Knowledge propagation is a necessity, both in academics and in the industry. The focus of this work is on how to achieve rapid knowledge propagation using collaborative study groups. The practice of knowledge sharing in study groups finds relevance in conferences, workshops, and class rooms. Unfortunately, there appears to be only few researches on empirical best practices and techniques on study groups formation, especially for achieving rapid knowledge propagation. This work bridges this gap by presenting a workflow driven computational algorithm for autonomous and unbiased formation of study groups. The system workflow consists of a chronology of stages, each made of distinct steps. Two of the most important steps, subsumed within the algorithmic stage, are the algorithms that resolve the decisional problem of number of study groups to be formed, as well as the most effective permutation of the study group participants to form collaborative pairs. This work contributes a number of new algorithmic concepts, such as autonomous and unbiased matching, exhaustive multiplication technique, twisted roundrobin transversal, equilibrium summation, among others. The concept of autonomous and unbiased matching is centered on the constitution of study groups and pairs purely based on the participants’ performances in an examination, rather than through any external process. As part of practical demonstration of this work, study group formation as well as unbiased pairing were fully demonstrated for a collaborative learning size of forty (40) participants, and partially for study groups of 50, 60 and 80 participants. The quantitative proof of this work was done through the technique called equilibrium summation, as well as the calculation of interstudy group Pearson Correlation Coefficients, which resulted in values higher than 0.9 in all cases. Real life experimentation was carried out while teaching ObjectOriented Programming to forty (40) undergraduates between February and May 2021. Empirical result showed that the performance of the learners was improved appreciably. This work will therefore be of immense benefit to the industry, academics and research community involved in collaborative learning.
The three key concepts that make up the title as well as content of this work are rapid knowledge propagation [
A research by [
As already stated, the major aim of this work is to present a new computational algorithm for rapid domain knowledge propagation through precisionbased study groups formation. This implies that the resulting study groups enforce autonomous and unbiased mixing or permutation of the participants, such that for any two learners Lx and Ly in a collaborative pair (Lx, Ly), there is synergy, such that the perceived knowledge gap of one partner is filled by the colleague. The general workflow [
The algorithmic Steps 1 and 2 constitute the preparatory stage, since they involve putting strategic procedures in place before the actual learning is kickstarted. The essence of Step 1 which is tagged [
S/N  Learner  Phone number  SGXXX  Exam score 

1  Daniel Labira  08029220192  SG1  
2  Jaru Fadama  08099976543  SG2  
3  Handy Galadima  07066513244  SG3  
4  Umaru Pope  09066500928  SG4  
5  Nnena Nwafe  07044313256  SG5  
6  Ogbodosa Malu  08077666777  SG6  
7  Matthew Jang  07011126536  SG7  
8  Gaya Duduya  09088766545  SG8  
9  Ifene Ruba  09042433251  SG9  
10  Salamatu Tayuta  08166544322  SG10  
11  Algafa Yande  09177652283  SG11  
12  Warru Zaga  07099287376  SG12  
13  Amara Dabere  08166555669  SG13  
14  Joseph Bravo  09011112222  SG14  
15  Thomas Zidiq  08055544433  SG15  
16  Omeruwa Wazobia  08099229292  SG16  
17  Balogun Zuwo  07052525252  SG17  
18  Ganji Sadiq  08088272799  SG18  
19  Yaro Usala  07079987765  SG19  
20  Oloma Moma  08033738000  SG20  
X 
X 
X 
X 

39  Sala Jaland  07044448888  SG39  
40  Power Roy  09053343784  SG40 
The algorithmic Steps 3, 4 and 5 consists of the first learning stage of the system workflow. During Step 3 tagged the
The workflow Steps 6 and 7 tagged
Issues  Details  Section 

Challenge 1  How many study groups SG1, S2, …. SGX where X is an integer, are most appropriate to be created for a learning population of cardinality P, where P is an even integer?  This issue was tackled in Section 3.3. 
Challenge 2  How many learning pairs should be created for each study group? What algorithmic steps will be used to achieve this?  This issue was also tackled in Section 3.3. 
Challenge 3  How does the computational algorithm ensure that the paring of learners in a study group is autonomous and unbiased, rather than being influenced by external views?  This has been explained in several sections, and especially Section 3.3. 
Challenge 4  What computational proof [ 
Tackled in proof and evaluation section. 
To tackle the enumerated challenges as part of workflow Steps 6 and 7, a sample architecture [
The first challenge is to decide on number of study groups to create, and to create same. This is part of the Step 6, tagged
The decision on number of study groups is achieved through a technique termed exhaustive multiplication [
Given the population size
In a similar way, the exhaustive multiplication tables for
A further and more detailed explanation of the use of exhaustive multiplication table will be based on study group population
The algorithmic steps for achieving this is as follows. To create S study groups out of P learners, then the number of learners L per study group is given by
Therefore, for P = 40 and S = 4, the number of learners in each study group is L = 40/4 which is 10. First, create 4 arrays, SG1, SG2, SG3 and SG4 as shown in
The assigned rankings are then used to generate the study groups by filling the four arrays in a twisted round robin pattern shown in
At this point, the Challenge 1 in
It is obvious that based on
In a similar way, the pairs are created for all the study groups SG1, SG2, SG3 and SG4 using the
After the formation of study groups and learning pairs, then comes the Collab Learning Stage of the system workflow, where “
The final stage in the general workflow is the loop stage. This is a point where the moderator, who is the overall lecturer, may decide to either continue with further sessions, or may terminate the workflow.
The evaluation of this work is based on two perspectives. One is through what is termed as equilibrium summation, and another is through the use of statistical correlation [
The major goal of this work is to ensure that learners are arranged in such a way that every learner with academic performance rating X is grouped with another learner with academic performance Y, such that there are visible equilibria in the summation for the entire study groups. This ensures that collaborative pairing enhances rapid knowledge propagation. The two important rules on summation in study groups are outlined as follows.
The sum of the positional constituents of every study group give an equal value. This is what is termed the equilibrium sum. For instance, for study group population P = 40, consisting of four distinct study groups, the value is 205 for all study groups. The flowchart for computing this value is shown in
Thus, the outcome from the implementation of the flowchart is as follows:
It is important to mention that a new flowchart symbol in form of solid cuboid was introduced in this work in order to effectively represent program loop. This is shown in
Apart from equilibrium at the study group level, there is also a unique sum of each of the pairs. The rule states that the sums of the positional contents of each pair in every study group in the entire population should be equal. For instance, for study group population P = 40, consisting of 4 study groups as shown in
Thus, the outcome from the implementation of the flowchart is as follows:
Correlation is a statistical measure of linear relationship between variables. A presentation technique known as scatter plot [
One of the ways to prove that the constitution of the study groups is near perfection is to compare the correlation coefficients of each study group with the rest of others, and to be sure that all resulting the correlation coefficients are close to +1. The Pearson Correlation Coefficient P_{C} is given by
where PC (X,Y) = Pearson Correlation Coefficient between variables X and Y,
X_{k} = values of the X variable, M_{X} = the mean values of variable X.
Y_{K} = values of the Y variable, M_{Y} = the mean values of variable Y.
The computation table for correlation P_{C} (SG1, SG2) is shown in
X = SG1  Y = SG2  M_{X}  M_{Y}  X − M_{X}  Y − M_{Y}  (X − X_{M})(Y − Y_{M})  (X − M_{X})^{2}  (Y − M_{Y})^{2} 

40  39  20.5  20.5  19.5  18.5  360.75  380.25  342.25 
33  34  20.5  20.5  12.5  13.5  168.75  156.25  182.25 
32  31  20.5  20.5  11.5  10.5  120.75  132.25  110.25 
25  26  20.5  20.5  4.5  5.5  24.75  20.25  30.25 
24  23  20.5  20.5  3.5  2.5  8.75  12.25  6.25 
17  18  20.5  20.5  −3.5  −2.5  8.75  12.25  6.25 
16  15  20.5  20.5  −4.5  −5.5  24.75  20.25  30.25 
9  10  20.5  20.5  −12  −11  120.75  132.25  110.25 
8  7  20.5  20.5  −13  −14  168.75  156.25  182.25 
1  2  20.5  20.5  −20  −19  360.75  380.25  342.25 
Therefore,
Furthermore, it can also be shown that P_{C}(SG1,SG3) = 0.9858, P_{C}(SG1,SG4) = 0.9674, P_{C}(SG2,SG3) = 0.9963, P_{C}(SG2,SG4) = 0.9850 and P_{C}(SG3,SG4) = 0.9961, all of which are very close to +1.
Furthermore, the Spearman’s Rank Correlation Coefficient S_{C} is given by
where D = difference in ranks of the two variables representing the two study groups being analyzed and n = number of participants in each study group.
It can be shown that D = 0 in each case, thus the resulting Spearman’s Rank Correlation Coefficient [
Based on the results of correlation coefficients, it implies that there is a very strong correlation coefficient between all the arrangements of the individual study groups SG1, SG2, SG3 and SG4. Similar correlation tests have been done after generating study groups for learners of populations size 50 with 5 study groups, 60 leaners with 3 study groups, 80 learners with 4 study groups, using this algorithm, and the resulting correlation coefficients were all very close to +1.
This research has presented a very unambiguous algorithm for achieving a rapid domain knowledge propagation using autonomous and unbiased matching based study groups. The result of the experiment was visualized [
There was a very significant [
In conclusion, this work has presented an innovative algorithm on how to create study groups, and pairs so as to achieve rapid knowledge propagation. A number of new concepts and computational techniques have evolved from this work, as contributions to knowledge. The work has been presented in a very unambiguous manner, with explicit and annotated workflows, flowcharts, among others. Mathematical proofs as well statistical correlations were also exploited for further evaluation of the work, with very impressive outcomes. The outcome of the final experimental run in this research shows about 97.5% success and 2.5% failure. Consequently, future research will focus on performing further investigative study on other factors that affect performance in collaborative study groups, especially in respect of the 2.5% of the participant such as SG2, who failed to perform as brilliantly as others. Future research will also involve running the experiment in a large scale [