Toward uncertainty of weighted networks: An entropy-based model

Accepted Manuscript Toward uncertainty of weighted networks: An entropy-based model Likang Yin, Yong Deng

PII: DOI: Reference:

S0378-4371(18)30613-7 https://doi.org/10.1016/j.physa.2018.05.067 PHYSA 19607

To appear in:

Physica A

Received date : 4 July 2017 Revised date : 11 April 2018 Please cite this article as: L. Yin, Y. Deng, Toward uncertainty of weighted networks: An entropy-based model, Physica A (2018), https://doi.org/10.1016/j.physa.2018.05.067 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Toward uncertainty of weighted networks: An entropy-based model Likang Yina,b , Yong Denga,∗ a

Institute of Fundamental and Frontier Science, University of Electronic Science and Technology of China, Chengdu, 610054, China b School of Hanhong, Southwest University, Chongqing 400715, China

Abstract Measuring the uncertainty is of both theoretical value and practical interest in the network science. The previous studies focus on measuring the uncertainty of the entire networks. However, how to measure the uncertainty of individuals is still an open issue. To address this issue, the “asking for help” example is used to model the user behaviors and explain the mechanism of the proposed method. In this paper, we develop three heuristic rules to measure the utility of adjacent neighbors to each ego in the networks. Then, the fuzzy systems theory is used to convert the utility of each neighbor into the membership functions. Next, we derive the uncertainty of each node based on the Shannon entropy. Our result demonstrates the overall uncertainty of the networks, and also the uncertainty for the individual nodes. Moreover, our model also reflects the uncertainty of nodes for choosing to strengthen or weaken the existed links between their neighbors with the evolution of networks. Instead of forming new links but changing the existed relationship between nodes, we consider the proposed uncertainty measure may suggest a crucial property of the networks on the opposite side of link prediction. Keywords: Uncertainty; Link prediction; Entropy; Weighted networks

1. Introduction From ancient prophets to modern scientists, making predictions is always an attractive activity throughout the history [1]. However, the uncertainty is ubiquitous in any predictive models, and how to handle the uncertainty is still an open issue. Luckily, the human-beings are skilled in abstraction, and the scientists have developed many mathematical tools to model the predictive problems, such as time-series uncertainty models [2, 3], fuzzy systems methods [4, 5] and the network approaches [6, 7]. These models are similar to black boxes, and they act in a manner to change the key of prediction models from “how it works” to “what are the outputs”. Specifically, the time-series models focus on the change of different ∗ Corresponding author: Yong Deng, Institute of Fundamental and Frontier Science, University of Electronic Science and Technology of China, Chengdu, 610054, China. Tel(Fax): 86-28-61830858 Email address: [email protected]; [email protected] (Yong Deng)

Preprint submitted to Physica A

May 16, 2018

states, and the network models encapsulate the problem into a system that consists of nodes and links [8]. Moreover, in the networks, the nodes indicate the variables in the practical problems and the links stand for the relationships between each variable [9]. However, the links represent different meanings in the different networks [10, 11]. For example, in the social networks, the nodes represent the individuals and the links may denote the communication between two individuals [12]. While in the traffic networks, the nodes represent the different locations, the link may represent the traffic flow. Therefore, to unify the notation of nodes and the meaning of links in different networks, we use “egos” to denote the nodes, and the links represent the degree of correlations between two “egos”. The network models are efficient tools to address the complex relationships between two “egos” [13]. The most well-known network is the Internet, and it is often investigated with the online social networks [14, 15]. Moreover, there are some other interesting networks, such as traffic networks [16, 17], financial networks [18], and academic research networks [19]. In those networks, many interesting and crucial issues are constantly emerging, such as network reconstruction [20, 21], and simulating the virus attack [22] and dynamic process [23], and link prediction problems [24, 25]. Moreover, both the network structure [26] and the semantic relationship [27] are crucial to these practical issues. Quantitatively measuring how the nodes interacted with each other is of great importance since it affects how information is propagated, the disease is spread, the knowledge is organized and discovered, and recommendation is established and personalized [28, 29]. Despite the extensive investigation into various factors that influence the network structure and relationships between pairs of nodes [30], quantitative assessments of uncertainty that give rise to macroscopic patterns characterizing structural regularities remain limited. To address this issue, the previous study [31] considers that the predictability is actually an inherent property of network itself, and they used structural consistency to measure the predictability of the entire networks [1]. The structure and rules of ego networks [32, 33] have been extensively studied, and the ego networks can be applied to many research areas, such as analyzing the research impact [34], identifying the human disease modules [34], and investigating emotional closeness [35]. However, how to measure the uncertainty of each individual “ego” (the node) in the networks is still an open issue. For example, in the real world, if an individual gets into trouble, the selection strategy of seeking help might be like “I need to find my parents/close friends or a very powerful person to rescue me”. How to measure this kind of uncertainty of the egos under this kind of circumstance in the social networks? Moreover, the real world can be more complex since preference (e.g., choose his friends or a more powerful person) of individuals is mixed. However, how good is “very good”? This is where the game theory leaves the practitioners, and this is where the fuzzy systems theory comes to play. The study is to address this very practical and straightforward problem that is “Whom to ask for help?” in the weighted networks. The basic idea of the proposed uncertainty model is to convert the local information of egos into utility functions, which are used to denote the preference of the ego to each selection strategy. Therefore, we first define the possible selection strategies based on the game theory [36, 37]. Then, the utility functions are normalized to derive the membership functions. Next, we obtain the uncertainty of 2

the membership functions based on Shannon entropy [38, 39, 40]. Moreover, to show the connections and differences between the proposed uncertainty and accuracy of link prediction methods, we compare the results with the three common link prediction methods in seven well-known networks. Finally, based on the uncertainty degree of each ego, we can derive the overall uncertainty of the entire networks. 2. Preliminaries 2.1. Data pretreatment The real world is not only filled with uncertainty [41, 42], but is also very complex with many factors affect each other [43, 44]. Each fact in the complex systems is interacting with each other fact or depending on the status of other facts [45, 46]. To model the complexity, many methodologies are presented such as Hierarchy model [47, 48, 49, 50]. However, compared with hierarchy model, complex network models are paid more and more attention due to the efficiency to explain the complexity. Different complex networks are sued to model different complicated systems. In this paper, we choose seven representative networks from real network data, including: (i) NetSci-A co-authorship network of scientists working on network theory and experiment [51]; (ii) USAir-The US Air transportation system [6]; (iii) Yeast-A protein-protein interaction network in budding yeast [52]; (iv) Router-A symmetrized snapshot of the structure of the Internet at the level of autonomous systems [53]; (v) Political Blogs (PB)-A network of the US political blogs [54]; (vi) Adolescent health [55] is a directed network which was created from a survey which was presented in 1994-1995. In this network, the nodes represent the adolescents, and the links from node vi to vj means the adolescent vi choose another adolescent vj to be his/her friend. Moreover, the weights of links indicate more interactions between two adolescents; (vii) King James-King James [56] is an undirected network which contains both names and the occurrences of the King James bible. Moreover, the basic topological features of the seven real networks are shown in Table 1. To test the accuracy of link prediction method, the data set L (the set of links) is divided into two parts, one is the training set LT and another part is the probe set LP . Obviously, ∪ ∩ L = LP LT and LP LT = ∅ (1) The training set contains the 90% of the whole data, the 10% of the whole data is the probe set. Both training set and the probe set are divided randomly with maintaining the connectivity of the whole network simultaneously. Moreover, only the information of LT is allowed to be used to compute the performance score Sxy . While the probe set, LP , is used for testing and no information therein is allowed to be used for prediction.

2.2. Evaluation Metrics In this paper, the general measurement method called Area Under the receiver operating characteristic Curve (AUC) [58] is used to measure the accuracy of the proposed method and compare the result with a bunch of existing prediction methods. In the field of Signal 3

Table 1: The basic topological features of nine real networks. N is the total numbers of nodes and M is the total numbers of links. C denotes average clustering coefficient and A denotes average path length. D denotes the average weighted degree. L is the number of self-loops in the networks [57]. Network NetSci USAir Yeast Router Political Blogs King James Adolescent

N 1589 332 2361 5022 1222 1776 1985

M 2742 2126 7182 6258 19021 18264 18274

C 0.791 0.749 0.368 0.033 0.360 0.741 0.334

A 1.994 2.564 4.648 3.973 3.390 3.676 6.277

D 0.815 0.462 3.042 1.246 15.565 9.250 14.857

L 0 0 536 0 0 0 0

Detection Theory (SDT), the Receiver Operating Characteristic (ROC) is often used to evaluate the effectiveness of classification algorithm. Likewise, we use the AUC to measure the accuracy of link prediction [59, 6]. Namely, ′

′′

n + 0.5n AUC = n

(2)

where n is times of independent comparisons, n′ is the times that the missing link having a higher score and n′′ is times that the missing link and nonexistent link having the same score. This measurement acts like a randomly chosen missing link (e.g. a link in S P ) is given a higher score than a randomly selected nonexistent link. For the general cases, if a prediction model has a better effect than choosing links randomly, the AUC should be larger than 0.5 and vice versa. Generally speaking, the degree to which the accuracy exceeds 0.5 indicates how much the algorithm performs better than pure chance. 2.3. Link prediction methods in weighted networks However, many of the natural networks are weighed by some built-in attributes [60, 61, 62], such as the communication frequency between two friends in social networks, the carbon flow between species in food webs or the amount of traffic load along connections in transportation networks. Here, we use three common link prediction method in weighted networks, and the WCN, WAA, and WRA are used as the benchmarks [63]. In fact, WCN, WAA, and WRA are extended similarity indices of CN [64], AA [65], and RA [66] in weighted networks [67] respectively. Notably, the weak ties phenomenon does exist in the weighted networks [68]. Also, the weighted algorithms with a free parameter significantly improved the performance of previously weighted methods. Namely, ∑ W CN = w(x, z)α + w(z, y)α , (3) Sxy z∈Oxy

4

W AA Sxy =

∑ w(x, z)α + w(z, y)α , log(1 + s(z)) z∈O

(4)

xy

W RA Sxy =

∑ w(x, z)α + w(z, y)α s(z) z∈O

(5)

xy

where the Oxy is the set of common neighbors of node pair (x, y), the w(x, z) is the weight ∑ of the link between x and z, and s(x) = z∈Γ(x) w(x, z)α . Moreover, when α = 0, the s(x) is the degree of node x, and the indices degenerate to the unweighted cases. When α = 1, the indices are equivalent to the simply weighted indices. The optimal values of α are smaller than 1 in most of the weighted networks. 3. Proposed Method The recent development of mathematical tools and the expanding availability of massive databases have allowed researchers to investigate the property of social networks and make explanations for human behavior. Social and organizational network analysis has become a hot research topic that involves multi-discipline. Developing a practical and efficient tool to analyze the type of “egos” in the social networks is becoming more and more important. The uncertainty of the individuals can be used in many research fields, such as personalized recommendation, predicting the possibility of crime, and detect the change of value orientation in the social networks. Entropy is widely used in many application systems [69, 70], and we shall show how to build the proposed entropy-based model in this section. The first problem is: given the local information (the degree of neighbors and the weight of each link), how to measure the uncertainty of any one node? The investigation of this paper starts with this simple question. Let us transfer the mathematical problem into a more practical case. Assume a man gets in trouble in the society, which adjacent neighbor he would ask for help? The first idea is choosing his kin or his close friend (kin selection). However, there exist the common phenomena in the society that “birds of a feather flock together,” which means the trouble of this man probably is another kind of trouble to his kin. In this context, this individual will tend to find a more powerful person to ask for help even this powerful man has a weak connection with this man. A step further, what drives the other people willing to help you (even your best friend)? The answer is reciprocity. People need to confirm that there a possibility of long-term cooperation and mutual benefits. To sum up, there are three basic states of the “ask for help” model. The first heuristic rule is: whether this individual knows about this person. Next, the second heuristic rule is: whether this person has enough ability to help you. And finally, the third heuristic rule is: whether this person is willing to help you. All the three basic heuristic rules are based on the assumption that all egos are rational and only do what is better for themselves. In this case, game theory provides a theoretical framework for interpreting carefully new interactions between individuals. Moreover, game theory is also used to understand the cooperative behavior of the egos, and the evolution 5

of the population and reveals the competition between the selfish egos at the bottom and the cooperative behaviors existing in real life. While the network models provide the mathematical tools that can quantitatively analyze the change of the variables and present a precise result for our problem. Based on the comparison of the likelihoods of the currently observed network driven by various mathematical tools, W. Qiang et al. [71] proposed a unified method to evaluate those different network models. Moreover, Q. Zhang et al. [72] used both likelihood analysis and link prediction methods to investigate multiple evolution mechanisms of complex networks, and the results demonstrated that the likelihood analysis methods performed much better with predicting the evolution in a very high accuracy. Now, we shall show how to transfer the fuzzy and linguistic words under the frame of game theory into precise and mathematic words based on three heuristic rules. Heuristic Rule 1: Kin Selection Kin selection [73] frequently occurs in food webs and biological networks. The reason for this phenomena is very intuitive: your kin probably is in the same class (measured by the degree) with you, and there is the strong tie (measured by the weight of the link) between you and your kin. This relationship provides opportunities for future cooperation and most of the traditional society featured by this characteristics. Moreover, the direct contact also provides opportunities for those people to ask their kin. To a certain extent, the utility of kin selection reveals the probability that this person comes to ask for help. In the kin kin selection, the utility of the adjacent node j to the ego node i is Ui,j , which can be defined as 2 n · wij n · wij · ∑n = ∑n ( j=1 wij )2 j=1 wij j=1 wij

wij kin Ui,j = ∑n

(6)

where the wij is the weight of the link between node i and node j, and the n is the number of all adjacent nodes of ego i. This utility function considers two parts of the local information, w w ×n the first part ∑n ijwij is the proportion of the total weight, and another part ∑nij wij considers j=1 j=1 the rank of the total weight. Heuristic Rule 2: Resource Selection Resource selection is ubiquitous in the scale-free network. The resource (measured by the degree) of the entire network is determined by a relatively small number of highly connected nodes, and the network is often characterized by the power-law degree distribution. Moreover, the probability that a node has k links follows P (k) ˜ k − γ, where γ is the degree exponent. This preferential attachment rule for organizing links between nodes results in nodes are highly connected are statistically more significant than in a random graph. In this real society, people are likely to ask the acquaintance who controls the relatively high resource for help. To some degree, the utility of resource selection reveals the probability that the ability of the adjacent neighbor to get the thing done. In the resource selection, resource . Namely, the utility of the adjacent node j to the ego node i is Ui,j n · d2j dj · n · ∑n = ∑n ( k=1 dk )2 k=1 dk k=1 dk

dj resource = ∑n Ui,j

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(7)

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Figure 1: The illustration of the three heuristic rules. This figure is taken from part of the real network. The size of the ego denotes the relative degree of the ego. Different colors are used to explain the relationship (i.g., close friend) between two egos. The link between two egos denote the preference of the selected ego (marked in orange color). The preference of each adjacent nodes is marked in different colors. where dk denotes the degree of the adjacent node k, and the n represents the number of all adjacent nodes of ego i. This index reflects the proportion of resource among all neighbors of the ego i, and it would prefer the adjacent node which has a higher degree. Heuristic Rule 3: Reciprocity Selection Reciprocity selection [74] is ubiquitous in the social networks. The principle of reciprocity selection is based on the contract for exchanging the social interests. All behaviors of people are used to establish long-term and mutually beneficial relationships, and therefore people can not do anything without a powerful strength. Recall that the resource selection suggests the resource of society are gathered in the powerful people (i.e., the stronger gets stronger), however, in the reciprocity selection, the weaker gets weaker. That is to say, the main difference between the two selection ways is the selected subject (the node). To some extent, the utility of reciprocity selection reveals the probability that the degree of the certain ego is willing to offer help. In the reciprocity selection, the utility of the adjacent node j to the reciprocity ego node i is Ui,j , which can be defined as di · n n · d2 · ∑n = ∑n i 2 ( k=1 dk ) k=1 dk k=1 dk

di reciprocity Ui,j = ∑n

(8)

where dk is the degree of the adjacent node k of node j (node j is the adjacent node of ego i), and n denotes the number of all adjacent nodes of node j. Notably, the reciprocity selection is similar to the resource selection, the major difference between the two rules is the subject of the degree calculation. The resource selection denotes the rank of adjacent nodes of ego i, while the reciprocity selection considers the rank of ego i of the adjacent nodes. In other words, the resource selection considers the “who you are”, while the reciprocity selection considers the “who I am.” It is apparent that our heuristics are based on the local information that consists of degree and weights, and the illustration of the three heuristics is shown in Figure 1. Since the proposed uncertainty measurement requires only the basic information of the networks, 7

the proposed method is a general method. Moreover, it is also a very efficient algorithm to calculate the overall uncertainty of any kind of weighted network since we can quickly obtain the result using the matrix operations. Furthermore, the proposed method is practical and adaptive to modern technology, because the distributed computing can be used to solve the adjacent blocking matrix respectively when handling large-scale social network. Actually, before calculating the utility functions, we could do a for-loop to calculate the sum of the weights (or the degree) of each node in the network. In other words, we can trade space for time. As shown in our pseudo codes, we have one nesting for-loops (one is from 1 to n, and the other one is from index i to n), and all the rest operations are merely constant. Therefore, the time complexity of our three heuristic rules is O((n2 ) / 2) ≈ O(n2 ). While the space complexity is a different story. As we mentioned before, we calculate the weights of all links and the degree of each node in the network, thus we need an extra space to store the vector (1 × n). Then, we need to initialize a N × N null matrix which is as the same size with the input N × N target network, and keeping the temporal variables (i.e., the results of utility function) does not matter to the final result. Therefore, the space complexity of our methods is O( n1 + n + n + 1) ≈ O(n). And next, we will show how to transfer those heuristics into the frame of fuzzy systems theory. But first, we shall define a meta-rule called Chain Rule for our heuristics. In fact, the chain rule draws the inspiration from the real chain rule in the differential equation, and it provides the opportunity for combining any pair of those utility functions or even three of them. The chain rule is used to combine the utility functions under different heuristic rules. Namely, Ui,j = Ui · Uj

(9)

where the Ui,j denotes the combination of utility functions of heuristic rules i and j. Now we shall show how to transfer the three heuristics into the frame of fuzzy systems theory and measure the uncertainty of any one node of the network. Recall that we have defined three heuristics to transfer the local information to the utility functions. The utility function denotes the degree that you can get the thing done. A simple example is that if a man is a social butterfly who has many friends, but nobody is an especially good friend to him, it is difficult to predict his actions/choices when he gets in trouble. In this paper, the fuzzy systems theory is used to handle this kind of problems. Assume each node has its own belief and distributes it to each neighbor according to the utility functions with the sum of components equals to 1. Each neighbor is transferred to the focal element of the membership function, and the utility of each neighbor is transferred into the belief of the membership function. This step can be treated as the normalization process according to the utility function. Namely, Ui (j) mi (j) = ∑N k=1 Ui (k)

(10)

where N is the number of the neighbors. In fact, the utility functions Ui (j) denotes the utility of node j to node i. 8

Table 2: The uncertainty of networks compare with the optimal parameter α of three weighted measurements under AUC and Precision (Precision values are in brackets). The mean of AUC and Precision values are obtained by the mean of 100 independent realizations. Notably, the value of heuristic rules denotes the uncertainty of the networks.

Network USAir NetSci Adolescent KingJames PB Router Yeast

WCN 0.9532 (0.5820) 0.9911 (0.8707) 0.7734 (0.2825) 0.9854 (0.4570) 0.9223 (0.3988) 0.6522 (0.091) 0.7348 (0.1746)

WAA 0.9655 (0.6232) 0.9914 (0.9722) 0.7724 (0.2995) 0.9851 (0.5426) 0.9262 (0.3718) 0.6526 (0.1052) 0.7359 (0.2244)

WRA 0.9721 (0.6348) 0.9915 (0.9694) 0.7738 (0.3365) 0.9857 (0.7040) 0.9265 (0.2733) 0.6530 (0.0846) 0.7358 (0.1826)

Rule 1

Rule 2

Rule 3

1.8852

1.6988

1.3446

1.2018

1.1417

1.213

1.693

1.7435

1.5136

2.2244

1.2817

1.3354

3.3422

2.5413

1.8898

0.6096

0.2459

0.2883

1.6196

1.137

0.9378

In the field of information science, the powerful tool, called Shannon entropy, is used to measure the uncertainty of the membership function. It reads H=−

N ∑

θi logb θi

(11)

i=1

where N is the number of basic states∑in the Frame of Discernment (FOD) , θi is the probability of state i appears satisfying N i=1 pi = 1. 4. Measuring the uncertainty in weighted networks

The game theory and uncertainty measures provide the practical tools to analyze the relationship of egos. In the first part of this section, we shall illustrate our primary results based on the various networks, and we shall see the overall uncertainty of different weighted networks. Then, based on the uncertainty result, we can obtain the distribution of human behavior under each combination of heuristics. Next, in order to show the proposed method is consistent with existing prediction method, we compare the uncertainty of network with the result of the three well-known link prediction methods. Finally, we proposed an optimal uncertainty measurement that can beat anyone combination of the heuristics. Let us back what interests us, the overall uncertainty of each network is shown in Figure 2. Recall that the combination of different heuristics is feasible using the chain 9

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Figure 2: The overall uncertainty of 7 networks which consists both weighted networks and unweighted networks. The value denotes the relative degree of preference to each heuristics. The rule 1 denotes the heuristic 1, and the rule 1+2 represents the combination of heuristics 1 and 2 based on our chain rule. rule Equation 9, and the combination is the relatively independent integrity which does not equal to the sum of its parts. The most interesting aspect of this figure is the fact that different network indeed has different organization and property under our three heuristics. Some networks (e.g., US Airports) has the lower uncertainty under a particular heuristic (i.e., the rule 3), which suggests that the network may follow the certain rule to construct and organize the structure. While the uncertainty of other networks under different heuristic is close to each other, and any one of the singleton heuristics has higher uncertainty than the combination. The possible reason for this phenomena is that the relationship between the egos follows serveral heuristics simultaneously, in other words, there might be various factors influence the egos to make decisions. The Figure 3 is used to illustrate the distribution of the uncertainty of links under three basic heuristics of the three networks. It is apparent that the result of Figure 3 is also consistent with Table 2. For instances, the network of Router prefers the heuristic rule 2, then the heuristic rule 3, and the heuristic rule 1 has the overall highest uncertainty. In the Router network, most of the links are marked by red color under heuristic rule 1, while many links are marked the blue color under heuristic rule 2, and numerous links are characterized in the gradient color (marked by yellow and green) under the heuristic rule 3. However, the situation is not the same story when comparing the network of politicalblogs with Router. As Figure 2 shows, there is a significant difference among the various heuristic rules. For example, the uncertainty of the combination of heuristic rule 1 and heuristic rule 3 is still slightly larger than the singleton heuristic rule 3. The uncertainty of each combination of heuristic rule 3 is higher than the singleton heuristic rule 3 itself, which means that the egos in the network follow only one rule, which is the reciprocity selection. We shall show how to quantify the preference of all heuristics in the next section. The Figure 3 provides the results obtained from the preliminary analysis of uncertainty. 10

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Figure 3: The distribution of uncertainty of links under three basic heuristics of the three selected networks. Each color dot denotes the degree of uncertainty. The links with higher uncertainty are marked by red color, and the links with lower uncertainty are marked by blue color. This figure shows the difference between different heuristic rules of the same network, and it also shows the difference of distribution of the uncertainty in different networks.

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Notably, in the social network Adolescent, the nodes represent the teens and the links from node vi to vj means the adolescent vi choose another adolescent vj to be his/her friend, and the link weights indicate the frequency of interactions between two adolescents. From the figure we can see that, the resource selection has the highest uncertainty, and kin selection has the lowest uncertainty. Since the uncertainty of the ego represents the probability that the ego behave like to the certain heuristics. The result suggests that when the egos of the selected Adolescent network get in trouble, they tend to help those egos which has a larger degree in their friendship. Meanwhile, the egos slightly prefer the reciprocity in their social relationships, while they do not care much about their kin relationships. The possible reason for this phenomena is the fact that the egos of the selected Adolescent network pay more attention to exchange the advantage (i.e., the reciprocity), they care about whether their effort would pay back. What stands out in the figure is the uncertainty of the network of Router. The Router network has the overall lowest uncertainty, and this interesting phenomenon results from the most of the nodes in the Router network has only one adjacent node. From the view of Shanon entropy, this phenomenon makes most of the nodes has the lowest uncertainty since the egos do not have the secondary choice. However, even the Router actually has the lowest uncertainty, this kind of point-to-point network has the inferior performance when predicting the missing links. One possible reason is that, to a certain node, this networks is very sparse, so it is significantly difficult to identify the missing links. Another interesting finding is that the precision of link prediction method is similar to the precision rate, while our model is similar to the recall rate. It is the perfect situation that both the precision rate and the recall rate are high, which indicates that the network is organized by some perceptible regulations, but in fact, the two index (the accuracy of link prediction and the uncertainty) are contradictory in some networks. A similar example in data mining field is that the accuracy of the link prediction alike to the accuracy rate while our uncertainty method is similar to the recall rate of the organizations of the network. The comparison between the link prediction method and the proposed uncertainty is illustrated in Table 2. As we can see, the correlation between the precision of link prediction and the proposed uncertainty method is entirely negative in Router network. Our model demonstrates that the Router has the lowest uncertainty, however, it also has the worst performance in predicting the missing links. Meanwhile, there are some other networks organized by the perceptible regulations, such as the network of Net-science and the Yeast network. In the next section, we shall present a novel view about uncertainty from the fuzzy system theory, and the optimal heuristics will be proposed to analyze the distribution of egos intensively. 5. View From Fuzzy Systems Fuzzy systems theory provides an efficient way to model fuzzy, linguistic information [75, 76, 77], which is heavily studied [78]. Recall we have mentioned that the lower uncertainty means the higher preference of the specific heuristic rules. Before we use fuzzy systems 12

theory to measure the degree of preference quantitatively, the optimal heuristic selection is required to present. The optimal heuristics take advantages of all the three heuristics (and all their combinations), then choose an optimal heuristics to make every ego in the network achieve its minimum uncertainty. Heuristic Rule 4: Optimal Selection Optimal selection is used to measure the uncertainty of the networks, in other words, it is the benchmark for the rest of the rules. The optimal selection reveals the lowest uncertainty of the networks. The utility function Ui of optimal selection can be defined as Uoptimal = min{a1 Ukin + a2 Uresource + a3 Ureciprocity }

(12)

where a1 , a2 , a3 are the boolean constants. It is apparent that the optimal selection is the optimal linear combination of three basic heuristics. The result of the optimal selection is shown in Table 3. The most interesting finding is that the difference between each basic heuristics and the optimal heuristics is also different. The main idea here is very intuitive: the difference indicates the preference of the network since the optimal selection reveals the lowest uncertainty of the networks, in other words, if the heuristics are very close to the optimal selection, the organization of the networks is more similar to the heuristics. Therefore, the difference between the basic heuristics and the optimal heuristics denotes the degree of preference. Recall the definition of the optimal selection, it acts in a manner to let each ego in the networks choose its optimal selection, and therefore, the optimal uncertainty of the network can be figured out since all egos are in the “optimal” states. Then, the number of optimal egos of different heuristics can be used to denotes the preference of the whole networks. We now define a term called the preference of the egos. For example, if the uncertainty of an ego under kin selection is higher than the uncertainty of the same ego under the resource selection, we claim that the ego ‘prefer’ to choose the resource selection. Here, the fuzzy systems theory is used to quantitatively measure the degree of preference of nodes to different rules (includes all the combinations of different rules). The membership function can be defined as m(i) = ni /K (13) where ni denotes the number of egos (nodes) which prefer the specific heuristic rule i (including the combined rules of different singleton rules), and K is the total number of the egos in the entire networks. By converting the distance between the uncertainty of optimal heuristics and the other three basic heuristics to the membership functions, the fuzzy system of the distribution of preference is constructed. The result of the distribution of preference is shown in Table 3. Notably, the focal elements with its cardinality larger than one (e.g., the {kin, resource} and the {kin, resource, reciprocity}) denotes that the preference of egos is mixed with multiple heuristics. Moreover, in the frame of fuzzy systems theory, each focal element (i.e., the heuristics) is exclusive to each other. The seven mutually exclusive and exhaustive focal elements of the membership function represent the distribution of seven different types of 13

Table 3: The overall preference and membership (membership values are in brackets) of 7 networks of different heuristic rules. The preference is obtained by the converting the distance between optimal selection and other heuristic rules. The membership function quantitatively analyzes the preference of each heuristic rules. The rule 1 denotes the kin selection, the rule 2 denotes the resource selection, and the rule 3 denotes the reciprocity. The combination of different rules is represented by the multiple operator.

USAir NetSci Adolescent KingJames PB Router Yeast

Rule 1 1.8852 (0.0873) 1.2018 (0.1168) 1.9680 (0.0458) 2.2244 (0.0240) 3.3422 (0.0579) 0.6096 (0.1250) 1.6196 (0.0979)

Rule 2 1.6988 (0.1009) 1.1417 (0.1670) 2.4864 (0.0288) 1.2817 (0.2065) 2.5413 (0.1552) 0.2459 (0.1421) 1.1370 (0.1485)

Rule 3 Rule 1+2 1.3446 1.3706 (0.2229) (0.2696) 1.2130 1.0224 (0.1279) (0.1954) 2.1895 1.8515 (0.1027) (0.3205) 1.3354 1.1428 (0.2473) (0.2511) 1.8898 2.3961 (0.2790) (0.1946) 0.2883 0.2459 (0.1700) (0.1412) 0.9378 1.1364 (0.2052) (0.1430)

Rule 1+3 1.5172 (0.1461) 1.1223 (0.1404) 1.7116 (0.4145) 1.3506 (0.2137) 1.8925 (0.1873) 0.2883 (0.1689) 0.9314 (0.1959)

Rule 2+3 2.2250 (0.0828) 1.3203 (0.1245) 2.7213 (0.0241) 2.1997 (0.0295) 3.2209 (0.0622) 0.4573 (0.1263) 1.4626 (0.1034)

Rule 1+2+3 Optimal rule 1.7645 0.9944 (0.0904) 1.1535 0.9614 (0.1280) 1.9289 1.4912 (0.0636) 1.9299 0.8741 (0.0309) 3.0548 1.6854 (0.0639) 0.4573 0.1782 (0.1265) 1.4189 0.8316 (0.1060)

preference. It is apparent in Table 3, the overall preference of some networks concentrates on a certain heuristics. Specifically, the network of the Adolescent prefers the reciprocity selection, which has the highest degree of confidence in its membership functions. The result is also consistent with the experiment result in the previous section. However, it is somewhat surprising that no preferred heuristics was found in this condition. The possible reason is the fact that the uncertainty of different heuristics is very close to each other. The inference of this phenomenon might be twofold. In the first case, the network itself is considered to be irregular and difficult to predict, which has been intensively discussed in the previous research [1]. However, this paper suggests another possible explanation: both the three basic heuristics and all the combination of the heuristics do not fit with the networks, and there exists another reasonable and suitable heuristics to describe the relationship between egos better. This hypothesis brings the initial problem of measuring uncertainty to the issue of the open world. More prior information of the networks is required to figure out how does the egos interact with each other and what are their preference. In this case, luckily, the feature of stimulating result is very apparent. First, the membership function of the preference is useless for decision-making since the belief of the membership function is evenly assigned to each heuristics. Second, the uncertainty of the networks is remarkably high under every heuristic rule. The reason for the phenomenon is also intuitive, the uncertainty of the heuristics is obviously high since the network is not consistent with the heuristics. Moreover, since all the heuristics disaccord with the network itself, the slight change between each heuristics is hard to detect, and finally, it leads to assigning the belief evenly to each focal 14

element. To conclude, fuzzy systems theory provides the mathematical tool to analyze the preference of each heuristics in the network quantitatively. Moreover, with the development of society and technology, the structure and organization of the networks are in a state of flux, and the relationship between egos becomes more and more complex and unpredictable. In this context, the view from fuzzy systems is the breakthrough since it can determine whether the heuristics are consistent with the structure of the networks. 6. Conclusion “Main story” of this paper is trying to tell how to measure the uncertainty of the network and how to apply the uncertainty to both theory and practice. In order answer this question, we first proposed a game model which consists three basic heuristics (i.e., kin selection, reciprocity selection, resource selection) and their combinations (using the chain rule) to model the possible relationships between egos in the networks. Then, the well-known Shannon entropy is used to measure the uncertainty of the egos according to the necessary local information (degree and weight of each neighbor), and the lower entropy implies stronger self-consistency of the networks. The uncertainty degree is also used to denote the preference of the heuristics of the networks, and therefore, the proposed uncertainty measure is a novel method to analyze the composition of the networks. Moreover, we compare the proposed method with the accuracy of three well-known link prediction methods with the free parameters. The result suggests an interesting phenomenon that our uncertainty degree is marginally correlated to the accuracy of link prediction, and it implies that the two different methods describe two distinct aspects of the organization of the network respectively. However, the accuracy of our model, to some extent, depends on the initial setting of the heuristics. Therefore, we convert the preference of the network to the membership functions using the fuzzy system theory, and the improper configuration of heuristics can be detected by analyzing the distribution of the membership function. Inspired by the results, we proposed a new viewpoint that the uncertainty degree, as the inherent attribution of the networks, is also limited by the setting of heuristics. This paper is a preliminary study to measure the uncertainty of the weighted networks. The main contribution of this paper is using the Shannon entropy to calculate the uncertainty under different heuristics. The setting of heuristics differs from network to network. How to apply the uncertainty of the networks to other research fields, it remains to be further investigated. Acknowledgment The authors greatly appreciate the anonymous reviews’ suggestions and the editor’s encouragement. The work is partially supported by National Natural Science Foundation of China (Grant Nos. 61573290, 61503237).

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 The paper attempts to quantify the uncertainty of nodes in the weighted networks.  Three heuristic rules are proposed to measure the utility of adjacent nodes.  The proposed predictability may suggest some properties of the networks on the opposite side of link prediction.