Structural robustness of networks with degree-degree correlations between second-nearest neighbors

Yuka Fujiki; Stefan Junk

doi:10.1371/journal.pone.0336970

Abstract

We numerically investigate the robustness of networks with degree-degree correlations between nodes separated by distance l = 2 in terms of shortest path length. The degree-degree correlation between the l-th nearest neighbors can be quantified by Pearson’s correlation coefficient r_l for the degrees of two nodes at distance l. We introduce l-th nearest-neighbor correlated random networks (l-NNCRNs) that are degree-degree correlated at less than or equal to the l-th nearest neighbor scale and maximally random at farther scales. We generate 2-NNCRNs with various r₁ and r₂ using two steps of random edge rewiring based on the Metropolis-Hastings algorithm and compare their robustness against failures of nodes and edges. As typical cases of homogeneous and heterogeneous degree distributions, we adopted Poisson and power law distributions. Our results show that the range of r₂ differs depending on the degree distribution and the value of r₁. Moreover, comparing 2-NNCRNs sharing the same degree distribution and r₁, we demonstrate that a higher r₂ makes a network more robust against random node/edge failures as well as degree-based targeted attacks. This behavior was observed in nearly all simulated cases, except for highly assortative power-law networks, where the relationship is more complex.

Citation: Fujiki Y, Junk S (2025) Structural robustness of networks with degree-degree correlations between second-nearest neighbors. PLoS One 20(12): e0336970. https://doi.org/10.1371/journal.pone.0336970

Editor: Tao Fu, Beijing University of Technology, CHINA

Received: December 20, 2024; Accepted: November 2, 2025; Published: December 5, 2025

Copyright: © 2025 Fujiki, Junk. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: See Table 2 and references.

Funding: This work was supported by a Grant-in-Aid for Scientific Research (No. 23K13010 and 23K12984) from the Japan Society for the Promotion of Science.

Competing interests: No authors have competing interests.

1 Introduction

Many complex networks in the real world share a common property that the distribution of the number of edges (degree) from each node is approximated by a power-law function, which is known as the scale-free property [1]. The degree correlation among nodes in such networks, particularly nearest-neighbor degree correlation (NNDC), influences network behavior, including robustness to node and edge failures, the spread of infections, and oscillator synchronization [2–4].

Additionally, recent studies have shown that the majority of complex networks in the real world exhibit long-range degree correlations (LRDCs) at greater distances than next-nearest neighbors [5]. For example, airlines often use a “hub-and-spoke" model and protein-interaction networks are structured into functional modules consisting of a high-degree “hub” protein and low-degree peripheral proteins[6]. They have typically negative NNDCs, yet can exhibit diverse second-neighbor structures depending on how hubs are surrounded by and connected via low-degree nodes. To illustrate such mechanisms, Fig 1 presents schematic examples where networks with identical NNDCs differ in LRDCs at distance l = 2 due to different global arrangements.

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Fig 1. Examples of small networks with identical NNDC but different LRDCs.

(a) Low-degree nodes are clustered around high-degree hubs, but the average degree of neighbors differs between hubs. This asymmetry leads to different r₂’s despite r₁ being fixed, where NNDC is quantified by r₁ and LRDC at distance l = 2 is quantified by r₂, both defined in Eq (1). (b) In the first row, hubs are not directly connected and are separated by a relatively long distance, suppressing indirect high-degree interactions and leading to a negative r₂.

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While some LRDCs can be explained as “extrinsic" LRDCs resulting from the propagation of the NNDCs, for the most part, these are “intrinsic" LRDCs that are not merely consequences of the NNDCs [7]. The effects of LRDCs at distance l>1 remain poorly understood, except for some special cases. For example, it is known that the shortest path length between high-degree nodes (hub) influences network functions and dynamical properties. Greater hub separation can reduce traffic congestion in transport networks [8], suppress pandemic onset in epidemic networks [9], and shape information flow in brain networks [10]. Repulsive correlation between hubs is also closely related to fractal structure [11,12], which is associated with anomalously slow diffusion and a fragile nature under node removal. Furthermore, transsortative structures (degree correlations between two neighbors of a node) can amplify the majority illusion [13]. However, a comprehensive understanding of their effects on network structure and dynamics, even for the simplest case l = 2, is still lacking. Towards understanding the general relationship between network properties and LRDCs, in this study, we numerically investigate the influence of LRDC at l = 2 on the structural robustness, which is one of the most fundamental properties of networks.

For the purpose of this work, robustness refers to the ability of a network to maintain its overall structural integrity against node or edge removal [14–18]. Positive (negative) NNDCs are known to make the network more robust (weaker) and tend to delay (advance) the loss of global connectivity, i.e., global connectivity is lost after a larger (smaller) percentage of random failures or targeted attacks [19–22]. This robustness concept can be interpreted as assessing the stability of network functioning, or the difficulty of eradicating infectious diseases and misinformation spreading on the network. For instance, robustness reflects the ability to deliver messages despite failures in communication networks. Also, in epidemic processes, node and edge removal can represent vaccination and the suppression of transmission pathways, respectively. Understanding the effect of LRDCs on the network robustness provides valuable insights into the resilience of networks and should lead to more effective system design.

As the framework for this investigation, we introduce l-th nearest-neighbor correlated random networks (l-NNCRNs) that are degree–degree correlated up to the l-th nearest-neighbor distance and maximally random at any further scale. Formally, LRDCs at distance l are characterized by the conditional probability that the degrees of two nodes are k and given that the nodes are separated by l steps. When we compare properties of l-NNCRNs with the same correlations at distances , any differences in their properties can be attributed to intrinsic LRDCs at distance l, making this framework a convenient tool for isolating their effects.

We numerically implement this framework in the case l = 2 for a representative choice of correlations, the strength of which is quantified and controlled by Pearson’s correlation coefficient r_l for . We generate 2-NNCRNs with various r₁ and r₂ using two steps of random edge rewiring based on the Metropolis-Hastings algorithm. Our results demonstrate, firstly, that the correlations at distances l = 0 and l = 1, i.e. the degree distribution and r₁, place some constraints on the range of r₂. We then perform random node or edge removal and degree-based targeted attack on networks and quantify the robustness based on the percolation critical point f_c and the robustness measure R introduced by Schneider et al. [23]. Both measures assess slightly different aspects of robustness, as will be explained in Sect 3.2.

Qualitatively, our results suggest that both f_c and R shift with r₂ even when r₁ is fixed, demonstrating that LRDCs influence network robustness. Somewhat surprisingly, the effect of r₂ on robustness follows a similar trend to that of r₁. The influence of r₂ on network robustness is weaker than that of r₁, but it is generally on a comparable order. Networks with larger r₂ tend to be more robust against both random node/edge failure and degree-based targeted attacks when the robustness is quantified by f_c. Using another measure of robustness, R, networks with larger r₂ are more fragile against random node/edge failure but remain more robust against degree-based targeted attacks.

To clarify how degree correlations at distance l = 2 affect network robustness, we decompose 2-NNCRNs into k-cores and confirm that a strongly positive r₂ induces a core, which is as robust as the core induced by r₁. This k-core perspective, together with the superedge interpretation given at the end of Sect 4, explains the similarity in the effects of r₁ and r₂ observed in our results. Finally, we examine the values of r₂ exhibited by real networks and the possibility for optimization of robustness based on LRDCs. Beyond the structural robustness, a complete picture of real-world dynamics requires addressing further aspects, such as the efficiency of information flow or robustness against cascading failures. We briefly comment on these concepts in Sect 5.2. We also outline potential extensions of our framework to higher-order correlations () and to more complex failure scenarios.

The manuscript is organized as follows. In Sect 2 we define l-NNCRNs by quantifying LRDCs using and r_l. Moreover, we explain the algorithm for generating 2-NNCRNs with various values of r_l using random edge rewiring based on the Metropolis-Hasting algorithm. Sect 3 presents the numerical results obtained from our simulations, including the effects of LRDCs on network robustness and the relationship between r₁, r₂ and the degree distribution. Finally, in Sect 4 we summarize our key findings and in Sect 5 propose avenues for future research in elucidating the role of LRDCs in complex networks.

2 Numerical method

We numerically sample networks with various degree-degree correlations between nodes at distance in terms of shortest path length.

We propose l-th nearest neighbor correlated networks (l-NNCRNs), which are random at shortest path distances greater than l and which serve as an idealized model to study the relation between network robustness and LRDCs without other confounding factors. 2-NNCRNs are sampled using a three-step edge-rewiring algorithm to fix, successively, the degree sequence, the NNDCs and finally the LRDCs at distance l = 2. The definition and sampling method of l-NNCRNs are explained in the following two sections.

Note that in real-world networks, the effect of r₂ is complicated by the fact that r₁ by itself also influences network robustness and, on the other hand, constrains the range of r₂. In addition, robustness may be affected by intrinsic long-range correlations at distances or other complex structural features, such as multi-scale or non-Markovian dynamics. The 2-NNCRN is an idealized model designed to disentangle the specific effect of second-neighbor correlations from these confounding factors. By generating networks with an identical degree distribution and r₁ while systematically varying r₂, we isolate the effect of LRDCs at distance l = 2 and demonstrate that they have a profound effect on network robustness. To investigate the relation between the robustness and LRDCs at l = 2, we compare the effect of random node/edge removal and of degree-based targeted attack on the network structure in 2-NNCRNs with different degree correlations. The analysis of these properties is explained in the final section.

2.1 Definition of l-NNCRNs

To analyze LRDCs in a given network, we first need to understand maximally random networks that have the same nearest-neighbor correlations as the network [7]. We can use that model as a baseline to judge whether the degree correlations at greater distances than l = 1 are just extrinsically caused by the NNDCs or whether they are intrinsic. Extending this, we call a network ensemble l-NNCRN if it is degree-degree correlated at less than or equal to the l-th nearest neighbor distance and maximally random at any further scale. A nearest-neighbor correlated random network is thus a 1-NNCRN.

The conditional probability that two randomly chosen nodes separated at a distance l have degree k and describes the LRDCs at distance l. Since the conditional probability is a high-dimensional matrix and not easy to interpret, we use the Pearson’s correlation coefficient r_l for the degrees of two nodes at distance l as a convenient observable that quantifies the strength of the l-th nearest neighbor degree correlation. It is defined as

(1)

where the sum is over all possible degrees. It can also be represented as

(2)

Here, the sum is over all nodes, is the shortest path distance between i and j. The term M is defined as

(3)

where is Kronecker δ. In the case of l = 1, this quantity is known as assortativity [2] and it has been extended to arbitrary l>1 by [24,25].

Note that the NNDCs can induce extrinsic LRDCs at distance l = 2, so that the value of r₂ may differ from zero even if the network is a 1-NNCRN. For a given network, we thus define to be the average value of r₂ among networks with the same degree sequence and NNDC, as described by . We say that a network is intrinsically degree-degree correlated at l = 2 if the value of r₂ significantly differs from .

Several other network ensembles can also exhibit long-range degree correlations (LRDCs), such as stochastic block models, hierarchical modular networks [26], and network-of-networks frameworks [27]. These models generate LRDCs through distinct mechanisms, including community structure, hierarchical organization, and intermodular connectivity. While illuminating, their robustness properties often require model-specific analysis and are not directly comparable with the maximally random l-NNCRNs used in this study. A detailed comparison with these alternative ensembles, as well as with motif-based correlation measures like the dk-series [28], is provided in Supporting Information.

2.2 Edge rewiring algorithm

Edge rewiring algorithms are widely used to investigate graph structures [28–30]. Through randomization via rewiring, the algorithm can generate the most randomized network ensembles under specific constraints. In this study, given parameters J₁ and J₂, we generate 2-NNCRNs through the following sequence of steps based on the Metropolis-Hasting algorithm started with an initial network G₀: (1) Rewire edges while preserving the degree sequence to generate a 1-NNCRN G₁(see Algorithm 1). (2) Rewire edges in the resulting network while preserving the NNDC (i.e., ) to generate a 2-NNCRN G₂ (see Algorithm 2).

Note that when choosing edges, we (temporarily) equip each edge with an orientation. We also mention that the rewiring in Algorithm 1 is irreducible for the set of graphs with the same degree sequence and that in Algorithm 2 is as well for the NNDC [31,32]. Moreover, the reason why we include Step 2 in both algorithms it to ensure that the Markov chains are aperiodic and thus converge to equilibrium.

Algorithm 1 Generating a 1-NNCRN while preserving the degree sequence.

1: Increase time step by 1.

2: Return to step 1 with probability 1/2.

3: Randomly choose two edges and .

4: If rewiring and creates a loop or a multi-edge, return to step 1.

5: Compute the current value of r₁, the value of r₁ after rewiring and , and the transition probability .

6: With probability , rewire edges and .

7: Repeat steps 1–6 until the system reaches equilibrium.

Algorithm 2 Generating a 2-NNCRN while preserving the degree sequence and .

1: Increase time step by 1.

2: Return to step 1 with probability 1/2.

3: Randomly choose an edge (u₁, u₂).

4: Randomly choose another edge (, ) such that has the same degree as u₁.

5: If rewiring and creates a loop or a multi-edge, return to step 1.

6: Compute the current value of r₂, the value of r₂ after rewiring and , and the transition probability .

7: With probability , rewire edges and .

8: Repeat 1–7 until the system reaches equilibrium.

These algorithms generate a canonical ensemble of networks in the sense of statistical physics, which satisfies a constraint in terms of the mean value (soft constraint) and belongs to the general class of exponential random graph models [33–35].

More precisely, in Algorithm 1 the degree of each node does not change, so the degree sequence acts as a hard constraint. On the other hand, the rewiring generates a set of networks with different NNDCs but which is random at further distances, i.e., a 1-NNCRNs. The mean value of r₁ is a soft constraint in Algorithm 1 and in equilibrium a configuration G₁ (with the same degree sequence as the initial network G₀) appears with probability

(4)

where r₁(G) is the value of r₁ for G and Z₁ is the partition function

(5)

The mean value of r₁ is zero in the case J₁ = 0 and moves in the positive/negative direction with J₁. In statistical physics, these parameters are analogous to an inverse temperature. As they are tuning parameters without a direct interpretation themselves, our analysis focuses on the resulting observable correlation coefficients, r₁ and r₂. Empirically, we found that these parameters should be chosen to be proportional to the number of edges in the network for the rewiring to be effective. Moreover, when the network size is sufficiently large, the value of r₁ tends to be narrowly distributed around its mean value (notice that the error bars in Fig 2 are barely visible).

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Fig 2. The mean value of r₁ and r₂ of 2-NNCRNs.

Each data point is the average value over 100 configurations, and the error bar is the standard deviation. For r₁, the error bars are smaller than the symbol size in all cases and are therefore not visible in the plot. The r₁ and r₂ of power-law networks exhibit larger error bars than those of Poisson networks, which can be attributed to significant differences in maximum degrees across random seeds. The symbols vary based on the rewiring parameter J₂. For J₂ = −2M,−M,0,M,2M, the symbols are a sphere, a downward triangle, a star, an upward triangle, and a box, respectively, where M is the number of edges in each network. In particular, the star indicates J₂ = 0, where the value of reflects the extrinsic LRDC at l = 2 induced by r₁ in 1-NNCRNs. If r₂ is greater than/less than , it implies that an intrinsic positive/negative LRDC is present at l = 2.

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The idea behind Algorithm 2 is similar, but in addition to the degree sequence the condition that u₁ and have the same degree ensures that is preserved, and in particular that r₁ remains constant. When the system reaches equilibrium in Algorithm 2, a configuration G₂ with the same degree sequence and NNDC as G₁ appears with probability

(6)

where r₂(G) is the value of r₂ for G and is the partition function in the form of

(7)

The mean value of r₂ equals in the case J₂ = 0 and moves in the positive/negative direction in accordance with J₂. Both rewiring procedures satisfy the detailed balance condition. Thus, after successively applying Algorithms 1 and 2 to an initial network G₀ until equilibrium is reached, we obtain network G₂ with probability

(8)

We prepare two kinds of random networks as the initial network G₀, which prescribes the degree distribution of the network and which is preserved by Algorithms 1 and 2. The first is the Erdős-Rényi random graph model, whose degree distribution is approximated by a Poisson distribution when the network is large enough. We set the number of nodes and the average degree at and , respectively. The second is the configuration model [36] with a power-law degree distribution with the number of nodes , the power-law exponent , the minimum degree k_min = 2, and the structural cutoff [37], where c is the normalization constant.

While the former network belongs to the group of networks with homogeneous degree distribution, the later is a network with power-law degree distribution, which is one of the common features shared by many real-world complex networks. Empirically, power law exponents in the range are common. Such a heterogeneous degree distribution tends to amplify the effect of degree-degree correlations on robustness and other properties of networks. Hereafter, we refer to these networks as Poisson and power-law networks, respectively.

For each setup, we generate 100 independent realizations of G₀. From each G₀, we apply Algorithm 1 with five values of J₁ = −2M,−M,0,M,2M to generate 1-NNCRNs with varying r₁, where M is the number of edges. Next, for each of these 1-NNCRNs, we apply Algorithm 2 with five values of J₂ = −2M,−M,0,M,2M to control r₂, resulting in 2-NNCRNs with distinct parameter sets per initial G₀, which means that for each pair, we obtain 100 independently generated 2-NNCRNs.

Because each G₀ is sampled from a random network ensemble, it does not contain structural correlations. Also, the irreducibility of both rewiring algorithms ensures that the equilibrium network ensemble only depends on the initial state through the degree sequence [31,32]. We use rewiring steps to ensure convergence. Details on the convergence of r₂ and the scalability of our algorithm are provided in Supporting Information.

2.3 Analyzing the structural robustness

The structural robustness of networks assesses the ease with which the global connectivity of the network is maintained. There are several strategies for destroying networks and processes of how networks fail [38–41]. Here, we investigate the most straightforward mechanisms, namely random node/edge failures, as well as targeted attacks removing nodes in decreasing order of degree.

As the fraction p of remaining nodes/edges decreases, the size of the largest connected component in the network decreases as well. For the network to maintain its functionality, global connectivity is necessary. Thus the value of p at which the largest connected component collapses, the critical point p_c, serves as an essential indicator of robustness. We find it more intuitive to evaluate robustness using the fraction of removed nodes/edges that can be removed before collapse, , so that a larger value of f_c indicates that the network is more robust.

In practice, does not always rapidly decrease around a single value of p. More precisely, the network may contain a densely connected core whose percolation occurs later than the more loosely connected periphery, which results in multiple phase transitions [42]. To illustrate this phenomenon, we compute the susceptibility

(9)

The peak of the susceptibility indicates the location of critical points in a continuous phase transition and is often used to approximate p_c. However, in situations as described above, the susceptibility exhibits multiple peaks, whose relative heights can change as the size of the network increases. These peaks can also merge or become indistinguishable, making it difficult to assign a well-defined critical point. This makes the peak position of susceptibility unsuitable for quantifying robustness. The multiple peaks phenomenon can be seen in the bottom right panel in Fig 4.

In this study, we instead approximate the critical point p_c using a threshold defined by

(10)

where the largest component size falls below 1% of the total and use as a robustness measure. Our results are qualitatively unchanged for other small thresholds, as shown in Supporting Information. Note that if the threshold is set too large, the measure no longer captures the collapse point of the giant component. Instead, it begins to reflect the size of the giant component above the percolation critical point, which is captured by another robustness measure, R, introduced below.

Beyond the critical point, the size of the giant component as p varies also carries important information. Schneider et al. [23] proposed the area under the curve as a robustness measure. A larger value of S(p) indicates that the network is more robust for any fixed fraction p of remaining nodes/edges and R thus measures the average robustness away from p_c.

In summary, network robustness can be interpreted using two measures: and R. A larger value of indicates that the network can maintain its global connectivity under a higher fraction of node/edge removals. The robustness measure R, defined as the area under the curve S(p), reflects the average connectivity across all node/edge removal scenarios, with a larger R indicating greater robustness.

To evaluate and R in a given network, we use the method proposed by Newman and Ziff [43].

In addition to structural robustness measures, we also compute the network efficiency E(p), defined as the average of the inverses of shortest path lengths between node pairs that remain connected after a fraction p of nodes or edges are removed [44]. Although E(p) does not directly measure connectivity, it reflects a functional aspect of information propagation or transport efficiency. We include this metric as it provides a preliminary indication that second-neighbor degree correlations (r₂) have a non-trivial effect on functional robustness. Since this aspect is beyond the main scope of this study, we do not discuss the efficiency results in Sect 3, but refer to them briefly in Sect 5.2 as a direction for future work.

3 Result

In this section, we present the results of the numerical calculations based on the method described in Sect 2.2. First, we confirm that the algorithm works as desired, i.e., that the generated 2-NNCRNs exhibit various degree-degree correlations between both nearest-neighbor nodes and second-nearest-neighbor nodes. Next, we assess their robustness against random node/edge failures and degree-based targeted attacks.

3.1 Constraints on degree correlations in 2-NNCRNs

The purpose of this section is to give some details about the networks obtained by the rewiring procedure explained above, and in particular to explain how the parameters influence the range of correlations achievable with our method. More precisely, we confirm that degree distribution and r₁ impose constraints to r₂ and that the rewiring algorithm can sample network configurations with various r₂ and the same r₁.

Fig 2 depicts the average values of r₁ and r₂ of 2-NNCRNs generated by the rewiring method described in the previous section starting with an initial network characterized by (a) the Poisson degree distribution and (b) the power-law degree distribution. The confidence intervals defined as three times the standard error for the mean value of r₁ and r₂ corresponding to the different parameters do not overlap, confirming that our sampling yields statistically distinct network sets. As shown in Fig 2, the range of r₂ strongly depends on the degree distribution and r₁. Let us comment on a few trends:

First, we note that r₂ is shifted in the positive direction by larger absolute values of r₁. Such positive extrinsic LRDCs have also been observed in NNCRNs in [7] and [45] and are consistent with the positive correlation of degrees between nodes at even distances. To understand this effect, note that if r₁ is close to + 1, similar degrees tend to be adjacent, leading to similar degrees at l = 2 and resulting in higher values of r₂. On the other hand, if r₁ is close to –1, high degrees tend to be adjacent to low degrees and vice versa. Due to this alternating pattern, we again find that nodes at distance l = 2 have similar degrees, which results in high values of r₂.

Next, we observe that the range of r₂ is narrower if the absolute value of r₁ is large in the case of Poisson degree distributed networks (Fig 2(a)). For power-law networks, the converse is true but the effect is much weaker (Fig 2(b)). In other words, the range of r₂ values achievable by 2-NNCRNs strongly depends on the value of r₁ in Poisson networks, while the same trend is not observed for power-law networks. At present, we do not have a satisfying explanation for this behavior and we conclude that the relationship between r₁ and r₂ is complex and constrained by the degree distribution.

Finally, we mention that neither the range of r₁ nor of r₂ are symmetric. This is not surprising. For example, as explained above, both r₁ close to + 1 and r₁ close to –1 tend to induce large positive values for r₂, but the mechanism is different in both cases.

3.2 Random failure

Having confirmed our samples to be degree-degree correlated at distances l = 1 and l = 2 in various ways, we now investigate their robustness using the method described in Sect 2.3.

Figs 3 and 4 depict the relative sizes of the largest connected component and the susceptibility (see Eq (9)) as functions of the probability p that an edge remains.

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Fig 3. The S,

, and E dependence on the edge retention probability p during random edge failure in Poisson networks.

The set of 2-NNCRNs is the same as in Fig 2(a). The left, middle, and right panels correspond to rewiring parameters J₁ = −2M, 0, and 2M, respectively. The colors and types of the lines correspond to rewiring parameters J₂.

https://doi.org/10.1371/journal.pone.0336970.g003

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Fig 4. The S,

, and E dependence on the edge retention probability p during random edge failure in power-law networks.

Here we plot the same information as Fig 3 for the set of 2-NNCRNs from Fig 2(b).

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The solid black lines in Figs 3 and 4 represent the case of J₂ = 0, i.e., the 1-NNCRNs without intrinsic correlations at distance , as the baseline for comparison. With the exception of the rightmost panel of Fig 4, the peak position of susceptibility suggests that p_c increases with larger r₂-values, for fixed r₁.

An exception to this trend is observed in power-law networks with a strongly assortative structure (r₁ = 0.74), as shown in the bottom-right panel of Fig 4. As discussed in Sect 2.3, the susceptibility suggests the double-peak transition, which has been reported in networks with high clustering coefficients [42]. The double-peak transition is known to be characteristic of networks with a core-periphery structure. The peak at lower p appears consistently at p = 0.052 and is independent of r₂, while the peak at larger p varies with r₂. Notably, the direction of this variation is opposite to the trend seen in other cases, that is, larger r₂ shifts the second peak toward smaller p.

Note moreover that for different values of r₂ the relative order of S(p) flips at a certain crossing point, i.e., up to that point the giant connected component with large r₂ is larger than the one corresponding to the network with small r₂, while the reverse holds above the crossing point. In particular, for large value of p the giant component size grows more slowly and is smaller with larger r₂. The two regions before and after the crossing point are aggregated into a single number R. Thus, when discussing the relationship between 2NNCRNs and robustness, it is necessary to look at both measurements, and R.

The two measures and R used to characterize the robustness of 2-NNCRNs are derived from Figs 3 and 4 and the result is summarized in Fig 5. We observe that increases with r₂ except for the case of assortative power-law networks, which means that the effect of perturbing r₂ in the positive/negative direction away from is to make the network more robust/fragile. Conversely, R tends to decrease with increasing r₂, which means that r₂ has a negative effect on robustness quantified by R. Quantitatively, the size of the shift in the robustness measures resulting from a change in r₂ is comparable to the effect of a change in r₁, as explained in the caption of Fig 5. The shift induced by r₂ is especially large in cases where the range of r₂ is wide, such as Poisson networks with r₁ = 0.00 or power-law networks with r₁ = −0.74.

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Fig 5. Robustness of 2-NNCRNs against random edge removal.

The symbols indicate the parameter J₂ and are the same as those in Fig 2, whereas lines connect points corresponding to 2-NNCRNs with the same r₁ and the line type corresponds to J₁. Each data point is the average value over 100 configurations, and the error bar is the standard deviation. The effect of r₂ can be seen in the vertical spread of data points for a given line style, while the effect of r₁ is seen in the vertical distance between different line styles. Note that these two effects are often comparable in magnitude.

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We have also performed the same analysis for random node failure, instead of edge failure. The results are similar to edge failure and are summarized in Fig 6.

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Fig 6. Robustness of 2-NNCRNs against random node removal.

The line types and symbols are analogous those in to Fig 5. The stronger decrease in of assortative networks with r₁>0 in the top-right panel, compared to edge failure, is likely due to the reduced accuracy of as an estimator of p_c, especially when p_c is close to zero.

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3.3 Targeted attack

It is known that networks that are robust against random failures can still be vulnerable to targeted attacks, where nodes are removed in order of decreasing degree, in particular in random networks with power-law degree distribution [39,40]. On the other hand, when robustness is evaluated by f_c, it is possible to be robust against both random failure and targeted attacks in networks with positive r₁ [22]. Here, we investigate how the r₂-dependence of robustness against targeted attacks differs from the case of random failure.

Figs 7 and 8 depict how the giant component size S and the susceptibility depend on the fraction p of surviving nodes under a targeted attack on the same 2-NNCRNs used in Figs 3 and 4. Fig 9 contains the corresponding values of and R.

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Fig 7. The S,

, and E dependence on the node retention probability p during targeted attack on Poisson networks.

This plot replicates the information from Fig 3 (Poisson networks) for the case of targeted attack.

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Fig 8. The S,

, and E dependence on the node retention probability p during targeted attack on power-law networks.

This plot replicates the information from Fig 4 (power-law networks) for the case of targeted attack.

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Fig 9. Robustness of 2-NNCRNs against targeted attack.

The line types and symbols are analogous those in to Fig 5.

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We observe that increases with r₂ for a fixed choice of r₁ and that this behavior qualitatively matches our observation in the case of random edge failure. Quantitatively, the strength of the shift in in response to a change in r₂ is stronger than in the case of random failure. Moreover, as in the case of random failure, the size of the shift in response to a change in r₂ is somewhat weaker compared to a change in r₁.

On the other hand, as observed from the bottom panels in Fig 9, there is a slight increase in R with respect to r₂, which is more pronounced in power-law networks. We note that the direction of change in R in response is opposite to what we observed in random failure. This difference in behavior is due to the fact that, while there is still a crossing point between S(p) similar to the discussion in Sect 3.2, it occurs at a higher proportion S(p) of surviving nodes than in the case of random failure. This shift in the crosspoint occurs because targeted attacks rapidly destroy the network’s core, meaning any significant giant component only exists in a state of high fragmentation. In contrast, under random failure, the network degrades more smoothly, allowing the crosspoint to occur at a lower fraction of surviving nodes. Thus the values of R are influenced more by the behavior of the curves around p_c and we expect that the change in R more closely matches the change in .

4 Conclusion

In this study, we have conducted numerical investigations into the structural properties of networks exhibiting degree-degree correlations between second-nearest neighbors, i.e., long-range degree correlations (LRDCs) at shortest path distance l = 2.

To achieve this, we have introduced l-th nearest-neighbor correlated random networks (l-NNCRNs), which exhibit degree-degree correlations up to the l-th nearest neighbor scale while being maximally random at any further scale. We have generated 2-NNCRNs using a two-step algorithm based on random edge rewiring and the Metropolis-Hastings algorithm for two representative degree sequences: Poisson and power-law distributions. Quantifying the strength of the LRDC at distance l using Pearson’s correlation coefficient r_l, this two-step algorithm generates 2-NNCRNs with identical r₁ but varying r₂. Finally, we have investigated the robustness of the networks by simulating random node/edge failures and degree-based targeted attacks.

We have observed that the robustness quantified by is an increasing function of r₂ when r₁ is held constant in the almost all cases we simulated (Figs 5, 6 and 9). Thus, networks with smaller (larger) r₂ than have been observed to be more fragile (more robust) against both random failure and targeted attack. On the other hand, the effect of r₂ on network robustness, when measured by R, is more nuanced. In the case of bond percolation (Fig 5), R decreases as r₂ exceeds , indicating that the network becomes more fragile. However, in the case of targeted attacks (Fig 9), R remains constant or increases slightly with r₂. This contrasts with our findings when robustness is quantified by and can be explained by the crosspoint-effect mentioned in Sect 3.2.

To summarize, our numerical results suggest that network robustness is correlated with r₂ when r₁ is fixed, as shown in Table 1. The trends in the influence of r₂ on robustness are similar to those of r₁. As seen in Figs 5, 6, and 9, the magnitude of the effect caused by changes in r₂ is roughly half of that caused by r₁, except in the case of R during targeted attacks. This highlights the significant role of second-neighbor degree-degree correlations in network resilience.

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Table 1. Summary of effects of an increase in either r₁ or r₂ on

,

, and R.

https://doi.org/10.1371/journal.pone.0336970.t001

In conclusion, if r₁ is kept fixed, then r₂ has a similar effect on robustness as r₁. To give some explanation of this effect, we have plotted in Fig 10 the size of the k-core [46] in power-law networks with strongly positive r₁ or r₂ and we observe that both constraints lead to quite similar values. To interpret this, recall that the k-core is the part of the network where the giant component starts to emerge as p approaches p_c from below and is thus intimately related to the robustness of the network. Quantitatively, the effect of r₂ on robustness is smaller than r₁ but comparable.

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Fig 10. Size of k-cores in power-law networks with different degree correlations.

The values are averaged over 100 configurations. Symbols represent different rewiring parameters: solid squares for , open spheres for , and crosses for , which means that the networks are random networks, 1-NNCRNs, and 2-NNCRNs, respectively.

https://doi.org/10.1371/journal.pone.0336970.g010

A simple, intuitive picture for why the effects of r₂ on network robustness are so similar to those of r₁ can be formed by imagining that each path of length two in the original network is replaced by a single “superedge”. The nearest-neighbor correlation r₁ of this new superedge network would then correspond to the second-neighbor correlation r₂ of the original network. This analogy by itself cannot amount to a complete explanation, as the state of two superedges originating from the same node is not independent (they may share their first edge in the original graph). Although not a formal proof, this conceptual model strongly suggests that r₂ plays a structural role analogous to that of r₁ in determining network robustness.

5 Discussion

While our main results reveal that r₂ affects robustness qualitatively similar to r₁, they also point to several nuanced phenomena. In this section, we examine an exceptional behavior observed in assortative power-law networks, where the robustness measure shows an unexpected trend and double-peak transitions. Afterwards, we discuss the implications of our findings, including potential applications to real-world network design, and outline directions for future research involving long-range correlations and higher-order network structure.

5.1 Exceptional behavior in assortative power-law networks

Here, we discuss an exceptional trend in observed in assortative power-law networks under random failures, see Sect 3.2 . This behavior can be attributed to the fact that the theoretical percolation threshold p_c is close to zero in this regime. In such cases, defined by any fixed positive threshold x no longer approximates p_c, but rather reflects how quickly the giant component grows after p exceeds p_c. For uncorrelated random networks, the theoretical threshold is given by the Molloy–Reed criterion , which equals 0.07 for the power-law networks we treat here, but assortative correlations may push the threshold even closer to zero. This makes behave differently from its usual interpretation as an indicator of connectivity collapse.

Analyzing double-peak transitions helps us to understand this phenomenon. We extract the position p^* of both peaks and define a robustness-like quantity for each. Fig 11 shows resulting f^* values for both the first and second peaks as a function of r₂. Although the largest f^* corresponding to the connectivity of the core remains stable under changes in r₂, the smaller f^* corresponding to the periphery decreases as r₂ increases. This observation is consistent with the fact that also decreases with increasing r₂ in this regime, supporting the interpretation that both and the smaller f^* reflect the connectivity of the peripheral region rather than that of the core.

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Fig 11. Robustness-like quantity f^* evaluated by the position of the peaks of susceptibility under random edge removal for the core (upper lines) and the periphery (lower lines).

The set of networks is the same as the cases r₁ = 0.74 and 0.66 at the right top panel of Fig 5).

https://doi.org/10.1371/journal.pone.0336970.g011

Interestingly, the enhancement of the second peak as increases suggests a phenomenon specific to r₂ that is not observed for r₁. While r₂ has been observed to have qualitatively similar effects to those of r₁ throughout most of this study, this result highlights the distinct behavior of r₂ in the context of double-peak transitions. These findings open new directions for future research on how long-range degree correlations influence structural phase transitions in networks.

5.2 Implications and further directions

It is natural to expect that LRDCs at l = 2 also play an important role in the robustness of a real-world network. To investigate this, we considered 29 real-world networks from the dataset previously analyzed in [7], which were selected to represent a diverse range of systems under constraints such as the need for domain diversity, data accessibility, and computational feasibility. For these networks, Fig 12 summarizes the observed values of r₂ as well as the value of for a 1-NNCRN with the same . In our dataset, most of the real-world networks exhibit larger r₂ than , and thus our results suggest that intrinsic LRDCs at l = 2 contribute positively to their robustness. Our result also suggests the applicability of r₂ to control the robustness of real-world networks by changing r₂ from , which we will try to implement in future work. Note that the 2-NNCRN model discussed here is an idealization in the sense that we compare networks that only differ in their degree correlations, which will not necessarily be the case in real-world networks. Although our findings suggest the potential role of r₂ in enhancing robustness, whether and how this effect can be harnessed or observed in real-world systems remains to be verified. It remains an important problem for future work to quantify the influence of LRDCs in such networks.

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Fig 12. LRDCs at l = 2 in real networks in Table 2.

Solid squares represent the r₂-values for real-world networks. Open circles denote calculated as the average r₂ values over 100 realizations of the corresponding 1-NNCRNs, with error bars indicating the standard deviation.

https://doi.org/10.1371/journal.pone.0336970.g012

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Table 2. Properties of the real networks in Fig 12.

https://doi.org/10.1371/journal.pone.0336970.t002

As mentioned in Sect 1, in addition to the effect of LRDC on structural robustness, it is natural to wonder about its effect on functional aspects of robustness. As a preliminary step in this direction, we have analyzed the behavior of the network efficiency E(p) which is plotted in the bottom rows of Figs 3, 4, 8 and 9. The r₂-dependent trends in E(p) largely mirror those of the largest component size S(p), though they emerge at different stages: S(p) shows r₂-sensitivity near the critical threshold, while E(p) exhibits variation even at high p, where the network remains largely intact.

To better understand this, Fig 13 shows how and the network diameter depend on r₁ and r₂ in the absence of node/edge removal. We observe that both positive and negative shifts in r₁ or r₂ reduce E, although the underlying mechanisms differ. For strongly positive values, the reduction is associated with diameter expansion, consistent with enhanced core-periphery structure. For strongly negative values, diameter decreases, but the distribution of path lengths sharpens and shifts, implying homogenization of distances. In both cases, r₂ influences efficiency in ways similar to r₁. From the perspective of information spreading, these findings suggest that strong degree correlations (either positive or negative) can limit functional efficiency even before any failure occurs. However, as can be seen from Figs 3, 4, 8 and 9, below the critical pint p_c positive r₂ can help maintain higher efficiency in the network.

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Fig 13. Efficiency and diameter of 2-NNCRNs.

The line types and symbols are analogous to those in Fig 5.

https://doi.org/10.1371/journal.pone.0336970.g013

Furthermore, previous work [77] has shown that degree assortativity plays a key role in cascade onset and size. We observed a similarly nuanced impact in a model of cascading failures, hence we have decided that a detailed investigation into these functional aspects, while promising, is beyond the scope of the present study and leave it for future work.

The insights gained from our study have important implications for the construction of more robust social systems and the development of effective contact rules to mitigate the spread of infectious diseases. In this context, one might wonder whether the scenario of “targeted attack” is practically relevant, since it presumes complete knowledge of the network structure (e.g., the number of contacts of all individuals threatened by an infectious disease). However, recent results have shown that similar attack schemes based on much more limited, local information yield essentially the same phase transition for scale free networks [78]. This suggests that our findings for the idealized targeted attack are likely to be robust.

Our results show that r₂ significantly affects network robustness, with its positive impact explained by the k-core structure induced when . Given that nonzero r₁ has emerged as a fundamental property of network structure, influencing not only network robustness but also various dynamic processes and functions, it is likely that r₂ plays a similar role. Further investigations are needed to explore the influence of r₂ on other dynamic processes.

Regarding future directions of research, it is natural to wonder whether correlations at distance l = 3 and larger can play a similar role than those at distance l = 2. Conceptually, we expect that the importance of these correlations is quite small because, in a small-world network, the diameter scales like and thus the correlations up to distance l encompass correlations between almost all nodes, even for very small values of l. Moreover, our approach based on network rewiring becomes challenging even in the case l = 3, since no irreducible method to choose two edges preserving second-nearest neighbor degree correlations is known. Moreover, note that in Algorithm 2 the most time-consuming step is to compute the updated value , which requires an update to all neighbors of and can thus be done in time , see also Supporting Information. In the case l = 3, we need to consider nodes up to distance 2 from and thus the time-complexity will be much higher (we expect it is be , depending on the implementation of the local update).

Supporting information

S1 Appendix. Additional analyses and methodological details.

Contains a detailed discussion of the scalability of the rewiring algorithm, other ensembles with long-range degree correlations as well as the effect of threshold value.

https://doi.org/10.1371/journal.pone.0336970.s001

(PDF)

Acknowledgments

The authors thank S. Mizutaka, T. Hasegawa, and K. Yakubo for fruitful discussions. The authors are grateful to two anonymous referees for their careful reading and valuable feedback.

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Figures

Abstract

1 Introduction

2 Numerical method

2.1 Definition of l-NNCRNs

2.2 Edge rewiring algorithm

2.3 Analyzing the structural robustness

3 Result

3.1 Constraints on degree correlations in 2-NNCRNs

3.2 Random failure

3.3 Targeted attack

4 Conclusion

5 Discussion

5.1 Exceptional behavior in assortative power-law networks

5.2 Implications and further directions

Supporting information

S1 Appendix. Additional analyses and methodological details.

Acknowledgments

References