Approaching the bi-objective critical node detection problem with a smart initialization-based evolutionary algorithm

Introduction

In recent years, complex networks have received a lot of attention due to their applicability in various domains. Several optimization problems were studied within complex networks like community detection (Fortunato, 2010), maximal influence node detection (Kempe, Kleinberg & Tardos, 2003) and link prediction (Liben-Nowell & Kleinberg, 2007). All of the aforementioned problems reveal major insights into the networks studied.

Identifying critical nodes (the critical node detection problem, or CNDP) in a complex network is a crucial task. The base problem consists of minimizing pairwise connectivity by removing a subset of K nodes in a given graph. In Arulselvan et al. (2009) it was proven to be an NP-hard problem.

The general formulation of the problem is Lalou, Tahraoui & Kheddouci (2018): given a G = (V, E) graph and a connectivity metric λ, find the set of nodes S⊆V such that G[V∖S] satisfies the metric λ. This metric is usually defined as an objective function that needs to be optimized (for example, maximize the number of components, minimize the component size, etc.).

CNDP has a wide field of applicability, for example in social network analysis (Fan & Pardalos, 2010), epidemic control (Tao, Zhongqian & Binghong, 2006), network immunization (Kuhlman et al., 2010) and biological networks (Liu et al., 2020). Several algorithms were designed for the CNDP. The majority of the exact methods are based on the integer linear programming formulation of the problem (Summa, Grosso & Locatelli, 2012). In Addis, Summa & Grosso (2013), a dynamic programming approach is proposed for a special class of graphs. As approximation algorithms, we can mention, for example, a simulated annealing algorithm (Ventresca, 2012). A thorough survey of existing methods for the CNDP can be found in Lalou, Tahraoui & Kheddouci (2018).

The CNDP has several variants exploring the connectivity metric λ. Other variants with constraints were introduced, such as the cardinality constrained critical node detection problem (CC-CNDP) (Arulselvan et al., 2011) and the component-cardinality-constrained critical node problem(3C-CNDP) (Lalou, Tahraoui & Kheddouci, 2016). One of the existing bi-objective variants of the CNDP is proposed in Li et al. (2019). In this variant the cost of removing the node counts. Another bi-objective variant proposed in Ventresca, Harrison & Ombuki-Berman (2018) is the base of our study (described in ‘The Bi-objective Critical Node Detection Problem’).

Evolutionary algorithms are powerful tools in optimization problems. Multi-objective optimization problems involve multiple objective functions which need to be optimized at the same time, so they can be used in real-world optimization problems. Because the BOCNDP is an NP-hard problem (Ventresca, Harrison & Ombuki-Berman, 2018), the use of the evolutionary algorithms is straightforward. To increase the performance of evolutionary algorithms, several techniques were designed, for example, hybridization, a special case of memetic algorithms which incorporates a local search in the initialization phase. Kazimipour, Li & Qin (2014) emphasizes the importance of population initialization techniques, introduces a new categorization, and mentions some concrete examples.

Due to the wide applicability of the critical node detection problem, this article introduces new smart initialization strategies that can be incorporated into any multi-objective optimization algorithm that treats the BOCNDP to increase its performance. These strategies can be used in other variants of the CNDP, or even for other computationally graphed theoretical problems because they take into account structural information about the network.

To summarise, the main goal of this article is as follows:

• a smart initialization which is based on a depth-first search: nodes lying on a path are chosen to be in the initial population;

• a random walk-based smart initialization strategy: a random walk is simulated on the graph, and nodes that appear more times in the walk are considered more important;

• a degree-based smart initialization strategy: nodes with a higher degree are more likely to be chosen in the initial population;

• statistical analysis of the three smart initialization strategies introduced here and their comparison with random initialization.

The rest of the article is organized as follows: In the second section, we describe the bi-objective critical node detection problem and the existing solving algorithms. In the third section, we present the proposed initialization algorithms. The next section describes the numerical experiments. The article ends with conclusions and further work.

The bi-objective critical node detection problem

Let G = (V, E) be an undirected graph, where V is the set of nodes, and E is the set of edges.

Let G = (V, E) be an undirected graph, where V is the set of nodes, and E is the set of edges. The bi-objective critical node detection problem was proposed in Ventresca, Harrison & Ombuki-Berman (2018) and consists of finding a fixed number of k nodes, which, if deleted from graph G, will optimize the following two objectives:

1. maximize the number of connected components;

2. minimize the variance of the cardinality of the connected components.

Formally the objectives are the following: (1)${\displaystyle max\left|H\right|,}$ (2)${\displaystyle minvar\left(H\right),}$ (3)${\displaystyle \text{such that}\sum _{i\in S}{w}_i\le W,}$

where w_i are the weights associated to the vertices of the graph and W > 0 is a constraint, H denotes $G\left[V\setminus S\right]$ the set of the connected components and var(H) denotes the variance of the cardinality of the connected components and can be calculated in the following way: (4)${\displaystyle \frac{1}{\left|H\right|}\sum _{h\in H}{\left(\left|h\right|-\frac{n^{\ast }}{\left|H\right|}\right)}^2,}$ where n^∗ = ∑_h∈H|h| is the number of nodes in $G\left[V\setminus S\right]$. The BOCNDP is different from the CNDP (Ventresca, Harrison & Ombuki-Berman, 2018).

Let us consider a simple example, the graph presented in Fig. 1. If we need to identify k = 2 critical nodes, then $S=\left\{2,3\right\}$ is the optimal solution. The $G\left[V\setminus S\right]$ will have 5 components, |H| = 5 and ${\displaystyle var\left(H\right)=\frac{1}{5}\cdot \left[{\left(1-\frac{13}{5}\right)}^2+4\cdot {\left(3-\frac{13}{5}\right)}^2\right]=\frac{16}{25}=0.64.}$

Because the BOCNDP is a relatively new problem formulation, the literature is not rich in proposed algorithms. In Ventresca, Harrison & Ombuki-Berman (2018), six existing multi-objective algorithms are used to solve the BOCNDP. In Li et al. (2019), a different variant of the BOCNDP, called Bi-CNDP, is introduced and studied with decomposition-based multi-objective evolutionary algorithms.

Evolutionary computation method

Evolutionary algorithms are powerful optimization tools, especially in multi-objective optimization problems. To increase the performance of these algorithms, hybrid versions are designed and analysed. Smart initialization of the population of an evolutionary algorithm can increase the performance of the algorithm significantly (Maaranen, Miettinen & Penttinen, 2007).

We present three strategies that can be used in the initialization phase of any multi-objective algorithm. The first one is based on a depth search algorithm, outlined in the algorithm 1. A depth-first search (DFS) algorithm is started with a random initial node, and every xth element will be added to the chromosome, where $x=\frac{\left|V\right|}{k}$, |V| is the number of nodes and k is the number of critical nodes.

The second initialization method is based on the degree distribution of the nodes. The first x nodes with the highest degree are set in the chromosome, and the rest of the k − x nodes are selected randomly, to preserve the stochastic nature of the initialization (Algorithm 2).

The third method is based on a random walk. We start the walk from a random node, t is the length of the walk and p_r is the probability to restart the walk. In each step, the decision is to continue the walk or to restart. If we fail to walk through k different nodes, the algorithm will restart from another initial point. In the walk, we keep counting how many times a node appears. The more times it appears, the higher the probability it is a gene in the chromosome. The main steps are presented in the algorithm 3.

These initialization strategies can be used in any kind of multi-objective evolutionary algorithm. The outline of the smart initialization-based algorithm is depicted in the algorithm 4.

 
_______________________ 
Algorithm 1 Depth-first search solution generator (DFS)___________________________ 
Require: G,k,x 
 1:  start ← SELECT(V ) 
  2:  S ← DFS(G,start) 
  3:  return S [::x]                                              ⊳ Take every xth element    

 
_______________________________________________________________________________________________________ 
Algorithm 2 Degree solution generator (Deg)__________________________________________ 
Require: G,k,x 
 1:  V ′ ← SORTED(V )    ⊳ Sort nodes according to their degree in DESC order 
  2:  S ← V ′ [:x]                         ⊳ Take first x nodes with the highest degree 
  3:  while |S| < k do 
 4:       node ← SELECT(V ′) 
  5:       if node / ∈ S then 
 6:            S ← S ∪{node} 
 7:       end if 
 8:  end while 
 9:  SHUFFLE(S) 
10:  return S_________________________________________________________________________ 

 
________________________________________________________________________________________ 
Algorithm 3 Random walk solution generator (RW)_________________________________ 
Require: G,k,t,pr 
  1:  visited ←∅ 
 2:  while True do 
 3:       core ← SELECT(V ) 
  4:       current ← core 
 5:       for i ← 1,t + 1 do 
 6:            if current ∈ visited then 
 7:                  visited[current] ← visited[current] + 1 
  8:            else 
 9:                  visited[current] ← 1 
10:            end if 
11:            restart ← RAND INT(1,100) 
12:            if restart ≤ pr then 
13:                  current ← core 
14:            else 
15:                  neighbors ← NEIGHBORS(G,current)⊳ Neighbors of the current 
    node 
16:                  current ← SELECT(neighbors) 
17:            end if 
18:       end for 
19:       if |visited|≥ k then 
20:            break 
21:       else 
22:            visited ←∅ 
23:       end if 
24:  end while 
25:  SORT(visited)      ⊳ Sort nodes in visited according to visits paid in DESC 
     order 
26:  return visited[:k]                         ⊳ Take the first k most visited nodes    

 
_______________________________________________________________________________________________________ 
Algorithm 4 Evolutionary algorithm with smart initialization_____________________ 
Require: G,k 
 1:  initialize1 population Ps; 
  2:  run  a  multi-objective  Pareto  based  optimization  algorithm2,   where 
     Pinitial = Ps; 
  3:  return Pareto front________________________________________________________________________ 
1for the initialization we use:  random initialization, depth-first search, degree, 
random walk; 
2e.g., NSGA-II, SPEA__________________________________________________________________________  

Numerical experiments

Results and discussion

In the case of synthetic benchmarks, we conducted ten independent runs for each initialization strategy (depth-first search, degree-based, random walk) and made comparisons with random initialization. Table 4 presents the mean values and the standard deviation of the hypervolume indicators. For a reference point, we set the nadir point of all unified Pareto fronts. We conducted a Wilcoxon sign rank nonparametric test for the hypervolume indicator reported by each method. The Wilcoxon sign rank assesses if there is a significant difference between the two sample means. An (*) is used to indicate the statistical significance of differences. All initialization strategies which are not statistically different from the best one are marked. Figure 2 presents the Pareto front obtained within a single run.

Table 5 describes the mean value and standard deviation obtained for the real datasets. Best results are marked with an (*).

Based on the results, we can draw a general conclusion for the synthetic benchmarks about the best initialization strategy. The structure of the graph determines which initialization is worth using, but based on the numerical experiments, all of them give better results than the random initialization. In the case of Barabási-Albert graphs, which contain hubs, the degree-based initialization gets the best result. Erds-Rényi graphs are random graphs, in which case the depth-first search algorithm seems to be best. For Forest-fire graphs, which are random graphs, the three proposed initialization types gave almost the same result. The Watts–Strogatz graphs have a dense structure, and the best results were provided by the random walk-based algorithm.

In the case of the real networks, all three proposed initialization strategies outperformed the random initialization. In most cases, the degree-based initialization seemed to give the best results.

Conclusions and further work

In this paper, we propose three smart population initialization methods for the BOCNDP problem. Numerical experiments show the effectiveness of the proposed approaches. All three methods outperformed the traditional random initialization.

As further work, other initialization strategies will be investigated and an adaptive algorithm can be developed to find the best initialization, taking into account the basic properties of the graph.

Supplemental Information