Optimization method for protecting the robustness of first-order nodes in complex networks

Chen, Mengjiao; Wang, Niu; Wei, Daijun

doi:10.1063/5.0225538

The robustness of complex networks remains a significant challenge in network science. This study proposes a method aimed at optimizing network robustness by introducing a protection mechanism for the first-order neighbor nodes of a failed node. The load adjustment parameter α and the protection strength parameter δ in the protection mechanism affect the threshold T_c have been determined through theoretical analysis. In addition, in the experiment of a scale-free network, it was proven that α and T_c exhibit a positive proportional relationship, while δ and T_c exhibit an inverse proportional relationship. Notably, the introduction of the protective mechanism consistently resulted in a lower T_c compared to scenarios without protection, validating its efficacy in preventing cascading failures. Finally, the robustness of empirical networks, which include the American Football network, Wikiquote Edits network, and Euroroads network, is compared before and after adding protection. The results demonstrate that the first-order neighbors of failed node are protected, which is an effective method for improving the robustness of complex networks.

I. INTRODUCTION

Complex networks, serving as a potent research tool, have permeated various disciplines, including biology,^1–3 engineering,^4,5 and finance.^6–8 Much of the infrastructure, such as power systems,^9,10 transportation systems,^11,12 and communication systems,^13,14 can be abstracted into complex networks. Cascading failure, driven by the failure of a few nodes in a complex network, may lead to the collapse of the entire network due to its inherent coupling relationships.^15,16 Once infrastructure networks fail in cascading, enormous losses will be caused to society. The cascading failure of complex networks is of great significance for the study of network robustness.^17,18 The competence of the network to maintain normal operation after encountering risks is known as the robustness of the network.^19,20 The research on cascading failure of complex networks about robustness has developed rapidly, and many excellent research results have been obtained.^21,22 The dynamics of cascading failures in complex networks have advanced to a mature stage, yet efforts to enhance network robustness are still in nascent phases.²³ Enhancing the robustness of complex networks assists in reducing the likelihood of network cascade failures. In infrastructure systems, the growth of robustness will enhance the ability of the system to cope with risks and lower losses.²⁴ Indeed, there are numerous real-world examples of complex systems suffering from cascading failures, leading to substantial human, material, and financial losses.^25,26 Thus, optimizing the robustness of the complex network is an open issue.

Some studies aim to enhance the robustness of networks by altering the structure of the network, involving operations such as adding, deleting, or rewiring internal nodes and edges.^27–30 Motter²⁷ showed that selective IR (intentional removals) can effectively inhibit the propagation of cascading failures in heterogeneous distributed networks. Cao et al.²⁸ utilized random links, betweenness centrality links, and low-ranking links to add edges between nodes, ensuring that the network maintains a high level of connectivity even after cascading failure occurs. Due to the capacity limitations of every node in the network, indiscriminate addition of edges is not feasible. It has been proven by researchers that removing edges can enhance the robustness of the network. Zhu et al.²⁹ employed the NNIP (Non-dominated Neighbor Immune Algorithm) to identify edges with varying operational costs that either inhibit or facilitate the NNIP propagation of failures on BA scale-free networks and random networks. By removing the edges that facilitate failure propagation before network attacks, this method effectively enhances the size of connected subgraphs after an attack on the network. Reconnecting edges involves modifying the original edge configuration of the network. When re-establishing connections within the network, the degree distribution of the network may change. Reconnecting edges is a widely employed approach to enhance network robustness. The smart rewiring named by Bai et al.³⁰ based on the reconnection of highly degree nodes can accelerate the robustness of the network.

There are studies that optimize the cascading failure model of networks in pursuit of achieving higher robustness.^31–34 The cascading failure model is a type of model used to describe how the failure of nodes or edges in a network system propagates internally and leads to cascading effects within the system.¹⁷ Wang³¹ presented a local mitigation strategy, in which when node i fails, its neighboring nodes will transfer their remaining capacity to node i, achieving the goal of mitigating node i’s failure. Tong et al.³² established an optimal probability allocation mechanism for redundant resources based on the load of each adjacent node. Ma et al.³³ gave a robustness optimization model for flexible distribution networks, which considers both system security and economy, taking into account the uncertainty of the source load to minimize losses. Liu³⁴ introduced a complex allocation strategy considering time-varying loads and time-varying residual capacity, which is more conducive to the rational utilization of scale-free network capacity.

However, these methods have some shortcomings in improving the robustness of complex networks. First, when changing the network structure to improve robustness, whether through the addition, removal, or reconfiguration of edges, it often results in the emergence of a recurring pattern known as the onion structure. In the onion structure, nodes with higher degrees tend to interconnect with each other, while nodes with lower degrees are distributed across concentric layers around the network’s central core.³⁵ The approach of altering the internal structure of a network often homogenizes networks with diverse structural characteristics, a transformation that may not be directly applicable to real-world systems. Second, the research on enhancing network robustness by modifying cascading failure models often overlooks the spatial distribution of nodes. Some studies on cascading failures have typically defined the initial capacity of a node solely based on its initial load, without taking into account the influence of the node’s spatial location on this initial capacity. It is worth noting that the spatial distribution of nodes carries significant importance in the context of infectious disease propagation. Typically, a node positioned at the center of a disease outbreak can result in the infection of a substantial portion of nodes within the network. Drawing a parallel, the cascading failure propagation process shares similarities with disease propagation, and the spatial placement of nodes also holds a pivotal role in the cascading effect. Finally, the above studies have given relatively little attention to the impact of overloaded nodes. In the case of overloaded nodes within the network, it is important to note that their overload does not automatically lead to immediate failure. Instead, over time, the network’s overloaded state tends to alleviate, providing an opportunity for these nodes to return to a normal operational state. In the process of disease transmission, the overload state can be manifested by a population that becomes infected with the disease but can recover to a healthy state.

Therefore, a method to optimize the robustness of complex networks is proposed in this paper. The main contributions of our method are as follows. First, the capacity of a node is defined by the initial load and cohesion, taking the location of the node into account. Second, a protective mechanism has been introduced for the first-order neighbor nodes of the initial failed node. This proactive measure is based on the understanding that cascading failures are often triggered by the overloading of first-order neighbor nodes due to the redistribution of the load from the failed node. The decision to protect first-order neighboring nodes can be justified from three aspects. Theoretically, first-order neighboring nodes are the closest to the node experiencing failure and are most likely to fail in the next stage. Protecting first-order neighboring nodes can significantly mitigate the occurrence of cascading failures. Practically, in the process of disease transmission, if an individual becomes infected, there is a high probability that those directly connected to them will also contract the disease. Individuals directly connected to the source of infection are manifested in the network as first-order neighbors. From the cost perspective, first-order neighbors are easily discovered and controlled, making protecting them the most cost-effective method. By minimizing the occurrence of failures among first-order neighbor nodes, the overall extent of cascading failures can be reduced. In addition, overloaded nodes are designed to distribute their excess load solely to neighboring nodes, ensuring their continued normal operation while simultaneously lowering the risk of neighboring node failures. Finally, the effectiveness and feasibility of this approach are validated through experiments conducted on undirected, unweighted, and connected BA scale-free networks as well as three empirical networks.

The remaining content of this article is organized as follows. Sec. II is the knowledge related to cohesion and cascading failure of networks. Section III introduces a cascading failure model after the addition of protection. Theoretical analysis and comparative experiment are given in Sec. IV. The simulation experiment on empirical networks is presented in Sec. V. Summary and future work are given in Sec. VI.

II. PRELIMINARIES

A. Cohesion of network and node

The cohesion of a network primarily depends on the connectivity among its nodes. A network with strong node connectivity has higher cohesion. In a social network, the more convenient the interactions between individuals and the smaller the number of people, the higher the cohesion of this social network. The cohesion of the network is related to the total number of nodes and the distances between nodes in the network, denoted as ∂[G],³⁶

\partial [G] = \frac{1}{N l} = \frac{N - 1}{\sum_{i \neq j} d_{i j}},

(1)

where N is the total number of nodes in the network, l is the average distance of the network, d_ij represents the shortest distance between node i and node j.

The cohesion of the network formed after the contraction of node i is referred to as the cohesion of node i, denoted as coh(i),

c o h (i) = \frac{1}{(N - k_{i}) l_{i}} = \frac{N - k_{i} - 1}{\sum_{m \neq n} d_{m n}},

(2)

where l_i is the average distance of the network after node i contracts. k_i is the degree of node i and d_mn represents the shortest distance between nodes m and n; when central nodes are contracted, the network’s average path shortens, the overall node count decreases, and network cohesion strengthens. The greater the cohesion of the network after node contraction, the more important the node becomes. Intuitively, node contraction can be understood as the process of merging the node itself with its neighboring nodes into a new node. The contraction process of the node is shown in Fig. 1. Given a network shown in Fig. 1(a). In Fig. 1(b), node i will contract together with its neighboring nodes (shaded area) into a new consolidated node. The network after the contraction of node i is displayed in Fig. 1(c). According to the definition of Eq. , the cohesion of the new network formed after the contraction of node i is defined as the cohesion of node i. Figure 1(c) is exactly the network formed after the contraction of node i. Therefore, in Fig. 1, the cohesion of node i can be calculated by calculating the cohesion of the network formed after the contraction of node i [i.e., Fig. 1(c)].

FIG. 1.

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The contraction process of node i. (a) An initial network. (b) Find the neighboring nodes of node i. (c) Contract node i and its neighboring nodes into a new node i′.

B. Cascading failure

In a network, the entire network failure caused by a small number of nodes or edge failures is called cascading failure of the network. The capacity load model is defined to analyze the cascading failure phenomenon in networks. In the capacity load model, the initial load and capacity of nodes need to be defined. The initial load of a node refers to the pressure it bears in its starting state. Node’s load is generally correlated with the node’s significance.³⁷ If a node’s load surpasses its capacity, the node will fail. For example, in a social network, each individual has a capacity to carry information. If the received information exceeds their processing capacity, it may lead to distortion of the information. An example of cascading failures occurring in a network is shown in Fig. 2. Figure 2(a) corresponds to the initial state. Node 0 fails, and its load is distributed to nodes 6 and 4. In Fig. 2(b), node 0 and its connecting edges are removed. Node 4 fails, and its entire load is transferred to nodes 8 and 9. In Fig. 2(c), node 4 and its connecting edges are removed. Nodes 8 and 9 fail due to overload, leading to the removal of nodes 8 and 9 along with their connecting edges. Node 8 has only one neighbor of node 9, and node 9 is failed. Therefore, the load of node 8 will no longer be passed to node 9. There are still three normal nodes 3, 5, and 6 in the neighbor nodes of node 9; thus, the load of node 9 will be allocated to nodes 3, 5, and 6. There are no failed nodes in the network in Fig. 2(d), and the network reaches a stable state.

FIG. 2.

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An example of cascading failures using the capacity-load model. (a) Initially removing node 0, node 0 distributes its own load to nodes 4 and 6. (b) Node 4 fails and passes its own load to nodes 8 and 9. (c) Nodes 8 and 9 fail, and node 9 distributes its own load to nodes 3, 5, and 6. (d) Nodes 3, 5, and 6 do not fail and there are no failed nodes in the network and the cascading process ends.

When simulating cascading failures within networks through a capacity load model, it is crucial to establish the initial load and capacity of nodes within the network. The load of a node i can be simply defined by its degree value as L_i,

L_{i} = k_{i}^{α},

(3)

where α is load adjustment parameters, α > 0; k_i is the degree of node i. An increased value of α will lead to a higher initial load for node i. Because the load distribution process continues through the network until the network stabilizes, the load on each node in the network is not static. The capacity of node i is represented by a simple linear relationship, denoted as C_i,³⁸

C_{i} = (1 + T) L_{i},

(4)

where T is a capacity adjustment parameter, T > 0. The larger the T, the larger the capacity of node i. After node i fails, load redistribution will be carried out to the neighboring nodes of the failed node according to the redistribution principles. Some research also allocates the load based on the proportional load of neighboring nodes, the load received by node j from node i is denoted as ΔL_ij,^38,39

Δ L_{i j} = L_{i} \frac{L_{j}}{\sum_{j \in Γ (i)} L_{j}},

(5)

where Γ(i) is the set of neighboring nodes of node i. L_j is the load of neighboring node j of node i.

III. THE PROPOSED METHOD

We propose a new capacity definition method and load allocation strategy. This new capacity definition takes into account the position of nodes. In our proposed method, the capacity of node i is denoted as

C_{i}^{'}

⁠,

C_{i}^{'} = (1 + T c o h (i)) k_{i}^{α},

(6)

Δ L_{i j}^{'} = L_{i} \frac{C_{j}^{'} - L_{j}}{\sum_{j \in Γ (i)} C_{j}^{'} - L_{j}} = L_{i} \frac{c o h (j) L_{j}}{\sum_{j \in Γ (i)} c o h (j) L_{j}},

(7)

where Γ(i) is the set of normal neighboring nodes of node i. L_i is the load of failed node i. L_j and

C_{j}^{'}

represent the load and capacity of neighboring nodes j, respectively. Note that unlike the load of the nodes, the capacity of each node in the network is fixed and is calculated according to Eq. . It is necessary to find out the set of normal neighbor nodes of node i before proceeding to the next step of load distribution because the load of the failed node will be distributed only to the normal neighboring nodes.

A novel load allocation strategy for first-order neighbor nodes of failed nodes is proposed in this paper. It is important to highlight that cascading failures often originate from the failures of first-order neighboring nodes. As a result, we introduce a safeguard mechanism for the first-order neighbor nodes of the failed node i by incorporating an overload mechanism into the cascading failure model. In real-world networks, nodes that exceed their capacity do not immediately experience failure. Instead, failure typically occurs due to prolonged overload that remains unresolved. For example, when users on social media platforms experience a large amount of interaction such as following, liking, and commenting in a short period of time, their personal information and published content may face a huge load. This may lead to insufficient server resources, delayed response times, and even render the user’s account or certain functions ineffective. Consequently, this article takes into account the recovery of nodes from an overloaded state to a normal state. The first-order neighbor nodes are classified into three categories,

\{\begin{aligned} L_{j} \leq (1 + T c o h (j)) k_{j}^{α}, n o r m a l s t a t e, \\ (1 + T c o h (j)) k_{j}^{α} \leq δ (1 + T c o h (j)) k_{j}^{α}, o v e r l o a d s t a t e, \\ δ (1 + T c o h (j)) k_{j}^{α} < L_{j}, f a i l e d s t a t e, \end{aligned}

(8)

where L_j and

(1 + T c o h (j)) k_{j}^{α}

are the initial load and capacity of node j, respectively; δ is the overload coefficient, δ > 1. The larger the value of δ, the stronger the node’s ability to handle excess load. In the cascading failure model with an added protection mechanism, the first-order neighbor nodes of the failed node are divided into three categories: normal state, overload state, and failed state, which is shown in Fig. 3. When node j is in the overloaded state, the node will allocate all the load exceeding its capacity to its neighbors according to the principle of remaining capacity allocation. When node j is in the failed state, all the load of the node j is distributed to the neighboring nodes m according to the allocation. The load received by node m from node j is denoted as

Δ L_{j m}^{'}

⁠,

\{\begin{aligned} Δ L_{j m}^{'} = (L_{j} - C_{j}^{'}) \frac{C_{m}^{'} - L_{m}}{\sum_{m \in Γ (j)} C_{m}^{'} - L_{m}}, n o d e j i s o v e r l o a d s t a t e, \\ Δ L_{j m}^{'} = L_{j} \frac{C_{m}^{'} - L_{m}}{\sum_{m \in Γ (j)} C_{m}^{'} - L_{m}}, n o d e j i s f a i l e d s t a t e, \end{aligned}

(9)

where Γ(j) is the collection of neighboring nodes of node j.

L_{j} - C_{j}^{'}

represents the excess load of first-order neighbor node j.

C_{m}^{'} - L_{m}

represents the remaining capacity of node m. Until no failed nodes appear, the network reaches stability. When allocating according to the remaining capacity of neighboring nodes, nodes with more remaining capacity will receive more load, while loads with less remaining capacity will receive less load. It is worth noting that the protective mechanism has been implemented only for the first-order neighboring nodes of the failed nodes. The majority of cascading failures in networks occur due to the widespread failures of first-order neighboring nodes. By reducing the number of failures among first-order neighboring nodes, it is possible to effectively control the scale of cascading failures. In addition, first-order neighbors are the nodes closest to the failed node, making their detection and control easier. Controlling first-order neighboring nodes is more efficient and cost-effective compared to controlling second-order or third-order neighbors. Therefore, we have implemented a protective mechanism only for the first-order neighboring nodes of the failed nodes. In other words, only the first-order neighbor node that fails first will have three states: normal state, overloaded state, and failed state. The proportion of remaining node S is used as a metric to gauge the robustness of the network,

S = \frac{N_{A}}{N},

(10)

where N_A is the number of remaining nodes in the network after cascading failure. N is the total number of network nodes.

FIG. 3.

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Schematic diagram of failure propagation. Green is the normal node, yellow is the overloaded node, and red is the failed node.

IV. THEORY AND EXPERIMENTS

A. Analysis and calculation

Assuming that node i has failed, the specific conditions that must be met to prevent the failure of its first-order neighbor j will be examined. The cascading failure threshold T_c and

T_{c}^{'}

that keep node j normal before and after adding a protection mechanism is discussed as follows:

Before adding protection mechanism
The initial load of node j plus the load obtained from allocation cannot be greater than its capacity C_j,
$L_{j} + Δ L_{i j} \leq C_{j} .$
(11)
Substituting Eq. (5) into formula (11),
$k_{j}^{α} + k_{i}^{α} \frac{k_{j}^{α}}{\sum_{m \in Γ (i)} k_{m}^{α}} \leq (1 + T c o h (j)) k_{j}^{α} .$
(12)
Simplification can be obtained as
$\frac{k_{i}^{α}}{c o h (j) \sum_{m \in Γ (i)} k_{m}^{α}} \leq T .$
(13)
Next, from the knowledge of probability theory,³⁹ we have
$\sum_{m \in Γ (i)} k_{m}^{α} = \sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} k_{i} \frac{k^{'} p (k^{'}) k^{' α}}{⟨ k ⟩} = \frac{k_{i} ⟨ k^{α + 1} ⟩}{⟨ k ⟩} .$
(14)
$\sum_{m \in Γ (i)} k_{m}^{α}$ is the sum of the αth power of the degrees of all neighboring nodes m to node $i . \sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} k_{i} \frac{k^{'} p (k^{'}) k^{' α}}{⟨ k ⟩}$ is a summation over all possible degrees k, where k_i is the degree of node i. k′p(k′) is the number of nodes of degree k′, and k′^α is the αth power of its degree. Dividing by ⟨k⟩ is for normalization. Based on the degree distribution p(k), we know that the percentage of nodes with degree k′ in the network is p(k′). Therefore, the degree of neighboring nodes, $\sum_{m \in Γ (i)} k_{m}^{α}$ can be written as a weighted sum through the degree distribution. $\sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} k^{'} p (k^{'})$ is the average degree ⟨k⟩ and can, therefore, be reduced to $\frac{k_{i} ⟨ k^{α + 1} ⟩}{⟨ k ⟩}$ ⁠.
The conditions that T needs to meet when node j remains normal,
$\frac{k_{i}^{α - 1} ⟨ k ⟩}{c o h (j) ⟨ k^{α + 1} ⟩} \leq T_{c} .$
(15)
The minimum T such that the network does not experience cascading failures is known as the threshold. In the end, the cascading failure threshold T_c is obtained in three different ranges α > 1, 0 < α < 1, and α = 1,
$T_{c} = \{\begin{cases} \frac{k_{\max}^{α - 1} ⟨ k ⟩}{c o h (j) ⟨ k^{α + 1} ⟩}, α > 1, \\ \frac{⟨ k ⟩}{c o h (j) ⟨ k^{2} ⟩}, α = 1, \\ \frac{k_{\min}^{α - 1} ⟨ k ⟩}{c o h (j) ⟨ k^{α + 1} ⟩}, 0 < α < 1 . \end{cases}$
(16)
A small T_c represents higher network robustness. From formula (16), as α increases, the robustness of the network decreases. The load regulation parameter α exhibits an inverse proportionality with the network robustness.
The proposed model—after adding protection mechanism
The initial load of node j plus the load obtained from allocation cannot be greater than its overload capacity,
$L_{j} + Δ L_{i j}^{'} \leq δ C_{j}^{'} .$
(17)
Substitute formulas (6) and (7) into Eq. (17),
$k_{j}^{α} + k_{i}^{α} \frac{C_{j}^{'} - L_{j}}{\sum_{m \in Γ (i)} C_{m}^{'} - L_{m}} \leq δ C_{j}^{'} .$
(18)
By simplifying Eq. (18), we can obtain
$\frac{1}{δ c o h (j)} + \frac{k_{i}^{α}}{δ \sum_{m \in Γ (i)} c o h (m) k_{m}^{α}} - \frac{1}{c o h (j)} \leq T .$
(19)
Based on the knowledge of probability theory,³⁹
$\begin{aligned} \sum_{m \in Γ (i)} c o h (m) k_{m}^{α} & = \sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} k_{i} \frac{k^{'} p (k^{'}) k^{' α} c o h (k^{'})}{⟨ k ⟩} \\ = \frac{k_{i}}{⟨ k ⟩} \sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} p (k^{'}) c o h (k^{'}) k^{' α + 1} \\ = \frac{k_{i}}{⟨ k ⟩} B, \end{aligned}$
(20)
where $B = \sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} p (k^{'}) c o h (k^{'}) k^{' α + 1}$ ⁠.
Taking the result into Eq. (19), we have
$\frac{k_{i}^{α - 1} ⟨ k ⟩}{δ B} + \frac{1}{δ c o h (j)} - \frac{1}{c o h (j)} \leq T_{c}^{'} .$
(21)
Similarly, the cascading failure thresholds T_c is obtained in different ranges α > 1, 0 < α < 1, and α = 1,

T_{c}^{'} = \{\begin{cases} \frac{k_{\max}^{α - 1} ⟨ k ⟩}{δ B} + \frac{1 - δ}{δ c o h (j)}, α > 1, \\ \frac{⟨ k ⟩}{δ B} + \frac{1 - δ}{δ c o h (j)}, α = 1, \\ \frac{k_{\min}^{α - 1} ⟨ k ⟩}{δ B} + \frac{1 - δ}{δ c o h (j)}, 0 < α < 1 . \end{cases}

(22)

In the above analysis, the factors affecting network robustness are determined. From formula (22), holding δ constant, as α increases, the robustness of the network decreases. Holding α constant, with an increase in δ, the network robustness increases.

Next, cascading thresholds with and without added protection will be compared. When α > 1, first compare the sizes of

\frac{k_{\max}^{α - 1} ⟨ k ⟩}{c o h (j) ⟨ k^{α + 1} ⟩}

and

\frac{k_{\max}^{α - 1} ⟨ k ⟩}{δ B}

⁠. Because

B = \sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} p (k^{'}) c o h (k^{'}) k^{' α + 1}

and δ > 1, we have

δ c o h_{\min} \sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} p (k^{'}) k^{' α + 1} \leq δ B \leq δ c o h_{\max} \sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} p (k^{'}) k^{' α + 1},

(23)

where coh_min and coh_max represent the minimum and maximum cohesions, respectively.

\sum_{k^{'} = k_{\min}}^{k^{'} = k_{\max}} p (k^{'}) k^{' α + 1} = ⟨ k^{' α + 1} ⟩

⁠; thus, we can get

δ c o h_{\min} ⟨ k^{' α + 1} ⟩ \leq δ B \leq δ c o h_{\max} ⟨ k^{' α + 1} ⟩ .

(24)

Because 0 < coh(j) < 1,

c o h (j) ⟨ k^{α + 1} ⟩ \leq δ B,

(25)

\frac{k_{\max}^{α - 1} ⟨ k ⟩}{δ B} \leq \frac{k_{\max}^{α - 1} ⟨ k ⟩}{c o h (j) ⟨ k^{α + 1} ⟩},

(26)

\frac{1 - δ}{δ c o h (j)} < 0 .

(27)

Finally, we can get

\frac{k_{\max}^{α - 1} ⟨ k ⟩}{δ B} + \frac{1 - δ}{δ c o h (j)} < \frac{k_{\max}^{α - 1} ⟨ k ⟩}{c o h (j) ⟨ k^{α + 1} ⟩} .

(28)

Above, we have completed the proof that the cascade failure threshold with protection added is less than the threshold without it for α > 1. The above proof also applies to the case when α < 1.

In the following, the cascade thresholds before and after adding protection when α = 1 will be compared. When α = 1, first compare the sizes of

\frac{⟨ k ⟩}{c o h (j) ⟨ k^{2} ⟩}

and

\frac{⟨ k ⟩}{δ B}

⁠. Equation is independent of the size of α; thus, we get

c o h (j) ⟨ k^{2} ⟩ \leq δ B .

(29)

From Eq. , we have

\frac{⟨ k ⟩}{δ B} + \frac{1 - δ}{δ c o h (j)} < \frac{⟨ k ⟩}{c o h (j) ⟨ k^{2} ⟩} .

(30)

The above completes the proof that when α = 1, the cascade threshold after adding protection is less than before adding protection. All symbols used in theoretical analysis are explained in Table I. The key parameters α and δ that affect the threshold T_c will be discussed in detail in the experiment.

TABLE I.

Meaning of parameters.

Symbol	Meaning
k_i	The degree of node i
coh(i)	The cohesion of node i
⟨k⟩	The average degree of the network
k_max	The maximum degree in initial network
k_min	The minimum degree in the initial network
L_i	The initial load of node i
C_i	The capacity of node i without the added protection mechanism
$C_{i}^{'}$	The capacity of node i with added protection mechanism
T	Capacity adjustment parameter
T_c	Capacity threshold without added protection mechanism
$T_{c}^{'}$	Capacity threshold with added protection mechanism
α	Load adjustment parameter
δ	Overload coefficient
N	The initial number of nodes in the network
N_A	Number of remaining nodes after cascading failure
S	The proportion of remaining nodes in the network

Symbol	Meaning
k_i	The degree of node i
coh(i)	The cohesion of node i
⟨k⟩	The average degree of the network
k_max	The maximum degree in initial network
k_min	The minimum degree in the initial network
L_i	The initial load of node i
C_i	The capacity of node i without the added protection mechanism
$C_{i}^{'}$	The capacity of node i with added protection mechanism
T	Capacity adjustment parameter
T_c	Capacity threshold without added protection mechanism
$T_{c}^{'}$	Capacity threshold with added protection mechanism
α	Load adjustment parameter
δ	Overload coefficient
N	The initial number of nodes in the network
N_A	Number of remaining nodes after cascading failure
S	The proportion of remaining nodes in the network

B. Experiments

In the cascading failure experiment, only the removal of a single node is considered, and only one node is removed at a time. The cascading failure model with the protection mechanism is shown in Algorithm 1. The BA network is a common type of network in the real world. Therefore, simulation experiments are conducted on BA scale-free networks. The experiment was repeated 100 times on an undirected, unweighted, and connected BA scale-free network, as shown in Fig. 4(a). From Fig. 4(a), the proportion of remaining nodes in the network fluctuates greatly before adding protection, and the fluctuating float of the percentage of remaining nodes in the network is significantly reduced after adding the protection mechanism. Most of the remaining node ratios of the network after adding protection are close to 1. The larger S is, the greater the proportion of remaining nodes in the network and the smaller the cascading failure size of the network. Taking S = 0.75 as the dividing line, S < 0.75 indicates that a larger scale of cascading failure has occurred in the network; S ≥ 0.75 indicates that the scale of network cascading failure is smaller and the network robustness is high.

ALGORITHM 1.

Cascading failure model with added protection mechanisms.

1: Input: Network G, parameters T, α, δ

2: Output: The proportion of remaining nodes in the network S

3: N ← the node of network G

4: Calculate all node’s cohesion coh(n), initial load L_n, and capacity

C_{n}^{'}

5: i ← Initial failure node and delete it

6: j ← A neighboring node of failed node i

7: Γ_i ← Set of all neighboring nodes of node i

8: Delete node i and distribute its load L_i to its neighbor nodes

9: while j ∈ Γ_i do

10: Calculate neighboring node’ load L_j after allocation

11: Consider overloading neighboring node j (Nodes that meet

L_{j} > C_{j}^{'}

⁠)

12: if

C_{j}^{'} < L_{j} \leq δ C_{j}^{'}

then

13: According to the first equation of Eq. to distribute the excess load to the neighbor nodes of node j

14: else

15: Delete node j and distribute its load to neighboring nodes according to the second equation of Eq. ; Determine whether the neighboring nodes of node j have failed. If any neighboring nodes have failed, execute the fifth line of code. Until there are no failed nodes in the network

16: end if

17: end While

18: Calculate the remaining number of nodes when the network maintains a steady state N_A

19: S ← N_A/N

20: Return S

FIG. 4.

View large Download slide

(a) 100 independent experiments comparing two strategies on BA scale-free degree network (N = 200). (b) This is a histogram plot of S at α = 0.5, δ = 1.1, and T = 1.8.

The distribution of the remaining node proportions in 100 experiments is presented in Fig. 4(b). From Fig. 4(b), when S < 0.75, in each interval, the proportion of remaining nodes before adding protection occurs more often than after adding protection. This observation suggests that the network’s robustness is lacking before the implementation of a protective mechanism, and there is a significant likelihood of experiencing widespread cascading failures. When S ≥ 0.75, the proportion of remaining nodes in the network with the added protection mechanism occurs significantly more often than before adding protection. The result with S ≥ 0.75 indicates a notable improvement in network robustness and a reduced likelihood of cascading failures following the implementation of the protective mechanism. On the whole, a significant majority of the remaining nodes in the network, before the addition of protection, are situated to the left of the demarcation line, reflecting a lower level of network robustness. However, after the inclusion of protection, the majority of remaining nodes in the network are now positioned to the right of the division line. These results in Fig. 4 affirm that the innovative protective mechanism introduced in our model effectively mitigates the extent of cascading failures within the BA network and enhances the proportion of nodes that remain operational.

1. The effect of α on network robustness

The influence of load adjustment parameter α on network robustness is considered. The experiment was carried out on a BA scale-free network with 800 nodes. We compared the proportion of remaining nodes in the network before and after implementing protection mechanism. When α = 0.5, 0.6, and 0.7, it becomes evident that there is a noticeable change in the proportion of remaining nodes before and after the introduction of the protection mechanism. The comparison of the remaining node proportions S before and after adding a protection mechanism is shown in Fig. 5. The results in Fig. 5 are based on the average of ten experiments.

FIG. 5.

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The effect of α on network robustness, δ = 1.2.

From Fig. 5, the proportion of remaining nodes after adding the protection mechanism is generally much higher than that before adding the protection mechanism. When T = 1.2, the proportion of remaining nodes S = 0 without the protection mechanism, while the proportion of remaining nodes after the protection mechanism is 0.3424, 0.7114, and 0.8888. When T = 2, no further cascading failures occur in the network after the addition of the protection mechanism; however, large-scale cascading failures are still occurring in the network prior to the addition of the protection mechanism, and the proportion of remaining nodes in the network is low. The reason is that after adding the protection, the overloaded node of the first-order neighbor of the network can transfer the excess load to the next-order neighbor node, thus reducing its burden and decreasing the probability of failure. When the proportion of first-order neighbor failures decreases, the scale of network failures decreases.

In Table II, the network after adding protection has reached stability and does not experience cascading failures at T = 4 and δ = 1.2; however, the proportion of remaining nodes in the network before adding protection is still low at α = 0.6 and α = 0.7. In Table III, the thresholds for cascading failures before and after the addition of the protective mechanism can be found. It is evident that, regardless of the value of α, when T = 4, the network’s proportion of remaining nodes remains relatively low before the protection mechanism is applied. Before the introduction of the protection mechanism, T ∈ [1, 4], the network does not exhibit a cascading failure threshold. After the addition of the protection mechanism, even for very small values of T, the network does not experience cascading failures. The significant reduction in the threshold underscores the effectiveness of the model proposed in this work in suppressing the occurrence of cascading failures.

TABLE II.

The proportion of remaining nodes S in the network under both strategies at T = 4 in Fig. 5.

	α = 0.5	α = 0.6	α = 0.7
Before	0.6332	0.3732	0.1822
After	0.998	0.998	0.998

TABLE III.

The cascading failure threshold T_c before and after the addition of the protection mechanism in Fig. 5.

	α = 0.5	α = 0.6	α = 0.7
Before	⋯	⋯	⋯
After	1.1	1.41	1.62

The effect of α on the proportion of remaining nodes in the network before and after adding protection is demonstrated in Fig. 6. In order to minimize errors, a total of ten experiments were performed. From Fig. 6, the adverse impact of increasing α on network robustness is evident. When α = 0.1, the proportion of remaining nodes after node removal in the network without protection is just 0.776, whereas in the network with added protection, it is 0.998. In contrast, when α = 0.67, the network without protection experiences a complete loss of remaining nodes S = 0 after node removal, while the protected network maintains S = 0.43, preserving nearly half of its nodes. This indicates that adding protection significantly enhances the network’s resilience to attacks. Moreover, the presence of various mutations in the protected network further underscores the pronounced negative effect of α on network robustness. There are a number of mutations present in the network with added protection, which also suggests that the negative effect of network robustness is significant.

FIG. 6.

View large Download slide

The proportion of remaining nodes in different α cases before and after adding the protection mechanism at T = 3 and δ = 1.2.

2. The effect of T on network robustness

The influence of capacity adjustment parameter T on network robustness is shown in Fig. 7. Compared with before and after the addition of the protection mechanism, the proportion of remaining nodes in the scale-free network in three cases of T = 2, T = 3, and T = 4. The results in Fig. 7 are based on the average of ten experiments.

FIG. 7.

View large Download slide

The effect of T on network robustness, δ = 1.2.

From Fig. 7, through the above analysis of load regulation parameters α, with the increase of α, the proportion of remaining nodes continues to decrease. Therefore, Fig. 7 overall maintains a declining trend. By observing the α, the proportion of remaining nodes after adding protection mechanism is higher than that without adding protection mechanism. As the capacity adjustment parameter T increases, the likelihood of sustaining a substantial proportion of remaining nodes in the network also rises. When α < 0.6, the network after adding the protection mechanism does not experience cascading failure and the proportion of remaining nodes is always closer to 1, regardless of the value of T. While the proportion of remaining nodes in the network before adding the protection is decreasing rapidly and reaches 0 at T = 2 and α = 0.68. Because the network is fragile at this time, the removal of the largest node can readily trigger the failure of other nodes in the network due to the excessive load. This amplifies the magnitude of cascading failures across the network and diminishes the proportion of remaining nodes. When α > 0.6, the network begins to encounter cascading failures. Despite this, the proportion of remaining nodes remains higher than the proportion of nodes that remained before the protection was added.

In Table IV, the proportion of remaining nodes in the network without an added protection mechanism is 0 at α = 0.8, regardless of the value of T. In the network with added protection mechanism, the proportion of remaining nodes remains at a high level at T = 3 and T = 4. The capacity adjustment parameter T is positively related to the network robustness. The proportion of remaining nodes in the network S is improved when increasing T.

TABLE IV.

The proportion of remaining nodes S in the network under both strategies at α = 0.8 in Fig. 7.

	T = 2	T = 3	T = 4
Before	0	0	0.0726
After	0.5496	0.8166	Red 0.986

The effect of T on the proportion of remaining nodes in the network before and after adding protection is demonstrated in Fig. 8. The results in Fig. 8 are based on the average of ten experiments. From Fig. 8, the proportion of remaining nodes in the network with added protection is always higher than that without. Moreover, when T = 4, the proportion of remaining nodes with added protection S = 0.978, and the proportion of remaining nodes without added S = 0.58. Therefore, the increase in the capacity parameter T can lead to a significant improvement in the robustness of the network with added protection.

FIG. 8.

View large Download slide

The proportion of remaining nodes in different T cases before and after adding the protection mechanism at α = 0.5 and δ = 1.1.

3. The effect of δ on network robustness

The effect of parameter δ on the proportion of remaining nodes in the network is shown in Fig. 9. From Fig. 9(a), δ = 1 means that no protection mechanism is added to the network. In most cases, the network has a low percentage of remaining nodes. At this point, the robustness of the network is poor, and removing the maximum degree node can easily lead to network collapse. At this point, the network has poor robustness and is prone to large-scale cascading failures. From Fig. 9(b), the first-order neighbors of the failed node have been protected at δ = 1.1, and in this case, the network has a high percentage of remaining nodes, implying that the robustness of the network is high and the probability of cascading failures occurring is small. From Fig. 9(c), the proportion of remaining nodes in the network remains at 0.998 no matter how T and α change at δ = 1.2, indicating that the removal of the node with the largest degree does not trigger a cascading failure and the robustness of the network is greatly improved. That is, the larger δ within a certain range, the smaller the cascading failure scale of the network, and the higher the robustness of the network. From Fig. 9, when a certain value of δ is taken, the network reaches a steady state without cascading failures. Therefore, the robustness of the network will not change significantly when δ increases based on this specific value. In theoretical analysis of Sec. IV, the larger δ, the smaller the cascading failure threshold T_c, and the higher the network robustness. This indicates that the results obtained from the experimental and theoretical analysis sections are consistent. The results show that even if δ takes a smaller value under our proposed protection mechanism, it makes the network have high robustness.

FIG. 9.

View large Download slide

The effect of δ on network robustness. (a) δ = 1. (b) δ = 1.1. (c) δ = 1.2.

4. The effect of node cohesion on network robustness

The effect of node cohesion on robustness is analyzed. Under two different mechanisms, we sorted the size of node cohesion in descending and ascending order and removed them in order, and the results are shown in Fig. 10. The results in Fig. 10 are based on the average of ten experiments. Before adding protection mechanisms, removing nodes in descending order first leads to cascading failure (Before, lar-34). After removing the tenth node, the remaining node proportion of the network drops to 0, indicating that the first ten nodes play an important role in the network. When nodes are removed in ascending order of cohesion (Before, low-34), after the network removes the 15th node, the proportion of remaining nodes in the network is 0. Compared to before adding protection mechanisms, the network robustness has greatly improved after adding protection mechanism. When removing nodes in descending order (After, lar-34), the remaining node proportion of the network drops sharply after removing 15 nodes. The remaining node proportion of the network reaches 0 after removing 15 nodes. After adding protection mechanisms, removing nodes in ascending order (After, low-34), when removing the 25th node, the remaining node proportion of the network drops sharply to 0. In addition, removing nodes with larger cohesion is easier to cause the network to collapse than removing nodes with lower cohesion, indicating that nodes with high cohesion also play an important role in the network. Also, this demonstrates that cohesion is meaningfully used to define node capacity.

FIG. 10.

View large Download slide

The effect of node cohesion on network robustness (α = 0.5, T = 1.2, and δ = 1.2).

V. EMPIRICAL NETWORKS

This paper compares the proportion of remaining nodes with and without added protection for three real networks. These networks all possess the characteristics of a BA scale-free network, and they are American Football network,⁴⁰ Wikiquote Edits network,⁴¹ and Euroroad network.⁴² The data information for these real networks is presented in Table V, where the visualizations of American Football Network and Euroroad Network is shown in Fig. 11.

TABLE V.

The statistical properties of the social network.

Dataset name	Number of nodes	Average degree
American Football network	82	3.81
Wikiquote Edits network	212	2.264
Euroroad network	1039	2.512

FIG. 11.

View large Download slide

(a) The illustration of the American Football network. (b) The illustration of the Euroroads network.

A. American Football network

The American Football network includes American football games between colleges in the IA region during the regular season of the fall of 2000. In this network, nodes represent teams, and edges represent matches between teams. The comparison chart illustrating the remaining node proportions before and after protection has been added in this network is displayed in Fig. 12.

FIG. 12.

View large Download slide

Comparison of the proportion of remaining nodes in the network before and after adding protective measures in the American Football network (α = 0.5, T = 1.5, and δ = 1.2).

From Fig. 12, in the American Football network, when the first eight nodes with large degrees are removed, whether a protection mechanism is added or not, the network will not fail in cascading. When the ninth node with a large degree value is removed, the network without a protection mechanism experiences cascading failure, and the proportion of remaining nodes decreases faster, and then maintains a uniform decline. This indicates that the first nine nodes play an important role in the network, and deleting these nodes will cause the network to lose most of its functionality. When the 32nd node is removed, the proportion of remaining nodes in the network tends to be close to 0. When the 12th node of the American Football network with the protection mechanism is removed, a small cascading failure occurs, and the proportion of remaining nodes decreases faster. Then, it maintained a uniform decline, and when the 32nd node was removed, the proportion of remaining nodes in the network was 0.39. It is nearly 40% more than the proportion of remaining nodes without the protection mechanism. Except for the first six nodes, the proportion of remaining nodes after adding protection is always higher than that without adding. In the experiment of the American Football network, the effectiveness of the protection mechanism proposed in this paper in improving network robustness is also proved.

B. Wikiquote Edits network

The Wikiquote Edits network is the editing network of Corsican Wikiquote. It includes users and pages from Corsican Wikiquote connected through editing events. Each edge represents an edit. In the Wikiquote Edits network, the comparison chart illustrating the remaining node proportions before and after protection has been added is displayed in Fig. 13. From Fig. 13, the robustness of the network with the added protection mechanism is significantly better than that of the network without the added protection mechanism in the Wikiquote Edits network. In the network without the protection mechanism, when the second node is removed, the proportion of remaining nodes in the network decreases sharply to 0.4857. This is because the robustness of the network is poor at this time, and the removal of the node with the largest degree value triggers the other nodes to fail as well, and the proportion of remaining nodes decreases. Thereafter, the removal of the network nodes causes the proportion of remaining nodes to decrease uniformly, and when the 42nd node in the network is removed, the proportion of remaining nodes in the network is 0.238, which indicates that the network has fewer remaining nodes to work properly. After adding the protection mechanism, the proportion of remaining nodes in the network is always higher than that of the network without the protection mechanism. When the fifth node is removed, the network undergoes a small cascade failure and the proportion of remaining nodes drops to 0.838. Since then, the proportion of remaining nodes in the network has been decreasing at a constant rate. When the 42nd node is removed, the proportion of remaining nodes is still as high as 0.569, which means that half of the nodes in the network are still functioning normally. This is because, as the number of failed nodes in the network grows, the number of protected first-order neighbor nodes also grows, and the nodes in the network are less likely to fail, and the robustness is enhanced, reducing the proportion of failed nodes. Experiments in the Wikiquote Edits network also prove the effectiveness of the method in this paper.

FIG. 13.

View large Download slide

Comparison of the proportion of remaining nodes in the network before and after adding protective measures in Wikiquote Edits (α = 0.5, T = 1.5, and δ = 1.2).

C. Euroroads network

The Euroroads network is an international road network mostly located in Europe, comprising 1039 nodes and 1305 edges, with an average degree of 2.512. The network is undirected; nodes represent cities and an edge between two nodes denotes that they are connected by a road. Simulation results of the cascading failure model, with and without protective measures, are illustrated in Fig. 14. From Fig. 14, it is evident that the proportion of remaining nodes S after adding protection consistently remains higher than the case without protection. When the first 11 nodes are removed from the network, the proportion of remaining nodes remains consistent in both scenarios. However, without protection, a significant decrease in the proportion of remaining nodes is observed upon the removal of the 12th node. Particularly, when the 14th node is removed, the proportion of remaining nodes drops sharply to 0.43. In contrast, in the cascading failure model with protection, after the removal of 14 nodes, the proportion of remaining nodes is 0.78. This indicates that the model with protective measures can suppress the outbreak of cascading failures, enhancing the network’s resilience against attacks.

FIG. 14.

View large Download slide

Comparison of the proportion of remaining nodes in the network before and after adding protective measures in Euroroads network (α = 0.5, T = 2 and δ = 1.2).

VI. CONCLUSION

The improvement of robustness in complex networks contributes to the protection of information within the network, ensuring the security of information and the reliability of network services. Thus, enhancing the robustness of complex networks is of utmost importance. An optimized cascade failure model is proposed with the aim of enhancing the robustness of complex networks. In the proposed model, particular attention is given to the overload state of nodes, and a protective mechanism is introduced by adding neighboring nodes of the initially failed nodes. Moreover, our study incorporates node cohesion when defining node capacity, accounting for each node’s specific position within the social network, which addresses a prior shortcoming in cascading failure research where node location was often overlooked. The key parameters influencing the robustness of complex networks are obtained under the proposed model through the theory section. The critical parameters are experimentally analyzed on a constructed unweighted, undirected, and connected BA scale-free network. In the empirical networks, three real networks were introduced and the robustness of each network was compared under different models. The results demonstrate that the model proposed in this paper effectively prevents the occurrence of cascading failures and enhances the robustness of complex networks. It is important to note that network cohesion alone may not provide precise insights into the exact positions of individual nodes. In addition, cohesion calculations rely on the network’s connectedness. Consequently, the approach presented in this paper may not be well-suited for disconnected networks. Therefore, a more precise robustness optimization strategy, which takes into account node locations and applies to disconnected networks, should be the focus of future research efforts.

SUPPLEMENTARY MATERIAL

The experimental data for Figs. 5 and 7 are included in the supplementary material.

ACKNOWLEDGMENTS

This work was supported by the National Social Science Foundation in China (Grant No. 23 & ZD115) and the Postgraduate Scientific Research Innovation Project of the School of Mathematics and Statistics at Hubei Minzu University, China (Grant No. STK2023011).

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Ethics Approval

The data used in this article are all sourced from open online data, and the data sources have been included in the Data Availability Statement.

Author Contributions

Mengjiao Chen: Conceptualization (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Niu Wang: Data curation (equal); Methodology (equal); Software (equal). Daijun Wei: Data curation (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The authors declare that the data supporting the findings of this study are available from http://konect.cc/networks/dimacs10-football/, http://konect.cc/networks/edit-cowikiquote/ and http://konect.cc/networks/subelj_euroroad/.

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© 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

Optimization method for protecting the robustness of first-order nodes in complex networks

I. INTRODUCTION

II. PRELIMINARIES

A. Cohesion of network and node

B. Cascading failure

III. THE PROPOSED METHOD

IV. THEORY AND EXPERIMENTS

A. Analysis and calculation

B. Experiments

1. The effect of α on network robustness

2. The effect of T on network robustness

3. The effect of δ on network robustness

4. The effect of node cohesion on network robustness

V. EMPIRICAL NETWORKS

A. American Football network

B. Wikiquote Edits network

C. Euroroads network

VI. CONCLUSION

SUPPLEMENTARY MATERIAL

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Ethics Approval

Author Contributions

DATA AVAILABILITY

REFERENCES

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Optimization method for protecting the robustness of first-order nodes in complex networks

I. INTRODUCTION

II. PRELIMINARIES

A. Cohesion of network and node

B. Cascading failure

III. THE PROPOSED METHOD

IV. THEORY AND EXPERIMENTS

A. Analysis and calculation

B. Experiments

1. The effect of α on network robustness

2. The effect of T on network robustness

3. The effect of δ on network robustness

4. The effect of node cohesion on network robustness

V. EMPIRICAL NETWORKS

A. American Football network

B. Wikiquote Edits network

C. Euroroads network

VI. CONCLUSION

SUPPLEMENTARY MATERIAL

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Ethics Approval

Author Contributions

DATA AVAILABILITY

REFERENCES

Citing articles via

Submit your article

Sign up for alerts

Related Content

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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