = r_{i,t} - \mu_{i,t}$ the residual of the return $r_{i,t}$ of commodity i at time t, being $\mu_{i,t}$ its conditional mean. We assume that the residual of the return of the generic ith commodity follows a GARCH(1,1) model, which can be written as follows

$$ \epsilon_{i,t} = \sigma_{i,t} z_{i,t} \quad (2) $$

where $(z_{i,t} : t \ge 0)$ is the innovation and it is an i.i.d. process with zero mean and unit variance, while $(\sigma_{i,t} : t \ge 0)$ is a univariate stochastic process that is independent from $z_{i,t}$ and whose form can be captured by the following:

$$ \sigma_{i,t}^2 = \xi_i + \alpha_i \epsilon_{i,t-1}^2 + \beta_i \sigma_{i,t-1}^2, \quad (3) $$

with $\xi_i > 0, 0 \le \alpha_i < 1, 0 \le \beta_i < 1$ and $\alpha_i + \beta_i < 1, \forall i = 1, \dots, N$. The GARCH parameters $\xi_i, \alpha_i$, and $\beta_i$ relate to different aspects of the volatility dynamics of commodity i. The intercept $\xi_i$ intervenes in the definition of the unconditional (i.e. long-run) volatility $\sigma_i$ and positively affects it, being $\sigma_i = \xi_i/(1 - \alpha_i - \beta_i)$; differently, $\alpha_i$ controls for the relevance of shocks in the past squared returns and it is useful for estimation purposes. In fact, the larger the $\alpha_i$ the larger the adjustment provided to the volatility estimates due to the past variation observed in the squared residuals of the returns. Finally, $\beta_i$ represents the relevance of past volatility in explaining the current volatility and it is the main feature characterizing the volatility process from a financial perspective. Indeed, a larger value of $\beta_i$ suggests the persistence of high or low volatility from a period to the following one. As the sum $(\alpha_i + \beta_i)$ approaches 1, we have that the effects of past shocks on current variance become stronger and more persistent in time.

The choice of GARCH(1,1) process is in line with a wide strand of literature on empirical applications, in that it approximates an ARCH($\infty$) process. We also point out that considering the GARCH process as the DGP for the returns volatility allows us to compute the distance between two commodities as the distance between two GARCH-processes (Caiado and Crato, 2010; Lúcio and Caiado, 2022; Otranto, 2008).

The GARCH-based distance can be obtained by noticing that the GARCH(1,1) process can be written as the following ARMA process of squared returns with AR coefficient equal to $(\alpha_i + \beta_i)$ and MA coefficient equal to $-\beta_i$:

$$ \epsilon_{i,t}^2 = \xi_i + (\alpha_i + \beta_i) \epsilon_{i,t-1}^2 - \beta_i \eta_{i,t-1} + \eta_{i,t}, \quad (4) $$

with $\eta_{i,t} = (\epsilon_{i,t}^2 - \sigma_{i,t}^2)$. As mentioned above, under stationarity and invertibility conditions of the process, $\epsilon_{i,t}^2$ in (4) admits the following AR($\infty$) representation (Otranto, 2008):

$$ \epsilon_{i,t}^2 = \frac{\xi_i}{1 - \beta_i} + \sum_{k=1}^{\infty} \pi_{i,k} \epsilon_{i,t-k}^2 + \eta_{i,t}, \quad (5) $$

where $\pi_{i,k} = \alpha_i \beta_i^{k-1}$.

Therefore, in analogy with Piccolo (1990), we can measure the dissimilarity between two GARCH processes through the Euclidean distance of their $\pi$'s coefficients. The squared distance between commodities i and j can be written as follows:

$$ \begin{aligned} \operatorname{vol} d_{i,j}^2 &= \sum_{k=1}^{\infty} (\pi_{i,k} - \pi_{j,k})^2 = \sum_{k=1}^{\infty} (\alpha_i \beta_i^{k-1} - \alpha_j \beta_j^{k-1})^2 \\ &= \alpha_i^2 \sum_{k=1}^{\infty} \beta_i^{2(k-1)} + \alpha_j^2 \sum_{k=1}^{\infty} \beta_j^{2(k-1)} - 2\alpha_i \alpha_j \sum_{k=1}^{\infty} \beta_i^{k-1} \beta_j^{k-1}. \end{aligned} \quad (6) $$

By the standard properties of the geometric series, the dissimilarity between i and j in (6) reduces to

$$ \operatorname{vol} d_{i,j}^2 = \frac{\alpha_i^2}{1 - \beta_i^2} + \frac{\alpha_j^2}{1 - \beta_j^2} - 2 \frac{\alpha_i \alpha_j}{1 - \beta_i \beta_j}. \quad (7) $$

Given a commodity i, we recall that the financial interpretation of $\beta_i$ is particularly relevant in the context of resilience assessment, as it measures the persistence of the volatility dynamics. Moreover, it is also directly related to the long-run unconditional volatility, so that an increase (decrease) of $\beta_i$ increases (decreases) the unconditional volatility. Therefore, analyzing the effects of shocks occurring on $\beta_i$ on the overall commodity market can be of relevance for understanding its resilience.

## 2.2. DCC model and correlation-based distance

The main drawback of the GARCH-based network model is that it does not account for commodities' correlations, which is important for studying commodities' network topology (Tabak et al., 2010). Moreover, although we study volatility dynamics through the GARCH process, the distance (7) does not depend on time. Another relevant way of studying commodity networks is through correlations, which we here assume to be dynamic.

The Dynamic Conditional Correlation model (Engle, 2002) takes advantage of the decomposition of the conditional covariance matrix at a given point in time t, $\Sigma_t$, into conditional standard deviations $D_t$ and conditional correlations $\Gamma_t$:

$$ \Sigma_t = D_t \Gamma_t D_t^T \quad (8) $$

Let $r_t$ be the N-dimensional vector of the returns at time t. We define $e_t = D_t^{-1} r_t$, the N-dimensional vector of devolatilized returns at time t. The DCC(1,1), model of Engle (2002) can be specified as follows:

$$ \begin{aligned} \Gamma_t &= Q_t^{-1} Q_t Q_t^{-1} \\ Q_t &= (1 - \phi - \theta) \Gamma + \phi (e_{t-1} - e_{t-1}') + \theta Q_{t-1} \end{aligned} \quad (9) $$

$$ Q_t^* = \operatorname{diag}(Q_t^{1/2}) $$

where $0 \le \phi \le 1$ and $0 \le \theta \le 1$, with $\phi + \theta < 1$; in detail, $\theta$ is a parameter governing the persistence of the correlation process, while $\phi$ represents how the realized correlations at time $t-1$ affect the estimates of the conditional correlation matrix at time t. A high $\theta$ indicates a great persistence of the correlation dynamics, while a large $\phi$ indicates that lagged realized correlations are particularly relevant to estimate the conditional correlations. Therefore, $\theta$ provides a financial interpretation in terms of the properties of the conditional correlation process, while $\phi$ is useful for modeling purposes. In fact, higher persistence in the correlations means that periods of large correlations are followed by periods with large correlations, while periods of low correlations are followed by periods of low correlations. We notice that the constraint $\phi + \theta < 1$ is useful for the stationarity of the conditional correlation process. The term $Q_t$ is the pseudo-conditional correlation matrix at time t, $\Gamma$ is the long-run correlation matrix, usually computed with the sample correlation estimator and $\Gamma_t$ is the conditional correlation matrix at time t. Interestingly, matrix $Q_t$ is called pseudo-correlation because the element on its main diagonal $q_{i,i,t}$ are close but not equal to 1 since devolatilized returns are used as input. Therefore, the correction
in (9) ensures that the diagonal elements of $\Gamma_t$ are exactly equal to 1. In the end, the off-diagonal elements of $Q_t$, namely $q_{i,j,t}$, are modeled as GARCH-type processes:

$$q_{i,j,t} = (1 - \phi - \theta)\rho_{i,j} + \phi e_{i,t-1}e_{j,t-1} + \theta q_{i,j,t-1}, \quad (10)$$

with $e_{i,t}$ the ith devolatilized return at time t and $\rho_{i,j}$ the unconditional (long-run) correlation coefficient between $e_{i,t}$ and $e_{j,t}$. Parameter estimation could be done by maximum likelihood and, as previously mentioned, the constraint $\phi + \theta < 1$ must hold. Given the parameter estimates, we can reconstruct the dynamic conditional correlation matrices for each time t. Therefore, following Mantegna (1999), we can measure the dissimilarity between two commodities i and j at time t using the correlation-based distance

$$\operatorname{corr} d_{i,j,t}^2 = 2(1 - \rho_{i,j,t}), \quad (11)$$

The static version of (11), where $\rho_{i,j,t}$ is replaced with the static unconditional correlation coefficient $\rho_{i,j} \forall t \in 1, \dots, T$, has been used by Tabak et al. (2010) for investigating the topology of the commodity market. The extension in (11) allows us to dynamically capture dissimilarities among the considered commodities.

## 2.3. Network models

We start from a graph $N = (V, E)$, where $V$ represents the set of $N$ nodes and $E$ is the set of $m$ links connecting couples of nodes.

From now on we will denote the generic node by $i$, with $i \in V$ or similarly $i = 1, \dots, N$, and the link $(i, j)$ denotes the connection between nodes $i$ and $j$. We assume that $V$ collects the considered $N$ commodities. We assume that the distance concepts in Section 2 describe the interconnections among the commodities, on the basis of the distance among the DGPs presented in Section 2.

Given $i, j \in V$, we denote the weight associated with the link between $i$ and $j$ by $w_{i,j}$. We assume that

$$w_{i,j} = \frac{1}{d_{i,j}^2 + 1}, \quad (12)$$

where $d_{i,j}$ is the generic distance between two commodities $i$ and $j$.

According to the definition of the weights in (12), a low value of $w_{i,j}$ means that node $i$ is quite dissimilar to node $j$ in terms of distance $d_{i,j}$. In contrast, a high value of $w_{i,j}$ is associated with a low distance between $i$ and $j$. We pose $d_{i,j}^2 = \infty$ when there is no link between $i$ and $j$ and, equivalently, $w_{i,j} = 0$. The weights in (12) form an $N$-squared weighted adjacency matrix $W = (w_{i,j}; i = 1, \dots, N, j = 1, \dots, N)$.

We denote $w_{i,j}$ in (12) by $\operatorname{vol}w_{i,j}$ when $d_{i,j}^2 = \operatorname{vol}d_{i,j}^2$, while we have $\operatorname{corr}w_{i,j}$ when $d_{i,j}^2 = \operatorname{corr}d_{i,j}^2$ (see formulas (7) and, (11), respectively).

## 2.4. Network shocks design

As already said above, the resilience of a network is its attitude to absorb an external shock. Therefore, the first step for defining the resilience measure is to conceptualize the shock in the network models described above. Specifically, we introduce volatility shocks (GARCH model) and dynamic correlation shocks (DCC model). The first type of shocks pertains to the persistence of dynamic or conditional volatility, whereas the second type is a shock concerning the volatility of correlation or conditional correlation.

The shock in the GARCH-based network is assumed to be a modification of the parameter that is the main driver of the volatility process' persistence. In detail, a shock is an increment (or decrement) of the volatility persistence parameter of $\beta_i$ due to external exogenous events. Thus, the shocked network is defined through the weights $\operatorname{vol}w$'s where $\beta_i$ is replaced by $\tilde{\beta}_i = \beta_i(1 + \zeta)$, where the real constant $\zeta$ is a shock of a given magnitude. The selection of the magnitude of $\zeta$ has to respect the condition $\alpha_i + \tilde{\beta}_i < 1$. This explains that $\zeta$ depends on $i$. Such a dependence will be omitted when possible. Thus, we have that the

maximum shock allowed for each commodity is $\max(\zeta_i) < (1 - \alpha_i - \beta_i)/\beta_i$. Shocks larger or equal to this threshold are not considered in the paper as they would suggest explosive behavior of the volatility dynamics.

After the shock, the proximity distance between $i$ and $j$ becomes $\tilde{d}_{i,j}^2$ — where the dependence on shocked node and $\zeta$ is conveniently omitted. The network weights become equal to

$$\operatorname{vol} \tilde{w}_{i,j} = \frac{1}{\operatorname{vol} \tilde{d}_{i,j}^2 + 1}, \quad (13)$$

with

$$\operatorname{vol} \tilde{d}_{i,j}^2 = \frac{\alpha_i^2}{1 - \tilde{\beta}_i^2} + \frac{\alpha_j^2}{1 - \beta_j^2} - 2 \frac{\alpha_i \alpha_j}{1 - \tilde{\beta}_i \tilde{\beta}_j}. \quad (14)$$

The network associated to the shocked node $i$ — in brief, the shocked network — is then $\tilde{N}_{i,\zeta} = (V, \tilde{W}_{i,\zeta})$, where $\tilde{W}_{i,\zeta}$ collects all the shocked weights in (13).

We can easily extend this definition of shocks $\zeta$ to a more global shock on an entire sector of commodities. We denote the shock in this case by $\lambda$. In particular, we apply a shock to an entire sector of the analyzed commodities. We define the generic subset $A$ where $A \subseteq V$ and we collect the shocks to the commodities in $A$ as $\Lambda = \{\lambda_a : a \in A\}$. For the sake of simplicity and without losing in generality, we take $\lambda_a = \lambda, \forall a \in A$.

As for the case of single commodity, the parameter $\beta_a$ is replaced by $\tilde{\beta}_a = \beta_a(1 + \lambda)$, for all $a \in A$. The magnitude of $\lambda$ has to respect the condition $\alpha_a + \tilde{\beta}_a < 1$, for each $a \in A$. Therefore, we take

$$\lambda < \max_{a \in A} [(1 - \alpha_a - \beta_a)/\beta_a].$$

After the shock, as discussed above, we have a variation of the proximity distance. The resulting shocked network is $\tilde{N}_{A,\lambda} = (V, \tilde{W}_{A,\lambda})$.

When dealing with the correlation-based network, we assume a shock to the parameter $\theta$ with size $\mu$, so that one has $\tilde{\theta} = \theta(1 + \mu)$. In doing so, we provide a modification of the estimated magnitude of the conditional correlations between any couple of commodities $i$ and $j$ from $\rho_{i,j,t}$, that is the value of the conditional correlation before the shock, to $\tilde{\rho}_{i,j,t}$, that is the conditional correlation after shocking the persistency parameter $\theta$ of an amount equal to $\mu$. More in detail, we estimate the conditional correlation matrix at a given time $t$ using the shocked persistency parameter $\tilde{\theta}$,

$$\begin{aligned} \tilde{\Gamma}_t &= \tilde{Q}_t^{-1} \tilde{Q}_t \tilde{Q}_t^{-1} \\ \tilde{Q}_t &= (1 - \phi - \tilde{\theta}) \Gamma + \phi (e_{t-1} e'_{t-1}) + \tilde{\theta} \tilde{Q}_{t-1} \\ \tilde{Q}_t^* &= \operatorname{diag}(\tilde{Q}_t^{1/2}) \end{aligned} \quad (15)$$

where $\tilde{\Gamma}_t$ is the shocked correlation matrix. Consequently, the pairwise correlation coefficients of a single ith commodity with all the remaining $N-1$ are shocked due to the $\mu$-shock in the persistence of the conditional correlation. The new network weights after the shock are

$$\operatorname{corr} \tilde{w}_{i,j,t} = \frac{1}{\operatorname{corr} \tilde{d}_{i,j,t}^2 + 1}, \quad (16)$$

where

$$\operatorname{corr} \tilde{d}_{i,j,t}^2 = 2(1 - \tilde{\rho}_{i,j,t}), \quad (17)$$

and $\tilde{\rho}_{i,j,t}$ be the correlation coefficient between commodities $i$ and $j$ at time $t$ implied by the shocked correlation matrix (15). The shocked network — i.e., the network associated the shock of size $\mu$ — is then $\tilde{N}_\mu = (V, \tilde{W}_\mu)$, where $\tilde{W}_\mu$ collects all the shocked weights (see formula (16) for the correlation network). Therefore, the shocked network is defined through the weights $\operatorname{corr}w$'s where $\theta$ is replaced by $\tilde{\theta} = \theta(1 + \mu)$, where the real constant $\mu$ is a shock of a given magnitude. The selection of the magnitude of $\mu$ has to respect the condition $\phi + \tilde{\theta} < 1$, so that we have that the maximum shock allowed for the entire sample of commodities is $\max(\mu) < (1 - \phi - \theta)/\theta$.
## 2.5. Definition of resilience

The proposed resilience measure is based on a specific centrality measure, the weighted clustering coefficient. We adopt the definition of this nodal centrality measure in Onnela et al. (2005). Such a device is particularly appropriate in our context because it expresses the level of inclusion of a given node in the surrounding environment of its neighbors — in the entire network, in our situation. Indeed, the clustering coefficient $C_i$ of node $i$ is given by the ratio between the existing triangles formed by the links around $i$ and the hypothetical ones, where triangles are evaluated by including also the weights of the links. Therefore, the local clustering coefficient of a node ranges between 0 and 1, being close to 1 when the cohesiveness around $i$ is strong and close to 0 in the opposite case. For the weighted network N, the clustering coefficient $C_i$ for a node $i \in V$ is defined as follows

$$C_i = \frac{2}{(N-1)(N-2)} \sum_{j,k} (w_{i,j} w_{j,k} w_{k,i})^{\frac{1}{3}}. \quad (18)$$

The clustering coefficient $\bar{C}$ of the network is given by the mean of all its nodal clustering coefficients:

$$\bar{C} = \frac{1}{N} \sum_{i=1}^{N} C_i. \quad (19)$$

The rationale behind the proposed resilience measure is to compare the clustering coefficient of the network before and after the shock; the network is highly resilient if the community structure around the nodes is preserved. We define such a resilience measure in the cases of the shocks occurring on node $i \in V$ or on the subset of commodities $A \subseteq V$ as described in Section 2.4, as follows:

$$R_{i,\zeta} = |\tilde{C}_{i,\zeta} - \bar{C}|, \quad (20)$$

$$R_{A,\Lambda} = |\tilde{C}_{A,\Lambda} - \bar{C}|, \quad (21)$$

$$R_{\mu} = |\tilde{C}_{\mu} - \bar{C}|, \quad (22)$$

where $\tilde{C}_{i,\zeta}$, $\tilde{C}_{A,\Lambda}$, $\tilde{C}_{\mu}$ are the network clustering coefficients of the shocked networks $\tilde{N}_{i,\zeta}$, $\tilde{N}_{A,\Lambda}$, $\tilde{N}_{\mu}$, respectively.

By (20), (21) and (22) and by the definition of the clustering coefficient in (18) we have that the resilience measure always ranges in $[0, 1]$; when it assumes value zero, then the community structure of the network has not been perturbed by the shock and the clustering coefficients of the shocked and unshocked networks take the same value. On the contrary, an unitary value of the resilience measure points to the highest scarcity in absorbing the exogenous shock.

# 3. Empirical analysis

## 3.1. Data

For the empirical analysis we consider daily returns of $N = 16$ commodities from October 4th 2005 to November 18th 2022. Commodities can be classified under different categories and, in the empirical analysis, we selected some of the most economically and financially impactful categories of commodities (Tabak et al., 2010), namely energy commodities (natural gas; gasoline, diesel, and WTI); metal commodities (aluminum, copper, gold, nickel, silver, and zinc); and agricultural (coffee, corn, cotton, sugar, soybeans, and wheat).

Fig. 1 shows the time series of the commodities returns; we note some specific subperiods when the volatility level is quite high. The more remarkable oscillations occur during major economic crises such as subprime mortgages, sovereign debt, or the more recent COVID-19 crisis. It is interesting to note the exceptionally high returns volatility associated with natural gas, wheat, and nickel during the Russo-Ukrainian war.

<caption>Table 1</caption>

Summary of the key descriptive statistics for the daily returns of all the considered commodities. Returns are expressed in %.

<table><thead><tr><th>Commodity</th><th>T</th><th>Mean</th><th>sd</th><th>Median</th><th>Min</th><th>Max</th><th>Skew</th><th>Kurtosis</th></tr></thead><tbody><tr><td>Natural gas</td><td>4424</td><td>-0.02</td><td>3.42</td><td>0.00</td><td>-30.05</td><td>38.17</td><td>0.49</td><td>8.56</td></tr><tr><td>Gold</td><td>4424</td><td>0.03</td><td>1.14</td><td>0.02</td><td>-9.82</td><td>8.62</td><td>-0.34</td><td>5.62</td></tr><tr><td>WTI</td><td>4424</td><td>0.02</td><td>2.67</td><td>0.03</td><td>-28.22</td><td>31.96</td><td>0.12</td><td>19.64</td></tr><tr><td>Soybeans</td><td>4424</td><td>0.02</td><td>1.53</td><td>0.05</td><td>-11.09</td><td>6.48</td><td>-0.80</td><td>5.34</td></tr><tr><td>Corn</td><td>4424</td><td>0.03</td><td>1.88</td><td>0.00</td><td>-26.86</td><td>12.76</td><td>-0.99</td><td>14.28</td></tr><tr><td>Copper</td><td>4424</td><td>0.02</td><td>1.75</td><td>0.00</td><td>-11.51</td><td>11.64</td><td>-0.12</td><td>3.90</td></tr><tr><td>Silver</td><td>4424</td><td>0.02</td><td>2.09</td><td>0.03</td><td>-19.55</td><td>12.20</td><td>-0.82</td><td>6.84</td></tr><tr><td>Aluminum</td><td>4424</td><td>0.01</td><td>1.49</td><td>0.00</td><td>-11.40</td><td>9.31</td><td>-0.21</td><td>3.24</td></tr><tr><td>Zinc</td><td>4424</td><td>0.02</td><td>2.21</td><td>0.00</td><td>-31.69</td><td>20.99</td><td>-0.90</td><td>20.12</td></tr><tr><td>Diesel</td><td>4424</td><td>0.01</td><td>2.23</td><td>0.02</td><td>-24.75</td><td>12.36</td><td>-0.80</td><td>10.31</td></tr><tr><td>Nickel</td><td>4424</td><td>0.01</td><td>2.66</td><td>0.00</td><td>-31.31</td><td>49.14</td><td>1.05</td><td>40.13</td></tr><tr><td>Wheat</td><td>4424</td><td>0.02</td><td>2.10</td><td>0.00</td><td>-11.30</td><td>19.70</td><td>0.23</td><td>3.55</td></tr><tr><td>Sugar</td><td>4424</td><td>0.01</td><td>2.06</td><td>0.00</td><td>-12.37</td><td>13.06</td><td>-0.05</td><td>3.12</td></tr><tr><td>Gasoline</td><td>4424</td><td>0.01</td><td>2.69</td><td>0.05</td><td>-38.54</td><td>22.40</td><td>-1.17</td><td>21.50</td></tr><tr><td>Coffee</td><td>4424</td><td>0.01</td><td>1.99</td><td>0.00</td><td>-11.09</td><td>11.79</td><td>0.15</td><td>1.83</td></tr><tr><td>Cotton</td><td>4424</td><td>0.01</td><td>1.84</td><td>0.00</td><td>-27.29</td><td>10.53</td><td>-0.95</td><td>14.12</td></tr></tbody></table>

Specifically, during the subprime mortgage crisis (2007–2008), the collapse of the U.S. housing market and the resulting financial instability triggered a recession that significantly reduced industrial activity and energy demand. This led to a sharp drop in oil prices, while metals like copper and aluminum suffered from diminished industrial use. Agricultural commodities saw mixed effects; essentials like wheat remained relatively stable due to inelastic demand, while others faced price declines. Looking at the sovereign debt crisis (2010–2012) in the Eurozone caused uncertainty in financial markets, depressing demand for metals and energy as industrial activity in Europe slowed. However, agricultural commodities maintained a degree of resilience, supported by growing demand in emerging economies like China and India. The COVID-19 pandemic (2020) disrupted global supply chains and caused unprecedented volatility. Energy commodities like oil plummeted as lockdowns reduced transportation and industrial activity, while metals showed diverging trends: gold soared as a safe haven, while industrial metals initially declined but recovered with economic stimulus efforts. Agricultural commodities experienced price surges due to supply chain bottlenecks and heightened demand for staple foods. Finally, the Russo-Ukrainian War (2022) has driven extreme volatility, particularly for natural gas, nickel, and wheat. Europe's dependence on Russian energy spiked gas prices, nickel markets reacted to supply threats from Russia, and wheat prices surged as Ukraine, a global breadbasket, faced export disruptions. These crises underscore the sensitivity of commodity markets to both economic and geopolitical shocks.

Table 1 provides an overview of the main descriptive statistics for the selected sample of commodities returns. The distributions of the daily returns are quite heterogeneous in terms of variability and shape. Indeed, despite all commodities show almost zero-mean returns over the considered time period, some differences in terms of volatility can be highlighted. For example, natural gas is the commodity showing the largest standard deviation, along with gasoline and WTI. Therefore, it seems that energy commodities show the largest volatility compared to the others. Viceversa, gold is confirmed as a safe haven asset with the lowest standard deviation. Furthermore, gold is the commodity with the lowest minimum and maximum returns, while energy commodities like natural gas and WTI show much larger variations. Moreover, most of commodities show negative skewness and large kurtosis, thus suggesting heavy-tailed returns distribution.

## 3.2. Networked commodities

To investigate the resilience of the selected commodities to volatility shocks, we need to define the volatility network structure. As explained in Section 2, the network is based on the proximity matrix associated to the distance in (7), which can be inferred by the GARCH(1,1)
<figure><img src="image_4.png" /><figcaption>Fig. 1. Daily commodity return's time series. Returns are expressed in %.</figcaption></figure>

parameters estimated for each commodity shown in Table 2. Table 2 also shows the DCC parameters $\phi$ and $\theta$, which are common to all the sets of considered commodities.

Considering GARCH parameters, Table 2 highlights that most commodities show persistent volatility dynamics, with $\beta$ larger than 0.9. Notable exceptions are WTI and gasoline, which present values below such a threshold. Nickel is the commodity with the lowest level of persistence.

Fig. 2 shows our basic network in which the nodes represent our 16 raw materials and the thickness of the arcs is a mirror of the connection strengths $w$'s. The distances of the nodes are determined by the Fruchterman-Reingold (FR) algorithm (see Fruchterman and Reingold, 1991), which is designed for weighted graphs and works by trying to position the nodes so that the most strongly connected nodes are close to each other in space. The basic idea is that stronger connections correspond to stronger attractive forces, leading nodes to be placed closer together. The size of the nodes is given by the clustering coefficient level normalized to the data in Table 3.

In particular, Table 3 shows that the raw materials are strongly related to each others, with a local clustering coefficient very high in all the cases. This means that this category of commodities forms a strong community in terms of volatility similarity. Fig. 2 leads to an efficient visualization of the structure of the network, with a clear

<table><caption>**Table 2**
Estimated DCC and GARCH parameters.</caption><thead><tr><th rowspan="2">Commodity</th><th colspan="3">GARCH</th><th colspan="2">DCC</th></tr><tr><th>α</th><th>β</th><th>ξ</th><th>φ</th><th>θ</th></tr></thead><tbody><tr><td>Natural gas</td><td>0.07</td><td>0.92</td><td>0.11</td><td>0.01</td><td>0.97</td></tr><tr><td>Gold</td><td>0.04</td><td>0.95</td><td>0.01</td><td>0.01</td><td>0.97</td></tr><tr><td>WTI</td><td>0.10</td><td>0.89</td><td>0.09</td><td>0.01</td><td>0.97</td></tr><tr><td>Soybeans</td><td>0.06</td><td>0.93</td><td>0.03</td><td>0.01</td><td>0.97</td></tr><tr><td>Corn</td><td>0.06</td><td>0.93</td><td>0.05</td><td>0.01</td><td>0.97</td></tr><tr><td>Copper</td><td>0.05</td><td>0.95</td><td>0.02</td><td>0.01</td><td>0.97</td></tr><tr><td>Silver</td><td>0.05</td><td>0.94</td><td>0.04</td><td>0.01</td><td>0.97</td></tr><tr><td>Aluminum</td><td>0.04</td><td>0.95</td><td>0.02</td><td>0.01</td><td>0.97</td></tr><tr><td>Zinc</td><td>0.04</td><td>0.96</td><td>0.01</td><td>0.01</td><td>0.97</td></tr><tr><td>Diesel</td><td>0.09</td><td>0.89</td><td>0.07</td><td>0.01</td><td>0.97</td></tr><tr><td>Nickel</td><td>0.10</td><td>0.88</td><td>0.11</td><td>0.01</td><td>0.97</td></tr><tr><td>Wheat</td><td>0.06</td><td>0.93</td><td>0.05</td><td>0.01</td><td>0.97</td></tr><tr><td>Sugar</td><td>0.04</td><td>0.95</td><td>0.04</td><td>0.01</td><td>0.97</td></tr><tr><td>Gasoline</td><td>0.11</td><td>0.86</td><td>0.20</td><td>0.01</td><td>0.97</td></tr><tr><td>Coffee</td><td>0.04</td><td>0.95</td><td>0.07</td><td>0.01</td><td>0.97</td></tr><tr><td>Cotton</td><td>0.04</td><td>0.95</td><td>0.03</td><td>0.01</td><td>0.97</td></tr></tbody></table>

interpretation of the graph. Indeed, as already mentioned above, the FR algorithm takes links weights and clustering coefficients into account
<figure><img src="image_5.png" /><figcaption>Fig. 2. GARCH-based network — no shocks.</figcaption></figure>

<table><caption>**Table 3**
Clustering coefficient of the GARCH-based network without shock.</caption><thead><tr><td>Commodity i</td><td>Clustering coefficient C<sub>i</sub></td></tr></thead><tbody><tr><td>Natural gas</td><td>0.84</td></tr><tr><td>Gold</td><td>0.84</td></tr><tr><td>WTI</td><td>0.75</td></tr><tr><td>Soybeans</td><td>0.86</td></tr><tr><td>Corn</td><td>0.86</td></tr><tr><td>Copper</td><td>0.85</td></tr><tr><td>Silver</td><td>0.86</td></tr><tr><td>Aluminum</td><td>0.83</td></tr><tr><td>Zinc</td><td>0.81</td></tr><tr><td>Diesel</td><td>0.78</td></tr><tr><td>Nickel</td><td>0.75</td></tr><tr><td>Wheat</td><td>0.86</td></tr><tr><td>Sugar</td><td>0.84</td></tr><tr><td>Gasoline</td><td>0.72</td></tr><tr><td>Coffee</td><td>0.81</td></tr><tr><td>Cotton</td><td>0.84</td></tr><tr><td>Avg.</td><td>0.82</td></tr></tbody></table>

during the physical simulation, by hypothesizing that attractive and repulsive forces act between the nodes. We observe that four nodes are rather isolated from the others: gasoline, nickel, diesel, and WTI. Differently, gold and wheat seem to be in the center of the network.

In the case of DCC, we construct the correlation network based on the estimated conditional correlation matrices, using the proximity measure in (11). In this case, we obtain *T* adjacency matrices, following (16). To study resilience, we shock the *θ* parameter associated with the DCC which leads to *T* shocked networks. Obviously, parameters *φ* and *θ* are constant over the commodities in the DCC case.

**Fig. 3** — Panel (a) shows the temporal evolution of the local clustering coefficients of the *N* commodities in the network, implied by the estimated correlations with the DCC, given the persistence level shown in Table 3. Fig. 3 — Panel (b) shows the temporal trajectory of the global clustering coefficient. We can note that the local clustering coefficients and the global one have rather similar patterns.

In particular, Fig. 3 — Panel (a) highlights that commodities became more interconnected during the 2008 financial crisis, afterward an

initial bottom peak occurred when the real estate bubble burst. Moreover, peaks of the clustering coefficient are observed in both directions in a short time window during COVID-19 and the Russo-Ukrainian war. These different trends suggest different responses from agents operating in the commodities market. Indeed, during the subprime mortgage crisis, a peak in the clustering coefficient is determined by an increase in commodity correlations. In the case of COVID-19 and the Russo-Ukrainian war, however, we observe greater uncertainty among operators, leading to some instability in the network. Moreover, in 2020 we observe an additional spike in the global clustering coefficient, in correspondence of the negative future oil prices. It is known, indeed, that this event hurt the commodity correlations (Ahonen et al., 2022; Corbet et al., 2021a,b).

### 3.3. Resilience analysis

In the next subsections, we present two different shocks, about volatility dynamics and conditional correlations, as explained in Section 2.

#### 3.3.1. Volatility shocks

We first performed shocks of different entities on the volatility-based network. Fig. 4 shows the resilience of the networked commodity market when each node is shocked according to its own feature. Indeed, the maximum level of shocks magnitude can differ between different commodities, as explained in Section 2.4. For example, Zinc has the lowest maximum amount of shock, while Gasoline tolerates the maximum shock possible among all the commodities. The remaining commodities, however, share similar amounts of the maximum shock of persistence.

Fig. 4 highlights that when the shock is negative, i.e. we observe a reduced volatility persistence of the *i*th commodity, the network turns out to be less disrupted. Differently, when increasing the volatility persistence, we have a relevant change in the structure of the network. Therefore, the function summarizing how the size of the shock affects the resilience is either “U” or “V” shaped.

In particular, we notice that some agricultural commodities, such as Cotton, Corn and Wheat, have “U-shaped” resilience functions while
<figure><img src="image_6.png" /><figcaption>Fig. 3. Panel (a) displays the level of the local clustering coefficient for commodities. Panel (b) shows the time series of the global clustering coefficient of the network.</figcaption></figure>

most energy and metals commodities are characterized by “V-shaped” curves. Interestingly, similarities can also be highlighted between commodities belonging to different sectors. For example, Gold, Silver, and Sugar show similar patterns, in that the network is stable to a low amount of persistence shocks, while the resilience index spikes for increasing shocks. Another example is Nickel, showing a “V-shaped” curve similar to those of Diesel and WTI.

In sum, Fig. 4 shows that in scenarios where the volatility persistence differs significantly from the original value, we observe an increased difference in the network structure. However, increasing the $\beta$ coefficient leads to a larger difference in the network structure compared to a reduction of the persistency in the GARCH dynamics.

While analyzing microscopic shocks on single commodities provides detailed information about specific market behavior, understanding the impact of volatility persistence shocks within sectors enables a more comprehensive assessment of market resilience and systemic risks. Indeed, studying shocks in volatility persistence within a given sector offers valuable insights into the broader dynamics of commodity markets, surpassing the narrow focus on individual commodity returns behaviors.

On the one hand, by studying sector-level shocks, researchers can better comprehend spillover effects and contagion risks, which are vital for effective risk management and policy interventions. Indeed, sectors often exhibit stronger interconnectedness, with commodities serving as crucial inputs across multiple industries. A shock in volatility persistence within a sector can propagate through supply chains, affecting related industries and amplifying its overall impact on the economy. On the other hand, sector-level analysis allows for a more nuanced understanding of market dynamics, considering shocks driven by factors such as regulation, and technological advancements that

are specific to only sectors of commodities. Moreover, considering macroeconomic implications, the analysis of common shocks in the energy sector provides insights into how fluctuations in energy markets might impact the overall economy.

Fig. 5 highlights three different shocks, one for each commodity sector considered in the paper. In particular, in panel (a) of Fig. 5 we display shocks for the energy sector, while in panel (b) we display shocks for the metals sector, and panel (c) the shocks for the agriculture sector. The y-axis represents the percentage change in resilience measure and the shaded band corresponds to the standard deviation of the resilience for each sector.

Let us consider shocks to all the energy commodities simultaneously first. Fig. 5 — Panel (a) illustrates that inducing a shock across the entire energy market produces a symmetric effect on the resilience measure. This symmetry implies that both a reduction and an increase in the volatility persistence of energy commodities lead to perturbations in the network structure. It is important to highlight that the relatively small magnitude of perturbation is a reflection of the modest scale of the shock under consideration. For example, let us examine a 0.5% increase or reduction in the volatility persistence dynamics driven by $\beta$, as well as in the unconditional volatility given by $\xi/(1-\alpha-\beta)$. As the shock has a small magnitude, we expect a similarly small perturbation. In particular, the maximum shock magnitude is determined by the minimum among the maximum shocks allowed among the commodities within the sector under consideration. Specifically, the threshold for the energy sector is set by the commodity Natural Gas, which sets the maximum level at 0.009.

We then consider Panel (b) of Fig. 5, where a shock common to the metals commodities is assumed. When we shock the metals sector, we observe a similar behavior compared to the energy sector; in fact,
<figure><img src="image_7.png" /><figcaption>Fig. 4. Resilience (y-axis) given a shock $\bar{\beta}_i$ (x-axis). The maximum shock allowed for each commodity $i$ is $max(\zeta_i) < (1 - \alpha_i - \beta_i)/\beta_i$, thus ensuring that $\alpha_i + \beta_i < 1$. Shocks larger than this threshold are not considered as they would suggest explosive behavior of the volatility.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5. We here show three different shocks. In panel (a) we display shocks for the energy sector. In panel (b) we display shocks for the metals sector. In panel (c) we display shocks for the agriculture sector. The y-axis represents the percentage change in the resilience measure. The shaded band corresponds to the standard deviation of the resilience for each sector.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Plot of the global clustering coefficient of the networks over time. The time series in black represents the value of the global clustering coefficient when the network is not shocked. The time series in red represents the value of the global clustering coefficient when the network is shocked at its minimum. The time series in blue represents the value of the clustering coefficient over time when the network is shocked at its maximum. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

resilience follows an almost identical pattern. The main difference is due to the limit imposed by the constraint related to the maximum shock associated with the features of the individual commodities. In this sense, the energy sector can cause greater disruptions compared to the metals sector. Indeed, the maximum shock allowed in metals sector is given by the commodity Zinc, which sets the threshold for the entire sector to 0.002. Moreover the metal sector impact on resilience in absolute term is lower respect to the energy sector.

Finally, we study the resilience of the commodity market to shocks in the agricultural commodities, as shown in Fig. 5 — Panel (c). While in the first two cases we see a similar pattern, when shocking the agricultural sector, we observe a different behavior or shape. In the two previous cases, we observe a linear relationship between the applied shock and the perturbation of the network. However, for the agricultural sector, the effects on the network appear to be more smoothed out. The maximum shock allowed in metals sector is given by the commodity Sugar, which sets the threshold for the entire sector to 0.008. When shocked, the agricultural sector perturbs the network to a lesser extent than the energy sector but to a greater extent than the metal sector.

The Fig. 5 also allows us to hypothesize about the importance of sectors within the commodity network. As observed, the energy sector appears to have the most significant impact on the stability of the network, followed by the agricultural sector, and finally the metal sector. According to this interpretation, shocks related to energy induce greater uncertainty among investors, rendering the commodity market more unstable.

The changes of network resilience and, consequently, of the network structure, are conditioned by several factors. The increase/decrease of the persistence parameter $\beta$ causes heterogeneous effects on the network. This occurs both in the case of positive and negative shocks either for shocks on specific commodities as well as for shocks on a specific sector. We notice that, in general, a large positive shock of $\beta$ leads to remarkable variations in the resilience level. Such variations are more evident for some commodities — like Gasoline and Coffee.

These $\beta$ shocks are reflected on the global clustering coefficient and consequently on the adopted resilience measure. For example, in the case of commodity-specific shocks, an increase in the volatility persistence could cause a decrease in the level of the global clustering coefficient because the $N-1$ existing links in the network weaken. Conversely, in the case of sector-based shock, an increase in $\beta$ could cause either an increase or decrease in the level of the global clustering

coefficient. On one side, the links between the commodities within the shocked sector and those belonging to other — non-shocked — sectors weaken. On the other side, we have that the connectedness between the commodities belonging to the sector under shock strengthens and a more cohesive cluster could be created in the network structure. Therefore, in this second case, the exact effect of the persistence shock on the network structure depends on the complex interaction between the increase in $\beta$, the structure of the financial network, and the dynamics of correlation between assets. In general, an increase in $\beta$ could favor greater cohesion among similar commodities, but other factors that could influence the clustering coefficient could also be considered.

### 3.3.2. Correlation shocks

Differently from what was done in the previous section, here we shock the network based on correlations between commodity returns. In this specific case, we modify the correlation by shocking the parameter $\theta$, as reported in Section 2.4. By definition of the DCC methodology, by shocking parameter $\theta$ leads to a time-dependent modification of the commodity networks.

Fig. 6 shows the level of the global clustering coefficient when the commodity network is not shocked (time series in black) when we apply the minimum shock of parameter $\theta$ (time series in red), which is $\mu = -0.010$, and finally when we apply the maximum allowed shock (time series in blue), which is $\mu = 0.016$. We recall that the maximum shock allowed is $max(\mu) < (1-\phi-\theta)/\theta$. It is quite interesting to observe the dynamics over the time of the global clustering coefficient. In fact, while for the minimum shock we do not see significant deviations in the global clustering coefficient of the unshocked network, the same cannot be said if we apply the maximum shock allowed. Despite the two shocks not being very high, the variations of the global clustering coefficient are considerable. In particular, we observe that the value of the global clustering coefficient of the network shocked to its maximum possible is consistently below than that of the unshocked network. The fact that an increase in the persistence of conditioned dynamic correlation leads to a decrease in the global clustering coefficient suggests that there is a weakening of connections among all possible triangulations of the network. This implies that, on average, the local clustering coefficient weakens, which is indicative of a generalized decrease of the weights $\omega_{ij}$'s with respect to their unshocked values. Indeed, a lower clustering coefficient might indicate a reduced tendency for commodities to move together (on average) in strongly correlated groups. This
<figure><img src="image_10.png" /><figcaption>Fig. 7. Panel (a) visualizes the time series percentile change of resilience for four different levels of shocks. The blue line represents the maximum allowed shock, the red line represents the minimum shock, the orange line represents the shock identified at the 25th percentile of the shock distribution, and the green line represents that at the 75th percentile. Panel (b) visualizes the percentage change of the average level of resilience for each analyzed shock level. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

might suggest greater diversification in portfolios, where assets have less tightly correlated relationships. Furthermore, the returns of certain commodities may be more or less correlated over time compared to others, leading to a more heterogeneous distribution of the network's conditioned dynamic correlations. This could be due to sector-specific factors, economic cycles, or specific events.

These results are confirmed by Fig. 7. In Panel (a) of the figure, we display the time series of the percentage variations in resilience over time for four different shock levels, while in Panel (b) we visualize the average percentage variations of the resilience based on the applied shock. Specifically, Fig. 7 — Panel (a) highlights the percentage variations in resilience over time for the maximum shock level (0.016) in blue, minimum shock level (−0.010) in red, and for the 25th and 75th percentiles of the shock distribution, namely Q1 in orange (−0.004) and Q3 in green (0.010). Panel (a) thus depicts how the level of resilience evolves over time in response to the aforementioned shocks.

As explained in Section 2, the DCC model generates multiple networks corresponding to the considered *T* time periods. Panel (a) therefore highlights the percentage change in resilience at each time *t*. We can observe that the greatest network disruption occurs when applying the maximum shock. Furthermore, comparing the Q3 and the minimum possible shock, which are symmetric (i.e., ±1%), we notice that the positive shock induces a greater disruption to the network compared to the negative shock. This result appears consistent with that shown in Fig. 6. Indeed, positive shocks seem to have a more pronounced effect on the level of the global clustering coefficient. In this case, we can assert that an increase in conditional dynamic persistence leads to a redistribution of weights within the structure, making it more homogeneous and consequently causing a weakening of the global clustering

coefficient, which is reflected in the adopted resilience measure.

The curve in Panel (b) confirms what has been previously stated, namely that a positive shock to the persistence of dynamic conditional correlations results in a more pronounced disturbance of the network. Such a curve is *V*-shaped, meaning that positive shocks to the correlation persistency generate larger perturbation to the network structure, as happens with most commodity shocks in the volatility dynamics. These results are relevant for both financial analysts and the industrial sector, as we provide valuable insights to understand and prevent potential generalized shocks in the commodity market.

## 4. Robustness analyses

In what follows we provide additional analyses to investigate the robustness of our main results. First, we consider if our results are robust to the distributional assumptions on the returns. To this aim, we relax the normality assumption and consider GARCH models with heavy-tailed distributed returns. Second, we consider shocks to the unconditional correlation matrix, rather than the conditional covariance estimated with DCC.

### 4.1. Volatility shocks with heavy-tails

The GARCH model estimated with a t-Student distribution serves as a robustness check against possible misspecification of the model due to the normal distribution assumption for the innovations. By relaxing the normality assumption and adopting the t-Student distribution, the model accounts for the heavy tails observed in the return series, capturing the extreme movements frequently present in commodity
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** GARCH-based network under the assumption of heavy-tailed returns — no shocks. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<table><caption>**Table 4**
Estimated GARCH parameters and the Clustering coefficients of the associated network.</caption><thead><tr><th rowspan="2">Commodity</th><th colspan="3">GARCH</th><th rowspan="2">Clustering coefficient C<sub>i</sub></th></tr><tr><th>α</th><th>β</th><th>ξ</th></tr></thead><tbody><tr><td>Natural gas</td><td>0.07</td><td>0.92</td><td>0.14</td><td>0.85</td></tr><tr><td>Gold</td><td>0.03</td><td>0.96</td><td>0.01</td><td>0.84</td></tr><tr><td>WTI</td><td>0.09</td><td>0.90</td><td>0.07</td><td>0.76</td></tr><tr><td>Soybeans</td><td>0.05</td><td>0.94</td><td>0.02</td><td>0.89</td></tr><tr><td>Corn</td><td>0.06</td><td>0.92</td><td>0.05</td><td>0.86</td></tr><tr><td>Copper</td><td>0.04</td><td>0.96</td><td>0.01</td><td>0.87</td></tr><tr><td>Silver</td><td>0.03</td><td>0.96</td><td>0.02</td><td>0.86</td></tr><tr><td>Aluminum</td><td>0.05</td><td>0.95</td><td>0.02</td><td>0.89</td></tr><tr><td>Zinc</td><td>0.03</td><td>0.97</td><td>0.01</td><td>0.83</td></tr><tr><td>Diesel</td><td>0.07</td><td>0.92</td><td>0.04</td><td>0.83</td></tr><tr><td>Nickel</td><td>0.05</td><td>0.93</td><td>0.08</td><td>0.88</td></tr><tr><td>Wheat</td><td>0.05</td><td>0.94</td><td>0.04</td><td>0.89</td></tr><tr><td>Sugar</td><td>0.03</td><td>0.96</td><td>0.03</td><td>0.85</td></tr><tr><td>Gasoline</td><td>0.08</td><td>0.90</td><td>0.14</td><td>0.80</td></tr><tr><td>Coffee</td><td>0.03</td><td>0.96</td><td>0.04</td><td>0.81</td></tr><tr><td>Cotton</td><td>0.05</td><td>0.94</td><td>0.04</td><td>0.89</td></tr></tbody></table>

returns (Gorton et al., 2013). The estimated parameters α and β, under t-student GARCH, shown in Table 4, indicate the persistence of high volatility, and the results remain consistent with those obtained under the normal specification. The parameters are slightly higher than those seen in Table 3; this difference may be attributed to the t-Student distribution's ability to capture extreme values.

Fig. 8 shows the unshocked version of the t-student GARCH-based network. Under these conditions, the network is very similar to the one generated under normal returns, except for the commodity Nickel, which appears to have a much larger size. Given the graphical representation of the network, this indicates that Nickel has a significantly higher local clustering coefficient, highlighting a greater cohesion among its adjacent nodes.

In Fig. 9, we observe similar behaviors when individual commodities are shocked. Notably, the maximum shocks are limited compared to the normalized case. This is due to the higher value of the β parameter in the case of heavy tails, which imposes a more stringent restriction on

the maximum shocks. Despite this, we observe extremely similar patterns in terms of network resilience. This similarity is further confirmed in Fig. 10, where shocks across different sectors are visualized. The most significant differences are observed in the energy sector, where the network appears more resilient when heavy tails are considered. However, this holds only for negative shocks, while for positive shocks the opposite applies. For agriculture sector, we find that the network is less resilient to negative shocks than to positive one. This suggests that the distributional assumption may affect the network's response to shocks in specific sectors. However, the main results are not affected much by the distributional assumption. This reinforces the robustness of our conclusions.

## 4.2. Unconditional correlation shocks

To test the robustness of our findings, we constructed an unconditional network where the correlations are assumed to be constant over time. This contrasts with the conditional network, which incorporates dynamic correlations based on the DCC model. The unconditional network, characterized by fixed correlations, offers a representation similar to the one based on volatility-based distance, as seen in the GARCH model, where the temporal variable is effectively removed. This approach allows for the analysis of the network structure without considering the dynamic changes in correlations over time, providing a simplified point of view of the relationships between commodities. Although simpler compared to the approach employing dynamic correlations, it has been widely used in previous research.

In the analysis of the unconditional network, it is noteworthy that the links are much stronger between commodities within the same sector (e.g., Diesel, Gasoline, WTI, or Aluminum, Copper, etc.). This result, shown in Fig. 11, is consistent with the findings of Tabak et al. (2010), which observe similar topological patterns in commodity networks. Fig. 11 leads to an efficient visualization of the structure of the network, with a clear interpretation of the graph.

For the unconditional correlation, we apply shocks between -20% and +20%, as taking the maximum shock approach (where the correlation is driven to -1 or +1) results in unreasonably large shocks that are unrealistic for many commodities (e.g., a shock of +4000%).
<figure><img src="image_12.png" /><figcaption>Fig. 9. Resilience (y-axis) given a shock $\tilde{\beta}_i$ (x-axis), under the assumption of heavy-tailed returns. The maximum shock allowed for each commodity $i$ is $max(\zeta_i) < (1 - \alpha_i - \beta_i)/\beta_i$, thus ensuring that $\alpha_i + \beta_i < 1$. Shocks larger than this threshold are not considered as it would suggest explosive behavior of the volatility.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. We here show three different shocks, under the assumption of heavy-tailed returns. In panel (a) we display shocks for the energy sector. In panel (b) we display shocks for the metals sector. In panel (c) we display shocks for the agriculture sector. The y-axis represents the percentage change in resilience measure. The shaded band corresponds to the standard deviation of the resilience for each sector.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Unconditional correlation-based network (e.g. see Tabak et al., 2010) - no shocks. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 12.** Resilience (y-axis) given a shock $\bar{p}_i$ (x-axis) to the unconditional correlation coefficient.</figcaption></figure>

Therefore, we use more reasonable shock levels for each pairwise correlation, ensuring that the analysis reflects plausible extreme events while maintaining a realistic representation of the data. Fig. 12 displays the resilience results to shocks of single commodities. Here, we observe identical patterns across all commodities, with symmetric V-shaped responses when applying the shocks. However, there are distinct behaviors observed within the energy commodities, which exhibit a different response compared to other commodities. Specifically, the energy commodities demonstrate a higher degree of resilience when both negative and positive shocks are applied. This resilience remains strong up to a certain threshold of positive shock (around 10%), beyond which we see a sharp increase in the response. This suggests that while the energy commodities show stability in moderate shock scenarios, extreme positive shocks lead to significant changes in their behavior.

Finally, Fig. 13 displays the network resilience when shocks to the entire sector of commodities are considered. Results present a clear divergence in the resilience measures across the three commodity sectors. The resilience index for the energy sector is approximately ten times larger than those for the metals and agriculture sectors thus suggesting that the network is not much resilient to shocks involving the entire energy sector. However, the energy sector shows a flat trend up to a certain shock level, beyond which resilience decreases significantly. This behavior suggests that high values on the resilience measure indicate a significant change in the network, which occurs when the correlation between commodities increases substantially.

In particular, for large positive correlation increases between all energy commodities (e.g., as seen in geopolitical events like the Ukraine war), the index value spikes dramatically. This sharp increase does not
<figure><img src="image_16.png" /><figcaption>Fig. 13. We here show three different shocks to the unconditional correlation coefficient. In panel (a) we display shocks for the energy sector. In panel (b) we display shocks for the metals sector. In panel (c) we display shocks for the agriculture sector. The y-axis represents the percentage change in the resilience measure. The shaded band corresponds to the standard deviation of the resilience for each sector.</figcaption></figure>

occur to the same extent for the other sectors. While a similar pattern is observed in the metals and agriculture sectors, the magnitude of the response is much smaller — approximately ten times less than in the energy sector.

This pattern suggests that when shocks lead to an increase in correlation across energy commodities (such as in the case of large-scale geopolitical events), the energy market — under the unconditional correlation network structure — is less resilient to such shocks. In contrast, the metals and agriculture sectors appear to be more resilient to such shocks, with their network structures remaining more stable even in the face of significant changes in pairwise correlations.

## 5. Conclusions

This paper offers a view of the commodities market along with its inner structure of interconnections. It deals with the analysis of the resilience of such a system, on the basis of links and shocks driven by volatility and correlation. The analysis is carried out by including the time-evolution of the commodities' financial features. To this end, we move from a DGP perspective associated with GARCH-based and Dynamic Conditional Correlation-based assumption, by using some key instruments of complex network theory. Therefore, different from previous papers dealing with unconditional correlation-based similarity networks, we introduce a novel approach for evaluating systemic

resilience, where the network of commodities is constructed on the basis of some statistical features. The resilience is then evaluated by considering shocks to the statistical process adopted for the construction of the network. This approach, which can be easily generalized to alternative statistical models, allows the study of systemic resilience from a different perspective.

Results suggest the presence of regularities and deviations in terms of the ability of the considered networks to absorb external solicitations — hence, validating the heterogeneity of the commodities in terms of sector and financial characteristics. Interestingly, the proposed methodology can be applied to describe how the networked commodities interact with relevant financial and macroeconomic variables, along with the stability properties of such a unified framework. In this respect, the resilience of a wide socio-economic context might be efficiently explored, to provide crucial insights on possible policies to be implemented for fostering the stability of macroeconomic systems. The heterogeneous nature of the involved entities could lead to important advancements also in the theoretical context of complex networks field. This challenging topic is already in our research agenda.

Moreover, although this paper primarily focuses on resilience to volatility and correlation shocks, we recognize that the role of trading volume shocks, especially during socio-economic turmoil, represents an important yet underexplored area deserving of future research. In fact,
changes in trading volumes can provide critical insights into market liquidity and the responsiveness of nodes within the network. Network structure including trading volumes have been considered in *Brida et al.* (2016) for the case of the European stock market, but for a different aim than our study. Previous studies have documented a well-established empirical relationship between trading volumes and volatility Volume shocks may influence financial stability through several channels. For example, *Jones et al.* (1994) finds that the volatility of the daily price could be explained by the daily number of trades and the average trade size. *Chan and Fong* (2000) show that order imbalance is a major source of this relationship. Similar evidence has been more recently found for the commodity market (*Go and Lau*, 2020; *Wellenreuther and Voelzke*, 2019). Empirical evidence suggests that, during socio-economic turmoil, abnormal changes in volume could amplify market volatility, leading to heightened price swings and spillover effects across financial markets. Furthermore, abrupt changes in volume may reflect shifts in investor sentiment (*Liu*, 2015), such as panic selling or speculative buying, which can exacerbate instability and trigger feedback loops in pricing or spillovers into related assets. Persistent deviations in trading volumes might also signal emerging risks or structural changes within the system, potentially impacting overall financial stability. However, this last channel is less explored, although it may be motivated by the known effect of trading volumes on volatility. Empirical evidence suggests that, during socio-economic turmoil, abnormal changes in volume can amplify market volatility, leading to greater price fluctuations and spillover effects across financial markets (*Go and Lau*, 2020; *Wellenreuther and Voelzke*, 2019). Moreover, sudden volume changes may reflect shifts in investor sentiment, including panic selling or speculative buying (*Liu*, 2015), which can exacerbate market instability and potentially trigger feedback loops in pricing or spillovers into related assets. Additionally, persistent deviations in trading volumes may indicate emerging risks or structural changes within the system, a channel that, while less explored, could be particularly important for understanding long-term financial stability.

Despite these potential insights, the direct influence of trading volumes on systemic stability remains insufficiently understood and would benefit from a dedicated study. Future research could explore how volume shocks interact with volatility and correlation shocks, enriching our understanding of the mechanisms driving financial fragility, contagion, and resilience in interconnected markets. This would not only complement the findings presented in this paper but also offer a more nuanced view of how liquidity dynamics and investor behavior impact the stability of commodity networks, particularly during periods of economic disruption.

Incorporating trading volumes into the analysis could significantly improve the current framework and provide deeper insights into the interplay between volumes, volatility, and correlations. We believe that exploring this avenue in future work could broaden the applicability of our methodology, offering a more comprehensive understanding of the resilience of commodity markets and their role in global financial stability. We hope that this research will inspire future studies, contributing to a better understanding of how volume shocks influence systemic risk and financial stability in commodity markets.

### CRediT authorship contribution statement

**Roy Cerqueti:** Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Raffaele Mattera:** Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Saverio Storani:** Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.

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