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Match making in complex social networks
Applied Mathematics and Computation 371 (2020) 124928
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Match making in complex social networks Fubing Mao a,b, Lijia Ma c, Qiang He d,∗, Gaoxi Xiao b a
Service Computing Technology and System Lab, Cluster and Grid Computing Lab School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan 430070, China School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore c College of Computer Science and Software Engineering, Shenzhen University, Shenzhen 518060, China d College of Information Science and Engineering, Northeastern University, Shenyang 110819, China b
a r t i c l e
i n f o
Article history: Received 6 August 2019 Revised 22 October 2019 Accepted 17 November 2019
Keywords: Complex network Match making Strongest partner Romantic partner Social system
a b s t r a c t Match making is of significant importance in some social systems. People may need to seek for romantic partners, teammates, collaborators, etc. In this paper, we propose a minimalist framework of match making in complex networks. Specially we adopt a simple model where each individual would greedily seek for making a match with the strongest partner within his/her social connection range. We explore a few matching schemes including greedy mode, roulette wheel selection mode and completely random mode on different networks. We also investigate when social systems become more densely connected, how the match making process would be affected. Our observations show that, in a more densely connected social network, individuals’ efforts for seeking for matches with the strongest partners would be more likely to end up with matching with someone similar to themselves. Meanwhile, the cost of such an approach may be quickly increased. The implications of such observations in real-life systems and open problems are briefly discussed. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Match making is the process for matching people into pairs. It could happen for finding partners for romantic relation, sports games, business collaboration, etc [1–7]. While “match making” may have different meanings in different scenarios, in this work, match making means that each individual shall try to form up a pair with one, and only one, neighbor he/she has. To represent this in a social network model, each node is seeking for a match with one and only one of its adjacent nodes. Though it has been of significant research interest to study on how people search for potential partners, sending out requests and finally reaching or failing to make a match [8–12], it is not well understood (i) how the social connections that individuals have affect their match making results; and (ii) when social systems become more densely connected (which is believed to be the case in many, though not all, modern social systems) [13–15], how the match making process would be affected. These open problems motivate the research work reported in this paper. Specifically, we propose a minimalist framework by studying match making following a simple greedy rule in complex social networks, with the main focus on evaluating the effects of network topology and connection density on match making in social networks. ∗
Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang 110819, China. E-mail addresses: [email protected] (F. Mao), [email protected] (Q. He), [email protected] (G. Xiao).
https://doi.org/10.1016/j.amc.2019.124928 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
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F. Mao, L. Ma and Q. He et al. / Applied Mathematics and Computation 371 (2020) 124928
Existing research on match making problem has been mainly focusing on its applications in different areas, mainly including a few aspects as follows. (1) Marriage Problem: This may be the most well-known match making problems [8,9,11,16–20]. Different matching strategies and systems have been proposed for helping find good romantic partners, e.g., in online dating systems [8], for finding good partners in offline system [9], and for finding life-long partners for marriage [11,16]. (2) Teamwork Problem: A matching model was presented in [21] considering skills and socio-emotional factors and their roles in supporting the composition of an efficient team. And an agent-based middleware was proposed in [22] for job matchmaking in a teleworking community. (3) Online Games: A matchmaking service in multiplayer online games was proposed in [23]. Another discussion was on constraints and policies of game design for allowing public transport commuters to engage in multiplayer games [24]. (4) Marketing: For example, a system was proposed for matchmaking in an electronic marketplace [25]. (5) Matching Service Request: Service composition is integrated into service discovery and matchmaking for matching service requests [26]. A hybrid web service matchmaker which analyzes the signature and specification of different web services was presented in [27]. And a privacy preserving matchmaking system was constructed in geosocial networks with untrusted servers [28]. (6) Several Areas in the Field of Social Physics: The problem of how cooperative behavior can emerge in the real life captures much attention [29–31]. For example, a co-evolutionary model of SPD (spatial version of Prisoner’s Dilemma) [30,32] was proposed for investigating the problem. SPD offers two strategies: C (Cooperation) and D (Defection) [29,30,32]. In SPD, not only the agent’s strategy (either cooperation or defection), but also the spatial structure (agent’s link) would be time-evolved obeying to a certain update rule [29–32]. While such studies help enhance the roles of match making in various applications, they did not study the effects of social connections on efficiency and results of social match making. We study match making in complex networks within a minimalist framework. Specifically, we adopt the assumption that each individual shall always try to make a match with the best individual among all his/her connections as far as the selected candidate is still available. While it is understood that such a greedy approach may not be the approach that people actually adopt in most applications, evaluating the cost and matching results of adopting such an approach may help us better understand why some “practical” approaches are adopted in social systems, and how the evolution of modern social systems may drive the changes of match making activities. Our studies reveal that, in a more densely connected social network, individuals’ efforts to seek for the strongest partners would be more likely to end up with matching with someone similar to themselves. Meanwhile, the cost of match making may be increased with the connection density. The implications of such observations in real-life systems and open problems will be briefly discussed. The main contributions of our paper are as follows. • We propose a minimalist framework of match making in complex networks. • We adopt a simple model where each individual would greedily seek for making a match with the strongest partner within his/her social connection range. Our proposed match making has different meanings that each node is seeking for a match with one and only one of its adjacent nodes. • We explore a few matching schemes including greedy mode, roulette wheel selection mode and completely random mode on different networks. • We investigate when social systems become more densely connected, how the match making process would be affected. We also briefly discuss the implications of our observations in real-life systems. The rest of the paper is organized as follows. We first introduce our network based minimalist framework in Section 2. Then we present the experiment results and discussions in Section 3. Finally, we conclude our work and remarks in Section 4.
2. Network based minimalist framework The minimalist framework of match making we shall adopt in this study is as follows. In a network G with n nodes, each node has a fitness value within the range [0, 1]. Here, the fitness value represents the value of the node which is similar to the value of a person in the real life. Since we investigate the match making of people in real life, in our paper each node has a fitness value in the networks. Actually, the fitness value can be any number no less than 0. For convenient discussion and investigation, we normalize the range and make the range be [0, 1]. At the beginning, all the nodes in the network are unmatched nodes. Then each node shall iteratively bid for making a match with the fittest node in its neighborhood area. Specifically, in each iteration, a single matching request is sent out by every unmatched node to one of its unmatched neighbors. A match is made when two nodes happen to send matching requests to each other. Once a match is made, the two matched nodes will not join the later iterations of match making. The detailed introduction to our proposed framework is in Section 2.1.
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Algorithm 1: Network Based Minimalist Framework of Match Making. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35:
Input: Given a network G with n nodes and m edges Output: The set of matched pairs and related results (the avgfitvaluegap among all the matched pairs etc) define totalfitvaluegap to record the sum of the absolute fitvaluegap between matched nodes. Its initialized value is 0. define avgfitvaluegap to record the average value of the absolute values of the fitness value gaps between matched nodes (Initialized value is 0). define a map variable socialmatchnodeidAndreqid to record the nodeid and its requested nodeid define a map variable nodeidAndreqid to temporarily record the nodeid and its requested nodeid (request to match with another node) define a map variable matchpairmap to record the nodeid and its matched nodeid assign values from [0, 1] randomly to nodes of G respectively //the value is fitness value define a set tmpvisitedset to record the nodes already visited define a variable nofoundflag to check whether a matched pair is founded. If cannot find a pair, then stop. The initialized value of nofoundflag to is 0, (1: is not found. 0: found) while !nofoundflag do set the map nodeidAndreqid to be empty//traverse all the nodes in G and find each node’s neighbor nodes which are not matched for each node in G for each node in G do if the node is not visited then find the node’s neighbor nodes which are not matched, and record them end if if the match mode is greedy mode then find the node with the max fitness value from the current node’s neighbor node set record the current node id and the finding/requested node id (current node’s neighbor node with the max fitness value) in nodeidAndreqid end if end for if nodeidAndreqid is empty (cannot find the matched pairs) then set the variable nofoundflag to be 1 //cannot found the new match pair. stop end if if nodeidAndreqid is not empty then for each element in map nodeidAndreqid do if nodes A sends request to match B and B also sends request to match A at the same time then define a variable and let fitvaluegapofpairAB = the absolute difference value between the two matched nodes A and B let totalfitvaluegap = the sum of all the fitvaluegapofpairAB end if end for end if end while let avgfitvaluegap = totalfitvaluegap / (total number of matched pairs) return
2.1. Our proposed minimalist framework of match making Our proposed network based minimalist framework of match making is presented in Algorithm 1. Lines 1 and 2 present the input and output of the framework. Lines 3–10 introduce the initialization of the network and the variables for recording data. From line 11–33, the algorithm find matched nodes and record the matched pairs. Specifically, lines 13–21 collect the nodes which send requests to match a node from their unmatched neighbors, and lines 22–32 obtain the matched pairs where the nodes send requests to each other at the same time. Line 34 acquires the average value of the fitness value gaps between all the matched nodes. Line 35 indicates the algorithm finishes. The time complexity and space complexity of our proposed framework are O(n2 m) and O(nm), respectively where n represents the number of the nodes and m stands for the number of edges in G. Within the same framework, we shall consider two slightly different approaches as follows. The first one is termed as greedy mode, where in each iteration, each unmatched node shall find among all its unmatched adjacent nodes the one with the highest fitness level and send a request to it. When there is a tie, i.e., there exist multiple unmatched adjacent nodes with the same highest fitness value, break it randomly. The process shall repeat until all the nodes are matched or no further matches can be made any more. As we could see, in the greedy mode, each node aggressively seeks for the best possible match, largely regardless of the cost that may occur (measured by the number of iterations) before a match is made. Similar to but less aggressive than the greedy mode, in a probabilistic mode (termed as roulette wheel selection mode), the probability an unmatched node is selected by one of its unmatched neighbors as a possible candidate for match making is proportional to its fitness value. 2.2. Fitness value gap between matched nodes We briefly introduce the fitness value gap. The fitness value of each node in the network follows a uniform distribution in [0, 1]. In a complete network, the absolute value of the fitness value gap between two matched nodes of the random mode can be analyzed as follows. For two independent matched nodes with fitness values x and y respectively, both x and
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y obey the uniform distribution in [0, 1]. f(x) shown in (1) and f(y) shown in (2) are the density distributions of x and y, respectively. Let variable z = |x − y|. F(z) represents the distribution of z. Thus, F (z ) = P (|x − y| ≤ z ). when z < 0, F (z ) = 0. 1 x−z 1−z 1 when z ≥ 1, F (z ) = 1. When 0 < z < 1, F (z ) = P (|x − y| ≤ z ) = 1 − z dx 0 dy − 0 dx x+z dy = 1 − (1 − z )2 = 2z − z2 . Thus, we obtain the distribution of F(z) listed in (3). We discuss the fitness value gap more in Section 3.2.
f (x ) =
1 0
f (y ) =
1 0
0
(1)
0
(2)
F (z ) =
0 2z − z 2 1
z≤0 0
(3)
3. Simulation results and discussion 3.1. Experiment setup We implement our minimalist framework of match making in the C++ programming language and perform our experiments on a HP Z420 workstation with Intel(R) Xeon(R) CPU 3.5 GHz and 16 G RAM. Our numerical simulations are mainly conducted on Erdos–Renyi (ER) network model [33,34] and scale-free (SF) network model [35,36]. In the network models, the number of nodes is 20 0 0. In the initial state, all nodes are unmatched with fitness values uniformly distributed between [0, 1]. We implement the greedy mode and the roulette wheel selection mode respectively. Further, a random mode is implemented as a benchmark case, where each node randomly chooses an adjacent unmatched node to send a matching request. We also conduct simulations on real-life social networks such as Orkut [38,39] and LiveJournal [38,39] and an autonomous system AS-733 [38]. Unless otherwise specified, we run 10 times with different random seeds for each experiment and present the average results. 3.2. The relation between average nodal degree and average fitness value gap We explore the variation of the average fitness value gaps when we adopt different average nodal degrees in different networks. In our work, the fitness value gap of a pair of matched nodes means the absolute value of the difference between the two nodes’ fitness values and the average fitness value gap is the mean value of all the fitness value gaps of matched nodes. We consider the ER and SF network models and adopt the greedy mode, probabilistic (roulette) mode and completely random mode for match making in each iteration. Fig. 1 shows the variation of the average fitness value gap with the increase of the average nodal degree in the ER and SF networks under the three different match making modes, respectively. The curves with circular marks (blue color) in Fig. 1(a) and (b) show the variations of the average fitness value gap with the increase of the average nodal degree in the ER network model and SF network model under the greedy mode, respectively. In the ER model, the blue curve with circular marks in Fig. 1(a) shows that when the average nodal degree increases from 5 to 1999 which forms into a complete graph, the average fitness value gap of all the matched nodes decreases from 0.1958 to 0.0 0 05. Further, when the average nodal degree increases from 5 (87% nodes matched) to 200 (nearly 100% nodes matched), the average fitness value gap decreases from the 0.1958 to 0.0068. In this stage, the average fitness value gap decreases quickly. After that, the gap decreases slowly but steadily with an increasing average nodal degree. The reason is that when the average nodal degree is low, the chance for two nodes with similar fitness values to be directly connected is low, which makes the fitness value gap relatively large. With an increasing average nodal degree the average fitness value gap decreases finally, reaching an exhausted level; meanwhile the match making percentage increases to be close to 100%. In the SF model, the blue curve with circular marks in Fig. 1(b) shows that when the average nodal degree increases from 4 to 1999 (a complete graph), the average fitness value gap of all the matched nodes decreases from 0.2813 to 0.0 0 05. Further, when the average nodal degree increases from 4 (73% nodes matched) to 380 (nearly 100% nodes matched), the average fitness value gap decreases from the 0.2813 to 0.0051. The observations and the main reasons remain largely the same as those in ER networks. The main difference between the results in ER and SF models under greedy mode is that the average fitness value gap in the ER model is smaller than that in the SF model under the same average nodal degree. This is because the ER model allows a higher match making percentage, which helps lower the average value gap. The red curves with rectangular marks in Fig. 1(a) and (b) show the variations of the average fitness value gap with the increase of the average nodal degree in the ER network model and SF network model under the probabilistic (roulette) mode, respectively. In the ER network model, the curve in Fig. 1(a) shows that when the average nodal degree increases from 5 to 1999, the average fitness value gap stays within the range from 0.2499 to 0.290. When the average nodal degree increases from 5 (87% nodes matched) to 100 (nearly 100% nodes matched), the average fitness value gap only slightly decreases from the 0.290 to 0.2543. In this stage, the trend shows that it decreases relatively quickly. After that, the fitness value gap decreases steadily but very slowly when the average nodal degree continues to increase. This is because the average fitness
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Fig. 1. Average nodal degree (X axis) and average fitness value gap (Y axis) under greedy mode, roulette mode and random mode in ER and SF networks. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
value gap approaches to 0.250 rather than 0 with an increasing average nodal degree. This can be easily understood. In a complete network with greedy matching operation, it is trivial to prove that when the network size goes to infinity, the average fitness gap value as the fittest node always match with the second fittest node, and the third fittest node matches with fourth fittest node, and so on. The average fitness gap value of the random mode can be approximately calculated as (4).
1 0
1 0
|x − y| dxdy =
1 0
1 x
(y − x ) dydx +
1 0
0
x
(x − y ) dydx =
1 1 1 + = 6 6 3
(4)
The average fitness gap value of the probabilistic (roulette) mode is more difficult to be calculated. For two nodes with fitness values x and y respectively, in the first round, the chance they choose each other is proportional to xy where both x and y follow a uniform distribution in [0, 1]. In later iterations, while the probability that two nodes with fitness values x and y make a match remains to be proportional to xy, the distribution of fitness value of unmatched nodes however changes in each iteration; nodes with higher fitness value have a higher chance to match earlier. While it may not be easy to derive a concise theoretical analysis on the average fitness value gap for such a case, it is reasonable to expect that the gap value should be somewhere between 0 and 1/3. In the SF network model, the red curve in Fig. 1(b) shows that when the average nodal degree increases from 4 to 1999 (a complete graph), the average fitness value gap stays within the range from 0.2499 to 0.3128. When the average nodal degree increases from 4 (73% nodes matched) to 380 (nearly 100% nodes matched), the average fitness value gap decreases from the 0.3128 to 0.2517. In this stage, the trend shows that it decreases quickly. After that, the fitness value gap decreases steadily when the average nodal degree continues to increase. The reason is the same as those in ER networks. The two network models do not lead to big difference in the average fitness value gap. The reason is that when we choose the probabilistic (roulette) mode to find nodes to match, the network models impact the matched pairs of the nodes a little. The green curves with triangular marks in Fig. 1(a) and (b) show the variations of the average fitness value gap with the increase of the average nodal degree in ER and SF network models under the completely random mode, respectively. In the ER network model, the figure shows that when the average nodal degree increases from 5 to 1999 (a complete graph), the average fitness value gap stays within the range from 0.3273 to 0.3374. In the SF network model, this value stays with the range of 0.3279 to 0.3375. The average value gap basically remain the same when the average nodal degree increases. This is not a surprise as the fitness value gap of a random matching operation is largely independent of the average number of neighbors each node has. Note that the two different network models basically do not impact the average fitness value gap under the random mode. This can be explained. For n independent variables x1 , x2 , x3 , . . . , xn within the range of [0, 1], the mean absolute difference (MAE) of any two randomly selected variables can be calculated as about 0.3333 using (5). The detailed procedure to get this value is described as follows.
MAE =
n n 1 xi − x j 2 n i=1 j=1
(5)
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Fig. 2. Average nodal degree (X axis) and total iteration time (Y axis) for finishing match under greedy mode, roulette mode and random mode in ER and SF networks. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Assume x1 and x2 are randomly chosen from [0, n] where n is integer and n > 0. The absolute difference between any two variables x1 , x2 is represented by ∇ x. ‘∇ x = 1’ has 2n possibilities. ‘∇ x = 2’ has 2(n − 1 ) possibilities. ‘∇ x = 3’ has 2(n − 2 ) possibilities. ‘∇ x = k’ has 2(n − k + 1 ) possibilities (k > 3). ‘∇ x = n’ has 2 possibilities. The total number of possibilities is n(n + 1 ) since 2n + 2(n − 1 ) + 2(n − 2 ) + · · · + 2 = n(n + 1 ). We use p(i) to represent the probability to get the value i. Thus, p(1 ) = 2n/[n(n + 1 )]. p(2 ) = 2(n − 1 )/[n(n + 1 )]. p(k ) = 2(n − k + 1 )/[n(n + 1 )]. p(n ) = 2/[n(n + 1 )]. The mathematical expectation of ∇ x is that E∇ x = ni=1 i × p(i ) = 1 × 2n/[n(n + 1 )] + 2 × 2(n − 1 )/[n(n + 1 )] + ... + n × 2/[n(n + 1 )] = (n + 2 )/3. If n = 65535, E∇ x = (65535+2)/3 = 21846. Since x1 and x2 are within the range of [0, 1], we get the value 0.3333 (21846/65535 = 0.3333). It is interesting to observe that with a large enough average number of connections per node, the fitness value gap decreases to 0.0 0 05. For 20 0 0 nodes, with their fitness value uniformly randomly distributed between 0 and 1, the average gap between two adjacent fitness values is roughly at the same scale, meaning that almost every node is matched with another node with fitness value as close to its own value as possible. An “unbalance” match between two nodes with a relatively large difference in their fitness values has a virtually zero chance to happen. When the social systems are becoming more densely connected, the chance of “making an unbalanced match” decreases quickly. 3.3. The relation between average nodal degree and total iteration time for finishing matching nodes We explore the variation of total iteration time for finishing matching nodes when we adopt different average nodal degrees in different networks. We also consider the ER and SF network models, and adopt the greedy mode, probabilistic (roulette) mode and completely random mode for finding the unmatched nodes to match, respectively. In our work, finishing matching means that we cannot make pairs from unmatched nodes any more. The total iteration time means the minimum iteration time for reaching maximum matched pairs under a certain average nodal degree. Fig. 2 shows the variation of the total number of iteration time with the increase of the average nodal degree in ER and SF networks under the three modes, respectively. The two blue curves with circular marks in Fig. 2(a) and (b) show the trends about the variations of the total number of iteration time with the increase of the average nodal degree in ER and SF network models under greedy mode, respectively. The trends show that with an increasing average nodal degree the total number of iteration time gradually increases. They are similar and the trends of growth are both gradually slowing down. If the average nodal degree is large enough, all the nodes finally make pairs. The reason is that with an increasing average nodal degree, each node has more directly adjacent
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nodes and the chance for two nodes match each other at the same time becomes smaller which increases the total number of iterations. When the average nodal degree reaches to a certain stage, the increased number of iteration time becomes small. The main difference between the results in ER and SF networks is that the total number of iteration time in the ER network is larger than the total number of iteration time in the SF network under the same average nodal degree. The reason is that SF network is more densely than the ER network which it helps make pairs more quickly. The red curves with rectangular marks in Fig. 2(a) and (b) show the trend of the total number of iteration time with the increase of average nodal degree in ER and SF network models under probabilistic (roulette) mode respectively. Both trends show that with an increasing average nodal degree, the total number of iteration time gradually increases. The main difference between them is that when the average nodal degree is very small, the total number of iteration time in the ER model is larger than the number of iteration time in the SF model. When the average nodal degree is large, the total number of iteration time in the ER model is smaller than the number of iteration time in the SF model under roulette mode. The possible reason is that probability (roulette) mode is more quickly to finish matching in ER model than SF model. The green curves with triangular marks in Fig. 2(a) and (b) show the trends of the total number of iteration time with the increase of average nodal degree in ER and SF networks under completely random mode, respectively. It has the same situation with that of the probabilistic (roulette) mode. Compared to the results in the ER network model under the three different match making modes, we can observe that the total number of iteration time for finishing matching in greedy mode under the same average nodal degree is the smallest. The total number of iteration time for finishing matching in probability (roulette) mode and completely random mode are basically the same. The reason is that greedy mode is a deterministic approach and other two modes are probabilistic approaches. Compared to the results in SF network model under the three different match making modes, it has the similar situations and reasons. How quickly the number of matched nodes increases in each iteration will be discussed later.
3.4. The relation between average nodal degree and the percentage of matched nodes in each iteration We perform experiments to investigate the relation between the average nodal degree and the percentage of matched nodes in each iteration under a certain average nodal degree. During the iteration, if the procedure cannot find any new matched pairs, the procedure will stop. The results is shown in Fig. 3. In the figures, Avg_node_degree represents the average nodal degree. We consider the ER and SF network models and also adopt the greedy mode, probabilistic (roulette) mode and completely random mode to find unmatched nodes for matching, respectively. In our work, the percentage of the matched nodes in each iteration is the value of the number of matched nodes in each iteration divided by the total number of nodes under the certain average nodal degree. We use representative figures to show the experiment results and other figures are similar to them. Fig. 3(a)–(d) is the results in the ER model under greedy mode and the iteration time starts from 1. With an increasing average nodal degree, the percentage of matched nodes at the first few of iterations decreases. When the average nodal degree is small, the overall trend is that the percentage of the matched nodes in each iteration decreases gradually and finally reaches 0 which represents all the nodes are matched or it cannot find any new matched pairs as shown in Fig. 3(a). When the average nodal degree is large enough, the percentage of matched nodes in each iteration fluctuates and is small as shown in Fig. 3(b) and (c). For the extreme case shown in Fig. 3(d), when the network is a complete graph, the percentages of matched nodes in different iterations are the same. The reason is that when the average nodal degree is small, the chance for two nodes to make a pair is high. It has large percentage of matched nodes at the beginning. The matching process repeats for the unmatched nodes and the number of unmatched nodes decreases which makes the percentage of matched nodes in each iteration decrease. When the average nodal degree is large enough, the chance for two nodes to make a pair is low which makes the percentage of matched nodes in each iteration fluctuate. The results in the roulette mode and the completely random mode have the similar situation with greedy mode. The main difference is that when the network is a complete graph, the percentages of matched nodes in different iterations under greedy mode are the same and the percentages of matched nodes in each iteration under roulette mode and completely random mode are basically the same since greedy mode is deterministic and other two modes are probabilistic. Furthermore, the overall trends are similar and the fluctuations are not the same due to the different modes for finding nodes to match. The situations and trends in the SF model under the three different match making modes are similar to the situations and trends in ER model under the three modes, respectively. The main difference between the results in the ER model and SF model under different modes is that the trends waves are different due to different network models. The percentages of matched nodes in each iteration under a complete graph implicitly show that in real-life system, when the connection grows enough, the possibility to find a partner with larger fitness value to make a pair becomes larger. However, the total number of iteration time (cost) for finishing matching increases largely or unacceptable. Our observations show that when the average nodal degree is small, the majority part of the matching is done in the first few iterations. The reasons are given as follows. • The number of nodes which can finally make pairs is small when the number of average nodal degree is small. • When the average nodal degree is small, it only takes a few of iteration for finishing matching.
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(a)
(b)
(c)
(d)
Fig. 3. ER network: the iteration time (X axis) and the percentage of matched nodes in each iteration (Y axis) under different average nodal degrees (Greedy mode).
3.5. The relation between the average nodal degree and the percentage of total matched nodes We also conduct experiments to explore the variations of the percentage of total matched nodes with the increase of the average nodal degree in the ER and SF network models under different modes for finding nodes to match. Fig. 4 shows the variation of the percentage of total number of matched nodes with the increase of the average nodal degree in the ER and SF network models under the greedy, probabilistic (roulette) and completely random modes, respectively. The blue curves with circular marks in Fig. 4(a) and (b) show the variations of the percentage of total number of matched nodes with the increase of the average nodal degree in the ER and SF network models under greedy mode. The trends show that with an increasing average nodal degree, the percentage of the total number of matched nodes gradually increases. In the ER model, when the average nodal degree increases from 5 to 100, the percentage of matched nodes increases sharply from 86.80% to 99.30%. When the average nodal degree continues to increase, the percentage of matched nodes changes a little since it is close to 100%. In SF model, when the average nodal degree increases from 4 to 176, the percentage of matched nodes increases sharply from 72.90% to 99.40%. When the average nodal degree continues to increase, the percentage of matched nodes also changes a little since it is close to 100%. The red curves with rectangular marks in Fig. 4(a) and (b) show the variations of the percentage of total number of matched nodes with the increase of the average nodal degree in ER and SF network models under the probabilistic (roulette) mode respectively. In the ER model, when the average nodal degree increases from 5 to 100, the percentage of matched
F. Mao, L. Ma and Q. He et al. / Applied Mathematics and Computation 371 (2020) 124928
(a) ER network: Average nodal degree (X ax-
(b) SF network: Average nodal degree (X ax-
is) and Percentage of matched nodes (Y axis)
is) and Percentage of matched nodes (Y axis)
(Greedy mode, Roulette mode and Random
(Greedy mode, Roulette mode and Random
mode).
mode).
9
Fig. 4. Average nodal degree (X axis) and the percentage of matched nodes (Y axis) under greedy mode, roulette mode and random mode in ER and SF networks. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
nodes increases sharply from 87.40% to 99.30%. When the average nodal degree continues to increase, the percentage of matched nodes changes a little since it is close to 100%. In the SF model, when the average nodal degree increases from 4 to 176, the percentage of matched nodes increases sharply from 73.40% to 99.40%. When the average nodal degree continues to increase, the percentage of matched nodes also changes a little since it is close to 100%. The green curves with triangular marks in Fig. 4(a) and (b) show the variations of the percentage of total number of matched nodes with the increase of the average nodal degree in ER and SF network models under the completely random mode respectively. In the ER model, when the average nodal degree increases from 5 to 100, the percentage of matched nodes increases sharply from 87.70% to 99.50%. When the average nodal degree continues to increase, the percentage of matched nodes changes a little since it is close to 100%. In the SF model, When the average nodal degree increases from 4 to 176, the percentage of matched nodes increases sharply from 73.0% to 99.40%. When the average nodal degree continues to increase, the percentage of matched nodes also changes a little since it is close to 100%. When the average nodal degree is large enough, the percentage of matched nodes can reach 100%. The main difference between them is that when the percentage of matched nodes does not reach 100%, the percentage of total number of matched nodes in the ER model is larger than the percentage of total number of matched nodes in the SF model under the same average nodal degree and it reaches 100% with smaller average nodal degree compared to the result in the SF model. The reason is that the connection in the ER model is more even than SF model which helps make all the nodes match. Compared to the results in the ER network model under three different modes, they basically have the similar variations since the network model and average nodal degree are the same, and we also repeat to find nodes to match until all the nodes are matched or there is not any new matched pair. Thus, the percentages of matched nodes are basically the same. It also has the same situation in that of the SF network under the different modes. 3.6. Real-life network and an autonomous system We conduct experiment on real-life networks: Orkut, LiveJournal, AS-733. The average fitness value gaps between matched nodes are 0.075, 0.2747 and 0.3333 in Orkut under greedy mode, roulette mode and completely random mode, respectively. The average fitness value gaps for LiveJournal are 0.2197, 0.3047 and 0.3333 under the three match modes, respectively. In an autonomous system namely AS-733, the average fitness value gaps are 0.3416, 0.3246 and 0.3349 under the three modes, respectively. The results show that in the greedy mode under the Orkut network, they overall have smaller gaps between matched nodes compared with the LiveJournal and the AS-733. The reason is that they have about 94% matched nodes in Orkut, but they only have about 75% and 31% matched nodes in LiveJournal and AS-733, respectively and also the average nodal degree in AS-733 is 4 which is very small. The results under real-life networks also show that if the network is densely connected and has a large enough average nodal degree, one node would be more likely to end up with matching with the one similar to itself.
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4. Conclusion In this paper, we proposed a minimalist framework of match making in complex networks, especially we adopted a simple model where each individual would greedily seek for making a match with the strongest partner within his/her social connection range. We explored the variations of the average fitness value gap between matched pairs of nodes, the total number of iterations to finish matching, the percentage of matched nodes in each iteration and the percentage of total matched nodes with the increase of the average nodal degree. We also considered the extreme case considering the complete graph (network). The experiment results revealed that in a more densely connected social network, individuals’ effort to seek for matches with the strongest partners would be more likely to end up with matching with someone similar to themselves. Meanwhile, the cost of such an approach may be quickly increased. In real-life system, when the average nodal degree was large enough, the cost was unacceptable and even if they had very large connectivity, the possibility for all the nodes to find the strongest partners was very low since the average fitness value gap was small. As for the future work, we are interested in the two-hop matching, the cost for making all the nodes match and opinion formation [37]. Acknowledgments This work was partially supported by Ministry of Education, Singapore, under contract MOE2016-T2-1-119 and was partially supported by National Natural Science Foundation of China, under research grant 61803629, the Fundamental Research Funds for the Central Universities, China, under Grant N171706002. References [1] Wikipedia, Matchmaking, 2017, https://en.wikipedia.org/wiki/Matchmaking. [2] D. Ariely, The Upside of Irrationality: The Unexpected Benefits of Defying Logic at Work and at Home, Harper, 2010. [3] R. Nayak, M. Zhang, L. Chen, A social matching system for an online dating network: a preliminary study, in: Proceedings of the 2010 IEEE International Conference on Data Mining Workshops, 2010, pp. 352–357. [4] S.M. Wagner, Partners for business-to-business service innovation, IEEE Trans. Eng. Manag. 60 (1) (2013) 113–123. [5] C.E. Sleep, J.A. Lavner, J.D. Miller, Do individuals with maladaptive personality traits find these same traits desirable in potential romantic partners? Pers. Individ. Dif. 119 (2017) 317–322. [6] G. Seidman, A. Roberts, V. Zeigler-Hill, Narcissism and romantic relationship presentation on social media: the role of motivations and partner attractiveness, Pers. Individ. Dif. 149 (2019) 21–30. [7] J.V. Ouytsel, E.V. Gool, M. Walrave, K. Ponnet, E. Peeters, Exploring the role of social networking sites within adolescent romantic relationships and dating experiences, Comput. Hum. Behav. 55 (2016) 76–86. [8] D. 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Vattani, Human matching behavior in social networks: an algorithmic perspective, PLoS ONE 7 (8) (2012) 1–9. [14] Y.-J. Bae, J. Cha, H. Lee, B. Lee, S. Baek, S. Choi, D. Jang, Social networks and inference about unknown events: a case of the match between Google’s Alphago and Sedol Lee, PLoS One 12 (2) (2017) 1–25. [15] Q. He, X. Wang, Z. Lei, M. Huang, Y. Cai, L. Ma, TiFim: a two-stage iterative framework for influence maximization in social networks, Appl. Math. Comput. 354 (2019) 338–352. [16] X. Zang, T. Yamasaki, K. Aizawa, T. Nakamoto, E. Kuwabara, S. Egami, Y. Fuchida, You will succeed or not? Matching prediction in a marriage consulting service, in: Proceedings of the 2017 IEEE Third International Conference on Multimedia Big Data (BigMM), 2017, pp. 109–116. [17] N. Guner, Y. Kulikova, J. Llull, Marriage and health: selection, protection, and assortative mating, Eur. Econ. Rev. 104 (2018) 138–166. [18] O.P. Attanasio, K.M. Kaufmann, Education choices and returns on the labor and marriage markets: evidence from data on subjective expectations, J. Econ. Behav. Organ. 140 (2017) 35–55. [19] N.G. Vaillant, V. Harrant, Determinants of the likelihood of finding the right partner in an arranged marriage: evidence from a french matchmaking agency, J. Socio. Econ. 37 (2) (2008) 657–671. Special Issue on the Household Economy. [20] R. Holzman, D. Samet, Matching of like rank and the size of the core in the marriage problem, Games Econ. Behav. 88 (2014) 277–285. [21] F.E. De Oliveira Da Silva, C.L.R. Motta, F.M. Santoro, C.E.T. De Oliveira, A social matching approach to support team configuration, in: Proceedings of the 15th International Conference on Groupware: Design, Implementation, and Use, CRIWG’09, 2009, pp. 49–64. [22] K. Sugawara, Y. Yu, B. Ragsdale, H. Hara, T. Kinoshita, Design of an agent-based middleware for job matchmaking in teleworking community, in: Proceedings of the Second International Joint Conference on Autonomous Agents and Multiagent Systems, AAMAS ’03, 2003, pp. 1128–1129. [23] M. Véron, O. Marin, S. Monnet, Matchmaking in multi-player on-line games: studying user traces to improve the user experience, in: Proceedings of Network and Operating System Support on Digital Audio and Video Workshop, NOSSDAV ’14, 2014. 7–7 [24] N. Zhang, Y. Lee, R.K. Balan, Matchmaking game players on public transport, in: Proceedings of the 2nd Workshop on Mobile Gaming, MobiGames ’15, 2015, pp. 31–36. [25] T.D. Noia, E.D. Sciascio, F.M. Donini, M. Mongiello, A system for principled matchmaking in an electronic marketplace, in: Proceedings of the 12th International Conference on World Wide Web, WWW ’03, 2003, pp. 321–330. [26] U. Küster, B. König-Ries, M. Stern, M. Klein, DIANE: an integrated approach to automated service discovery, matchmaking and composition, in: Proceedings of the 16th International Conference on World Wide Web, WWW ’07, 2007, pp. 1033–1042. [27] S.S.R. Abidi, A. Daniyal, S.F. Mehdi, Web service matchmaking using a hybrid of signature and specification matching methods, in: Proceedings of the 2014 IEEE/WIC/ACM International Joint Conferences on Web Intelligence (WI) and Intelligent Agent Technologies (IAT), Vol. 2, 2014, pp. 266–273. [28] Q. Dong, D. Huang, Privacy-preserving matchmaking in geosocial networks with untrusted servers, in: Proceedings of the 2017 IEEE 37th International Conference on Distributed Computing Systems (ICDCS), 2017, pp. 2591–2592. [29] J. Tanimoto, Dilemma solving by the coevolution of networks and strategy in a 2 × 2 game, Phys. Rev. E 76 (21126) (2007) 1–20. [30] J. Tanimoto, A study of a quadruple co-evolutionary model and its reciprocity phase for various prisoner’s dilemma game, Int. J. Mod. 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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Match making in complex social networks Fubing Mao a,b, Lijia Ma c, Qiang He d,∗, Gaoxi Xiao b a
Service Computing Technology and System Lab, Cluster and Grid Computing Lab School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan 430070, China School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore c College of Computer Science and Software Engineering, Shenzhen University, Shenzhen 518060, China d College of Information Science and Engineering, Northeastern University, Shenyang 110819, China b
a r t i c l e
i n f o
Article history: Received 6 August 2019 Revised 22 October 2019 Accepted 17 November 2019
Keywords: Complex network Match making Strongest partner Romantic partner Social system
a b s t r a c t Match making is of significant importance in some social systems. People may need to seek for romantic partners, teammates, collaborators, etc. In this paper, we propose a minimalist framework of match making in complex networks. Specially we adopt a simple model where each individual would greedily seek for making a match with the strongest partner within his/her social connection range. We explore a few matching schemes including greedy mode, roulette wheel selection mode and completely random mode on different networks. We also investigate when social systems become more densely connected, how the match making process would be affected. Our observations show that, in a more densely connected social network, individuals’ efforts for seeking for matches with the strongest partners would be more likely to end up with matching with someone similar to themselves. Meanwhile, the cost of such an approach may be quickly increased. The implications of such observations in real-life systems and open problems are briefly discussed. © 2019 Elsevier Inc. All rights reserved.
1. Introduction Match making is the process for matching people into pairs. It could happen for finding partners for romantic relation, sports games, business collaboration, etc [1–7]. While “match making” may have different meanings in different scenarios, in this work, match making means that each individual shall try to form up a pair with one, and only one, neighbor he/she has. To represent this in a social network model, each node is seeking for a match with one and only one of its adjacent nodes. Though it has been of significant research interest to study on how people search for potential partners, sending out requests and finally reaching or failing to make a match [8–12], it is not well understood (i) how the social connections that individuals have affect their match making results; and (ii) when social systems become more densely connected (which is believed to be the case in many, though not all, modern social systems) [13–15], how the match making process would be affected. These open problems motivate the research work reported in this paper. Specifically, we propose a minimalist framework by studying match making following a simple greedy rule in complex social networks, with the main focus on evaluating the effects of network topology and connection density on match making in social networks. ∗
Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang 110819, China. E-mail addresses: [email protected] (F. Mao), [email protected] (Q. He), [email protected] (G. Xiao).
https://doi.org/10.1016/j.amc.2019.124928 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.
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Existing research on match making problem has been mainly focusing on its applications in different areas, mainly including a few aspects as follows. (1) Marriage Problem: This may be the most well-known match making problems [8,9,11,16–20]. Different matching strategies and systems have been proposed for helping find good romantic partners, e.g., in online dating systems [8], for finding good partners in offline system [9], and for finding life-long partners for marriage [11,16]. (2) Teamwork Problem: A matching model was presented in [21] considering skills and socio-emotional factors and their roles in supporting the composition of an efficient team. And an agent-based middleware was proposed in [22] for job matchmaking in a teleworking community. (3) Online Games: A matchmaking service in multiplayer online games was proposed in [23]. Another discussion was on constraints and policies of game design for allowing public transport commuters to engage in multiplayer games [24]. (4) Marketing: For example, a system was proposed for matchmaking in an electronic marketplace [25]. (5) Matching Service Request: Service composition is integrated into service discovery and matchmaking for matching service requests [26]. A hybrid web service matchmaker which analyzes the signature and specification of different web services was presented in [27]. And a privacy preserving matchmaking system was constructed in geosocial networks with untrusted servers [28]. (6) Several Areas in the Field of Social Physics: The problem of how cooperative behavior can emerge in the real life captures much attention [29–31]. For example, a co-evolutionary model of SPD (spatial version of Prisoner’s Dilemma) [30,32] was proposed for investigating the problem. SPD offers two strategies: C (Cooperation) and D (Defection) [29,30,32]. In SPD, not only the agent’s strategy (either cooperation or defection), but also the spatial structure (agent’s link) would be time-evolved obeying to a certain update rule [29–32]. While such studies help enhance the roles of match making in various applications, they did not study the effects of social connections on efficiency and results of social match making. We study match making in complex networks within a minimalist framework. Specifically, we adopt the assumption that each individual shall always try to make a match with the best individual among all his/her connections as far as the selected candidate is still available. While it is understood that such a greedy approach may not be the approach that people actually adopt in most applications, evaluating the cost and matching results of adopting such an approach may help us better understand why some “practical” approaches are adopted in social systems, and how the evolution of modern social systems may drive the changes of match making activities. Our studies reveal that, in a more densely connected social network, individuals’ efforts to seek for the strongest partners would be more likely to end up with matching with someone similar to themselves. Meanwhile, the cost of match making may be increased with the connection density. The implications of such observations in real-life systems and open problems will be briefly discussed. The main contributions of our paper are as follows. • We propose a minimalist framework of match making in complex networks. • We adopt a simple model where each individual would greedily seek for making a match with the strongest partner within his/her social connection range. Our proposed match making has different meanings that each node is seeking for a match with one and only one of its adjacent nodes. • We explore a few matching schemes including greedy mode, roulette wheel selection mode and completely random mode on different networks. • We investigate when social systems become more densely connected, how the match making process would be affected. We also briefly discuss the implications of our observations in real-life systems. The rest of the paper is organized as follows. We first introduce our network based minimalist framework in Section 2. Then we present the experiment results and discussions in Section 3. Finally, we conclude our work and remarks in Section 4.
2. Network based minimalist framework The minimalist framework of match making we shall adopt in this study is as follows. In a network G with n nodes, each node has a fitness value within the range [0, 1]. Here, the fitness value represents the value of the node which is similar to the value of a person in the real life. Since we investigate the match making of people in real life, in our paper each node has a fitness value in the networks. Actually, the fitness value can be any number no less than 0. For convenient discussion and investigation, we normalize the range and make the range be [0, 1]. At the beginning, all the nodes in the network are unmatched nodes. Then each node shall iteratively bid for making a match with the fittest node in its neighborhood area. Specifically, in each iteration, a single matching request is sent out by every unmatched node to one of its unmatched neighbors. A match is made when two nodes happen to send matching requests to each other. Once a match is made, the two matched nodes will not join the later iterations of match making. The detailed introduction to our proposed framework is in Section 2.1.
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Algorithm 1: Network Based Minimalist Framework of Match Making. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35:
Input: Given a network G with n nodes and m edges Output: The set of matched pairs and related results (the avgfitvaluegap among all the matched pairs etc) define totalfitvaluegap to record the sum of the absolute fitvaluegap between matched nodes. Its initialized value is 0. define avgfitvaluegap to record the average value of the absolute values of the fitness value gaps between matched nodes (Initialized value is 0). define a map variable socialmatchnodeidAndreqid to record the nodeid and its requested nodeid define a map variable nodeidAndreqid to temporarily record the nodeid and its requested nodeid (request to match with another node) define a map variable matchpairmap to record the nodeid and its matched nodeid assign values from [0, 1] randomly to nodes of G respectively //the value is fitness value define a set tmpvisitedset to record the nodes already visited define a variable nofoundflag to check whether a matched pair is founded. If cannot find a pair, then stop. The initialized value of nofoundflag to is 0, (1: is not found. 0: found) while !nofoundflag do set the map nodeidAndreqid to be empty//traverse all the nodes in G and find each node’s neighbor nodes which are not matched for each node in G for each node in G do if the node is not visited then find the node’s neighbor nodes which are not matched, and record them end if if the match mode is greedy mode then find the node with the max fitness value from the current node’s neighbor node set record the current node id and the finding/requested node id (current node’s neighbor node with the max fitness value) in nodeidAndreqid end if end for if nodeidAndreqid is empty (cannot find the matched pairs) then set the variable nofoundflag to be 1 //cannot found the new match pair. stop end if if nodeidAndreqid is not empty then for each element in map nodeidAndreqid do if nodes A sends request to match B and B also sends request to match A at the same time then define a variable and let fitvaluegapofpairAB = the absolute difference value between the two matched nodes A and B let totalfitvaluegap = the sum of all the fitvaluegapofpairAB end if end for end if end while let avgfitvaluegap = totalfitvaluegap / (total number of matched pairs) return
2.1. Our proposed minimalist framework of match making Our proposed network based minimalist framework of match making is presented in Algorithm 1. Lines 1 and 2 present the input and output of the framework. Lines 3–10 introduce the initialization of the network and the variables for recording data. From line 11–33, the algorithm find matched nodes and record the matched pairs. Specifically, lines 13–21 collect the nodes which send requests to match a node from their unmatched neighbors, and lines 22–32 obtain the matched pairs where the nodes send requests to each other at the same time. Line 34 acquires the average value of the fitness value gaps between all the matched nodes. Line 35 indicates the algorithm finishes. The time complexity and space complexity of our proposed framework are O(n2 m) and O(nm), respectively where n represents the number of the nodes and m stands for the number of edges in G. Within the same framework, we shall consider two slightly different approaches as follows. The first one is termed as greedy mode, where in each iteration, each unmatched node shall find among all its unmatched adjacent nodes the one with the highest fitness level and send a request to it. When there is a tie, i.e., there exist multiple unmatched adjacent nodes with the same highest fitness value, break it randomly. The process shall repeat until all the nodes are matched or no further matches can be made any more. As we could see, in the greedy mode, each node aggressively seeks for the best possible match, largely regardless of the cost that may occur (measured by the number of iterations) before a match is made. Similar to but less aggressive than the greedy mode, in a probabilistic mode (termed as roulette wheel selection mode), the probability an unmatched node is selected by one of its unmatched neighbors as a possible candidate for match making is proportional to its fitness value. 2.2. Fitness value gap between matched nodes We briefly introduce the fitness value gap. The fitness value of each node in the network follows a uniform distribution in [0, 1]. In a complete network, the absolute value of the fitness value gap between two matched nodes of the random mode can be analyzed as follows. For two independent matched nodes with fitness values x and y respectively, both x and
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y obey the uniform distribution in [0, 1]. f(x) shown in (1) and f(y) shown in (2) are the density distributions of x and y, respectively. Let variable z = |x − y|. F(z) represents the distribution of z. Thus, F (z ) = P (|x − y| ≤ z ). when z < 0, F (z ) = 0. 1 x−z 1−z 1 when z ≥ 1, F (z ) = 1. When 0 < z < 1, F (z ) = P (|x − y| ≤ z ) = 1 − z dx 0 dy − 0 dx x+z dy = 1 − (1 − z )2 = 2z − z2 . Thus, we obtain the distribution of F(z) listed in (3). We discuss the fitness value gap more in Section 3.2.
f (x ) =
1 0
f (y ) =
1 0
0
(1)
0
(2)
F (z ) =
0 2z − z 2 1
z≤0 0
(3)
3. Simulation results and discussion 3.1. Experiment setup We implement our minimalist framework of match making in the C++ programming language and perform our experiments on a HP Z420 workstation with Intel(R) Xeon(R) CPU 3.5 GHz and 16 G RAM. Our numerical simulations are mainly conducted on Erdos–Renyi (ER) network model [33,34] and scale-free (SF) network model [35,36]. In the network models, the number of nodes is 20 0 0. In the initial state, all nodes are unmatched with fitness values uniformly distributed between [0, 1]. We implement the greedy mode and the roulette wheel selection mode respectively. Further, a random mode is implemented as a benchmark case, where each node randomly chooses an adjacent unmatched node to send a matching request. We also conduct simulations on real-life social networks such as Orkut [38,39] and LiveJournal [38,39] and an autonomous system AS-733 [38]. Unless otherwise specified, we run 10 times with different random seeds for each experiment and present the average results. 3.2. The relation between average nodal degree and average fitness value gap We explore the variation of the average fitness value gaps when we adopt different average nodal degrees in different networks. In our work, the fitness value gap of a pair of matched nodes means the absolute value of the difference between the two nodes’ fitness values and the average fitness value gap is the mean value of all the fitness value gaps of matched nodes. We consider the ER and SF network models and adopt the greedy mode, probabilistic (roulette) mode and completely random mode for match making in each iteration. Fig. 1 shows the variation of the average fitness value gap with the increase of the average nodal degree in the ER and SF networks under the three different match making modes, respectively. The curves with circular marks (blue color) in Fig. 1(a) and (b) show the variations of the average fitness value gap with the increase of the average nodal degree in the ER network model and SF network model under the greedy mode, respectively. In the ER model, the blue curve with circular marks in Fig. 1(a) shows that when the average nodal degree increases from 5 to 1999 which forms into a complete graph, the average fitness value gap of all the matched nodes decreases from 0.1958 to 0.0 0 05. Further, when the average nodal degree increases from 5 (87% nodes matched) to 200 (nearly 100% nodes matched), the average fitness value gap decreases from the 0.1958 to 0.0068. In this stage, the average fitness value gap decreases quickly. After that, the gap decreases slowly but steadily with an increasing average nodal degree. The reason is that when the average nodal degree is low, the chance for two nodes with similar fitness values to be directly connected is low, which makes the fitness value gap relatively large. With an increasing average nodal degree the average fitness value gap decreases finally, reaching an exhausted level; meanwhile the match making percentage increases to be close to 100%. In the SF model, the blue curve with circular marks in Fig. 1(b) shows that when the average nodal degree increases from 4 to 1999 (a complete graph), the average fitness value gap of all the matched nodes decreases from 0.2813 to 0.0 0 05. Further, when the average nodal degree increases from 4 (73% nodes matched) to 380 (nearly 100% nodes matched), the average fitness value gap decreases from the 0.2813 to 0.0051. The observations and the main reasons remain largely the same as those in ER networks. The main difference between the results in ER and SF models under greedy mode is that the average fitness value gap in the ER model is smaller than that in the SF model under the same average nodal degree. This is because the ER model allows a higher match making percentage, which helps lower the average value gap. The red curves with rectangular marks in Fig. 1(a) and (b) show the variations of the average fitness value gap with the increase of the average nodal degree in the ER network model and SF network model under the probabilistic (roulette) mode, respectively. In the ER network model, the curve in Fig. 1(a) shows that when the average nodal degree increases from 5 to 1999, the average fitness value gap stays within the range from 0.2499 to 0.290. When the average nodal degree increases from 5 (87% nodes matched) to 100 (nearly 100% nodes matched), the average fitness value gap only slightly decreases from the 0.290 to 0.2543. In this stage, the trend shows that it decreases relatively quickly. After that, the fitness value gap decreases steadily but very slowly when the average nodal degree continues to increase. This is because the average fitness
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Fig. 1. Average nodal degree (X axis) and average fitness value gap (Y axis) under greedy mode, roulette mode and random mode in ER and SF networks. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
value gap approaches to 0.250 rather than 0 with an increasing average nodal degree. This can be easily understood. In a complete network with greedy matching operation, it is trivial to prove that when the network size goes to infinity, the average fitness gap value as the fittest node always match with the second fittest node, and the third fittest node matches with fourth fittest node, and so on. The average fitness gap value of the random mode can be approximately calculated as (4).
1 0
1 0
|x − y| dxdy =
1 0
1 x
(y − x ) dydx +
1 0
0
x
(x − y ) dydx =
1 1 1 + = 6 6 3
(4)
The average fitness gap value of the probabilistic (roulette) mode is more difficult to be calculated. For two nodes with fitness values x and y respectively, in the first round, the chance they choose each other is proportional to xy where both x and y follow a uniform distribution in [0, 1]. In later iterations, while the probability that two nodes with fitness values x and y make a match remains to be proportional to xy, the distribution of fitness value of unmatched nodes however changes in each iteration; nodes with higher fitness value have a higher chance to match earlier. While it may not be easy to derive a concise theoretical analysis on the average fitness value gap for such a case, it is reasonable to expect that the gap value should be somewhere between 0 and 1/3. In the SF network model, the red curve in Fig. 1(b) shows that when the average nodal degree increases from 4 to 1999 (a complete graph), the average fitness value gap stays within the range from 0.2499 to 0.3128. When the average nodal degree increases from 4 (73% nodes matched) to 380 (nearly 100% nodes matched), the average fitness value gap decreases from the 0.3128 to 0.2517. In this stage, the trend shows that it decreases quickly. After that, the fitness value gap decreases steadily when the average nodal degree continues to increase. The reason is the same as those in ER networks. The two network models do not lead to big difference in the average fitness value gap. The reason is that when we choose the probabilistic (roulette) mode to find nodes to match, the network models impact the matched pairs of the nodes a little. The green curves with triangular marks in Fig. 1(a) and (b) show the variations of the average fitness value gap with the increase of the average nodal degree in ER and SF network models under the completely random mode, respectively. In the ER network model, the figure shows that when the average nodal degree increases from 5 to 1999 (a complete graph), the average fitness value gap stays within the range from 0.3273 to 0.3374. In the SF network model, this value stays with the range of 0.3279 to 0.3375. The average value gap basically remain the same when the average nodal degree increases. This is not a surprise as the fitness value gap of a random matching operation is largely independent of the average number of neighbors each node has. Note that the two different network models basically do not impact the average fitness value gap under the random mode. This can be explained. For n independent variables x1 , x2 , x3 , . . . , xn within the range of [0, 1], the mean absolute difference (MAE) of any two randomly selected variables can be calculated as about 0.3333 using (5). The detailed procedure to get this value is described as follows.
MAE =
n n 1 xi − x j 2 n i=1 j=1
(5)
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Fig. 2. Average nodal degree (X axis) and total iteration time (Y axis) for finishing match under greedy mode, roulette mode and random mode in ER and SF networks. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Assume x1 and x2 are randomly chosen from [0, n] where n is integer and n > 0. The absolute difference between any two variables x1 , x2 is represented by ∇ x. ‘∇ x = 1’ has 2n possibilities. ‘∇ x = 2’ has 2(n − 1 ) possibilities. ‘∇ x = 3’ has 2(n − 2 ) possibilities. ‘∇ x = k’ has 2(n − k + 1 ) possibilities (k > 3). ‘∇ x = n’ has 2 possibilities. The total number of possibilities is n(n + 1 ) since 2n + 2(n − 1 ) + 2(n − 2 ) + · · · + 2 = n(n + 1 ). We use p(i) to represent the probability to get the value i. Thus, p(1 ) = 2n/[n(n + 1 )]. p(2 ) = 2(n − 1 )/[n(n + 1 )]. p(k ) = 2(n − k + 1 )/[n(n + 1 )]. p(n ) = 2/[n(n + 1 )]. The mathematical expectation of ∇ x is that E∇ x = ni=1 i × p(i ) = 1 × 2n/[n(n + 1 )] + 2 × 2(n − 1 )/[n(n + 1 )] + ... + n × 2/[n(n + 1 )] = (n + 2 )/3. If n = 65535, E∇ x = (65535+2)/3 = 21846. Since x1 and x2 are within the range of [0, 1], we get the value 0.3333 (21846/65535 = 0.3333). It is interesting to observe that with a large enough average number of connections per node, the fitness value gap decreases to 0.0 0 05. For 20 0 0 nodes, with their fitness value uniformly randomly distributed between 0 and 1, the average gap between two adjacent fitness values is roughly at the same scale, meaning that almost every node is matched with another node with fitness value as close to its own value as possible. An “unbalance” match between two nodes with a relatively large difference in their fitness values has a virtually zero chance to happen. When the social systems are becoming more densely connected, the chance of “making an unbalanced match” decreases quickly. 3.3. The relation between average nodal degree and total iteration time for finishing matching nodes We explore the variation of total iteration time for finishing matching nodes when we adopt different average nodal degrees in different networks. We also consider the ER and SF network models, and adopt the greedy mode, probabilistic (roulette) mode and completely random mode for finding the unmatched nodes to match, respectively. In our work, finishing matching means that we cannot make pairs from unmatched nodes any more. The total iteration time means the minimum iteration time for reaching maximum matched pairs under a certain average nodal degree. Fig. 2 shows the variation of the total number of iteration time with the increase of the average nodal degree in ER and SF networks under the three modes, respectively. The two blue curves with circular marks in Fig. 2(a) and (b) show the trends about the variations of the total number of iteration time with the increase of the average nodal degree in ER and SF network models under greedy mode, respectively. The trends show that with an increasing average nodal degree the total number of iteration time gradually increases. They are similar and the trends of growth are both gradually slowing down. If the average nodal degree is large enough, all the nodes finally make pairs. The reason is that with an increasing average nodal degree, each node has more directly adjacent
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nodes and the chance for two nodes match each other at the same time becomes smaller which increases the total number of iterations. When the average nodal degree reaches to a certain stage, the increased number of iteration time becomes small. The main difference between the results in ER and SF networks is that the total number of iteration time in the ER network is larger than the total number of iteration time in the SF network under the same average nodal degree. The reason is that SF network is more densely than the ER network which it helps make pairs more quickly. The red curves with rectangular marks in Fig. 2(a) and (b) show the trend of the total number of iteration time with the increase of average nodal degree in ER and SF network models under probabilistic (roulette) mode respectively. Both trends show that with an increasing average nodal degree, the total number of iteration time gradually increases. The main difference between them is that when the average nodal degree is very small, the total number of iteration time in the ER model is larger than the number of iteration time in the SF model. When the average nodal degree is large, the total number of iteration time in the ER model is smaller than the number of iteration time in the SF model under roulette mode. The possible reason is that probability (roulette) mode is more quickly to finish matching in ER model than SF model. The green curves with triangular marks in Fig. 2(a) and (b) show the trends of the total number of iteration time with the increase of average nodal degree in ER and SF networks under completely random mode, respectively. It has the same situation with that of the probabilistic (roulette) mode. Compared to the results in the ER network model under the three different match making modes, we can observe that the total number of iteration time for finishing matching in greedy mode under the same average nodal degree is the smallest. The total number of iteration time for finishing matching in probability (roulette) mode and completely random mode are basically the same. The reason is that greedy mode is a deterministic approach and other two modes are probabilistic approaches. Compared to the results in SF network model under the three different match making modes, it has the similar situations and reasons. How quickly the number of matched nodes increases in each iteration will be discussed later.
3.4. The relation between average nodal degree and the percentage of matched nodes in each iteration We perform experiments to investigate the relation between the average nodal degree and the percentage of matched nodes in each iteration under a certain average nodal degree. During the iteration, if the procedure cannot find any new matched pairs, the procedure will stop. The results is shown in Fig. 3. In the figures, Avg_node_degree represents the average nodal degree. We consider the ER and SF network models and also adopt the greedy mode, probabilistic (roulette) mode and completely random mode to find unmatched nodes for matching, respectively. In our work, the percentage of the matched nodes in each iteration is the value of the number of matched nodes in each iteration divided by the total number of nodes under the certain average nodal degree. We use representative figures to show the experiment results and other figures are similar to them. Fig. 3(a)–(d) is the results in the ER model under greedy mode and the iteration time starts from 1. With an increasing average nodal degree, the percentage of matched nodes at the first few of iterations decreases. When the average nodal degree is small, the overall trend is that the percentage of the matched nodes in each iteration decreases gradually and finally reaches 0 which represents all the nodes are matched or it cannot find any new matched pairs as shown in Fig. 3(a). When the average nodal degree is large enough, the percentage of matched nodes in each iteration fluctuates and is small as shown in Fig. 3(b) and (c). For the extreme case shown in Fig. 3(d), when the network is a complete graph, the percentages of matched nodes in different iterations are the same. The reason is that when the average nodal degree is small, the chance for two nodes to make a pair is high. It has large percentage of matched nodes at the beginning. The matching process repeats for the unmatched nodes and the number of unmatched nodes decreases which makes the percentage of matched nodes in each iteration decrease. When the average nodal degree is large enough, the chance for two nodes to make a pair is low which makes the percentage of matched nodes in each iteration fluctuate. The results in the roulette mode and the completely random mode have the similar situation with greedy mode. The main difference is that when the network is a complete graph, the percentages of matched nodes in different iterations under greedy mode are the same and the percentages of matched nodes in each iteration under roulette mode and completely random mode are basically the same since greedy mode is deterministic and other two modes are probabilistic. Furthermore, the overall trends are similar and the fluctuations are not the same due to the different modes for finding nodes to match. The situations and trends in the SF model under the three different match making modes are similar to the situations and trends in ER model under the three modes, respectively. The main difference between the results in the ER model and SF model under different modes is that the trends waves are different due to different network models. The percentages of matched nodes in each iteration under a complete graph implicitly show that in real-life system, when the connection grows enough, the possibility to find a partner with larger fitness value to make a pair becomes larger. However, the total number of iteration time (cost) for finishing matching increases largely or unacceptable. Our observations show that when the average nodal degree is small, the majority part of the matching is done in the first few iterations. The reasons are given as follows. • The number of nodes which can finally make pairs is small when the number of average nodal degree is small. • When the average nodal degree is small, it only takes a few of iteration for finishing matching.
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(a)
(b)
(c)
(d)
Fig. 3. ER network: the iteration time (X axis) and the percentage of matched nodes in each iteration (Y axis) under different average nodal degrees (Greedy mode).
3.5. The relation between the average nodal degree and the percentage of total matched nodes We also conduct experiments to explore the variations of the percentage of total matched nodes with the increase of the average nodal degree in the ER and SF network models under different modes for finding nodes to match. Fig. 4 shows the variation of the percentage of total number of matched nodes with the increase of the average nodal degree in the ER and SF network models under the greedy, probabilistic (roulette) and completely random modes, respectively. The blue curves with circular marks in Fig. 4(a) and (b) show the variations of the percentage of total number of matched nodes with the increase of the average nodal degree in the ER and SF network models under greedy mode. The trends show that with an increasing average nodal degree, the percentage of the total number of matched nodes gradually increases. In the ER model, when the average nodal degree increases from 5 to 100, the percentage of matched nodes increases sharply from 86.80% to 99.30%. When the average nodal degree continues to increase, the percentage of matched nodes changes a little since it is close to 100%. In SF model, when the average nodal degree increases from 4 to 176, the percentage of matched nodes increases sharply from 72.90% to 99.40%. When the average nodal degree continues to increase, the percentage of matched nodes also changes a little since it is close to 100%. The red curves with rectangular marks in Fig. 4(a) and (b) show the variations of the percentage of total number of matched nodes with the increase of the average nodal degree in ER and SF network models under the probabilistic (roulette) mode respectively. In the ER model, when the average nodal degree increases from 5 to 100, the percentage of matched
F. Mao, L. Ma and Q. He et al. / Applied Mathematics and Computation 371 (2020) 124928
(a) ER network: Average nodal degree (X ax-
(b) SF network: Average nodal degree (X ax-
is) and Percentage of matched nodes (Y axis)
is) and Percentage of matched nodes (Y axis)
(Greedy mode, Roulette mode and Random
(Greedy mode, Roulette mode and Random
mode).
mode).
9
Fig. 4. Average nodal degree (X axis) and the percentage of matched nodes (Y axis) under greedy mode, roulette mode and random mode in ER and SF networks. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
nodes increases sharply from 87.40% to 99.30%. When the average nodal degree continues to increase, the percentage of matched nodes changes a little since it is close to 100%. In the SF model, when the average nodal degree increases from 4 to 176, the percentage of matched nodes increases sharply from 73.40% to 99.40%. When the average nodal degree continues to increase, the percentage of matched nodes also changes a little since it is close to 100%. The green curves with triangular marks in Fig. 4(a) and (b) show the variations of the percentage of total number of matched nodes with the increase of the average nodal degree in ER and SF network models under the completely random mode respectively. In the ER model, when the average nodal degree increases from 5 to 100, the percentage of matched nodes increases sharply from 87.70% to 99.50%. When the average nodal degree continues to increase, the percentage of matched nodes changes a little since it is close to 100%. In the SF model, When the average nodal degree increases from 4 to 176, the percentage of matched nodes increases sharply from 73.0% to 99.40%. When the average nodal degree continues to increase, the percentage of matched nodes also changes a little since it is close to 100%. When the average nodal degree is large enough, the percentage of matched nodes can reach 100%. The main difference between them is that when the percentage of matched nodes does not reach 100%, the percentage of total number of matched nodes in the ER model is larger than the percentage of total number of matched nodes in the SF model under the same average nodal degree and it reaches 100% with smaller average nodal degree compared to the result in the SF model. The reason is that the connection in the ER model is more even than SF model which helps make all the nodes match. Compared to the results in the ER network model under three different modes, they basically have the similar variations since the network model and average nodal degree are the same, and we also repeat to find nodes to match until all the nodes are matched or there is not any new matched pair. Thus, the percentages of matched nodes are basically the same. It also has the same situation in that of the SF network under the different modes. 3.6. Real-life network and an autonomous system We conduct experiment on real-life networks: Orkut, LiveJournal, AS-733. The average fitness value gaps between matched nodes are 0.075, 0.2747 and 0.3333 in Orkut under greedy mode, roulette mode and completely random mode, respectively. The average fitness value gaps for LiveJournal are 0.2197, 0.3047 and 0.3333 under the three match modes, respectively. In an autonomous system namely AS-733, the average fitness value gaps are 0.3416, 0.3246 and 0.3349 under the three modes, respectively. The results show that in the greedy mode under the Orkut network, they overall have smaller gaps between matched nodes compared with the LiveJournal and the AS-733. The reason is that they have about 94% matched nodes in Orkut, but they only have about 75% and 31% matched nodes in LiveJournal and AS-733, respectively and also the average nodal degree in AS-733 is 4 which is very small. The results under real-life networks also show that if the network is densely connected and has a large enough average nodal degree, one node would be more likely to end up with matching with the one similar to itself.
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4. Conclusion In this paper, we proposed a minimalist framework of match making in complex networks, especially we adopted a simple model where each individual would greedily seek for making a match with the strongest partner within his/her social connection range. We explored the variations of the average fitness value gap between matched pairs of nodes, the total number of iterations to finish matching, the percentage of matched nodes in each iteration and the percentage of total matched nodes with the increase of the average nodal degree. We also considered the extreme case considering the complete graph (network). The experiment results revealed that in a more densely connected social network, individuals’ effort to seek for matches with the strongest partners would be more likely to end up with matching with someone similar to themselves. Meanwhile, the cost of such an approach may be quickly increased. 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