Robustness Analysis of Bipartite Task Assignment Networks: a Case Study in Hospital Logistics System

Received March 17, 2019, accepted April 18, 2019, date of publication May 1, 2019, date of current version May 16, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2914258

Robustness Analysis of Bipartite Task Assignment Networks: A Case Study in Hospital Logistics System

QI XUAN 1,2, (Member, IEEE), YEWEI YU2, JINYIN CHEN2, ZHONGYUAN RUAN 3, ZHONG FU4, AND ZICONG LV4 1Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China 2College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China 3College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China 4Hangzhou Curetech Co., Ltd., Hangzhou 310011, China Corresponding author: Zhongyuan Ruan ([email protected]) This work was supported by the National Natural Science Foundation of China under Grant 61572439, Grant 11605154, and Grant 61502423.

ABSTRACT Task-oriented social systems involve workers and tasks, which can typically be modeled by bipartite networks. The absence of workers may cause some tasks cannot be finished in time, thus hampering the smoothing operation of the whole system. In this study, we focus on the robustness analysis of such bipartite task-oriented social networks under the absence of some workers. In particular, taking hospital logistics systems as an example, we propose four attack strategies: efficiency-based attack, centrality-based attack, diversity-based attack, and influence-based attack. The experiments on nine real hospital logistics systems show that these systems are especially vulnerable to the diversity-based attack. Different hospitals may present different patterns: some hospital logistics systems are relatively robust against these attacks while others are more vulnerable. The former hospitals are relatively small in size and adopt a global task assignment mode in which the workers can be easily replaced by others. On the contrary, the workers in the latter ones are highly professional and irreplaceable due to the large scale of the hospitals and the adoption of a regional task assignment mode. We also design two task reassignment mechanisms to model the cascading failure of such systems. In this case, the working load plays an important role in the robustness of these systems. Moreover, we find that the robustness and efficiency seem to be negatively correlated with each other, and how to balance the two in real systems still remains an open problem.

INDEX TERMS Bipartite network, network robustness, node centrality, logistics system, social network, task assignment, diversity.

I. INTRODUCTION of the system, so as to improve the efficiency of workers and Networks are ubiquitous [1]–[5]. Many real systems can be the robustness of the whole system. viewed as complex networks, such as World Wide Web In the hospital logistics system, factors such as talent com- (WWW), Internet, social network etc. The hospital logis- petition between hospitals and personal career development tics system is such a system where the workers and tasks of workers often lead to the personnel flow, which has a constitute a bipartite network. With the continuous reform certain impact on the robustness of the system. Furthermore, of the hospital management system, hospital logistics has higher ability matches better salary, so accurately measuring become an important part of the development of hospitals the importance of workers is also conducive to the estab- today, which is the necessary guarantee to ensure the normal lishment of the incentive mechanism. Properly evaluating operation of the hospital. The research on hospital logistics the importance of workers is particularly important for the system could help to index and quantize the operation status system robustness and employee development. In the sce- nario of hospital logistics system, workers and tasks need to complement each other. As such a complex system runs, The associate editor coordinating the review of this manuscript and approving it for publication was Ning Weng. it’s a challenge to evaluate every worker in the system.

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Performance appraisal methods are important to employ- so on [10], [37], [44]. Starting from the emerging hospital ees [6]. The conventional way is to use the fuzzy approach, logistics system, this paper constructs the worker-task bipar- which appears to be subjective [7], [8]. Specifically, we trans- tite network and quantifies the robustness of the task assign- late the issue of measuring the robustness and person- ment network, so as to optimize the hospital logistics system. nel importance of hospital logistics systems into network The proportion of isolated nodes related to the largest con- attack and node importance ranking. Many methods have nected component is selected in order to make the network been proposed for network attack. Related work based robustness have practical significance. on network metrics is quite mature, including degree, The main contributions of this paper are as follows k-core decomposition, betweenness centrality [9] and so on. : We collect the task data in nine hospitals and establish In addition, some network dismantling methods [10] are pro- • nine bipartite task assignment networks to describe their posed, which rely on collective influence [11], decycling structures. and tree breaking [12], articulation points [13] or network We propose four strategies (including efficiency-based, communities [14]. For the purpose of personnel evaluation, • centrality-based, diversity-based and influence-based) we put forward several attack strategies that are specifi- to attack task assignment networks; and compre- cally suitable for the task assignment situation. By studying hensive experiments show that centrality-based and this, we want to identify the vertices so that the most cru- diversity-based attacks are more effective than the oth- cial elements of the system can be carefully protected [15]. ers. This article starts from multiple dimensions to achieve a We introduce two task reassignment mechanisms when more detailed assessment of workers. First of all, from the • some workers are absent, which facilitates the study on perspective of traditional characteristics, we use task time, cascading failure of task assignment networks. difficulty and other factors to get the evaluation indexes We explain why the attack effects vary on different hos- related to worker efficiency. Secondly, from the perspective • pitals, which may provide suggestions for managers to of network topology, we rank the importance of worker design appropriate logistics systems for hospitals based nodes in the network. It not only has important theoret- on their own demands, i.e., depending on whether they ical research significance [16], but also has a wide range focus on efficiency or robustness. of practical applications [17]–[19]. Ranking in bipartite net- work has its practical significance, which can be applied to The rest of this paper is organized as follows. In Sec. II, the following examples: problems and solvers, products and we give the four attack strategies based on the efficiency, consumers [20]. We consider the eigenvector centrality [20] centrality, diversity, and influence, respectively, propose a to evaluate the importance of a worker in the bipartite new metric for measuring the robustness of the bipartite net- network. And then the diversity index is also used to work and also introduce our task reassignment mechanisms. evaluate the worker, which is usually used for ecological In Sec. III, we introduce our dataset collected from nine species [21], [22]. Finally, the network dismantling method is hospitals, and perform comprehensive experiments on testing used to quantify the influence of workers, so as to reasonably the robustness of the logistics systems of these nine hospitals. evaluate their value. We then make a brief discussion in Sec. IV, and conclude the The robustness of the hospital logistics system affects the paper in Sec. V. normal operation of actual transportation tasks. The mea- surement of system robustness is not only of great signif- II. METHOD icance to the whole system, but also has a certain impact Considering a hospital logistics system, we establish a task on personnel evaluation and task assignment in the hospital assignment network containing two types of nodes, workers logistics system. With the continuous development of science and tasks. In particular, we give the following definition and technology, more and more attention has been paid to Definition 1: A task assignment network is a weighted: the network resilience— the robustness of a system under bipartite network, denoted by G (P, T , E, W ), where various attacks [15], [23]–[26], since robustness is a basic P p , p , , p denotes the= set of workers, T requirement for the smooth operation of a system. Research 1 2 m t ,=t , { , t ···denotes} the set of tasks, E denotes the set= on network robustness involves various fields such as trans- 1 2 n of{ links··· between} the workers and tasks, and W [w ] portation network [27], [28], power network [29], [30] and ij m n is a link weight matrix. We assume there is a link= between⇥ social network [31]–[36]. The existent quantified indexes of worker pi and task tj, i.e., eij E, if wij > 0, while they are robustness often include validity, connectivity, failover and 2 disconnected if wij 0. reversibility [37], [38]. Many previous efforts have compared = the robustness of random network [39], [40], small world network [41] and scale-free network [42] [43] under differ- A. ATTACK STRATEGIES ent attack strategies. There are many classification stan- To address the robustness of a hospital logistics system, dards for network robustness measurement, and the common we build a bipartite network based on the task assignment. measures include the relative size of the largest connected For example, as shown in Fig. 1 (a), the network contains component, the average backward geodesic distance and three workers and four tasks. If a task is assigned to a worker,

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and workers, respectively, and also an adjacency matrix 3 representing the allocation of tasks to workers, the following assumptions are then made to create a rating system [20]: Task ratings are additive with respect to worker ratings, • and vice versa. The contribution of worker pi to the rating of task tj is • proportional to the rating of task and is inversely propor- tional to the weight between them, i.e., wij, meaning that fewer workers can do harder tasks. FIGURE 1. (a) Task assignment network. There are three workers, namely The contribution of task tj to the rating of worker pi • p1, p2 and p3, and four tasks, namely t1, t2, t3 and t4. The width of line indicates the frequency that the task is assigned to the worker. (b) Task is proportional to the rating of worker and the weight reassignment. When worker p3 is absent, task t3 is reassigned to p1 and between them, meaning that more central workers can p2 since these two workers have experience on this task, therefore, do more tasks. the weights between them, i.e., w13 and w23, are updated. Based on these assumptions, we can get the rating vectors by the following equations when the system is convergent : we establish a link between them. Each link has a weight, y 3x, (2) representing the times the task is assigned to the worker. = x µ3y, (3) Normally, workers may be absent with almost equal = probability, due to illness etc. We consider this case as ran- where the parameters and µ are used to normalize the rating dom attack. More interestingly, we should consider the sit- vectors, and the constraint is x y 1. uations that workers may be agitated to leave their posts, k k=k k= especially when there are strong competition among differ- 3) DIVERSITY-BASED ATTACK ent labor dispatching companies and hospitals. In this case, Generally, for workers, due to different working habits and it is quite important for the managers to measure the impor- preference, their ability to cope with different tasks may tance, or irreplaceability, of different workers in ensuring the vary from each other. If a worker can deal with more tasks smooth operation of hospital logistics systems. in a specific period of time, we think this worker is more important in keeping the stability of a system, since he/she 1) EFFICIENCY-BASED ATTACK can better fill the vacancy caused by the absence of other The most common metric to measure the importance of a workers. To measure the task diversity of a worker, we use the worker is efficiency. In a task assignment system, the effi- Shannon Entropy [47], [48], which appears to be a commonly ciency of a worker is always related to the number of tasks used diversity index. Differently, we modify the Shannon and task execution time, which could be defined as : Entropy by multiplying a fraction of tasks done by the worker, n Tij wij which is defined as 'i ( ), (1) : = T ⇥ s s j j i si 1 X Di ij log2( ), (4) = n ij where T represents the average execution time of task t , s i 1 j j i X= represents the total number of tasks executed by worker pi, where n is the total number of tasks, si is the number of tasks Tij represents the average execution time of worker pi on the performed by worker pi, and ✓ij is calculated by task tj, and wij represents the number of times that worker pi : wij performs task tj. Based on this definition, we assume that if ij , (5) the execution time of a worker on a task is smaller than the = sj average execution time of the task, his/her efficiency on this meaning the proportion of times that task tj was performed task is relatively higher. To perform the attack, we first rank by worker pi over the total times it was performed. It can be the worker nodes based on their efficiency, from high to low, found that, for a worker, diversity could be related to degree, and then remove them one by one. i.e., the worker of higher degree must involve more tasks and thus generally have higher diversity. 2) CENTRALITY-BASED ATTACK The importance of a worker is not only reflected in himself, 4) INFLUENCE-BASED ATTACK but also in the tasks he does. In most cases, the more important In recent years, research on network dismantling has con- the task is, the more it can highlight the value of the worker tinued to deepen, the essence of which is to identify the from the side. Eigenvector centrality has been widely used most critical nodes in a given network [10]–[14]. This is in webpage ranking for search engine [45] and other rating a measure different from the standard network metrics. By systems [20], [45], [46]. Recently, Daugulis proposed a gen- comparing the impact of different nodes on the network eralized eigenvector centrality to rank the nodes in the bipar- connectivity, the sequence of nodes that makes the network tite graph [20]. If we define rating vectors x and y for tasks collapse fastest is found. Similarly, after dismantling the

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task assignment network, the order of different workers is remaining workers who have done this task before. For obvious. When a worker is not in the position, if the normal instance, as shown in Fig. 1 (b), e.g., when p3 leaves, the part operation of the system is greatly damaged, then he is quite of task t3 that p3 is responsible for, need to be reassigned important to some extent. Morone and Makse introduced the to p1 and p2. Then, the weight of link between p1 and t3, concept of collective influence(CI) to better understand the i.e., w13, and that between p2 and t3, i.e., w23, should be relationship between the influence of a node and its topologi- updated. In particular, suppose worker pk is absent, then the cal structure [11], [49]–[51]. According to the reference [11], assignment of task tj is updated by the collective influence of node i is defined as : wij ✓ijwkj, i k CI`(i) (ki 1) (kj 1), (6) wij + 6= ; (9) = = (0, i k. j @Ball(i,`) = 2 X where ✓ is a function of w , which means that the reassign- where ki represents the degree of the node i, and Ball(i, `) is ij ij the frontier of the ball which is centered on node i with radius ment of task tj is determined by the experience of workers `. Inevitably, the implementation of the algorithm needs to on this task. In reality, there might be several different ways calculate the CI of each node i and update the network by to reassign the task, that is, the function may have different removing the node with the largest CI value. The process of forms, such as the two listed as follows. One takes all. In the traditional hospital logistics sys- removing nodes and recalculating CI until the giant compo- • nent is destroyed achieves the network dismantling. tem, head nurse may decide which worker takes over the released task tj. In this case, the task may be reas- B. METRIC FOR NETWORK ROBUSTNESS signed to a single worker for convenience, i.e., we first The evaluation of network robustness for a task assignment randomly choose one of workers who have experience network is critical when it may be confronted by various on task tj, and let him/her take over the part that pk was attacks. The factors such as resignation largely affect the responsible for. In other words, ✓ij is either 1 or 0, and stability of the system. The robustness measure R [10], [52] is the probability that pi is chosen to take all is defined as often used to measure the performance of the network, which 1 is calculated by the relative size of the largest connected , i k & wij > 0 p(✓ij 1) nj 1 6= ; (10) component. R is defined as = = 8 <0, i k wij 0. m n = | = 1 + R s(Q), (7) where nj is the number: of workers that have experience = m n Q 1 on task tj. + X= Experts take more. The experience of worker pi on task where m n is the number of nodes in the network, s(Q) is • t is measured by the mean times per day for the task, the size of+ the largest connected component after removing Q j which is in proportion to the link weight w . In this case, nodes. ij we have In the actual task assignment network, not only the connec- wij tivity of the network needs to be concerned, but also the ratio , i k of isolated task nodes should be paid attention to. Compared ✓ij i k wij 6= ; (11) = 8 6= to the former, the ratio of isolated nodes is more directly 0, i k.

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also investigate its monthly or seasonal variation. The difference between the departure time and the arrival time is used to define the task execution time of the worker on the task. On the worker side, for each record, there is a worker • ID and thus we can measure the efficiency, centrality, diversity and influence of workers.

B. TASK ASSIGNMENT NETWORK PROPERTIES In a hospital logistics system, we think that two records share the same pair of places, denoted by A and B, no matter whether it is from A to B or from B to A, correspond to the same task. The task involves the same group of medical work- ers and the same distance, so the difficulty is similar. Now, we have our task nodes, based on which we can establish the bipartite task assignment network. Based on this method, we construct bipartite task assignment networks for the logis- tics systems of nine hospitals in Zhejiang Province, China. Several basic attributes of these networks are presented in TABLE 1. In particular, these attributes include: the number of workers m, the number of tasks n, the number of links E , the diameter D, the average shortest path length l , the aver-| | age degree k , the density ⇢, the assortativity coefficienth i r, the averageh clusteringi coefficient C, the average clustering coefficient for workers CP, the average clustering coefficient FIGURE 2. (a) Dispatching center interface. The dispatching center is the control center for the comprehensive dispatching of logistics transport for tasks CT , the average closeness centrality CC, the average tasks, which realizes the overall management of the unassigned, degree centrality DC and the average betweenness centrality assigned and completed tasks. (b) Mobile phone interface. According to the task lists released by the task dispatching center of the hospital, BC. Note that here the clustering coefficient for a bipartite workers perform tasks at the required time and place. They need to scan network is defined as [53]: two-dimensional codes on their mobile phones to record the details of the tasks, including the task location, time, tools, content and so on. u N(N(v)) cuv cv 2 (12) = P N(N(v)) to April 2018. With the deepening of hospital logistics where N(N(v)) is the 2nd-order neighbor set of v by excluding informatization, the dispatch and execution of tasks can itself, and cuv is the pairwise clustering coefficient between be recorded by computers and mobile phones. According nodes u and v. The average clustering coefficient for the to the task lists released by the task dispatching center whole network then is calculated by of the hospital, workers use the mobile APP to scan the 1 two-dimensional codes to record the details of their tasks. C cv. (13) Different two-dimensional codes are set in each department = m n + X of the hospital. When the workers perform tasks, they can Then, we can get the average clustering coefficient only for scan the code on their mobile phones at the required places to the workers CP and for the tasks CT , respectively. record the location, time, tools and content of the tasks. Data We can find that there are some structural differences can be stored in the database for subsequent analysis. The among different hospitals in their task assignment networks, interface of dispatching center and mobile phone is shown reflecting the differences of the actual hospital scale and in Fig. 2. In this paper, our research mainly focuses on the logistics transportation mode, which may have certain influ- properties related to workers and tasks. The details of the ence on the network robustness. dataset are introduced as follows. To study the evolution of these networks, we establish the On the task side, for each record, we have follow- daily task assignment network, and find that most attributes • ing information: the departure place, the destination, keep steady with time on the whole. However, when we look the departure time and the arrival time. We use the pair at the evolution in a week, we find the distinct difference of departure and destination places to characterize tasks. between working days and weekend, as shown in Fig. 3. Note Both departure time and arrival time are recorded in the that here we only present the trends of four hospitals, they format of ‘‘year/month/day hour:minute:second’’ (e.g., are similar for the others. This is quite reasonable since there 2017/8/3 9:30:23), which provides the opportunity to must be more various tasks, and also each worker should be not only study the efficiency of a worker on a task but responsible for more tasks in working days.

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TABLE 1. Basic attributes of the bipartite task assignment networks in the nine hospitals.

FIGURE 4. The robustness as functions of the fraction of removed nodes under random attack for the nine hospitals. The solid line is the average robustness under the random attack, and the banded shadow area represents the error range for multiple experimental results.

FIGURE 3. The change of average degree with time for the daily task assignment networks of four hospitals. Here, we choose the date from 2017-09-02 to 2017-09-16 to present the distinct difference of network structure between working days and weekend.

C. NETWORK ROBUSTNESS ANALYSIS As mentioned above, we are more concerned with the impact of isolated tasks on the robustness of hospital logistics sys- tems, so we choose I to measure the robustness of the system. The two robustness metrics have a certain correlation, so the curves obtained are basically the same. In order to make the FIGURE 5. The topological structure of task assignment networks for results clearer, we only select I to show in the figure. hospital #3 (left) and hospital #9 (right), in which only the top 50 tasks In order to compare the robustness of the logistics systems with the largest number are selected. in several hospitals, we randomly remove the worker nodes in a task assignment network with the fraction of removed nodes in this hospital, there are clear division of labor due to the varied from 10% to 100%. For each fraction, the experiment large scale of the hospital, i.e., some workers are responsible is repeated 1000 times and the robustness of the network is for certain tasks in fixed regions for a relatively long time. recorded. The results are shown in Fig. 4, where we can find Therefore, when certain workers are absent, these tasks may that, in general, the system becomes less robust as more and be isolated soon, leading to relatively poor robustness of the more worker nodes are removed. task assignment network. For hospital #9, more workers are More interestingly, we find that although most hospitals responsible for each task due to the small scale of the hospital. present quite similar trends, hospital #3 seems quite different Therefore, when a worker is absent, the others can easily from the others. When we look into the task assignment take over the tasks, leading to relatively strong robustness of networks, the workers in hospital #3 are mainly responsible the task assignment network. For the sake of clarity, the task for 5-10 tasks which account for the vast majority of daily assignment network consisting of the top 50 tasks with the work. When some workers are removed, the robustness of the largest number is taken as an example. Fig. 5 shows the topol- network decreases almost linearly. This is mainly because, ogy structure of the network for Hospital #3 and Hospital #9.

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FIGURE 6. The robustness of the network as functions of the fraction of removed nodes under various attacks for the nine hospitals.

Now, let’s attack the task assignment networks based on to evaluate workers nowadays. As for the network disman- the efficiency, centrality, diversity, and influence of work- tling method, its effect is not bad. It belongs to the medium ers, respectively, and the results are shown in Fig. 6. As grade, next to the centrality-based and diversity-based we can see, for most cases, the robustness of the network attacks. Moreover, we can see that centrality-based and decreases quite slowly as a small fraction of worker nodes diversity-based attacks are quite close to each other in most are removed. However, when the fraction of removed worker cases, except for hospital #3. Since in the attack, we first nodes reaches a certain threshold, the robustness of the net- sort all the worker nodes based on their efficiency, centrality, work may decrease quite fast as more worker nodes are diversity, or influence, and then remove nodes from large to removed. This is reasonable, since almost all the task assign- small, we thus further investigate the Spearman correlation ment networks of hospital logistics systems have certain between pairwise of them. The correlation results are pre- redundancy, i.e., these networks are still connected as a whole sented in TABLE 2, where we can see that centrality and when a small number of workers are absent. However, most diversity indeed have quite high correlation in most cases. of them will be broken when a large number of worker nodes However, they are quite different for hospital #3, where are removed, in this case, the hospital logistics systems may workers have better division of labor, and thus the task not run regularly. assignment network has sparser structure, as shown in Fig. 5. In the five attack strategies including random attack, In addition, there is no significant relationship between centrality-based and diversity-based attacks behave better the efficiency-based attack and other methods, because the than the other three, as show in Fig. 6, i.e., the robustness efficiency-based indicator evaluates the workers from the of the network decreases much faster under centrality-based non-topological perspective such as task location and time. or diversity-based attack as the fraction of removed worker According to the changing trends of the curves in Fig. 6, nodes increases. Interestingly, we find that efficiency-based the nine hospitals can be roughly divided into two groups. attack is close to, and sometimes even worse than, random One group of hospitals have relatively robust logistics sys- attack, indicating that removing efficient workers may not tems, such as hospital #2, hospital #5 and hospital #6. For influence the connection of the whole task assignment net- these hospitals, the task assignment network maintain good work, i.e., the efficiency has less effect on robustness of the connectivity even when 50% workers are removed by any whole system, although efficiency is still the main metric attack strategy. In fact, these hospitals adopt a global task

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TABLE 2. The correlation analysis between importance metrics. attack effect as the strategy to remove worker nodes. The worker nodes are deleted in descending order according to the diversity. Fig. 7 compares the three cases: two consider cascading failure while the other doesn’t. As we can see, the network robustness decreases as more worker nodes are removed. Without considering the cascade failure, the curve changes relatively smoothly, while in the case of cascading failure, the network quickly collapses when a part of nodes are removed. This phenomenon is quite similar to the cascad- ing failure in traffic networks and power networks [58], [59]. Comparing the two task reassignment mechanisms, it is obvi- assignment mode, and thus workers can easily fill the vacancy ous that the network will collapse faster in the case where when some are absent. The logistics systems of the other one person takes all tasks. More interestingly, we find that groups of hospitals are of less robustness, since these hospi- the task assignment networks of hospital #1, hospital #6, tals tend to adopt a regional task assignment mode, i.e., they hospital #8 and hospital #9 collapse especially fast when we have relatively better division of labor, especially for hospital consider cascading failure, i.e., most task nodes are isolated #3, where the workers are responsible for the certain tasks of when less than half worker nodes are removed, while they the corresponding regions. are relatively robust without considering cascading failure, i.e., we need remove more than 80% worker nodes to make most task nodes isolated. This might be because these hos- D. CASCADING FAILURE pitals have relatively heavy tasks, which may lead to more Cascading failures often occur in complex networks. Previous critical cascading failure in task assignment networks. studies have investigated cascading failures in coupled map lattices with different topologies [54]. The north American power grid is one of the classic examples of cascading failure IV. DISCUSSION in which the entire network is disrupted as a few nodes A. SPECIALIST OR GENERALIST? are removed [55]. Many subsequent studies also analyzed As we can see in the above analysis, in some hospitals, the node load [56] and redistribution rules in the cascade workers are responsible for a small fraction of tasks, we call models [57]. In this paper, based on the removal of worker them specialists; while for the others, workers are responsible nodes in the task assignment network, the cascading failure for a large fraction of tasks, we call them generalists. Fig. 8 model is constructed by determining the node load and estab- gives the average fraction of tasks per worker in each hos- lishing the reassignment mechanism. In many cases, when pital, where we can see that the average fraction of tasks in workers are absent, their tasks need to be reassigned to the different hospitals indeed varies significantly, e.g., this value others. However, when the workers undertake more and more in hospital #2 is as high as 64%, while in hospital #3 it is less tasks, they may be overloaded and thus their efficiency will than 10%. largely decrease. To simulate such situations, we turn to the The average fraction of tasks for each hospital reflects cascading failure model, i.e., the failure of some nodes may the transportation mode of the logistics system in the hospi- lead to the failure of others, and so forth, eventually leading tal. Choosing specialists or generalists needs to be adapted to the collapse of the entire system. Factors influencing the to different actual situations. Based on our field investiga- network cascade effect include: tion, the regional transportation mode needs more specialists, which may be adopted in relatively large comprehensive Initial load on the nodes. Here, we choose the workers • hospitals, while the global transportation mode needs more and their long-term responsible tasks to establish the task generalists, which is adopted more frequently in small and assignment network, and use the average numbers of medium-sized hospitals. tasks per day as the initial load of the worker nodes. The relationship between node capacity and initial load. • B. EFFICIENCY OR ROBUSTNESS? The capacity of a worker node is the maximum load We always expect that robustness and efficiency can be both that this worker can withstand, and the node capacity guaranteed in a system. However, in real situation, this might is usually proportional to the initial load. In this paper, not be always true. In this study, we find that regional trans- we specify that the node capacity is three times of the portation mode can improve the efficiency for those large initial load. hospitals, but it may reduce the robustness of the whole Task reassignment rule after node failure. When a • logistics systems; while the global transportation mode can worker node fails, we reassign the tasks according to increase the robustness of the logistics systems of those Eq. (9), Eq. (10) and Eq. (11) to the workers who are small or medium-sized hospitals, but it may slightly decrease familiar with these tasks. their efficiency. How to achieve a balance between efficiency According to the comparison of different attack strate- and robustness is always an important issue in practical gies above, we choose the diversity-based attack with better applications.

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FIGURE 7. The comparison of whether to consider cascading failures or not. Without considering the cascade failure, the curve changes gently. In contrast, in the case of cascading failure, the network quickly crashes when partial nodes are removed.

transportation mode is more specialized, workers can be more focused, and thus their efficiency could be improved [60]. Otherwise, workers need to pay attention to a large number of tasks, which may lead to low efficiency. Therefore, for the hospital logistics system, workers should be familiar with different task types as much as possible, but they can be responsible for part of the tasks in a regional manner, which ensures that the efficiency is as high as possible under the premise of system stability. How to balance the two still remains an open problem, and left for future works.

V. CONCLUSION To the best of our knowledge, this is the first work FIGURE 8. The average fraction of tasks (blue histogram) and average efficiency (green curve) for nine hospitals. From the change trend of to address the robustness of bipartite task-oriented social broken line, the trend of the two curves is opposite. networks, under the absence of workers. We define a metric to measure the robustness of such a bipartite We find that the average efficiency and the average fraction network, and meanwhile propose four attack strategies, of tasks are negatively correlated with each other, as shown including efficiency-based attack, centrality-based attack, in Fig. 8, i.e., if a hospital has relatively high average effi- diversity-based attack and influence-based attack. We find ciency, it always has relatively small average fraction of that, by comparison, most hospital logistics systems are tasks. Since the fraction of tasks is positively correlated with more vulnerable to diversity-based attack, especially for the diversity of workers defined by Eq. (4), based on the those large-scale hospitals adopting regional task assignment experimental results, we thus can conclude that the efficiency mode, since they have relatively better division of labor and robustness are indeed negatively related in hospital logis- and thus the workers are relatively irreplaceable. Moreover, tics systems. This is reasonable, considering that, when the we design two task reassignment mechanisms to model the

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cascading failure of hospital logistics systems. The results [18] X. Song, X. Wang, A. Li, and L. Zhang, ‘‘Node importance evaluation show that the workers in a hospital have higher working load, method for highway network of urban agglomeration,’’ J. Transp. Syst. Eng. Inf. Technol., vol. 11, no. 2, pp. 84–90, Apr. 2011. the hospital will be more vulnerable to diversity-based attack. [19] S. Vitali, J. B. Glattfelder, and S. Battiston, ‘‘The network of global The balance between efficiency and robustness is especially corporate control,’’ PLOS ONE, vol. 6, no. 10, Oct. 2011, Art. no. e25995. important here since these two seems to be negatively corre- [20] P. Daugulis, ‘‘A note on a generalization of eigenvector centrality for lated with each other in our study. bipartite graphs and applications,’’ Networks, vol. 59, no. 2, pp. 261–264, Mar. 2012. In the future, our network robustness metric, attack strate- [21] A. J. Hamilton, ‘‘Species diversity or biodiversity?’’ J. Environ. 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[43] Y. Li, J. Wu, and A.-Q. Zou, ‘‘Effect of eliminating edges on robustness YEWEI YU received the B.S. degree from of scale-free networks under intentional attack,’’ Chin. Phys. Lett., vol. 27, the Changzhou Institute of Technology, China, no. 6, Jun. 2010, Art. no. 068901. in 2016. She is currently pursuing the M.S. degree [44] R. Albert, H. Jeong, and A.-L. Barabási, ‘‘Error and attack tolerance of with the College of Information Engineering, complex networks,’’ Nature, vol. 406, no. 6794, pp. 378–382, Jul. 2000. Zhejiang University of Technology, Hangzhou, [45] S. Brin and L. Page, ‘‘The anatomy of a large-scale hypertextual China. Her current research interests include data Web search engine,’’ Comput. Netw. ISDN Syst., vol. 30, nos. 1–7, mining, machine learning, and social networks. pp. 107–117, Apr. 1998. doi: 10.1016/S0169-7552(98)00110-X. [46] C. Bergstrom, ‘‘Eigenfactor: Measuring the value and prestige of scholarly journals,’’ College Res. 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E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 70, no. 5, in physics from Guizhou University, in 2008, and Nov. 2004, Art. no. 056113. the Ph.D. degree in physics from East China Nor- [55] R. Kinney, P. Crucitti, R. Albert, and V. Latora, ‘‘Modeling cascading mal University, Shanghai, China, in 2013. He is failures in the North American power grid,’’ Eur. Phys. J. B-Condensed currently a Lecturer with the College of Computer Matter Complex Syst., vol. 46, no. 1, pp. 101–107, 2005. Science and Technology, Zhejiang University of [56] J. F. Zheng, Z.-Y. Gao, and X.-M. Zhao, ‘‘Modeling cascading failures in Technology. His current research interests include congested complex networks,’’ Phys. A: Statistical Mech. Appl., vol. 385, complex systems and complex networks. no. 2, pp. 700–706, Nov. 2007. [57] J.-W. Wang and L.-L. Rong, ‘‘Effect attack on scale-free networks due to cascading failures,’’ Chin. Phys. Lett., vol. 25, no. 10, p. 3826, Oct. 2008. [58] Z. Zhao, P. Zhang, and H. Yang, ‘‘Cascading failures in intercon- nected networks with dynamical redistribution of loads,’’ Phys. A, Statist. Mech. Appl., vol. 433, pp. 204–210, Sep. 2015. [59] J. Ma, S. Wang, Y. Qiu, Y. Li, Z. Wang, and J. S. Thorp, ‘‘Angle stability analysis of power system with multiple operating conditions considering ZHONG FU received the B.S. degree in education cascading failure,’’ IEEE Trans. Power Syst., vol. 32, no. 2, pp. 873–882, technology from the Zhejiang University of Tech- Mar. 2017. nology, Hangzhou, China, in 2007. He turned out [60] Q. Xuan, A. Okano, P. Devanbu, and V. Filkov, ‘‘Focus-shifting patterns to be the Vice President of ENJOYOR Company of OSS developers and their congruence with call graphs,’’ in Proc. 22nd Limited. He is currently a General Manager with ACM SIGSOFT Int. Symp. Found. Softw. Eng., Nov. 2014, pp. 401–412. the Hangzhou Curetech Co., Ltd. He is mainly responsible for the decision-making of the com- pany’s new product development direction and QI XUAN received the B.S. and Ph.D. degrees business management mode. in control theory and engineering from Zhejiang University, Hangzhou, China, in 2003 and 2008, respectively. He was a Postdoctoral Researcher with the Department of Information Science and Electronic Engineering, Zhejiang University, from 2008 to 2010, and a Research Assistant with the ZICONG LV received the B.S. degree from Anhui Department of Electronic Engineering, City Uni- Medical University, in 2014. He is currently the versity of Hong Kong, Hong Kong, in 2010. From Product Director of the Hangzhou Curetech Co., 2012 to 2014, he was a Postdoctoral Fellow with Ltd., with more than six years of software devel- the Department of Computer Science, University of California at Davis, opment experience and more than three years of Davis, CA, USA. He is currently a Professor with the Institute of Cyberspace medical product development experience. He is Security, and the College of Information Engineering, Zhejiang University of responsible for the overall core technology of the Technology, Hangzhou. His current research interests include network-based company, mainly organizing the formulation and algorithm design, social network data mining, social synchronization and implementation of major technical decisions and consensus, reaction-diffusion network dynamics, machine learning, and technical programs. computer vision.

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