The critical node detection problem in hypergraphs using weighted node degree centrality

Weighted node degree centrality

The weighted node degree centrality was proposed in Kapoor, Sharma & Srivastava (2013), as a natural extension of traditional centrality measures to hypergraphs. The following section will provide a short summary of the proposed metric and how it applies to our study.

Definition 4 disregards the strength of the ties between nodes, so a weighted degree centrality is proposed to mitigate this problem.

Multiple hyperedge weights can be described, all of them make use of the frequency of the hyperedge’s appearance or its multiplicity (m_j) and the cardinality of the hyperedge (c_j), the importance of multiplicity and cardinality can be demonstrated in example 1. The following weights are used in the application:

• Constant: w_j = 1

• Frequency based: w_j = m_j

• Newman’s definition of ties strength in collaboration networks: $w_j=\frac{m_j}{\left|c_j\right|-1}$

• Network theory: $w_j=1-{\left(1-\frac{1}{c_j-1}\right)}^{m_j}$

In any one instance of the algorithm, a singular weighting scheme is used at a time, incorporating itself into the fitness function. The goal of the fitness calculation is the minimization of the average weighted node degree centrality in the network, after the removal of a set of nodes. A more formal definition of the problem is given in Definition 6.

Encoding

The created population for the GA contains a fixed number pop_size of individuals. These individuals in turn contain k number of nodes that are “proposed” as a possible critical node list. Each node is represented by its integer index in the node dictionary, in turn, meaning, that nodes represented with labels can also be processed.

Fitness

The fitness value for each individual is calculated using the average weighted node degree centrality in the hypergraph $\mathcal{H}$ after the removal of the selected individuals’ nodes from the total node dictionary. This removal modifies the structure of $\mathcal{H}$, in order to not lose information, the fitness calculation is done on a copy of the original hypergraph. Multiple weight calculations are considered, and all of them are described in detail at the weighted node degree centrality explanation.

Crossover

The crossover process is described in Algorithm 2. It samples a subset of the total population with size tournament_size, we then sort the selected subset and use the two best individuals, according to their fitness, and we initiate the crossover process using them. The crossover is done by zipping the two individuals’ node lists and then separating them, we attribute special attention to the correctness of the results, i.e., nodes can not repeat in a correct node list. The two new individuals are appended to the child list and the process is repeated tournament_count times. After crossover completion, the child population is pre-evaluated, in order to skip the evaluation at later steps of the algorithm.

Mutation

The mutation operator simply selects an individual from the child population and replaces one of its nodes with a new node, taking into account the correctness criteria stated at the crossover operator. The newly mutated individual is then inserted back into the child population. The new individual is also pre-evaluated.

Selection

The algorithm combines the original population and any newly-created child population, including any possible mutations, after which we trim this new set to the original pop_size using elitism (keeping the best individuals), this will become the new population for the next iteration of the algorithm (a (μ + λ) selection scheme is used). The selection process also involves an evaluation step at the end, in case any individual needs re-evaluation.

Population size

The population size parameter refers to the number of individuals considered in each generation of the genetic algorithm. Generally, a larger value provides better results, but having too large of a value inquires a negative trade-off of performance while also providing diminishing returns. Population size also factors into the number of tournament rounds used in the creation of the new population in each generation, meaning that having an excessively large population provides an even bigger penalty in speed. The population size parameter has been chosen as 50 from the pool of 20, 50, and 100 after a round of parameter-setting runs on a few synthetic networks, that have shown no significant difference in results with the 50 and 100 cases, so we opted for the smaller one, to decrease step times.

Mutation chance

The mutation chance or p_mut gives a percentage chance of a mutation occurring during a genetic algorithm generation. Each generation can have at most one mutation occurring and if a mutation is required then the child selected for mutation and the node replaced by the mutation is selected at random, without any bias, from the children created through the crossover tournament and the nodes that are not present in the selected child respectively. The impact that p_mut has on the algorithm is twofold. Firstly a high mutation chance has an easier time escaping local optimums and moving our search along, resulting in fewer required generations. Secondly, a higher chance of mutation provides volatile results. p_mut was also chosen from a pool of values: 1%, 2% or 5% as a result of the same parameter selection process. The value chosen was 5% because it did the best out of all tested values.

Generation count

This parameter refers to the number of generations of the GA that we process. This value started out as high as 10,000 but after a few test runs we identified that no more than 200 generations are necessary in most cases that we have studied, so this number was chosen as a stable point and it was not subject to the parameter selection process.

Probability of selection into the crossover process

The probability of selection into the crossover process or p_cross for short factors into the number of tournament rounds and the number of parents selected to participate in the tournament. This probability should be high, but not 100% in order to exclude some parents from participating and in turn ensure higher variability. This number was also chosen from a pool of possibilities: 60% or 80%. Testing has shown a marginal improvement for the 80% case.

Tournament size

A round of the crossover tournament always combines two separate individuals to create two new children, but the process of selecting the two participating parents is not trivial. We need to sample the two best parents from those that were selected to participate in the current round, the number of participating parents in any round is the number given by the tournament size parameter. This number was also chosen from a pool of possibilities: 2, 3, or 4. Observations after several parameter-setting runs show, that anything above a value of 2 will provide very similar results. Finally, a value of 3 was chosen.

Type of weighted centrality

The algorithm also receives as a parameter, the desired type of weighted centrality from the supported list presented in the description of the weighted centrality measure. For the parameter-setting scenario of the GA, a simple constant weight is used to eliminate any variation that this measure would introduce.

Networks

Some baseline synthetic networks were constructed using LFR benchmarks (https://www.santofortunato.net/resources). These graphs, described in Table 1, were smaller in scale and were mainly used as both proof of concept about the correctness of the algorithm with the different weight types and as a benchmarking tool, for the parameter-setting part of the research process.

In addition to synthetic networks, two real-world networks were used as proof of concept, the data sets were used in Chodrow, Veldt & Benson (2021) and were derived from Congressional data compiled by Charles Stewart and Jonathan Woon. In these networks, nodes represent either members of the US House of Representatives in the first network or members of the US Senate in the second one; while hyperedges correspond to committee memberships. Some basic information about these networks is presented in Table 2.

Results

Firstly some results on the synthetic networks will be given. The results will not be analyzed in terms of meaning, given the fact, that the data is artificially generated and does not correspond to anything real-world related. The results provided in Figs. 2 and 3 are averages of ten runs, for six similarly generated community networks.

Regarding the results presented in Figs. 2 and 3, we see that the total fitness value of the population and the fitness value of the best individual in the population simultaneously and suddenly lower to a point, typically reached at the 25–50 generation mark, after which a stagnation period is entered. The latest changes happened at around generation 150, which is why a generation number of 200 was chosen. In a local conclusion, it can be said, that the algorithm works in terms of lowering the desired fitness value, indifferent of the used weight type or the network analyzed, which means that this algorithm as a tool of analysis can be useful and can provide interesting results.

Comparison between heuristic and GA

Validating the usefulness of the results obtained by the proposed genetic algorithm should be a priority. Since the approach of critical node detection on hypergraphs is a fairly new field of study, no real benchmarks are set. Nevertheless, a comparison of some kind should be considered. This paper presents a comparison between a proposed heuristic algorithm and our GA.

Table 1:

Synthetic networks and basic properties.

Hypergarph	|X|	|D|	rank	mean edge size	median edge size
lfr_150_m01_on70_om2	150	150	7	3.86	4
lfr_150_m02_on70_om2	150	150	6	3.69	3
lfr_150_m03_on70_om2	150	150	6	3.76	3
lfr_150_m01_on50_om2	150	150	7	4.00	4
lfr_150_m02_on50_om2	150	150	6	4.04	4
lfr_150_m03_on50_om2	150	150	6	3.90	4

DOI: 10.7717/peerjcs.1351/table-1

Table 2:

Real networks and basic properties.

Hypergarph	|X|	|D|	rank	mean edge size	median edge size
house-committees	1290	341	82	34.8	40
senate-committees	282	315	31	17.2	19

DOI: 10.7717/peerjcs.1351/table-2

The proposed heuristic works by removing a set of nodes S′ with the size of k, having the largest weighted node degree centrality, and then calculating the average weighted node degree centrality. k is the same number of nodes removed by the GA in any fitness calculation, and it is the same k as the number of critical nodes from the CNDP. This heuristic was chosen since our goal with the GA is similarly to minimize the average weighted node degree centrality. Logically, we could say, that by removing the largest contributors to the average of weighted node degree centrality in the original network, the new average would be strictly better, than removing any other combination of nodes. This comparison proves the contrary since the set of truly critical nodes will further lower the average, and since the GA gives better results, we can consider them nodes that are more similar to critical nodes.

Comparisons were made with four synthetic networks, on all weight types, taking results from the average of ten GA runs and comparing them with the result given by the heuristic on the same networks, an overall improvement of 2.57% was observed with differences ranging from similar results, all the way up to an improvement of 14.14%. In Fig. 4 a comparison is presented on a typical synthetic network on the different weight types, with results of the GA presented as the blue lines, while the green lines represent the heuristic result for the same network and weight type.