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A Constrained Cooperative Coevolution Strategy for Weights Adaptation Optimization of Heterogeneous Epidemic Spreading Networks

In this paper, the dynamic constrained optimization problem of weights adaptation for heterogeneous epidemic spreading networks is investigated. Due to the powerful ability of searching global optimum, evolutionary algorithms are employed as the optimizers. One major difficulty following is that the dimension of the weights adaptation optimization problem is increasing exponentially with the network size and most existing evolutionary algorithms cannot achieve satisfied performance on large-scale optimization problems. To address this issue, a novel constrained cooperative coevolution (C^3) strategy which can separate the original large-scale problem into different subcomponents is tailored for this problem. Meanwhile, the ϵ constraint-handling technique is employed to achieve the tradeoff between constraint and objective function. To validate the effectiveness of the proposed method, some numerical simulations are conducted on a Bárabasi-Albert network.

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1 Introduction

Epidemic spreading over complex networks [1] has attracted lots of attention since the pioneering work [2] of Daniel Bernoulli in 1760. Many researches are focused on the mathematical modeling of disease spreading process, classical epidemic models such as the susceptible-infected-susceptible (SIS) [3, 4, 5, 6], and the susceptible-infected-recovered (SIR) [7] model have been well studied for decades. Since the spreading of disease (such as AIDS, SARS, etc.) may cause numerous damages to the human society, developing control policies for epidemic spreading process is of great significance, with potential applications in public health. As pointed out in [8], the two common strategies to suppress epidemic spreading scale are increasing the recovery rate and decreasing the infection rate. For example, in [9], the PID control laws are implemented for the classical SIR model where the vaccination rate is the control variable. Despite the innovative results, these studies did not take the “budget” or the so-called “control cost” into consideration, which are commonly existed in real-world scenarios. In recent years, optimal control of epidemic spreading [10, 11, 12, 13] has gained more and more attentions. For example, in [14], an optimal control strategy is designed for vaccination, quarantine and treatment actions.

Besides from the above control strategies focusing on the epidemic spreading parameters, recently another control strategy which aims at adjusting topology of the underlying network was investigated [15]. Acting as the “bridge” for epidemic spreading from infected individuals to healthy ones, the network topology determines the epidemic transmission efficiency. A quantitative parameter that defines the strength of two nodes in a network is the value of weight between them. Since the physical meaning of the weight can be described by the contact frequency of two individuals, the intuitive idea of controlling the weights is more natural and practical than controlling the spreading parameters. In [16], an individual-based weight adaptation mechanism in which individuals’ contact strength is adaptable depending on the level of contagion spreading over the network is proposed. An optimal control formulation is also presented to address the trade-off between the global infected level and the local weight adaptation cost corresponding to the topology of the underlying contact network. In both [15] and [16], the objective function contains both the infection cost and control cost. However, the problem of minimizing the infection cost with given fixed number of budget, i.e. control cost has not been studied, which is more practical than the unconstrained optimization problems. This motivates us to investigate the constrained optimization problem of weights adaptation.

Meanwhile, when solving the optimal control problems in both [15] and [16], the forward-backward sweep method (FBSM) [17]

is used to find the numerical solutions, which is an indirect method itself. The intuitive idea of FBSM is that the initial value problem of the state equations is solved forward in time, using an estimate for the control and adjoint variables. Then the adjoint final value problem is solved backwards in time. The complicated procedures add several difficulties to the optimal problem, which can be summarized as follows:

  • Need to compute various partial derivatives of the Hamiltonian and solve additional differential equations, which introduce more errors for the optimization problem.

  • Need to make an initial guess of the adjoint variables, the sensitivity of the method to changes in initial guesses.

In [17], the authors found that the FBSM method fails to terminate under some circumstances.

Evolutionary algorithms (EAs) [18, 19] which are inspired by nature, have shown powerful searching ability for the global optimum with few restrictions. Also, since EAs are direct methods, the implementations are much simpler than other conventional methods. As one of the most powerful evolutionary algorithm, differential evolution (DE) [20, 21, 22]

has shown superior performance over other heuristic algorithms on very complex searching and optimization problems. The control parameters of DE are few and it is highly efficient

[20]. This motivates us to solve the constrained optimization problem based on DE. However, since the weights adaptation involves all links in the network for a given period of time [16], the dimension of the optimal solution is relatively high. And solving this kind of large-scale optimization problem [23] is a challenging problem in the community of evolutionary computation. In addition, how to achieve the tradeoff between constraint and objective function is another difficulty.

Motivated by the above considerations, a dynamic constrained optimization problem of weights adaptation for heterogeneous epidemic spreading networks based on SIS model is formulated. To deal with the high-dimensional optimization problem, a novel constrained cooperative coevolution () strategy which can separate the original high-dimensional search space into some low-dimensional ones by random grouping strategy is proposed. The constraint-handling technique is employed to achieve the tradeoff between constraint and objective function. Moreover, as a commonly used variant of the classical DE, the differential evolution with neighborhood search (NSDE) [24] algorithm is employed as the optimizer for these sub-problems.

The main contributions of this paper can be summarized as follows:

  • A dynamic constrained optimization problem of weights adaptation for heterogeneous epidemic spreading networks is formulated.

  • Evolutionary computation techniques with strong search ability as well as quite small amount of demands are applied to solve this kind of problem.

  • A novel constrained cooperative coevolution () strategy is tailored for this real-world large-scale optimization problem.

The rest of this paper is organized as follows. The model description and problem formulation are given in Section 2. Differential evolution is briefly introduced in Section 3. In Section 4, the methodology is given in detail. Some numerical experiments are presented in Section 5. Finally, this paper is concluded in Section 6.

2 Model Description and Problem formulation

2.1 Heterogeneous Weighted SIS-based Network Model

In the heterogeneous weighted SIS-based network, all nodes can be classified into two possible states according to their health status, that is, susceptible and infected. Susceptible individuals can be infected by infected individuals through the links between them and in state

. Meanwhile, infected individuals can be cured and become susceptible again and in state . To be more specific, every node at time

is infected with probability Pr

and susceptible with probability Pr. At each time , a node can only be in either of these two states, thus PrPr.

Then the following heterogeneous weighted SIS-based network model is obtained from the N-intertwined mean-field approximation (NIMFA) [25, 26, 16]:

(1)

where denotes the probability of node being infected at time . The infection and curing rates and for each node in the network are described by two independent Poisson processes; denotes the weight of edge from node to node ; is the number of nodes in the network.

Then (1) can be rewritten in the following compact form:

(2)

where and

To be noticed, the network considered here is a directed one and the weights of which are in association with the infected level in the network as it is shown in (2). Define for all node so that self-loop is not considered here.

2.2 Problem Formulation

As it is shown in (2), the infected level can be controlled by the weights adaptation in the network. However, bearing the cost of weights adaptation in mind, a natural dynamic constrained optimization problem is formed as follows:

(3)

where is the set of all admissible weights; denotes the infection cost function for each individual , the objective function denotes the total infection cost for all individuals in the considered period ; denotes the cost function for weights adaptation and denotes the initial weight at , is a constant that characterizes the maximum cost of weight adaptation, is the inequality constraint.

The dynamic constrained optimization problem is interpreted as: how to design the adaptive weights for all links from to such that the infected level in the network can be suppressed to the maximally extent under given budgets.

3 Differential Evolution

Differential evolution (DE) [27, 28] is used as a base optimizer to solve the dynamic constrained optimization problem in (3), which is arguably one of the most powerful stochastic real-parameter optimization algorithms in current use. Different from traditional evolutionary algorithms (EAs), DE uses difference of individual trial solutions to explore the objective function landscape. Typically, DE consists of four stages: initialization, mutation, crossover, and selection.

Step 1: Initialization
The ultimate goal of DE is to find a global optimum point in a -dimensional real parameter space . In the beginning, a randomly initiated population which consists of -dimensional individuals is generated. The subsequent generations in DE are denoted by . The population at generation is denoted as follows:

where is the

th target vector in current generation:

Initially (), the population should be uniformly randomized within the search space constrained by the maximum and minimum bounds: and . Hence the initialization is formulated as follows:

where

is a uniformly distributed number between

and .

Step 2: Mutation
The mutation operator aims to create a mutant vector for each target vector through utilizing the differential information of pairwise individuals. The following mutation operator is adopted in this paper.

DE/current-to-best/1:

where , is the mutant vector, and are mutually exclusive integers randomly chosen from , is the best individual in the current population, and the scaling factor is a positive control parameter for scaling the difference vectors.

Step 3: Crossover
The crossover operator is employed to enhance the diversity of the population. The trial vector is formed by exchanging components between the target vector and the mutant vector . The following binomial crossover is utilized:

if
otherwise.

where , , is a randomly chosen index from . is the crossover rate.

Step 4: Selection
In the selection step, the target vector is compared with the trial vector , the better one can be survived in the next generation

if
otherwise.

4 Methodology

4.1 Encoding Mechanism

As it is described above, the dimension of the decision space is related to the number of links in the network and the time length . To be more specific, for a directed network with nodes, we have

(4)
Fig. 1: Schematic of the encoding mechanism.

To better illustrate the encoding mechanism, a schematic graph is presented in Figure 1. The network consists of nodes, where blue and red nodes represent susceptible and infected individuals, respectively. denotes the th target vector in the th generation, with a dimension of . Each element of corresponds to a specific element of the weight matrix at time in the manner presented in Figure 1. Bearing the infection cost and the weights adaption constraint in mind, the weight matrix at time is evolved adaptively to balance the objective function and constraint. Considering the time length , the th target vector at th generation is formulated as:

To be noticed, the health status of all individuals in the network is varying from time to time. For example, as it is shown in Figure 1, node is infected at time while it is cured and become susceptible at time . Therefore, this is a dynamic optimization problem considering the interactions between epidemic spreading process and weights adaptation.

Remark 4.1

With this encoding mechanism, one inevitable problem is that the dimension of the decision space increases exponentially with the size of the network as it is shown in (4). Therefore, this constrained optimization problem (3

) may suffers from the “curse of dimensionality”, which implies that most of the EAs’ performance deteriorates rapidly as the increasing of the dimensionality of the search space.

4.2 Differential Evolution with Neighborhood Search (NSDE)

As a variant of classical DE introduced in 3, NSDE [24] is effective in escaping from local optima when searching in circumstances without knowing the preferred step size. The main difference between NSDE and DE is that the neighborhood search (NS) strategy is utilized, which is a typical technique in evolutionary programming (EP) [29]. To be more specific, the scaling factor in classical DE is replaced in the following manner:

if
otherwise.

where

is a Gaussian random number with mean 0.5 and standard deviation 0.5, and

is a Cauchy random variable with scale parameter

. In NSDE, the parameter was set to a constant number 0.5.

4.3 Constraint-handling Technique

The constraint-handling technique [30], which is adopted by the winner of IEEE CEC2010 competition, is utilized to compare two target vectors and . To be more specific, is better than if the following conditions are satisfied:

if (5a)
if (5b)
otherwise. (5c)

where denotes te degree of constraint violation on the constraint as follows:

In Eq. 5a, is designed to decrease with the increasing of the generation as follows [31]:

if
otherwise.

where is the maximal degree of constraint violation of the initial population; is a parameter to truncate the value of ; is set to be in this paper.

4.4 A Constrained Cooperative Coevolution () Strategy

In this subsection, a novel coevolution strategy is developed to solve this dynamic constrained optimization problem motivated by the DECC-G algorithm in [23]. The core of this coevolution strategy is “divide-and-conquer”. That is, the original -dimensional search space is divided into number of -dimensional ones by random grouping strategy, where .

The constrained cooperative coevolution () framework for this problem can be summarized as follows:

(1) Set to start a new cycle.

(2) Decompose an original -dimensional target vector into low-dimensional subcomponents with dimension randomly, i.e. . Here “randomly” indicates that each dimension in the original target vector has the same probability to be assigned into any of the subcomponents.

(3) Optimize the th subcomponent with NSDE and constraint-handling technique introduced in 4.2 and 4.3 for a predefined number of fitness evaluations (FEs).

(4) if then and go to Step 3.

(5) Stop if halting criteria are satisfied; otherwise go to Step (1) for the next cycle.

Remark 4.2

The probability of strategy to assign two interacting variables and into a single subcomponent for at least cycles is:

where is the total number of cycles and is the number of subcomponents. The proof can be referred to [23].

Given network size and , then . Select , then the number of subcomponents . When the number of cycles , we have:

These results demonstrate that strategy has relatively high probabilities to optimize interacting variables in a single subcomponent for at least one or two cycles.

4.5 NSDE under the framework

Assign NSDE as the base optimizer for the subcomponents, it is straightforward to obtain the NSDE with constrained cooperative coevolution algorithm, denoted as NSDE-. The pesudocode of NSDE- is given in Algorithm 1.

1 /*Initialization*/
2 ;
3 ;
4 for  do
5       ;
6       for  do
7             ;
8             ;
9             ;
10             /*Use sub-optimizer*/
11             ;
12             ;
13             ;
14            
15      
return
Algorithm 1 NSDE-

5 Numerical Experiments

To validate the effectiveness of the proposed method, a synthetic network is obtained as the underlying network. When solving the dynamic constrained optimization problem in (3), each optimization algorithm is used for 25 independent runs.

5.1 Datasets

For validation, one of the most famous synthetic complex network-the Bárabasi-Albert (BA) network [32] is used. When constructing the BA network, initially fully connected nodes are placed in the network. A new node is connected to existing nodes with probability proportional to the degree of them at each step [33]. A total number of nodes is fixed for the BA network in the simulations, originally this BA network is a bidirectional network.

The topologies of the BA network is presented in Figure 2, where circles represent nodes in the network and their sizes are proportional to degrees.

Table I below summarizes the topological features of the network, where denote the total number of nodes, average degree, average clustering coefficient and density of the network, respectively.

Network name
Bárabasi-Albert 20 8.5 0.519 0.447
TABLE I: Topological features of the analyzed network.
Fig. 2: Bárabasi-Albert network

5.2 Parameters Selection

For all experiments, the objective function for individual at time adopts the form:

Meanwhile, the cost function for weights adaptation for a pair of individuals and is:

The value of the constraint is selected as 700.

Based on the underlying network introduced in the previous subsection, some numerical simulations are conducted to illustrate the effectiveness of the proposed method (NSDE-).

The epidemic parameters used in the simulations are presented in Table II, where denote the initial infection state, infection rate, curing rate, and terminal time, respectively. These parameters are carefully chosen to make sure that the infected level is relatively high when there is no weights adaptation, hence the effects of weights adaptation on epidemic spreading can be easily observed.

Remark 5.1

As it was presented in [25], the sharp epidemic threshold was rigorously proven to be:

where

is the largest eigenvalue-the spectral radius of the adjacency matrix

. For the BA network, the epidemic threshold value is obtained as . Referring to the epidemic parameters in Table II, it is easily obtained that , hence the infected level is relatively high without weights adaptation.

Regarding to the algorithm parameters, the population size , the number of FEs and the crossover rate for both NSDE and NSDE- algorithms are the same. For NSDE- strategy, the dimension of the subcomponents is selected as .

Network name FEs Cr
Bárabasi-Albert 0.153 0.4 0.3 10 350 6.30e+06 0.9
TABLE II: Epidemic and algorithm parameters used in the simulations.

5.3 Results

(a) Fitness value
(b) Constraint violation
(c) Epidemic evolution
(d) Weights adapation
Fig. 3: Simulation results on BA network for 25 independent runs

For Bárabasi-Albert network, the simulation results are presented in Figure 3(a), 3(b), 3(c) and 3(d). Both the proposed NSDE- and the existing NSDE method are used. Meanwhile, two control groups are added for comparison, i.e. “No adaptation” and “constant adaptation” strategy. The formulation of these two strategies at time are as follows:

where

In this manner, these two strategies are employed as the baselines for comparison. “No adaptation” means that the weights remain unchanged in the considered time period, while “Constant adaptation” refers to a “discount” on the original weights with the budget fully used. For the epidemic parameters in Table II and , the constant adaptation ratio can be calculated as .

Figure 3(a) and 3(b) illustrate the evolution process of the mean best fitness value and constraint violation with respect to generations over 25 independent runs, respectively. For the “No adaptation” and “Constant adaptation” strategy, the solutions are determined initially. Hence the value of the objective function and constraint violation for these two strategies remain unchanged. For the “No adaptation” and “Constant adaptation” strategies, the constraint violation remains zero. Combing Figure 3(a) and 3(b), it is evident that the proposed NSDE- outperforms NSDE on BA network. Moreover, NSDE- outperforms both the “No adaptation” and “Constant adaptation” strategies.

Remark 5.2

One interesting phenomenon to be noticed in Figure 3(b) is that the evolution of constraint violation for NSDE- almost coincides with that of NSDE. However, the evolution of fitness value for these two algorithms are rather different. This is due to the use of constraint-handling technique in 4.3, which puts more emphasis on the constraint violation than the fitness value for earlier generations.

Figure 3(c) and 3(d) demonstrate the evolution process of two main indices, i.e. infected level and total weights over time, which characterize the epidemic spreading and weights adaptation level, respectively. The definition of theses two indices are as follows:

The physical meaning of the infected level and the total weights are the expectation of infection probability and total number of weights, respectively. In this manner, can be used as an index which reflects the current infected level among the whole population of individuals in the network. Similarly, reflects the weights variation of the network concerned. For infected level, “No adaptation” strategy achieves a relatively high level, which is used as a baseline for comparison as it is demonstrated in 5.2 and Remark 5.1. However, “Constant adaptation” strategy achieves a lower infected level than “No adaptation” strategy, which is mainly due to that the “Constant adaptation” strategy makes full use of the budget as it is shown in Figure 3(d). Referring to the total weights variation in Figure 3(d), it is obvious that remains unchanged at every time for “No adaptation” strategy. For “Constant adaptation” strategy, the value of total weights decreases to a lower value and keeps unchanged for the whole time period. However, for both NSDE and NSDE- strategy, the value of total weights experience a deceasing process first and restore to the initial value of time , which coincides with the results in [16]. The restoring process of for NSDE- is more moderate than that of NSDE, hence the infected level is lower at most time. One interesting observation is that the value of total weights for NSDE- drops to a lower value than that of the “Constant adaptation” strategy, and restores to the initial value at time , the infected levels indicate that this kind of “dynamically adaptation” is more effective in controlling the epidemic spreading scale.

In addition, To test statistical significance, the multi-problem Wilcoxon’s test

[31] are implemented to compare these methods, the results are shown in Table III. Once more, the results indicate that the proposed NSDE- is superior to the other competitors.

Algorithm Mean OFV Std Dev -value
NSDE- 106.4530 0.1506 -
NSDE 127.1566 1.3425 1.4157e-09
No adaptation 160.5280 0.00 9.7285e-11
Constant adaptation 126.8617 0.00 9.7285e-11
TABLE III: Wilcoxon’s test on BA network for 25 independent runs.

6 Conclusion

In this paper, a dynamic constrained optimization problem of weights adaptation for heterogeneous epidemic spreading networks is formulated. Combining constrained cooperative coevolution () strategy with NSDE, a novel NSDE- algorithm is tailored for this problem. Numerical experiments on a BA network showed the effectiveness of this algorithm. In future research, how to expand this algorithm for on-line implementation is an interesting yet challenging topic.

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