Analysis of temporal networks Name: Chen Xi ID number: 15307130444 Abstract: The study of complex networks has been going on for many years. But because of the difficulty of getting a lot of data, most of people used to study static networks in early stage so that they ignored one of the important ariables, time. However, in recent years, time dimension has become an important aspect when researchers study on networks. This report is aimed to study the properties of temporal networks and the influence of the time windows by analyzing some properties of two datasets in a static way after dividing the whole network into some subnetworks (which are called slices in the following article) Key words: temporal networks, time windows, clustering coefficient, shortest distance, small world, periodicity 1 Introduction There are so many networks in our life will change over time such as interaction among people, the deployment of wiFi access in a campus and so on. So we have to consider an additional dimension-time when we want to do some dynamical research. Different from static networks, temporal networks are those networks whose structure may change over time, which means the edges between nodes may appear or disappear during the evolution. At present, the research of temporal networks mainly includes network modeling, statistical characteristics analysis, propagation dynamics and related human behavior analysis on such networks. And one of the biggest differences between temporal and static networks is that the edges between vertices of temporal networks need not be transitive In static networks, whether directed or not, if A is directly connected to b and b is directly connected to C, then a is indirectly connected to C via a path over B. However, in temporal networks, if the edge(A, B)is active only at a later point in time than the edge(B, c), then a and c are disconnected, as nothing can propagate from a via B to C, which shows the importance of time order So the study on temporal network usually conducted on the premise of time-respecting path, which means that paths are usually defined as sequences of contacts with non decreasing times that connect sets of vertices 2. And for further research, some chers use community structures to divide the temporal networks, or study the whole network without any division. However, when it comes to division, it's not easy to make time windows as small as possible to reduce the influence of time order and make sure there is only once interaction between two nodes in a time window. and in the next part, this report will count the number of interaction times as the weight of an edge between two nodes, which makes every slice a weighted network By calculating some properties of networks to compare the evolution in different time windows and analyze results in combination with reality. What's more, the report will compare the results of temporal networks with ER model networks to get more conclusions 2. Methods The two datasets of this report are from Hypertext 2009 dynamic contact network and Hospital contact network (fromhttp:/www.sociopatterns.org/datasets/).TheformeronewascollectedduringtheacmHypertext2009 conference. about 110 Conference attendees volunteered to wear radio badges that monitored their face-to-face roximity from 8: 00am on Jun 29th 2009. And the latter is a temporal network of contacts between patients, patients and health-care workers(HCWs) and among HC Ws in a hospital ward in lyon, France, from Monday, December 6 2010 at 1: 00 pm to Friday, December 10, 2010 at 2: 00 pm, which included 46 HCWs and 29 patients. They are tab- separated lists representing the active contacts during 20-second intervals of the data collection. Each line has the form"tij, where i and j are the anonymous IDs of the persons in contact, and the interval during which this contact
Analysis of temporal networks Name: Chen Xi ID number: 15307130444 Abstract: The study of complex networks has been going on for many years. But because of the difficulty of getting a lot of data, most of people used to study static networks in early stage so that they ignored one of the important variables, time. However, in recent years, time dimension has become an important aspect when researchers study on networks. This report is aimed to study the properties of temporal networks and the influence of the time windows by analyzing some properties of two datasets in a static way after dividing the whole network into some subnetworks (which are called slices in the following article) Key words: temporal networks, time windows, clustering coefficient, shortest distance, small world, periodicity. 1. Introduction There are so many networks in our life will change over time such as interaction among people, the deployment of WiFi access in a campus and so on. So we have to consider an additional dimension-time when we want to do some dynamical research. Different from static networks, temporal networks are those networks whose structure may change over time, which means the edges between nodes may appear or disappear during the evolution. At present, the research of temporal networks mainly includes network modeling, statistical characteristics analysis, propagation dynamics and related human behavior analysis on such networks. And one of the biggest differences between temporal and static networks is that the edges between vertices of temporal networks need not be transitive. In static networks, whether directed or not, if A is directly connected to B and B is directly connected to C, then A is indirectly connected to C via a path over B. However, in temporal networks, if the edge (A, B) is active only at a later point in time than the edge (B, C), then A and C are disconnected, as nothing can propagate from A via B to C, which shows the importance of time order [1] . So the study on temporal network usually conducted on the premise of time-respecting path, which means that paths are usually defined as sequences of contacts with nondecreasing times that connect sets of vertices [2] . And for further research, some researchers use community structures to divide the temporal networks, or study the whole network without any division. However, when it comes to division, it’s not easy to make time windows as small as possible to reduce the influence of time order and make sure there is only once interaction between two nodes in a time window. And in the next part, this report will count the number of interaction times as the weight of an edge between two nodes, which makes every slice a weighted network. By calculating some properties of networks to compare the evolution in different time windows and analyze results in combination with reality. What’s more, the report will compare the results of temporal networks with ER model networks to get more conclusions. 2. Methods The two datasets of this report are from Hypertext 2009 dynamic contact network and Hospital contact network (from http://www.sociopatterns.org/datasets/ ). The former one was collected during the ACM Hypertext 2009 conference. About 110 Conference attendees volunteered to wear radio badges that monitored their face-to-face proximity from 8:00am on Jun 29th 2009. And the latter is a temporal network of contacts between patients, patients and health-care workers (HCWs) and among HCWs in a hospital ward in Lyon, France, from Monday, December 6, 2010 at 1:00 pm to Friday, December 10, 2010 at 2:00 pm, which included 46 HCWs and 29 patients. They are tabseparated lists representing the active contacts during 20-second intervals of the data collection. Each line has the form “t i j”, where i and j are the anonymous IDs of the persons in contact, and the interval during which this contact
was active is [t-20s, t]. If multiple contacts are active in a given interval, there will be multiple lines starting with the same value of t All the analysis process was conducted in C language. Firstly, it should be emphasized that the temporal networks here are simplified without direction, which is called weak connectivity according to Nicolas's definitions two vertices i and j of a temporal network are defined to be strongly connected if there is a directed, time-respecting path connecting i to j and vice versa, while they are weakly connected if there are undirected time-respecting paths from i to j andj to i, i. e the directions of the contacts are not taken into account 1. This report focused on the evolution of one dataset and used different time windows such as 3 hours 6 hours. 12 hours. 24 hours and the whole network-48 hours, then counted the number of interaction times and regarded them as the weight of an edge between two nodes. After dividing the temporal network into above slices, there may be no interaction in some slices in a day so it should be deleted. And in order to keep the unity of two time agglomerations(two days in this report ), the corresponding slices in another day should also be deleted. Then the rest of the subnetworks could be regarded as some static networks to be analyzed. and this report mainly calculated the degree distribution, average degree(named strength"in weighted networks), average clustering coefficient(Ci -((k11)j=1 N wi and average shortest distance(Floyd algorithm) network)with p-0.1 and calculated the same properties of network and used the average to make a comparis e? Secondly, in order to search the relationship between temporal network and other network models, this repor counted the number of nodes and total weight in every slices, then put them into a er model network(also a weight Thirdly, this report used the methods above to analyze another dataset and kept the time windows same, which could make a comparison in different temporal networks and help researcher know more about the evolution of temporal networks in different time windows 3. Results After programming and processing data, the related results are as follows (1) Figure I and Figure 2 are average results of three properties of temporal network and ER model network different times windows. It shows that no matter what kind of dataset it is, the clustering coefficient in temporal network is far more than that value in ER model while the shortest distance has opposite result. And the average degree in two kinds of network is almost the same. Figure 3 is the curves of three properties in different time windows from 3 hours to 48 hours and the red curve represents network in ACM Hypertext conference and the blue one is the network in hospital Time windows Average degree Chustering coefficient Shortest distance 2742641792 0.007807417 3 hours 25600183 7.822216083 Temporal 006850667 6 hours m 50.50066533 0.0000145 1047694533 Temporal 70.3812325 0.004379 1457543 2 hours 0.0000175 10704636 Temporal 139.1617645 0.0086045 2.511026 24 hours ER model 20995049 253.261261 48 hours 2118591 ER model 253.261261 0.000035 25.792138 (Figure 1. ACM Hypertext
was active is [ t – 20s, t ]. If multiple contacts are active in a given interval, there will be multiple lines starting with the same value of t. All the analysis process was conducted in C language. Firstly, it should be emphasized that the temporal networks here are simplified without direction, which is called weak connectivity according to Nicolas ’s definitions: two vertices i and j of a temporal network are defined to be strongly connected if there is a directed, time-respecting path connecting i to j and vice versa, while they are weakly connected if there are undirected time-respecting paths from i to j and j to i, i.e. the directions of the contacts are not taken into account [3] . This report focused on the evolution of one dataset and used different time windows such as 3 hours, 6 hours, 12 hours, 24 hours and the whole network-48 hours, then counted the number of interaction times and regarded them as the weight of an edge between two nodes. After dividing the temporal network into above slices, there may be no interaction in some slices in a day so it should be deleted. And in order to keep the unity of two time agglomerations (two days in this report), the corresponding slices in another day should also be deleted. Then the rest of the subnetworks could be regarded as some static networks to be analyzed. And this report mainly calculated the degree distribution, average degree (named “strength” in weighted networks), average clustering coefficient(𝑐𝑖 = 1 𝑘𝑖∗(𝑘𝑖−1) ∑ ∑ (𝑤𝑖𝑗 𝑚𝑤𝑗𝑘 𝑚𝑤𝑖𝑘 𝑚) 1 𝑁 3 𝑘=1 𝑁 𝑗=1 ) and average shortest distance (Floyd algorithm). Secondly, in order to search the relationship between temporal network and other network models, this report counted the number of nodes and total weight in every slices, then put them into a ER model network (also a weighted network) with p=0.1 and calculated the same properties of network and used the average to make a comparison. Thirdly, this report used the methods above to analyze another dataset and kept the time windows same, which could make a comparison in different temporal networks and help researcher know more about the evolution of temporal networks in different time windows. 3. Results After programming and processing data, the related results are as follows: (1) Figure 1 and Figure 2 are average results of three properties of temporal network and ER model network in different times windows. It shows that no matter what kind of dataset it is, the clustering coefficient in temporal network is far more than that value in ER model while the shortest distance has opposite result. And the average degree in two kinds of network is almost the same. Figure 3 is the curves of three properties in different time windows from 3 hours to 48 hours. And the red curve represents network in ACM Hypertext conference and the blue one is the network in hospital. (Figure 1. ACM Hypertext)
Time windows Average degree Chustering coefficientShortest distance 81202251 006054667 3 hours ER model 15904426128333 3592305081 0.006454875 6.921292375 6 hours 111.753307 ER me 111.7533068 0.00002075 41.28081713 12 hours Temporal 198071539 0.0063875 392359275 ER model 1980715393 0.0000435 43.635674 24 hours Temporal319.4871795 0.0026135 3.5081905 model 3194871795 00001853 48.7788385 Temporal 52983871 844 5981 48 hours 52983871 0.000024 43.679006 Figure 2. Hospital) window (2)From Figure 4 to Figure 7 are the curves of average clustering coefficient and average shortest distance of every slice in different time windows(3h, 6h, 12h)of two datasets. The blue curves are results of temporal networks and the red curves belong to ER model networks. The X axis is the label of time slice and the Y axis is property. And when it comes to clustering coefficient, the blue curves are higher than red curves and they are undulate obviously. But it is totally opposite when it comes to shortest distance conference(3h) conference(6h conference(12h) 0012 0.003 系列1一系列2 系列1一系列2 系列1一系列2 (Figure 4. ACM Hypertext, clustering coefficient) conference(3h) time 一系列1一系列 系列1一系列2
(Figure 2. Hospital) (Figure 3. Time windows) (2) From Figure 4 to Figure 7 are the curves of average clustering coefficient and average shortest distance of every slice in different time windows (3h, 6h, 12h) of two datasets. The blue curves are results of temporal networks and the red curves belong to ER model networks. The X axis is the label of time slice and the Y axis is property. And when it comes to clustering coefficient, the blue curves are higher than red curves and they are undulate obviously. But it is totally opposite when it comes to shortest distance. (Figure 4. ACM Hypertext, clustering coefficient)
(Figure 5. ACM Hypertext, shortest distance) septal(3h) hospital(6h) hospital(12h) 05 105 time 系列1一系列2 系列2 系列1一系列2 (Figure6.Hospital,clustering coefficient) hospital(3h) hospital(12h 80 4 50 系列1一系列 al, shortest distance) ()From Figure 8 to Figure 1l are degree distribution of every slice in time window of 12 hours of two datasets just use this time window as an example to get the curve). And Figure 12 is the degree distribution in time window of 48 hours, which is the whole network in this research. As shown in the pictures, the fluctuation of curves is different in every slice but there is something common in two datasets. For example, in ACM conference, slice I and slice 3 are much more undulate than slice 2 and slice 4. There is the same result in Hospital dataset. conference(12h. 1) conference(12h. 2) 0045 IILA (Figure 8. ACM Hypertext, 12 hours)
(Figure 5. ACM Hypertext, shortest distance) (Figure 6. Hospital, clustering coefficient) (Figure 7. Hospital, shortest distance) (3) From Figure 8 to Figure 11 are degree distribution of every slice in time window of 12 hours of two datasets (just use this time window as an example to get the curve). And Figure 12 is the degree distribution in time window of 48 hours, which is the whole network in this research. As shown in the pictures, the fluctuation of curves is different in every slice but there is something common in two datasets. For example, in ACM conference, slice 1 and slice 3 are much more undulate than slice 2 and slice 4. There is the same result in Hospital dataset. (Figure 8. ACM Hypertext, 12 hours)
conference(12h. 3) conference(12h. 4) 5 04 0025 0.25 0015 0005 50100150200250300350 (Figure 9. ACM Hypertext, 12 hours) hospital(12h. 1) hospital(12h. 2) 007 006 0025 005 003 0005 ( Figure 10. Hospital, 12 hour hospital(12.3) hospital( 12h. 4) 045 0035 0025 0015 001 100200300400500600700800 100 300 500600 ( Figure 11. Hospital, 12 hours)
(Figure 9. ACM Hypertext, 12 hours) (Figure 10. Hospital, 12 hours) (Figure 11. Hospital, 12 hours)
conference(48h) hospital(48h) 0018 0.02 ≥ 日0015 001 0004 0.005 (Figure 12. Two datasets, 48hours 4. Discussion (1)Small world According to the results above, especially from Figure I to Figure 7, they all shows that the average clustering coefficient in temporal network is larger than that in ER network(more than 100 times) while the average path distance is shorter(less than 1/4 times). Such properties are similar to the main idea in small world model. And taking these two datasets in reality into account, whether ACM Hypertext conference or hospital contacts, there are always some people who have acquaintances in the same network. So it is going to generate some shortcuts among interaction, which is exactly the concept of small world. And the results indicated that the temporal networks have the characteristic of small world, which could provide another way to study on dynamical research such as (2) Periodicity and paroxysm As shown in Figure 4 to Figure 7, the average clustering coefficient and average shortest distance in two datasets are fluctuating over time and the curves usually move back and forth between highest and lowest points which may show the paroxysm of the temporal networks. And this reveals that the past and the future in a temporal network are not independent, which is also called non- Markov property ). Taking the first picture in Figure 3 for example, the first peak appears when time slice equals I and it means 8: 00am-11: 00am in time window of 3 hours, which is the exact time for meeting, so the contacts among people are frequent and the chance to form clusters may increase, which can change clustering coefficient. But when time slice equals 5(about 23: 00pm), people may have a rest and the chance to form clusters may decrease. And the shortest distance has the similar analysis In Figure 8 to Figure ll, the degree distribution of different slices of two datasets are changed obviously over time but there are still some rules For instance both the first slice and the third slice are more undulate than the second slice and the forth slice in ACM Hypertext network and hospital network, which may show the periodicit of the temporal network Whether periodicity or paroxysm has illustrated that the interaction between nodes in a temporal network not all the same. It may be active sometimes but inactive in other intervals, and these changes eventually make up the whole network, which provides the further evidence of the importance of taking time dimension into account to do research on evolution of complex network ()The influence of different time windows According to the Figure l to Figure 3, especially in Figure 3, it is found that different time windows can have different values of properties. The average degree in two datasets are all increasing with the increase of time window and it is caused by the number of nodes in different time windows. However, the change of other two properties of two datasets are different. In ACM Hypertext conference, the curve of e clustering coefficient full of ups and downs sharply but the curve of shortest distance is almost decreasing with the increase of time
(Figure 12. Two datasets, 48hours) 4. Discussion (1) Small world According to the results above, especially from Figure 1 to Figure 7, they all shows that the average clustering coefficient in temporal network is larger than that in ER network (more than 100 times) while the average path distance is shorter (less than 1/4 times). Such properties are similar to the main idea in small world model. And taking these two datasets in reality into account, whether ACM Hypertext conference or hospital contacts, there are always some people who have acquaintances in the same network. So it is going to generate some shortcuts among interaction, which is exactly the concept of small world. And the results indicated that the temporal networks have the characteristic of small world, which could provide another way to study on dynamical research such as epidemic. (2) Periodicity and paroxysm As shown in Figure 4 to Figure 7, the average clustering coefficient and average shortest distance in two datasets are fluctuating over time and the curves usually move back and forth between highest and lowest points, which may show the paroxysm of the temporal networks. And this reveals that the past and the future in a temporal network are not independent, which is also called non-Markov property [4]. Taking the first picture in Figure 3 for example, the first peak appears when time slice equals 1 and it means 8:00am-11:00am in time window of 3 hours, which is the exact time for meeting, so the contacts among people are frequent and the chance to form clusters may increase, which can change clustering coefficient. But when time slice equals 5 (about 23:00pm), people may have a rest and the chance to form clusters may decrease. And the shortest distance has the similar analysis. In Figure 8 to Figure 11, the degree distribution of different slices of two datasets are changed obviously over time but there are still some rules. For instance, both the first slice and the third slice are more undulate than the second slice and the forth slice in ACM Hypertext network and hospital network, which may show the periodicity of the temporal network. Whether periodicity or paroxysm has illustrated that the interaction between nodes in a temporal network is not all the same. It may be active sometimes but inactive in other intervals, and these changes eventually make up the whole network, which provides the further evidence of the importance of taking time dimension into account to do research on evolution of complex network. (3) The influence of different time windows According to the Figure 1 to Figure 3, especially in Figure 3, it is found that different time windows can have different values of properties. The average degree in two datasets are all increasing with the increase of time window and it is caused by the number of nodes in different time windows. However, the change of other two properties of two datasets are different. In ACM Hypertext conference, the curve of average clustering coefficient is full of ups and downs sharply but the curve of shortest distance is almost decreasing with the increase of time
windows. And in hospital network, the curve of average clustering coefficient is almost decreasing while the curve of shortest distance is fluctuating slightly Even though the specific influence on properties of two networks are kind of different, the results still indicate that different time windows can influence the properties of a network because of the change of interaction among people over time, which means people should choose the proper time windows when they want to study on temporal network (4)Results and reality Results above are analysis of some data, and what's the relationship between such results and human activities in reality? First, what should be emphasized is that both of these datasets are"closed"systems that a group of individuals gathers and interacts in a repeated fashion 51 In ACM Hypertext conference, 110 attendees almost enter and leave the hall together every day. At that time, most of them will form some clusters which leads to the small world. And it is obvious that the trace of their behavior during the whole conference is regular such as meeting, having a meal or sleeping, which results in periodicity and paroxysm The similar explanation is also feasible for hospital network. According to the paper about this dataset (61 8: 00am--17: 00pm is the most crowded period in a day and 14037 contacts were recorded overall, 94. 1% of which during daytime, which leads to the result such as the first picture in Figure 6- the peak appears at slice 1, slice 6 that correspond to 13: 00-16: 00pm and 7: 00-10: 00am. In addition, the nurses or doctors have to go to the wards to visit their patients and take care of them every day so there must be interaction among them. above repeated situations in hospital can result in the properties of small world and periodic evolution ()limitations There are still some limitations in this report for some reasons. Firstly, regarding the interaction times as the weight of edges mainly focuses on the importance of contacts but ignores the time when these contacts happen, which is related to transition in temporal network again. It means the time window is not small enough so it must have some inevitable error when the subnetworks are analyzed in a static way. And such error about clustering coefficient may be reduced if use the definition of temporal-weighted clustering coefficient proposed by Jing Cui, Yi-Qing Zhang and Xiang Li l since it takes the existing duration of triangles into consideration Secondly, no matter how many nodes in the slice, the report al ways generated a ER network with probability quals 0 I while there is a threshold pcN which can decide the connectivity in ER model network, so it may lead to some errors. For example, when n equals 23 in ACM Hypertext the threshold is about 0. 13 but when N equals 96 it is about 0.048. So it indicates that the probability in different network should be different, too. But actually, since the report has averaged these results, the final error could be reduced. Thirdly, the number of participants whether in ACM Hypertext conference or in hospital are too small(about 110 and 75), which may make it more difficult to find some rules due to lack of a representative tendency such as Poisson or power law distribution. For example, Figure 8 to Figure 12 show that the tails of degree distribution in these temporal networks are not reduced to nearly zero but keep at a certain value or even go up, which seems to be close to power law. Nevertheless, the rest of the curve is much more random than expectation, which may lead to some confusion. And in fact, the types of degree distribution such as Poisson and power law are usually found on the premise of large amounts of data, so in this report the degree distribution is not only affected by weight of edges but also affected by the limitation of the number of data 5. Conclusion On the basis of two datasets that are" closed"systems, this report aims to study on some properties of temporal network and the influence of time windows in evolution. And by using the method that divide the datasets
windows. And in hospital network, the curve of average clustering coefficient is almost decreasing while the curve of shortest distance is fluctuating slightly. Even though the specific influence on properties of two networks are kind of different, the results still indicate that different time windows can influence the properties of a network because of the change of interaction among people over time, which means people should choose the proper time windows when they want to study on temporal network. (4) Results and reality Results above are analysis of some data, and what’s the relationship between such results and human activities in reality? First, what should be emphasized is that both of these datasets are “closed” systems that a group of individuals gathers and interacts in a repeated fashion [5] . In ACM Hypertext conference, 110 attendees almost enter and leave the hall together every day. At that time, most of them will form some clusters which leads to the small world. And it is obvious that the trace of their behavior during the whole conference is regular such as meeting, having a meal or sleeping, which results in periodicity and paroxysm. The similar explanation is also feasible for hospital network. According to the paper about this dataset [6] , 8:00am -- 17:00pm is the most crowded period in a day and 14037 contacts were recorded overall, 94.1% of which during daytime, which leads to the result such as the first picture in Figure 6 — the peak appears at slice 1,slice 6 that correspond to 13:00-16:00pm and 7:00-10:00am. In addition, the nurses or doctors have to go to the wards to visit their patients and take care of them every day so there must be interaction among them. Above repeated situations in hospital can result in the properties of small world and periodic evolution. (5) limitations There are still some limitations in this report for some reasons. Firstly, regarding the interaction times as the weight of edges mainly focuses on the importance of contacts but ignores the time when these contacts happen, which is related to transition in temporal network again. It means the time window is not small enough so it must have some inevitable error when the subnetworks are analyzed in a static way. And such error about clustering coefficient may be reduced if use the definition of temporal-weighted clustering coefficient proposed by Jing Cui, Yi-Qing Zhang and Xiang Li [7] since it takes the existing duration of triangles into consideration. Secondly, no matter how many nodes in the slice, the report always generated a ER network with probability equals 0.1 while there is a threshold 𝑝𝑐~ 𝑙𝑛𝑁 𝑁 which can decide the connectivity in ER model network, so it may lead to some errors. For example, when N equals 23 in ACM Hypertext the threshold is about 0.13 but when N equals 96 it is about 0.048. So it indicates that the probability in different network should be different, too. But actually, since the report has averaged these results, the final error could be reduced. Thirdly, the number of participants whether in ACM Hypertext conference or in hospital are too small (about 110 and 75), which may make it more difficult to find some rules due to lack of a representative tendency such as Poisson or power law distribution. For example, Figure 8 to Figure 12 show that the tails of degree distribution in these temporal networks are not reduced to nearly zero but keep at a certain value or even go up, which seems to be close to power law. Nevertheless, the rest of the curve is much more random than expectation, which may lead to some confusion. And in fact, the types of degree distribution such as Poisson and power law are usually found on the premise of large amounts of data, so in this report the degree distribution is not only affected by weight of edges but also affected by the limitation of the number of data. 5. Conclusion On the basis of two datasets that are “closed” systems, this report aims to study on some properties of temporal network and the influence of time windows in evolution. And by using the method that divide the datasets
into some subnetworks in different time windows and calculate some important properties in a static way then compare them with ER model network, this report has found that such systems have the feature of small world including large clustering coefficient and small shortest distance compared to ER network. Their evolution show he characteristics of periodicity and paroxysm, which is closely related to the daily activities of human such as meeting in ACM Hypertext and visiting the wards in hospital. And different time windows will influence the average degree, clustering coefficient, shortest distance in both of two datasets. What's more, this report has also nted out some limitations during research. For instance, the static way to analyze every slice which ignores the existing durations, the unchanged probability of ER model network and the size of the whole network are some aspects may lead to errors. Still, this report did something that combined the human interaction with abstract temporal network The interaction among people in the social life is always one of the heated topic which attracts lots of scientists or sociologists to study on. And time dimension is absolutely an important factor needed to consider. So he researches on temporal networks are not only analysis of some datasets but also modeling. Even though there is not a complete way to model temporal network at present, the related study is bound to continue and develop on the basis of analysis of datasets References: [1]. Petter Holme, Jari Saramaki. Temporal networks. Phys. Rep. 519,97-125(2012), DOl: 10. 1016/j. physrep 2012,03.001, Cite as arXiv: 1108. 1780 nlin.AOI [2].P. Holme, C E. Edling, F. Liljeros, Structure and time-evolution of an Internet dating community, Social Networks 26(2004)155- 174 [3].V. Nicosia, J. Tang, M. Musolesi, G. Russo, C. Mascolo, V. Latora, Components in time-varying graphs, e-print ar Xiv: 1 106.2134 1. Ingo Scholtes, Nicolas wider, Rene Pfitzner, Antonios Garas, Claudio Juan Tessone and Frank Schweitzer: SCHOLTES L, WIDER N, PFITZNER R, et al. Slow-down vs speed-up of diffusion in non- Markovian temporal networks]. Nature Communications, 2014, 5:5024.DO:10.1038/ comms6024 [5]. Lorenzo Isella, Juliette Stehle, Alain Barrat, Ciro Cattuto, Jean-Francois Pinton, and Wouter Van den Broeck. What's in a crowd? Analysis of face-to-face behavioral networks. Journal of Theoretical Biology 271(2011)166-180 [6].Philippe Vanhems, Alain Barrat, Ciro Cattuto, Jean-Francois Pinton, Nagham Khanafer, Corinne Regis, Byeul-a Kim, Brigitte C Nicolas Voirin, Estimating Potential Infection Transmission Routes in Hospital Wards Using Wearable Proximity Sensors PLos C [7].CUI J, ZHANG Y Q, LI X. On the clustering coefficients of temporal networks and epidemic dynamics[ C]/Proceedings of 2013 IEEE International Symposium on Circuits and Systems(ISCAS)...: IEEE, 2013: 2299 [8]. Tore Opsahl and Pietro Panzarasa(2009). "Clustering in Weighted Networks". Social Networks. 31(2): 155-163
into some subnetworks in different time windows and calculate some important properties in a static way then compare them with ER model network, this report has found that such systems have the feature of small world including large clustering coefficient and small shortest distance compared to ER network. Their evolution shows the characteristics of periodicity and paroxysm, which is closely related to the daily activities of human such as meeting in ACM Hypertext and visiting the wards in hospital. And different time windows will influence the average degree, clustering coefficient, shortest distance in both of two datasets. What’s more, this report has also pointed out some limitations during research. For instance, the static way to analyze every slice which ignores the existing durations, the unchanged probability of ER model network and the size of the whole network are some aspects may lead to errors. Still, this report did something that combined the human interaction with abstract temporal network. The interaction among people in the social life is always one of the heated topic which attracts lots of scientists or sociologists to study on. And time dimension is absolutely an important factor needed to consider. So the researches on temporal networks are not only analysis of some datasets but also modeling. Even though there is not a complete way to model temporal network at present, the related study is bound to continue and develop on the basis of analysis of datasets. References: [1].Petter Holme, Jari Saramäki. Temporal networks. Phys. Rep. 519, 97-125 (2012), DOI: 10.1016/j.physrep.2012.03.001, Cite as: arXiv:1108.1780 [nlin.AO] [2].P. Holme, C.E. Edling, F. Liljeros, Structure and time-evolution of an Internet dating community, Social Networks 26 (2004) 155– 174. [3].V. Nicosia, J. Tang, M. Musolesi, G. Russo, C. Mascolo, V. Latora, Components in time-varying graphs, e-print arXiv:1106.2134. [4]. Ingo Scholtes, Nicolas Wider, René Pfitzner, Antonios Garas, Claudio Juan Tessone and Frank Schweitzer: SCHOLTES I, WIDER N, PFITZNER R, et al. Slow-down vs. speed-up of diffusion in non- Markovian temporal networks[J]. Nature Communications, 2014, 5: 5024. DOI:10.1038/ncomms6024 [5].Lorenzo Isella, Juliette Stehlé, Alain Barrat, Ciro Cattuto, Jean-François Pinton, and Wouter Van den Broeck. What’s in a crowd? Analysis of face-to-face behavioral networks. Journal of Theoretical Biology 271 (2011) 166-180. [6].Philippe Vanhems, Alain Barrat, Ciro Cattuto, Jean-François Pinton, Nagham Khanafer, Corinne Régis, Byeul-a Kim, Brigitte Comte, Nicolas Voirin, Estimating Potential Infection Transmission Routes in Hospital Wards Using Wearable Proximity Sensors PLoS ONE 8(9), e73970 (2013) [7].CUI J, ZHANG Y Q, LI X. On the clustering coefficients of temporal networks and epidemic dynamics[C]//Proceedings of 2013 IEEE International Symposium on Circuits and Systems (ISCAS).[S.l.]:IEEE, 2013:2299. [8].Tore Opsahl and Pietro Panzarasa (2009). "Clustering in Weighted Networks". Social Networks. 31 (2): 155–163