Tensor-Based Link Prediction in Intermittently Connected Wireless Networks

T ensor-Based Link Prediction in In termitten tly Connected Wireless Net w orks Mohamed-Ha ykel ZA Y ANI , Vincent GA UTHIER , Ines SLAMA , Djamal ZEGHLACHE L ab. CNRS SAMO V AR UMR 5157, T ele c om SudParis, Evry, F r anc e Abstract Through sev eral studies, it has been highligh ted that mobilit y patterns in mobile netw orks are driv en by h uman b eha viors. This effect has b een particularly observ ed in in termittently connected netw orks lik e DTN (Delay T olerant Netw orks). Giv en that common so cial in tentions generate similar human b eha vior, it is relev ant to exploit this knowledge in the net work proto cols design, e.g. to identify the closeness degree b et ween tw o no des. In this pap er, w e prop ose a temp oral link prediction technique for DTN whic h quan tifies the behavior similarity b et w een eac h pair of no des and makes use of it to predict future links. Our prediction method k eeps trac k of the spatio-temp oral asp ects of no des b eha viors organized as a third-order tensor that aims to records the evolution of the net work top ology . After collapsing the tensor information, w e compute the degree of similarity for eac h pair of no des using the Katz measure. This metric giv es us an indication on the link o ccurrence b et ween t wo no des relying on their closeness. W e show the efficiency of this metho d by applying it on three mobility traces: tw o real traces and one synthetic trace. Through several simulations, we demonstrate the effectiv eness of the technique regarding another approach based on a similarity metric used in DTN. The v alidity of this metho d is prov en when the computation of score is made in a distributed w ay (i.e. with lo cal information). W e attest that the tensor-based tec hnique is effective for temporal link prediction applied to the intermitten tly connecte d net works. F urthermore, w e think that this technique can go b eyond the realm of DTN and we b elieve this can be further applied on every case of figure in whic h there is a need to deriv e the underlying social structure of a netw ork of mobile users. Keywor ds: Link prediction, wireless net works, in termittent connections, tensor, Katz measure, b ehavior similarit y, DTN 1. Introduction In recent y ears, extensive researc h has addressed chal- lenges and problems raised in mobile, sparse and intermit- ten tly connected net w orks (i.e. DTN). In this case, for- w arding pack ets tightly dep ends on contacts o ccurrence. Since the existence of links is crucial to deliver data from a source to a destination, the contacts and their prop erties emerge as a k ey issue in designing efficient comm unication proto cols [1]. Ob viously , the o ccurrence of links is led by the b ehavior of the no des in the netw ork [2]. It has b een widely shown in [3, 4] that human mobility is directed b y so cial inten tions and reflects spatio-temp oral regular- it y . A no de can follow other no des to a sp ecific lo cation (spatial lev el) and may bring out a b ehavior which may b e regulated b y a sc hedule (temporal lev el). The social in tentions that gov ern the b ehavior of mobile users hav e also b een observed through statistical analyses in [2, 5] by sho wing that the distribution of inter-con tact times follow truncated p ow er law. Email addresses: mohamed-haykel.zayani@telecom-sudparis.eu (Mohamed-Hayk el ZA Y ANI), vincent.gauthier@telecom-sudparis.eu (Vincen t GAUTHIER), ines.slama@telecom-sudparis.eu (Ines SLAMA), djamal.zeghlache@telecom-sudparis.eu (Djamal ZEGHLACHE) With the in tention of impro ving the p erformance of in- termitten tly connected wireless net work protocols, it is paramoun t to trac k and understand the b ehaviors of the no des. W e aim at prop osing an approac h that analyzes the net work statistics, quan tifies the social relationship b et ween each pair of no des and exploits this measure as a score whic h indicates if a link w ould o ccur in the im- mediate future. W e strongly b elieve that the social ties b et ween no des highly gov ern the status of a link and es- tablishes an indication for the link prediction: it w ould nev er o ccur if tw o no des ha ve no common social interac- tions and w ould rather b e effectiv e and lasting with more correlated moving patterns. In this pap er, we adapt a tensor-based link prediction algorithm successfully designed for the data-mining con- text [6, 7]. Our proposal records the netw ork structure for T time p erio ds and predicts links occurrences for the (T+1) th p erio d. This link prediction tec hnique is designed through tw o steps. First, tracking time-dependent net- w ork snapshots in adjacency matrices which form a ten- sor. Second, applying of the Katz measure [8] inspired from so ciometry . The link prediction technique computes the degree of b ehavior similarity of eac h pair of nodes rely- ing on the tensor obtained in the first step. A high degree of behavior similarit y means that the t wo no des hav e the Pr eprint submitted to Elsevier Mar ch 8, 2022 same “so cial” in tentions. These common inten tions are expressed b y the willingness to meet eac h other and/or b y similar moving patterns to visit a same lo cation. They also promote the link o ccurrence b et ween t wo socially close no des in the immediate future (prediction of the perio d T +1 after tracking the b ehaviors of nodes during T time p erio ds). W e further discuss ho w we design the tensor-based pre- diction metho d and detail the t wo main steps in order to ac hieve link prediction. On the one hand, w e describ e how to track the net work topology ov er time with a tensor. On the other hand, we explain how to compute and in ter- pret the Katz measure. W e then ev aluate the effectiv eness of predictability through several sim ulation scenarios de- p ending on the nature of the trace (real or synthetic), the n umber of recording p erio ds and the similarit y metric com- putation which can b e used in a cen tralized or distributed w ay . Besides, to the best of our kno wledge, this work is the first to p erform the prediction technique in a distributed w ay . The assessment of its efficiency can b e b eneficial for the improv emen t or the design of comm unication proto cols in mobile, sparse and intermitten tly connected netw orks. The paper is organized as follows: Section 2 presen ts the related w ork that highligh ts the growing in terest to the so cial analysis and justifies the recourse to the tensors and to the Katz measure to perform predictions. In Section 3, we emphasize the tw o main steps that characterize our prop osal. Section 4 details simulation scenarios used to ev aluate the tensor-based prediction approach, analyzes the obtained results, assesses its efficiency and proposes a discussion ab out the describ ed link prediction technique. Finally , we conclude the pap er in Section 5. 2. Related W ork The Social Net work Analysis (SNA) [9, 10] and ad- ho c net working hav e pro vided new persp ectives for the design of netw ork proto cols [11, 12, 13]. These proto- cols aim to exploit the so cial asp ects and relationship features b et ween the nodes. Studies conducted in the field of SNA hav e mainly fo cused on t wo kinds of con- cepts: the most w ell-known centralit y metrics suggested in [9, 14, 15, 16] and the communit y detection mechanisms prop osed in [17, 18, 19, 9]. F rom this persp ectiv e, several w orks hav e tried to develop syn thetic mo dels that aim to repro duce realistic moving patterns [3, 20]. Nonetheless, the study done in [1] has outlined that synthetic models cannot faithfully repro duce the human behavior b ecause these syn thetic mo dels are only lo cation-driv en and they do not trac k so cial inten tions explicitly . W e consider in this w ork the Time-V ariant Comm unity mobilit y model (TV C mo del) [3]. The TVC mo del dep ends on t wo main c haracteristics that influence the b ehavior of nodes: geo- graphical lo cation preferences and time-dep enden t b ehav- ior. This design tries to be closer to h uman-based b ehavior and implicitly repro duces the so cial asp ects that c harac- terizes ad-ho c net works. Nev ertheless, [10] has underlined the limits of these pro- to cols when the netw ork topology is time-v arying. The main drawbac k comes do wn to their inability to mo del top ology changes as they are based on graph theory to ols. Nev ertheless, the tensor-based approac hes ha ve b een used in some works to build statistics on the b eha viors of nodes in wireless net w orks o v er time as in [21]. Thakur et al. [4] ha ve also dev elop ed a mo del using a collapsed ten- sor that tracks user’s location preferences (characterized b y probabilities) with a considered time granularit y (week da ys for example) in order to considered the emergence of “b eha vior-aw are” delay toleran t net works. In this paper, w e propose a link prediction technique that tracks the temporal net work topology evolution in a tensor and computes a metric in order to c haracter- ize the so cial-based behavior similarity of eac h pair of no des. Some approaches hav e addressed the same prob- lem in data-mining in order to p erform link prediction. Acar et al. [6] and Dunlavy et al. [7] ha ve pro vided de- tailed methods based on matrix and tensor factorizations for link prediction in social netw orks such as the DBLP data set [22]. These metho ds hav e b een successfully ap- plied to predict a collab oration b etw een tw o authors rely- ing on the data set of the structures of relationships ov er time. Moreov er, they ha ve highligh ted the use of the Katz measure [8], which can b e seen as a similarit y metric, by assigning a link prediction score for eac h pair of no des. The efficiency of the Katz measure in link prediction has b een also demonstrated in [23, 24]. 3. Description of the T ensor Based Prediction Metho d It has been highligh ted that a human mobility pattern sho ws a high degree of temporal and spatial regularity , and eac h individual is characterized by a time-dependent mo- bilit y pattern and a trend to return to preferred lo cations [2, 3, 4]. In order to impro ve the design of wireless netw ork proto cols, and esp ecially the in termitten tly connected net- w orks, it is important to exploit this kno wledge since these in teractions usually hav e an impact on the net work struc- ture and consequently on the netw ork p erformance. Thus, in this paper, we propose an approach that aims to exploit similar b ehavior of no des in order to predict link o ccur- rence referring to the so cial closeness. Predicting future links based on their social closeness is a challenge that is w orth an inv estigation. Indeed, a go o d link prediction technique con tributes to improv e the opp ortunistic forw arding of pack ets and also enhances the deliv ery rate and/or decreases latency . Moreo ver, it helps to a void situations where pack ets encum b er the queue of the no des that are not able to forward them tow ards their final destinations. T o quantify the social closeness betw een each pair of no des in the net work, w e use the Katz measure [8] inspired from so ciometry . This measure aims at measuring the so- cial distance b et ween p ersons inside a so cial netw ork. W e 2 also need to use a structure that records link o ccurrence b et ween each pair of no des ov er a certain perio d of time in order to p erform the similarity measure computation. The records represen t the net w ork behavior statistics in time and space. T o this end, tensors are used. A tensor Z consists in a set of slices and eac h slice corresp onds to an adjacency m atrix of the netw ork trac ked ov er a giv en p erio d of time D . After the trac king phase, w e reduce the tensor into a matrix (or collapsed tensor) which expresses the w eigh t of eac h link according to its lifetime and its recen tness. A high w eight v alue in this matrix denotes a link whose corresponding nodes share an important degree of closeness. W e apply the Katz measure on the collapsed tensor to compute a matrix of scores S that not only con- siders direct links but also indirect links (multi-hop con- nections). The matrix of scores expresses the degree of similarit y of each pair of no des resp ecting to the spatial and the temp oral lev els. The higher the score is, the b et- ter the similarity pattern gets. Therefore, t wo no des that ha ve a high similarit y score are most likely exp ected to ha ve a common link in the future. 3.1. Notation Scalars are denoted by lo w ercase letters, e.g., a . V ectors are denoted by b oldface lo wercase letters, e.g., a . Matri- ces are denoted by boldface capital letters, e.g., A . The r th column of a matrix A is denoted by a r . Higher-order tensors are denoted by b old Euler script letters, e.g., T . The n th fron tal slice of a tensor T is denoted T n . The i th en try of a vector a is denoted by a ( i ), elemen t ( i, j ) of a matrix A is denoted by A ( i, j ), and elemen t ( i, j , k ) of a third-order tensor T is denoted b y T i ( j, k ). 3.2. Matrix of Sc or es Computation The computation of the similarity scores is modeled through t wo distinct steps. First, w e store the in ter- con tact b etw een no des in a tensor Z and reduce it to a matrix X called the collapsed tensor. In a second step, we compute the matrix of similarity scores S relying on the matrix X (cf. Fig. 1). 3.2.1. Col lapsing the data fr om the tensor W e consider that the data is collected into the tensor Z . The slice Z t ( i, j ) describ es the status of a link be- t ween a no de i and a node j during a time p erio d b etw een [ tD , ( t + 1) D [ where Z t ( i, j ) is 1 if the link exists during the time p erio d D and 0 otherwise. The tensor is formed b y a succession of adjacency matrices Z 1 to Z T where the subscript letters designs the observed perio d. T o collapse the data into one matrix as done in [6, 7], w e choose to compute the collapsed weigh ted tensor (which is more ef- ficien t than collapsed tensor as shown in [6] and [7]). The links structure is considered ov er time and the more recen t the adjacency matrix is, the more weigh ted the structure gets. X ( i, j ) = T X t =0 (1 − θ ) T − t Z t ( i, j ) (1) Where the matrix X is the collapsed w eigh ted tensor of Z , and θ is a parameter used to adjust the weigh t of recen tness and is b etw een 0 and 1. 3.2.2. Katz Me asur e The Katz measure, which is affiliated to sociometry , was first prop osed by Leo Katz in [8]. He considers a so cial net work as a undirected graph G = ( V , E ) where eac h v ertices V = { v 1 , v 2 , ..., v k } is a finite set of no de that represen t a p ersons and each edge E = { e 1 , e 2 , ..., e k } is a finite set of connection (or an endorsemen t) b etw een t wo p ersons. W e denote a subset P ` ( v i , v j ) = { e 1 , e 2 , ..., e ` } ⊂ E as a path of length ` b etw een no de v i and v j . The score that c haracterizes the couple ( v i , v j ) is defined by the w eight of paths P ` ( v i , v j ) connecting p erson v i to person v j , ∀ v ∈ V . Katz defined his metric b et ween t wo nodes as S ( i, j ) as depicted in Eq.(2). It is a function that decreases pro- p ortionally to the path length P ` ( v i , v j ). Katz did so in order to emphasize the fact that endorsemen ts strength fade ov er a successive c hains of recommendations. This measure can b e seen as a generalization of follo wers (as in Twitter) or an indegree measure. It indicates the n umber of votes of a p erson as well as the identit y of the voter (his vote is v aluable compared to the num b er of v otes he receiv es). This metric is widely used in studies which aim to predict links o ccurrence [6, 7, 24], especially in so cial net works as co-authorship communities as the DBLP [22] and arXiv [25] databases. Given that there are “so cial re- lationships” betw een no des in netw orks with intermitten t connections, it is c hallenging to exploit this measure and to apply it on collected data. Therefore, the Katz score of a link b et ween a no de i and a node j as given b y [8]: S ( i, j ) = + ∞ X ` =1 β ` P ` ( v i , v j ) (2) Where β is a user defined parameter strictly sup erior to zero and β ` is the weigh t of a ` hops path length. It is clear that the longer the path is, the low er the w eight gets. There is also another form ulation to compute Katz scores by means of collapsed w eighted tensor as detailed previously . Then, the score matrix S can b e rewritten as: S = + ∞ X ` =1 β ` · X ` = ( I − β · X ) − 1 − I (3) Where I is the identit y matrix and X is the obtained collapsed weigh ted tensor. In Fig. 1, w e provide an example that describes the t wo main steps of the link prediction technique. W e con- sider a netw ork of 4 no des whose top ology is dynamic ov er time. A t each p erio d t (from 0 to 3), each o ccurred link 3 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 0 1 1 1 1 0 0 1 1 1 0 0 1 0 1 0 1 1 1 2 1 0 0 0 2 0 0 0 1 2 0 0 1 1 2 1 0 0 1 3 1 0 0 1 3 1 0 0 1 3 1 1 0 0 3 1 0 0 0 4 1 0 1 0 4 1 1 1 0 4 0 1 0 0 4 1 1 0 0 1 2 3 4 1 0 1.512 2.952 2.152 2 1.512 0 0.8 2.44 3 2.952 0.8 0 1.152 4 2.152 2.44 1.152 0 1 2 3 4 1 0 0.0015 0.003 0.0022 2 0.0015 0 0.0008 0.0024 3 0.003 0.0008 0 0.0012 4 0.0022 0.0024 0.0012 0 1 2 3 4 t =0 1 2 3 4 t =1 1 2 3 4 t =2 1 2 3 4 t =3 (1) Collect the adjacency matrix over successive periods of time (2) Collapse the different slices into one matrix (cf. eq. 1) (3) Compute the Katz Scores (cf. eq. 3) Figure 1: Example of the matrix S computation is caught in the corresp onding adjacency matrix. All ad- jacency matrices form the tensor. The latter structure is used to determine the collapsed weigh ted tensor by com- puting Eq. (1) (by setting θ to 0.2) for each pair of no des. Then, the matrix of scores is computed b y applying Eq. (3) ( β is set to 0.001) on the collapsed weigh ted tensor. The measure go es b ey ond estimating a link weigh t be- t ween tw o no des. Indeed, it tak es into consideration all p ossible paths b et ween tw o no des and then quantifies the so cial relationship betw een them. As described previously , when tw o no des are connected through short paths, the score characterizing this pair is high. Hence, the score can b e treated as the node’s moving pattern similarity , in view of the fact that the no des conserve their vicin- it y (short paths). When t wo nodes share high score, this means that their b ehaviors are similar and that they are geographically quite close. Therefore, a link occurrence b et ween them is v ery likely . 3.3. Matrix of Sc or es Interpr etation The relationship b etw een eac h pair of nodes is expressed b y a score S ( i, j ), this score reflects the degree of similarit y b et ween node i and node j . As men tioned in the Katz mea- sure analysis, shorter paths lead to higher scores. Thus, t wo no des that share a high score are no des that are con- nected through short paths during some perio d of time and therefore ha ve similar b ehaviors (similar so cial inten- tions). The similarity here is related to common prefer- ences in spatial and temp oral space. Two no des maintain their connectivit y when they mov e in the same direction and at the same time. Therefore, these scores can b e con- sidered as indicators to a possible link existence in the future. Thus, the link prediction is done through measur- ing behavior similarity for each no des pairs in the matrix S . The computation of matrix S , as describ ed before, is done in a centralized w a y . It means that the matrix S is computed based on a full kno wledge of the netw ork top ology o ver time. This may not b e suitable with ad hoc wireless net w orks where no central entit y is considered and could in addition b e v ery costly . A distributed mecha- nism should then b e examined. In a distributed mecha- nism, eac h no de w ould apply the prediction metho d rely- ing only on information related to its nearest neigh b ors. It is paramount to remem b er that a Katz formulation giv es more weigh t to short paths and assigns low scores to long paths. Therefore, the scores with neighbors located at few hops a wa y should be sufficient and s trong enough com- pared to scores with further ones. This hypothesis will be discussed in Section 3. 4. Performance Ev aluation and Sim ulation Results T o ev aluate how efficien t is the tensor-based link pre- diction in intermitten tly connected wireless netw orks, w e consider three different traces (tw o real traces and one syn thetic trace). F or eac h scenario, we compute the cor- resp onding scores matrix S as describ ed earlier and assess the efficiency of the link prediction metho d through ev al- uation techniques. In the following, we firstly presen t the 4 T able 1: Ma jor P arameters Used in TVC Mo del Parameter V alue Simulation Area Edge Length 1000 meters Netw ork Nodes Number 100 nodes Netw ork Nodes Range 75 meters Netw ork Geographical Communities Number 2 Maximum No des Sp eed 15 m/s Minimum No des Sp eed 5 m/s Average No des Sp eed 10 m/s traces used for the link prediction ev aluation. Then, we exp ose the corresponding results and analyze the effectiv e- ness of the prediction metho d. 4.1. Simulation T r ac es W e consider three traces to ev aluate the link prediction approac h. Two of them are real traces and the third is syn thetic. W e exploit them to construct the tensor b y generating adjacency matrices (with different time perio d t : 5, 10, 30 and 60 minutes). At eac h case, w e trac k the required statistics ab out no des behavior within T perio ds. W e also consider the adjacency matrix corresp onding to the p erio d T +1 as a b enchmark to ev aluate Katz scores matrix. W e detail, in the follo wing, the used traces. • First T race: Dartmouth Campus trace: W e c ho ose the trace of 01/05/06 [26] and construct the tensor slices relying on SYSLOG traces betw een 8 a.m. and midday (4 hours). The n umber of no des is 1018. • Second T race: MIT Campus trace: W e fo cus on the trace of 07/23/02 [27] and consider also the ev en ts b et ween 8 a.m. and midday to build up the tensor. The num b er of nodes is 646. • Third T race: TVC Model trace: In this sc enario, w e use the trace generator prop osed by Hsu et al. [28] which repro duces the concept of the TVC mo del. W e consider a square simulation area with an edge length equal to 1000 meters and where 100 no des are in motion. W e randomly generate t wo locations as the no de’s geographical preferences and keep comm unity switc hing and roaming probabilities as in the example pro vided in the generator. As in the other scenarios, w e track no des b ehavior during 4 hours. T able 1 sum- marizes the main parameters considered in generating TV C Model traces. F or each scenario, w e generate adjacency matrices corre- sp onding to a differen t perio d t : 5, 10, 30 and 60 minutes. Then, to record the netw ork statistics o v er 4 hours, the tensor has respectively a n umber of slices T equal to 48, 24, 8 and 4 slices (for the case where t =5 minutes, it is necessary to hav e 48 p erio ds to cov er 4 hours). As men- tioned earlier, w e take into account b oth centralized and distributed cases for the computation of scores. • The Centralized Computation: The centralized w ay assumes that there is a cen tral entit y which has full kno wledge of the net w ork structure at eac h p erio d and applies Katz measure to the global adjacency ma- trices. • The Distributed Computation: Each no de has a limited kno wledge of the net work structure. W e assume that a node is a w are of its tw o-hop neigh- b orho o d. Hence, computation of Katz measures is p erformed on a lo cal-information-basis. In b oth cases, w e fix θ and β to 0.2 et 0.001 resp ectively . Later, we explain why w e choose these v alues. 4.2. Performanc e A nalysis As describ ed in the previous section, we apply the link prediction metho d to the three t yp es of traces while con- sidering the different tensor slice perio ds in b oth central- ized and distributed cases. In order to assess the efficiency of this metho d, w e consider several link prediction sce- narios (according to the trace, the tensor slice perio d and the scores computation wa y) and we use different ev alu- ation techniques (ROC curves, CDF curves, AUC metric and top scores ratio at T +1). W e detail in the follo wing the results obtained with each ev aluation technique and analyze the link prediction efficiency . 4.2.1. Analy sis of the ROC Curves Fig. 2, 4 and 6 depict the R OC curv es (Receiver Op- erating Characteristic) [29] for both distributed and cen- tralized computing approaches respectively obtained from Dartmouth Campus trace, MIT Campus trace and TVC mo del trace. F or each trace figure, (a), (b), (c) and (d) curv es corresp ond to a tensor slice time of 5, 10, 30 and 60 minutes respectively . W e first notice that, for all scenarios, the prediction of all links is quite efficien t, compared to the random guess (the curve’s b ends are at the upp er left corner). More- o ver, t wo other observ ations ha ve to b e mentioned in the case of real traces (Dartmouth Campus and MIT Campus traces). First, it is highlighted that the smaller the ten- sor slice (adjacency matrix) p erio d is, the more reliable the prediction gets. This observ ation is ob vious for tw o reasons. On the one hand, with a low tensor slice time, the probability of tracking a short and occasional contact b et ween t w o no des is not likely . On the other hand, record- ing four hours of statistics requires 48 adjacency matrices of 5-minutes p e rio ds instead of 4 matrices for 60-minutes p erio ds case. Thus, tracking a short con tact betw een t wo no des has less influence when the tensor slices are more n umerous. As an example, in the case where the tensor slice time is 5 min utes, a fleeting contact can b e caught b y one adjacency matrix among 48. Ho w ever, for the case where the slice time is 60 minutes, the fleeting con tact is trac ked b y one tensor slice among 4, which significantly giv es it more w eight compared to the former case. Hence, 5 short tensor slice p erio ds enable us to minimize the prob- abilit y of tracking a short contact existence and to restrict its impact. Short tensor slice p erio ds also allow us to b etter trac k the social interactions (meetings in a cafeteria, courses in an amphitheater, etc) b et ween no des whic h determine the o ccurrence of links. Successive adjacency matrices of 5 min utes giv e more accurate description of netw ork struc- ture o ver time as both analyzing and identifying these so- cial even ts are easier through smaller perio ds. The second observ ation concerns the similar results obtained at the centralized and distributed matrix of scores computation. In fact, the similarit y is higher when the paths considered b etw een a pair of nodes are short. Thereb y , paths that hav e more than tw o hops ha ve w eak er scores and so are less weigh ted compared to shorter ones. The distributed case assumes that eac h no de kno ws its neigh b ors at most at t wo hops. That is why distributed scores computation presen ts p erformances whic h are so similar to the cen tralized ones. Regarding the res ults obtained from the syn thetic trace (TV C mo del trace), it is obvious that there are no sig- nifican t differences b etw een the ROC curv es as the tensor slice p erio ds v aries (esp ecially for the scenarios where the p erio d t is higher than 5 min utes). On top of having the same p erformances with the t wo scores matrix computa- tion wa ys, c hanging the adjacency matrix time p erio d does not impact the link prediction efficiency . This observ ation could be explained by the conclusion dra wn by Hossmann et al. in [1] which outlines that lo cation-driv en mobility mo dels do not care ab out so cial inten tions. In addition, through the proposed behavior similarity metric b etw een a pair of no des, Thakur et al. [4] pro ve that TV C model lim- its mo ving patterns to visiting preferred lo cations and do not tak e care of any social co ordination. With TV C mo del, mo vemen t patterns are the same for all nodes (mo ving in to t wo geographical comm unities) and rep etitive (the chosen mo ving speed is b et ween 5 and 15 m/s with an av erage sp eed of 10 m/s). They are only regulated by geographi- cal preferences (eac h no de visits the preferred comm unity with an “individual willingness” and there is no correla- tion betw een its moving pattern and those of other nodes). Therefore, having sev eral tensor slices is not differen t from considering few er ones. Moreov er, the adjacency matrix at T +1 in each scenario is quite the same. 4.2.2. Anal ysis of the CDF Curves In order to highlight the impact of the choice of the p e- rio d t on the link prediction, we represent in Fig. 3, 5 and 7 the sk ewed Cum ulative Distribution F unction (CDF) of the scores obtained respectively for Dartmouth Campus trace, MIT Campus trace and TV C mo del trace (only strictly p ositive scores are considered). A t each trace’s CDF figure, (a) and (b) corresp ond to the distributed and cen tralized scores matrix computation resp ectively . The obtained results for real traces (Dartmouth Campus and MIT Campus) sho w that the spreading of distribution is T able 2: Ev aluation metrics for the prediction of all links ap- plied on Dartmouth Campus trace h h h h h h h h h h h h h h h Prediction Cases Metrics AUC T op Scores Ratio at T +1 Distributed Case and t =5 mins 0.9850 2944/3267 (90.11%) Centralized Case and t =5 mins 0.9844 2945/3267 (90.14%) Distributed Case and t =10 mins 0.9817 2866/3340 (85.80%) Centralized Case and t =10 mins 0.9826 2866/3340 (85.80%) Distributed Case and t =30 mins 0.9360 2758/3832 (71.97%) Centralized Case and t =30 mins 0.9324 2758/3832 (71.97%) Distributed Case and t =60 mins 0.9153 2928/4270 (68.57%) Centralized Case and t =60 mins 0.9069 2926/4270 (68.52%) narro wer when the p erio d t is larger. In fact, at a CDF with wider spreading (especially at the case of t =5 min), high scores that express link occurrence prediction are eas- ier to figure out. On the contrary , the in terv al of scores is narro w and so the score’s analysis is more imprecise. These results confirm the ones obtained through ROC curv es. While the CDF results of real traces lo ok similar, the ones of the syn thetic trace show that the tensor slice p erio d has a less significative impact. Indeed, the cum ulative dis- tribution functions are redundant at o ver 80% of obtained scores (when the scores are situated b et ween 0 and 3 . 10 − 3 ). This observ ation also applies to ROC curv es results. As a final note, we underline that the syn thetic trace CDF sho ws a higher p ercentage of weak scores than those of real traces. This obse rv ation explains the more limited prediction efficiency outlined with the TV C mo del trace. 4.2.3. Evaluation of the link pr e diction te chnique thr ough p erformanc e metrics As another ev aluation step, adapted metrics are used in order to further w eigh the p erformance of the prop osed link prediction technique. At this step, on top of ev alu- ating prediction of all links, we try to fo cus on assessing the efficiency of our technique in predicting new links that o ccurred for the first time at T +1 (while ignoring all previ- ously seen links). T o this end, we compute the Area Under the ROC Curv e metric (AUC metric) [29] which could b e considered as a go o d p erformance indicator in our case. Th us the top scores ratio metric at T +1 is also consid- ered. T o determine this metric, we compute the accu- rate num b er of links identified through the link prediction tec hnique. W e list, for each considered time p erio d, the n umber of existing links at p erio d T +1, which w e call L . Then, we extract the links having the L highest scores and determine the num b er of existing links in b oth sets. The ev aluation metrics are computed for all traces with differ- en t tensor slice p erio ds in b oth distributed and cen tralized scenarios. The results corresp onding to all links prediction are listed in T able 2 (Dartmouth Campus trace), T able 3 (MIT Campus trace) and T able 4. The results corresp ond- ing to new links prediction are listed in T able 5, T able 6 and T able 7(respectively for Dartmouth Campus, MIT Campus and TVC mo del traces). 6 (a) 5 minutes tensor slice perio d (b) 10 minutes tensor slice perio d (c) 30 minutes tensor slice perio d (d) 60 minutes tensor slice perio d Figure 2: ROC Curves for differen t prediction cases applied on Dartmouth Campus trace (a) Distributed computation of scores (b) Centralized computation of scores Figure 3: Cumulativ e distribution function of Katz scores obtained from Dartmouth Campus trace Regarding all links prediction results, we note, based on the high v alues of AUC metric (o v er than 0.9 at real traces) and top scores ratio obtained at T +1, that the prediction metho d is efficien t in predicting future links. Moreov er, we note that prediction is b etter when the tensor slice p eri- o ds are shorter. W e also observe that the cen tralized and distributed matrix of scores computation achiev e similar p erformances. In addition, the results related to the top scores metric attests to the fact that the prediction of all links is efficient (at least 68% of links are identified and this p ercen tage can exceed 90% in some cases) at b oth cen tralized and distributed scenarios. W e also note that the previous observ ation regarding the redundancy of the results as the tensor slice perio d v aries with the synthetic trace is confirmed. Indeed, The num ber of existing links at T + 1 is the same when the p erio d t is o ver 5 min utes. Moreo ver, AUC metric and top scores ratio has almost al- w ays the same v alue. Nonetheless, when the prediction only concerns new links, AUC metric v alues considerably decrease. This observ ation presumes that the prediction is not that accurate when only new links are considered. Giv en that new links are not trac ked b y the tensor, their scores are low (and even null). This in terpretation is sup- p orted by the top scores ratio at T + 1. In fact, the p er- cen tages of iden tified new links are very lo w (no more than 8% in the b est cases). Hence, the tensor-based link predic- 7 (a) 5 minutes tensor slice perio d (b) 10 minutes tensor slice perio d (c) 30 minutes tensor slice perio d (d) 60 minutes tensor slice perio d Figure 4: ROC Curves for differen t prediction cases applied on MIT Campus trace (a) Distributed computation of scores (b) Centralized computation of scores Figure 5: Cumulativ e distribution function of Katz scores obtained from MIT Campus trace tion technique is not efficien t when the prediction targets the o ccurrence of new links. This result is also highlighted in [6] and [7]. It is also imp ortant underline the fact that our mec ha- nism efficiency is dep endent of chosen v alues of θ and β . W e depict in Fig. 8(a) and Fig. 8(b) the top scores ratio at T + 1 and the A UC, resp ectively , obtained for different v alues of θ and β . W e can note that the v alues set to θ (i.e. 0.2) and to β (i.e. 0.001) enables us to reach a quite efficien t level of prediction. This results are relativ e to a prediction set p erformed on the MIT Campus trace with the distributed version of our metho d (as described in the Section 4.1). 4.2.4. Pr e diction Performanc e Comp arison b etwe en the T ensor-Base d T e chnique and the appr o ach of Thakur et al. W e aim through this subsection to compare our pro- p osal to another similar approac h (we use the distributed approac h to compute the scores). As we are designing a metric that expresses the degree of similarity of t wo no des, w e choose to compare the tensor-based tec hnique p erfor- mance to the one of the similarity metric suggested by Thakur et al. [4]. The latter metric measures the de- gree of similarity of the behaviors of t wo mobile nodes and the b ehavior of eac h no de is expressed b y an association matr ix . The columns of the matrix represen t the p ossi- 8 (a) 5 minutes tensor slice perio d (b) 10 minutes tensor slice perio d (c) 30 minutes tensor slice perio d (d) 60 minutes tensor slice perio d Figure 6: ROC Curves for differen t prediction cases applied on TVC mo del trace (a) Distributed computation of scores (b) Centralized computation of scores Figure 7: Cumulativ e distribution function of Katz scores obtained from TVC mo del trace ble lo cations that a no de can visit and the rows express time granularit y (hours, days, weeks, etc.). The dominant b eha vioral patterns are trac ked using the Singular V alue Decomp osition (SVD) [30]. F or more details ab out the similarit y metric computation, we refer the reader to [4]. W e compare the top scores ratio at T +1 and the area under the ROC curve metrics and we measure them for differen t tensor slice times or time gran ularities (5, 10, 30 and 60 min utes). F or this comparison, we use the MIT trace of 07/23/02 and trac k adjacency matrices and/or asso ciation matrices from 8 a.m. to 3 p.m.. The associated results for the top scores ratio at T +1 and for the area under the ROC curve are resp ectively depicted in Fig. 9(a) and Fig. 9(b) and the p erformance gap are resp ectiv ely displa yed in Fig. 9(c) and Fig. 9(d). W e firstly fo cus on the comparison according to the top scores ratio at T +1. W e underline that our proposal sho ws more efficien t prediction ability compared to Thakur et al. framework especially when the tensor slice time/time gran ularity tends to b e short. The different ”natures” of the metrics used in eac h approach explain the results ob- tained for the t wo sets of comparison. Indeed, the measure quan tifies the similarity of nodes based on their encoun ters and geographical closeness. In other w ords, the prediction measure cares ab out contacts (or closenesses) at (around) the same location and at the same time. Mean while, the 9 (a) Area Under the ROC Curv e (b) T op Scores Ratio at T +1 Figure 8: Prediction performance of the tensor-based technique distributed v ersion for different v alues of β and θ (a) Comparison according to the T op Scores Ratio at T +1 metric (b) Comparison according to the Area Under the ROC Curve metric (c) The T op Scores Ratio at T +1 metric gap b e- tw een the t wo approac hes (d) The Area Under the R OC Curve metric gap b e- tw een the t wo approac hes Figure 9: Prediction performance comparison betw een the tensor-based technique and the approach of Thakur et al. similarit y metric prop osed by Thakur et al. is defined as an asso ciation metric. Hence, it measures the degree of similarit y of b ehaviors of t wo mobile no des without neces- sarily seeking if they are in the same lo cation at the same time. As w e ha ve previously stated, the prediction p erfor- mance of the tensor-based link prediction metho d is better with shorter tensor slice times. Then, with a longer tensor slice time, the interpretation of netw ork statistics b ecomes less precise. This observ ation accoun ts for the prediction p erformance more comparable for both approac hes with a larger tensor slice time/time granularit y . On the other hand the comparison based on the AUC metric, w e remark that the tw o approaches show quasi- similar prediction efficiency with a slightly b etter p erfor- mance for our prop osal, mos tly b ec ause the o verwhelm- ing num b er of noneffective links in tro duce a bias in the 10 T able 3: Ev aluation metrics for the prediction of all links ap- plied on MIT Campus trace h h h h h h h h h h h h h h h Prediction Cases Metrics AUC T op Scores Ratio at T +1 Distributed Case and t =5 mins 0.9838 1922/2147 (89.52%) Centralized Case and t =5 mins 0.9842 1925/2147 (89.66%) Distributed Case and t =10 mins 0.9813 1867/2187 (85.36%) Centralized Case and t =10 mins 0.9807 1866/2187 (85.32%) Distributed Case and t =30 mins 0.9631 1757/2311 (76.02%) Centralized Case and t =30 mins 0.9618 1757/2311 (76.02%) Distributed Case and t =60 mins 0.9256 1803/2657 (67.85%) Centralized Case and t =60 mins 0.9361 1817/2657 (68.38%) T able 4: Ev aluation metrics for the prediction of all links ap- plied on TVC mo del trace h h h h h h h h h h h h h h h Prediction Cases Metrics AUC T op Scores Ratio at T +1 Distributed Case and t =5 mins 0.8851 717/931 (77.01%) Centralized Case and t =5 mins 0.8860 717/931 (77.01%) Distributed Case and t =10 mins 0.8409 750/1080 (69.44%) Centralized Case and t =10 mins 0.8401 749/1080 (69.35%) Distributed Case and t =30 mins 0.8412 757/1080 (70.09%) Centralized Case and t =30 mins 0.8424 757/1080 (70.09%) Distributed Case and t =60 mins 0.8388 755/1080 (69.90%) Centralized Case and t =60 mins 0.8399 755/1080 (69.90%) T able 5: Ev aluation metrics for the prediction of new links ap- plied on Dartmouth Campus trace h h h h h h h h h h h h h h h Prediction Cases Metrics AUC T op Scores Ratio at T +1 Distributed Case and t =5 mins 0.6671 1/144 (0.69%) Centralized Case and t =5 mins 0.6518 1/144 (0.69%) Distributed Case and t =10 mins 0.6759 1/184 (0.54%) Centralized Case and t =10 mins 0.6913 1/184 (0.54%) Distributed Case and t =30 mins 0.6469 20/684 (2.89%) Centralized Case and t =30 mins 0.6269 24/684 (3.50%) Distributed Case and t =60 mins 0.6472 51/1008 (5.05%) Centralized Case and t =60 mins 0.6115 58/1008 (5.75%) T able 6: Ev aluation metrics for the prediction of new links ap- plied on MIT Campus trace h h h h h h h h h h h h h h h Prediction Cases Metrics AUC T op Scores Ratio at T +1 Distributed Case and t =5 mins 0.6823 8/107 (7.47%) Centralized Case and t =5 mins 0.6921 8/107 (7.47%) Distributed Case and t =10 mins 0.7221 0/141 (0.00%) Centralized Case and t =10 mins 0.7121 4/141 (2.83%) Distributed Case and t =30 mins 0.6955 0/267 (0.00%) Centralized Case and t =30 mins 0.6843 0/267 (0.00%) Distributed Case and t =60 mins 0.6929 23/620 (3.70%) Centralized Case and t =60 mins 0.7383 25/620 (4.03%) calculations of the AUC metric. The reduced num b er of o ccurring links and the findings obtained for the first set of comparison explain the little AUC gap in fav or of the tensor-based link prediction approac h. T able 7: Ev aluation metrics for the prediction of new links ap- plied on TVC mo del trace h h h h h h h h h h h h h h h Prediction Cases Metrics AUC T op Scores Ratio at T +1 Distributed Case and t =5 mins 0.4954 0/76 (0.00%) Centralized Case and t =5 mins 0.4920 0/76 (0.00%) Distributed Case and t =10 mins 0.4758 2/131 (1.52%) Centralized Case and t =10 mins 0.4664 2/131 (1.52%) Distributed Case and t =30 mins 0.4730 2/131 (1.52%) Centralized Case and t =30 mins 0.4816 2/131 (1.52%) Distributed Case and t =60 mins 0.4583 2/131 (1.52%) Centralized Case and t =60 mins 0.4769 4/131 (3.05%) 4.3. Discussion In wireless net works and specifically in in termittently connected ones, it is imp ortant to exploit so cial relation- ships that influence no des mobilit y . T aking adv antage of the social asp ect within these netw orks could ensure a bet- ter routing strategy and therefore improv e the pack ets de- liv ery rate and reduce latency . Through our prop osal, we aim to track even tual similarities betw een mobilit y pat- terns of no des and wisely exploit them for a b etter link prediction. As seen earlier, link occurrence b etw een t w o no des is more lik ely when they hav e similar social b ehaviors. Then, iden tifying no des that hav e similar mobility pattern could help to predict effective links b et ween no des in the future. The more accurate the link prediction is the more opti- mized the routing sc heme could get. In fact, an efficient link prediction w ould help to make b etter decisions in the forw arding pro cess. F or example, a no de w ould rather decide to p ostp one sending a pac ket to an ev entual cur- ren t next hop b ecause the link prediction sc heme estimates that a b etter forwarder (closer to the destination for ex- ample) is going to app ear in the immediate future. Also, link prediction could prev ent buffer o verloading. Indeed, an ov erloaded node would rather drop a pack et if the link prediction sc heme indicates that there will not b e an y p os- sible route tow ard the destination in the future and before the pac ket’s TTL expires. Through this approach, w e can get quite efficient prediction results. As men tioned previously , the efficiency of the technique used can exceed 90% of identified links (with slice p erio d equal to 5 min utes). The link prediction relies on mea- suring the similarity of the mobility of nodes. Song et al. [31] ha ve in v estigated the limits of predictabilit y in h uman mobilit y . Relying on data collected from mobile phone carriers, they hav e found that 93% of user mobility is potentially predictable. The b est predictability percent- age reached by our approach agrees with the conclusion of Song et al.. W e ha ve also shown through the sim ulations that pre- diction efficiency is similar, for a specific scenario (t ype of trace and slice time perio d), in the case of both cen- tralized and distributed computation of Katz scores. As w e hav e explained, the distributed scheme is only able to 11 T able 8: Ev aluation metrics for distributed prediction scenar- ios of applied on MIT Campus trace h h h h h h h h h h h h h h h Prediction Cases Metrics AUC T op Scores Ratio at T +1 One-hop knowledge and t =5 mins 0.9747 1921/2147 (89.47%) Two-hops knowledge and t =5 mins 0.9838 1922/2147 (89.52%) One-hop knowledge and t =10 mins 0.9671 1865/2187 (85.27%) Two-hops knowledge and t =10 mins 0.9813 1867/2187 (85.36%) One-hop knowledge and t =30 mins 0.9406 1756/2311 (75.98%) Two-hops knowledge and t =30 mins 0.9631 1757/2311 (76.02%) One-hop knowledge and t =60 mins 0.8810 1789/2657 (67.33%) Two-hops knowledge and t =60 mins 0.9256 1803/2657 (67.85%) main tain high scores (link o ccurrence is likely) as no des record neighbors at one and tw o hops. The seeming lac k of information do es not infer on predicting effectiv eness. This observ ation also tallies with Acar and al. conclu- sion. Indeed, in the data mining context, they ha ve tried to mak e the metho d scalable and prop osed the T runcated Katz technique (expressed by eq. (11) in [6]). It con- sists in determining Katz scores replacing the collapsed w eighted tensor b y a lo w-rank appro ximation one. The results show that this latter technique retains high predic- tion efficiency . Hence, restricting the scores computation on most weigh ted links (in terms of recentness and dura- tion) does not incur dramatic consequences on prediction efficiency . W e hav e assumed for the computation of similarity scores using the distributed w ay that nodes kno w their t wo-hop neighbors. It is obvious that exc hanging infor- mation betw een no des about neighbors causes additional o verhead and consequently more solicited resources. F rom this persp ective, a question can b e highligh ted: w ould the tensor-based link prediction method remain effectiv e if the kno wledge of no des is limited to the direct neighbors? T o answ er to this question, w e tak e in to consideration the sce- nario where the distributed computation of scores is based on one-hop neigh b oring kno wledge and we compare it to the scenario whic h uses the tw o-hops knowledge. W e use the MIT Campus trace and trac k the netw ork topology during 4 hours (i.e. the trace of 07/23/02 from 8 a.m. to midda y) and we consider different tensor slice times. The comparison is made with the top scores ratio at T +1 and the AUC metrics. The results are rep orted in T able 8. When the knowledge is limited to the neighbors at one hop, the closeness only means that it exists a direct link b et ween tw o no des. This scenario do es not consider the relationships b etw een no des when they are separated by m ulti-hops paths. The results confirm that the prediction effectiv eness is lesser. Even if the top scores ratios at T +1 are close, the p erformance of the one-hop knowledge sce- narios are sligh tly w orse. The AUC metric attests also that the prediction is less efficien t in such cases. In fact, considering the tw o-hops kno wledge generates more signi- ficativ e true p ositive rate for the R OC curv e (expressed b y b etter top scores ratio at T +1) while the false p ositive rate remains practically the same (due to the ov erwhelm- ing n umber of noneffective links). Nevertheless, the b est scenario to retain is not obvious to identify if w e compare the cost of exchanging local information b et ween nodes to the cost of less efficient link prediction. F uture simu- lations and real deplo yments will enable us to determine whic h setting is preferable to consider. 5. Conclusion Human mobilit y patterns are mostly driven b y social in- ten tions and correlations in the behaviors of p eople form- ing the netw ork appear. These similarities quantifies the correlation b etw een the spatial level in terms of visited lo cations and the temp oral level regarding mobilit y cor- relation o ver perio d of time. The kno wledge ab out the b eha vior of no des greatly helps in improving the design of communication proto cols. Intuitiv ely , t w o no des that follo w the same so cial in tentions ov er time promote the o ccurrence of link in the immediate future. In this pap er, w e presented a link prediction tec hnique inspired from data-mining and exploit it in the context of wireless net works. Our con tribution in this pap er, as a new link prediction technique for the in termittently connected wireless net works, is designed through tw o ma jor steps. First, the net w ork topology is track ed o v er sev eral time p erio ds in a tensor. Secondly , after collapsing the struc- tural information, Katz measure is computed for each pair of no des as a score. A high score means similar mo ving patterns inferring the closeness of the no des and indicates that a link o ccurrence is lik ely in the future. Through the link prediction ev aluation, we hav e ob- tained relev ant results that attest the efficiency of our con tribution and agree with some findings referred in the literature. W e summarize them in the following p oin ts: • The tensor-based link prediction tec hnique is quite efficien t esp ecially when applied on real traces (Dart- mouth Campus and MIT Campus traces). The result are supp orted by the ROC curv es and the ev aluation metrics (AUC and T op Scores Ratio at T +1 metrics). • Applied on real traces, the prop osed prediction tec h- nique pro vides more accurate results with low er tensor slice (or tensor adjacency matrices) times. • The prediction results with the synthetic trace (TVC mo del trace) confirm the lack of social interactions. The inten tions of no de are only gov erned by the pre- ferred lo cations and do not correlate with the inten- tions of the other no des. • The link prediction metho d guarantees goo d p erfor- mance when prediction is applied to all links. Nev- ertheless, the prediction of new links (not o ccurring according to statistics and b y ignoring all links seen previously) is not accurate (very low AUC and top scores ratio at T +1 metrics). 12 • Applying the prediction technique in a distributed w ay (nodes kno ws only their neigh b ors at most at tw o hops) achiev es similar predicting p erformance com- pared to the use in cen tralized w a y (an entit y has full-kno wledge about net work structure ov er time). • The temporal tensor-based link prediction describ ed in this pap er is based on an encounter metric which tak es in to accoun t the o ccurring contacts at the same lo cation and at the same time. W e provide a p er- formance comparison with a similar approach built around an asso ciation similarity metric (that quan- tifies similarity based on preferred lo cations regard- less of time correlations) and show that our prop osal ac hieves b etter prediction results. Go o d link prediction offers the possibility to further im- pro ve opportunistic pack et forwarding strategies b y mak- ing b etter decisions in order to enhance the delivery rate or limiting latency . Therefore, it will b e relev an t to supply some routing proto cols with prediction information and to assess the contribution of our approach in enhancing the p erformance of the net work esp ecially as w e prop ose an efficien t distributed version of the prediction method. The prop osed tec hnique also motiv ates us to inquire into future enhancemen ts as a more precise tracking of the behavior of no des and a more efficien t similarity computation. 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