Encyclopedia of Social Network Analysis and Mining, Springer 2013

Transcription

1 Enccopedia of Socia Network Anasis and Mining Springer 2013 Muti-aered Socia Networks Piotr Bródka Przemsław Kazienko Wrocław Universit of Technoog Institute of Informatics Wrocław Poand Snonms Muti-aered Socia Networks aered socia network Muti-reationa socia network Mutidimensiona socia network Mutipe socia network Gossar SN socia network SSN - singe-aered socia network MSN muti-aered socia network Definition A socia network SN 1 is defined as a tupe <VE> where: V is a not-empt set of actors representing socia entities: humans organizations departments etc caed aso vertices nodes vertees members cases agents instances or points; E is a set of directed edges reations between actors caed aso arcs connections or ties where a singe edge is represented b a tupe <> ϵv and for two edges <> and < > if <> < > and = then Note that <> <> because we consider here directed socia networks Since socia networks usua represent one kind of reationships the are aso caed singeaered socia network SSN [Magnani 11] A muti-aered socia network MSN is a network etended to mutipe edges between pairs of nodes/actors It is defined as a tupe <V E > where: V is a not-empt set of actors socia entities; 1 In this essa a socia network SN is aso caed a singe-aered socia network SSN to distinguish it from a muti-aered socia network MSN see beow

2 Enccopedia of Socia Network Anasis and Mining Springer 2013 E is a set of tupes <> ϵv and for an two tupes <> < >E if <> < > = and = then ; is a fied set of distinct aers tpes of reationships Introduction It is quite obvious that in the rea word more than one kind of reationship can eist between two actors eg fami friendship and work ties and that those ties can be so intertwined that it is impossibe to anase them separate [Fienberg 85] [Minor 83] [Sze 10] Socia networks with more than one tpe of reation are not a compete new concept [Wasserman 94] but the were anased main at the sma scae eg in [McPherson 01] [Padgett 93] and [Entwise 07] Just ike in the case of reguar singe-aered socia network there is no wide accepted definition or even common name At the beginning such networks have been caed mutipe network [Hathornthwaite 99] [Monge 03] The term is derived from communications theor which defines mutipe as combining mutipe signas into one in such wa that it is possibe to separate them if needed [Hami 06] Recent the area of muti-aered socia network has started attracting more and more attention in research conducted within different domains [Kazienko 11a] [Sze 10] [Rodriguez 07] [Rodriguez 09] and the meaning of mutipe network has epanded and covers not on socia reationships but an kind of connection eg based on geograph occupation kinship hobbies etc [Abraham 12] Socia networks emerging from different tpes of socia media or socia networking sites are good eampes of muti-reationa networks A main reason for that is the fact that these sstems offer arge and diverse datasets incuding information about peope profies and their various activities that can be anased in depth Since this data refects users behaviours in the virtua word the etracted socia networks are caed onine socia networks [Garton 97] web-based socia networks [Gobeck 06] or computer-supported socia networks [Weman 96] Bibiographic data bogs photos sharing sstems ike Fickr e-mai sstems teecommunication data socia services ike Twitter or Facebook video sharing sstems ike YouTube Wikipedia and man more are the eampes of services providing data sources which ma be used b man researches to anase undering socia networks However this vast amount of data and especia its muti-reationa character simutaneous present new research chaenges reated to processing probems of this data [Domingos 03] Athough most of the eisting methods work proper for singe-aered networks there is a ack of weestabished toos for muti-aered network anasis Deveopment of new measures is ver important from the perspective of further advances in the web science as the muti-reationa networks can be found amost everwhere The are more epressive in terms of the semantic information and give opportunit to anase different tpes of human reationships [Rodriguez 09] Ke Points Each aer in the muti-aered socia network MSN corresponds to one tpe of reationships between peope Different reationships can resut from the character of connections tpes of

3 Enccopedia of Socia Network Anasis and Mining Springer 2013 communication channe or tpes of various coaborative activities that humans eg users of various IT services can perform within a given sstem or in a given environment The eampes of different reationships can be: friendship fami or work Different communication channes that resut in different tpes of connections are: emai echange VoIP cas instant messenger chats etc The separate reationship tpes can aso be defined based on users common activities within compe services ike in the photo pubishing service Since users ma pubish their photos comment pictures provided b others add some photos to their favourites then each such activit can refect various kinds of reationships: author-commentator commentator-commentator author-favourite etc Additiona reationships can aso possess semantic meaning as for eampe pubishing and commenting photos is a much more proactive action more semantic than just adding photos to favourites often utiized due to acquaintance with the author see [Kazienko 2011a] Another eampe where information about user activities has a cear semantic meaning can be an internet forum where peope who are ver active and post a ot of queries can be perceived as new to a fied On the other hand peope who comment a ot but do not post an queries can be seen as eperts in a forum domain Actors V and a edges E E from on one aer correspond to a simpe singe-aered socia network SSN= <V E {}> In genera a muti-aered socia network MSN=<VE> ma be represented b a mutigraph where mutipe reations are represented b mutiedge [Newman 10] Hence a the structura measures presented beow can aso be appied to other kinds of compe networks that are described b means of mutigraphs aer 1 friendsip v z t u aer 2 work v z t u aer 3 fami v z t u Muti-aered Socia Network MSN Figure 1 An eampe of the muti-aered socia network MSN

4 Enccopedia of Socia Network Anasis and Mining Springer 2013 In Figure 1 the eampe of three-aered socia network is presented The set of actors consists of {t u v z} so there are si actors in the network that can be connected with each other on three aers: 1 friendship 2 work and 3 fami On the aer 1 eight reationships tupes between actors: < 1 > < 1 > <z 1 > <z 1 > <z 1 > <uz 1 > <uv 1 > <vu 1 > can be distinguished 6 edges on aer 2 and 7 on aer 3 The muti-aered socia network from Fig 1 represents a quite dense fami network 3 that is simutaneous ess intensive coaboration network 2 and a bit more crowded friendship structure 1 Historica Background Socia networks with more than one kind of reationship between the same actors ma take man different names The most common one is Muti-aered or just aered socia networks [Bródka 12] [Geffre 09] [Hami 06] [Kenned 09] [Magnani 11] [Schneider 11] but aso Muti-reationa socia networks [Sze 10] Muti-dimensiona socia networks [Kazienko 11b] Mutidimensiona Tempora socia network [Kazienko 11c] or Mutivariate socia networks [Sze 10] are in use Additiona the researchers in the fied of muti-aered networks aso tr to deveop new modes of networks that capture not on the muti-aered profie of socia data Authors in [Cantador 06] proposed a muti-aered semantic socia network mode that enabes to investigate human interests in more detais than when the are anased a together Wong at a in [Wong-Jiru 07] propose muti-aered mode where on one aer describes reation between peope the other aers are processes appications sstems and a phsica network Despite the fact that man researchers investigate muti-aered socia networks there is surprising few definitions or descriptions going beond the statement It is the network where two nodes are connected b more than one connection reation tie Newman [Newman 10] defines such a network as a mutigraph with mutiedges between nodes A network is then represented b the adjacenc matri whose each ce A ij a mutiedge actua on preserves a simpe number of edges between node i and j The second possibe representation is an adjacenc ist where a mutiedge is represented b mutipe identica entries in the ist of neighbours eg an adjacenc ist vzz means that node has one edge to node v three edges to and two to z The main probem with Newman s definition is that he has never assigned an abes on those edges so in this mode the information about which edge represents what component singe-aered network is ost A different approach is presented b Magnani and Rossi in [Magnani 11] The describes two concepts: 1 Piar Muti-Network and 2 Muti aer Network M-Mode The first concept defines socia network with mutipe reation as a set of singe-aered networks {<V 1 E 1 > <V 2 E 2 > <V k E k >} and some mapping reation b means which a node in one network is mapped to another node in the second network It means that one node from one singe-aered network SN 1 can refect on one node in another singe-aered socia network SN 2 This case is tpica for most web-based services eg a Facebook account ma correspond to the Twitter account The second concept is amost the same but the mapping function is sight different ie man nodes users from one singe-aered socia network SN 1 can match to a singe node in another singe-aered socia network SN 2 For eampe if one singe-aered socia network SN 1 represents reations between co-workers from one compan and another network SN 2 characterizes reations between whoe departments in the

5 Enccopedia of Socia Network Anasis and Mining Springer 2013 same compan then man empoees working in a given department D man nodes in SN 1 are mapped to this singe department D one node in the second network SN 2 Kazienko at a in [Kazienko 11b] [Kazienko 11c] present et another mode of mutidimensiona tempora socia network which considers three distinct dimensions of socia networks: aer time and group dimension A the dimensions share the same set of nodes that corresponds to socia entities: singe humans or groups of peope A aer dimension describes a kinds of reationships between users of the sstem; a time dimension refects the dnamics of the socia network and a group dimension focuses on interactions within separated socia communities groups At the intersection of a this dimensions is a sma socia network which contains on one kind of interactions one aer for a particuar group in a given time snapshot This concept aows to anase sstems where peope are inked b man different reationship tpes aers in the aer dimension ike in compe socia networking sites eg Facebook It means peope ma be connected as friends via common groups ike it etc It can aso refer compe reationships within reguar companies: department coeagues best friends coeagues from the compan trip members of particuar project team etc Mutidimensionait provides an opportunit to anase each aer separate and at the same time investigate different aggregations over instances of the aer dimension For eampe et us consider a network composed of si aers three from the rea word: fami ties work coeagues and gm friends and three from the virtua word ie friends from Facebook and fiends from the MMORPG game as we as friends from some discussion forum Now one has man different possibiities for studies on such a network for eampe: i to anase each aer separate ii to aggregate aers from the rea word and compare them to the virtua word aers aggregation and fina iii to aggregate a aers together Additiona a time dimension provides possibiit to investigate the network evoution and its dnamics For eampe the anasis i how users neighbourhoods change when one of the neighbours eaves the network and how it affects the network in onger term ii how roes of group eaders eg project team eaders change over time are overtaken b different peope or iii how changes on one aer affect the other aers Fina the group dimension aows studing groups eisting within the socia network Using mutidimensionait not on the usua socia groups can be anased friend fami schoo work etc but aso groups created upon various member features ike gender age ocation etc Moreover the mode aows to compare the resuts of different communit etraction methods eg b means of graph-based socia communit etraction or tpica data mining custering To concude the mutidimensiona socia network enabes to anase a three dimensions at the same time eg how interaction on different five aers of two socia groups changes over three seected periods Rodriguez in [Rodriguez 07] defines a muti-reationa socia network as a tupe G = N E W where N is the set of nodes in the network E is a set of directed edges W is the set of weights associated with each edge of the network W = E However two ears ater in [Rodriguez 09] he defined the same network as M = V E where V is the set of vertices in the network E = {E 1 E 2 E m } is a fami of edge sets in the network and an E k V V:1 k m Each edge set in E has a different semantic interpretation Therefore it is crucia to define one genera concept of the muti-aered socia network The idea further presented is ver simiar to Piar Muti-Network presented in [Magnani 11] but it was introduced one ear earier in [Kazienko 10] The main difference is that instead of mapping function in this mode the set of nodes is unified and fied ie it is common for a aers The appropriate definition was presented above in the Definition Section

6 Enccopedia of Socia Network Anasis and Mining Springer 2013 Measures in Muti-aered Socia Network Despite the muti-aered nature of networks is an absoute natura concept researchers have refrained from anasing more than a singe aer at once for man ears For eampe Wasserman and Fraust [Wasserman 94] recommended that common centrait and prestige measures shoud be cacuated for each reation aer separate and suggested not to perform an aggregation of the reations Unfortunate the do not epain wh the advised such approach The possibe arguments for that are: i potentia oss of information which ma occur during aggregation process and ii not sufficient computationa power in 1994 However some reations are so intertwined that it is impossibe to anase them separate and if considered together the ma revea additiona vita information about the network Especia nowadas when it is so important to anase information diffusion in the goba network For eampe et us consider the foowing case: User A pubishes some information in this case a movie on YouTube Net user B who subscribes user A s channe on YouTube forwards this information and puts out the movie on the Facebook board Afterwards user C who is user B s Facebook friend and does not know user A at a watches this movie and sends it via Twitter to user D who knows neither A nor B If the data from each above sstems YouTube Facebook Twitter wi be anase as separate socia networks then it wi be hard to find how the information the movie has circuated from user A to user D However if the data from a three sstem coud be merged into one muti-aered socia network then the path from user A to user D can be quite easi traced The second scenario from the rea word is vira infection: A infects its spouse B then B goes to work and infects B s coeague C and fina C infects its spouse D at home Hence once again if on fami ties or work reations are anased it is hard to understand wh the disease has spread from A to D But if both networks are investigated simutaneous it is reative eas to spot the connection between A and D The data about actors activities and interactions coected in different IT sstems enabe to etract compe socia networks in which different tpes of reations eist A these reations can be anased separate or parae as the knowedge is not on hidden in individua aers but aso in cross aered reationships The information about what happens on one aer can infuence the actions on other ones So it is compete natura that recent scientists work ver hard to deveop a variet of measures and methodoogies which wi aow to anase muti-aered socia networks Therefore researchers tr to cope with muti-aered networks b anasing aers separate b means of eisting methods for singe-aered networks and then comparing the resuts using some correation measure eg Jaccard coefficient or cosine measure In [Sze 10] authors distinguished 6 different reation tpes between users of the massive mutipaer onine games First the anased the profie of each aer separate and afterwards the studied correations and overap between the etracted tpes of reations One of the interesting findings is that users tend to pa different roes in different networks From the structura perspective authors found that different tpes of interactions are characterised b different patterns of connectivit eg according to their stud power-aw degree distributions

7 Enccopedia of Socia Network Anasis and Mining Springer 2013 indicate aggressive actions Another eampe is the anasis on Fickr [Kazienko 11a] where authors distinguished as man as eeven tpes of reationships between users First the authors investigated aers separate and then used the correation measures to compare these networks Their main finding was that reationships ma be either semantica or socia driven Wong-Jiru at a in [Wong-Jiru 07] computed the number of metrics geodesic distance no of geodesic paths maimum fow point connectivit in/out-degree centrait coseness centrait fow betweenness centrait reachabiit densit node betweenness centrait and edge betweenness for each aer separate net the aggregated the network using weights caed composite network score assigned to each aer It is aso possibe to create man singe-aered socia networks from the muti-aered network and then to app some eisting methods to such structures In [Rodriguez 09] authors presented the path agebra to transform a muti-reationa network to man singereationa networks that are semantica-rich Another wa to dea with muti-aered networks is to deveop new methods for their anasis Such work is usua to some etent based on the eisting methods for singeaered networks The investigated topics in that research fied are among others: communit mining [Cai 05] [Mucha 10] ranking of network nodes [Zhuge 03] paths [Aeman-Meza 05] shortest paths [Bródka 11c] and unique paths finding [in 04] Magnani and Rossi in [Magnani 11] introduced the in-degree centrait and distance measures for their M-Mode Geffre et a in [Geffre 09] defined three measures: i terrorist socia connectedness across muti-aered affiiations ii their invovement in operation and iii their emergence during periods and at ocations of interest a in order to determine which terrorist is the critica one to cance trigger or impact the operations Kenned in [Kenned 09] aso tried to resove a simiar probem He deveoped methods which shoud be abe to identif potentia targets within a muti-aered socia network in order to cut the resource from them and maimize either the protection or disruption of the organization Further in this essa a number of the most popuar measures which are utiized to anase singe-aered structure were adapted to the muti-aered socia network These measures were previous presented anased and tested in [Bródka 10] [Bródka 11a] [Bródka 11b] [Bródka 11c] [Bródka 12] Muti-aered Neighbourhood To be abe to define compe measure for muti-aered socia networks a neighbourhood N of a given node on a given aer for muti-aered socia network MSN=<VE> shoud be carified as foows: N : E E Set N is equivaent to the simpe neighbourhood for reguar singe-aered SN Node\aer {uz} {uvz} {uvz} {z} {v} {vz} z {tu} {} {t} u {vz} {v} {}

8 Enccopedia of Socia Network Anasis and Mining Springer 2013 t {vz} {} {vz} v {tu} {u} {t} Tabe 1 Node neighbourhoods for each aer for MSN from Figure 1 A muti-aered neighbourhood of a given node with a minimum number of aers required α 1 α is a set of nodes which are neighbours of node on at east α aers in the MSN Five different versions of muti-aered neighbourhood ma be distinguished The first one is the muti-aered neighbourhood MN In α derived from the edges incoming to node in the foowing wa: MN In : E The vaue of MN In α denotes the set of neighbours that are connected to node with at east α edges ie on at east α aers of MSN For α=1 we need an edge on on one aer whie for α= if node MN In α then node must have edges to a given node on a eisting aers For the eampe MSN from Figure 1 MN In 1={uvz} MN In 2={uvz} MN In 3={z} Another muti-aered neighbourhood MN Out α respects on edges outgoing from node : MN Out : E For the MSN from Figure 1 we have MN Out 1={uvz} MN Out 2={uvz} MN Out 3={z} If we consider both incoming or outgoing edges on an aers then we obtain MN InOutAn α: MN InOutAn : E E Neighbourhood MN InOutAn α incudes nodes that have at east α incoming and α outgoing edges to and from node respective but these edges ma occur on different aers For the network from Figure 1 there are foowing sets: MN InOutAn 1={uvz} MN InOutAn 2={uvz} MN InOutAn 3={z} The net tpe of muti-aered neighbourhood MN InOut α is quite simiar to MN InOutAn α but it is more restrictive Each neighbour MN InOut α must have bidirectiona connections on at east α aers in MSN ie both the incoming and outgoing edge have to occur on the same aer to satisf the condition as foows: MN InOut : : E E In the eampe MSN Figure 1 we have MN InOut 1={uvz} MN InOut 2={vz} MN InOut 3={z} The fina fifth neighbourhood MNα is the east restrictive It takes into consideration an incoming or outgoing edges on an aer but the tota number of these aers shoud be at east α: MN : : E E

9 Enccopedia of Socia Network Anasis and Mining Springer 2013 In the eampe socia network the foowing neighbourhoods occur: MN1={uvz} MN2={uvz} MN3={uz} see Figure 1 Muti-aered neighbourhoods MN α for other nodes can be found in Tabe 2 Node\Metric MN 1 MN 2 MN 3 {uvz} {uvz} {uz} u {vz} {v} {} z {tu} {t} {} u {zv} {v} {} t {vz} {vz} {} v {tu} {tu} {} Tabe 2 Muti-aered neighbourhoods for a nodes from Figure 1 MNα is utiized for studies described in further sections of this essa However the other neighbourhood versions ma aso be used and it woud require on sma modifications of the measures proposed beow Note that there eists some correations between various neighbourhood versions: MN InOutAn α = MN In α MN Out α and MN InOut α MN InOutAn α MNα MN InOut α MN In α MNα and MN InOut α MN Out α MNα The smaest neighbourhood is MN InOut α whie the argest is MNα It means that MN InOut α is the most restrictive whereas MNα is the east Athough muti-aered neighbourhood is a structura measure it aso has a semantic meaning For eampe a arge MNα for a big vaue of α wi be an indicator that person is a communication hub Neighbourhood MN In α is more restrictive so it provides some more detaied semantic meaning Its arge vaue when α is high means that there is a ot of incoming interaction towards For eampe in the contet of the compan it wi mean that is a ine manager or another person whom a group of peope is reporting using different communication channes These channes can be treated as separate aers in the network On the other hand a high vaue of MN Out α for greater α points that person is probab responsibe for propagating information in the network eg a person responsibe for buetin distribution in the organisation It aso heps to investigate which communication channes are negected Such eampes can be mutipied Pease note that it can serve to investigate both positive and negative human behaviours For eampe a person within a compan with high MN Out α who is not responsibe for propagating information can be seen as a chatterbo who spend more time on communication than on doing the job Aso at the eve of the whoe network the muti-aered neighbourhood can be anased eg power-aw distribution of MNα depending on α means that not a tpes of reations are fu used and peope tend to focus on on few reation tpes and negect others A more indepth evauation of this measure can be found in [Bródka 12] Cross aered Custering Coefficient A cross aered custering coefficient CCCα was introduced and investigated in [Bródka 10a] and [Bródka 12] The idea was to aow cacuation of custering coefficient [Watts 98] for the muti-aered socia network MSN For a given node and s non-empt neighbourhood MNα cross aered custering coefficient CCCα is computed in the foowing wa:

10 where: Enccopedia of Socia Network Anasis and Mining Springer 2013 CCC in MN MN out MN 2 MN 216 inmnα the weighted indegree of node in the muti-aered neighbourhood MNα of node within the simpe singe-aered network <V E {}> ie within on one aer ; outmnα the weighted outdegree of node in the muti-aered neighbourhood MNα of node in the network containing on one aer If neighbourhood MNα={} then CCCα=0 The weighted indegree in MNα for a given node in the network <V E {}> containing one aer is the sum of a weights wz of edges <z> incoming to node from other nodes z that are from aer and beong to muti-aered neighbourhood MNα: in MN wz zmn 217 ikewise the weighted outdegree outmnα for a given node and neighbourhood MNα is the sum of a weights wz of the outgoing edges <z> that come from to s neighbours z on aer : out MN w z zmn 218 Note that if the sum of weights of outgoing edges for a given node is 1 this is the usua practice in socia network anasis see [Bródka 09] then CCCα is awas from the range [0;1] The vaue CCCα=1 happens when each neighbour MNα has outgoing reationships towards a other nodes zmnα and on to them The vaue of CCCα equas 0 when has on one neighbour ie MNα =1 or when none of the s neighbours MNα has no reationship with an neighbour zmnα For node t from the eampe socia network Figure 1 MNt1=MNt2={vz} but CCCt1=CCCt2=0 because there are no edges between v and z Due to MNt3={} we have CCCt3=0 For node z: MNz1={tu} MNz2={t} MNz3={} If weights of a edges equa 1 then CCCz1= 16 / 24 = 2 / 3 and CCCz2= 8 / 18 = 4 / 9 Since there is on one neighbour in MNz3 the vaue CCCz3=0 The formua for another measure muti-aered custering coefficient MCC as we as the two specia cases of cross aered custering coefficient CCCα were described in [Bródka 10a] One of them is muti-aered custering coefficient in etended neighbourhood MCCEN It is in fact the cross aered custering coefficient for on one aerα=1 ie MCCEN=CCC1 Muti-aered custering coefficient in reduced neighbourhood MCCRN presented in [Bródka 10a] is equivaent to the cross aered custering coefficient for a aers α= ie MCCRN=CCC Cross aered Custering Coefficient can aso be interpreted in the semantic contet For eampe peope who are in the professiona reationships eg co-workers prefer to communicate via e-mai as it enabes to keep track of their interactions It means that their custering coefficient wi be higher at one aer assuming that their neighbours communicate

11 Enccopedia of Socia Network Anasis and Mining Springer 2013 with each other than at the rest of the aers A different case occurs in tpica socia environments where peope tend to use different communication channes and it can resut in big vaue of CCC athough the custering coefficient for a singe aer ma not be particuar high It is a consequence of the fact that our neighbours can interact with each other using different communication aers phone ca e-mai tet messages etc Muti-aered Degree Centrait Apart from custering coefficient there eists another measure common used in socia network anasis degree centrait This measure for muti-aered socia network were introduced and eamined in [Bródka 11b] and [Bródka 12] Cross aered Degree Centrait The first muti-aered degree centrait is caed cross aered degree centrait CDC It is defined as a sum of edge weights both incoming to and outgoing from node towards mutiaered neighbourhood MNα divided b the number of aers and tota network members: CDC MN w where: w the weight of edge <> MN m 1 w Simiar to indegree and outdegree versions of degree centrait we can define cross aered indegree centrait CDC In α in the muti-aered socia network MSN: CDC In MN w m 1 and cross aered outdegree centrait CDC Out α: CDC Out MN w m 1 As in the case of the cross aered custering coefficient CCCα the vaue of CDCα direct depends on the parameter α which determines the muti-aered neighbourhood of a given socia network member Muti-aered Degree Centrait Version 1 The other three muti-aered degree centraities are not cacuated based on MNα but using the oca neighbourhood in particuar aer N The first of them MDC 1 is defined as a sum of s oca weighted degree centraities in each aer divided b the number of aers: 1 MDC N w N m 1 w The first muti-aered indegree centrait MDC 1In is defined as foows:

12 Enccopedia of Socia Network Anasis and Mining Springer m w MDC N In and the first muti-aered outdegree centrait MDC 1Out : 1 1 m w MDC N Out Muti-aered Degree Centrait Version 2 The net muti-aered degree centrait MDC 2 is a sum of s oca weighted degree centraities in each aer but in opposite to MDC 1 it is divided b the quantit of the union of s neighbourhoods from a aers MN m w w MDC N N Note that the union of neighbourhood sets from a aers for a given member is the same as the muti-aered neighbourhood MNα for α=1 ie 1 MN N The muti-aered indegree centrait MDC 2In is: MN m w MDC N In and the first muti-aered outdegree centrait MDC 2Out : MN m w MDC N Out Muti-aered Degree Centrait Version 3 The ast muti-aered degree centrait MDC 3 is quite simiar to MDC 2 but instead of the neighbourhood sets from a aers the sum of s oca weighted degree centraities in each aer is divided b the sum of neighbourhood quantities on each aer: N N N m w w MDC 1 3 The third version of first muti-aered indegree centrait MDC 3In :

13 Enccopedia of Socia Network Anasis and Mining Springer MDC In N m 1 w N and third version of first muti-aered outdegree centrait MDC 3Out : 3 MDC Out N m 1 w N Both Cross aered and Muti-aered Degree Centraities are hepfu in the interpretation of the socia network semantics The provide more information than MNα as the take into consideration not on the number of reations but aso their quait ie connection strengths Muti-aered Edge Muti-aered Path and Shortest Path Discover in MSN The ast two measures important for socia network anasis are coseness centrait and betweenness centrait Unfortunate the are quite compicated because to cacuate them the shortest paths are needed That is wh it became necessar to deveop an approach which wi aow to cacuate shortest paths in MSN This approach was introduced in [Bródka 11c] To etract the shortest paths from MSN some eisting agorithms can be utiized However since there is no singe edge between pairs of members in MSN it is required to cacuate the distance between an two neighbours in MSN Basica in most of the agorithms the shortest path is etracted based on the cost of transition from one node to another negative connection ie the greater cost the onger path vaue On the other hand in most socia networks the edges reations are considered to be positive so the epress how cose the two nodes are to each other the greater weight the shorter path That is wh each tpica positive weight w[01] assigned to edges in MSN has to be converted to negative distance vaues b subtracting w from one Figure 2 If there is no reationship between node and node then w = 0

14 Enccopedia of Socia Network Anasis and Mining Springer 2013 w Positive to negative 1 - w Figure 2 Transformation of positive coseness of the socia reationship into negative distance strangeness The distance strangeness d between and in MSN aggregated over a aers is cacuated using the weights w[01] of a eisting edges <> from to ie from a aers The sum of these weights is subtracted from the number of aers - and normaized b the tota number of aers in MSN as foows: d 1 w w 1 w Of course if MSN has been buit based on negative reations ike: apath hatred hostiit etc the weight does not have to be subtracted from 1 to obtain the distance because it aread refects the distance Muti-aered Edge Once we have the distance characterized the muti-aered edge ME between two nodes can be defined Overa three different versions of muti-aered edge ME ma be distinguished: 1 A muti-aered edge ME based on the number of aers is computed in foowing wa: ME : : A muti-aered edge ME from to eists if there are at east α edges <> from to in MSN ie : α The weight of such a muti-aered edge ME is equa to the distance from to d 2 A muti-aered edge ME d based on the distance between users is computed as foows:

15 Enccopedia of Socia Network Anasis and Mining Springer 2013 ME d : d 235 A muti-aered edge ME d from to eists if the distance from to is ower than or equa β: d β The weight of such an edge ME d is equa to the distance from to d 3 A muti-aered edge ME cacuated based on both the number of aers and the distance between users is cacuated in the foowing wa: ME : : d 236 A muti-aered edge ME from to eists if there eist at east α edges <> from to α in the MSN and simutaneous the distance from to is not : greater than β ie d β The weight of such a muti-aered edge ME is equa to the distance from to d Muti-aered Path The muti-aered path is a ist of nodes and from each of these nodes there is a muti-aered edge to the net node in the ist The ength of such path from node to v is a sum of distances of a muti-aered edges from the path The shortest muti-aered path is the path with the smaest ength SSPv among a paths eisting from node to v in muti-aered socia network Shortest Path Cacuation Now the concept presented above can be appied to cacuate shortest paths in two different approaches The first approach is to cacuate muti-aered edges and their ength at the beginning of cacuation Net an common used shortest path cacuation agorithm can be utiized in the same wa ike for reguar socia networks For eampe the tpica Dijkstra agorithm provides a shortest paths with their engths SSPv outgoing from a singe node If appied to muti-aered edges it is as foows: DAP Dijkstra Agorithm with Preprocessing Input: MSN=<VE> V source node Output: the ist of shortest paths outgoing from to a vv and their engths SSPv Cacuate distances dz for each zv Cacuate a muti-aered edges ME P= {} empt set /* P set of nodes aread processed */ T=V /* T set of nodes to process */ SSPv= for a vv SSP=0 pred=0 /* predv predecessor of seected */ /* node v on the path from to v */

16 Enccopedia of Socia Network Anasis and Mining Springer 2013 whie PV do begin end v=argmin{sspv vt} P:=Pv T:=T\v if SSPv= then end whie for wn out v do if SSPw> SSPv+ dvw then begin SSPw:= SSPv+ dvw predw=v end The second approach is to modif the seected agorithm and process edges on the f This approach is sight better because additiona muti-aered measures can be added inside for eampe muti-aered neighbourhood An eampe of the Dijkstra agorithm modified into muti-aered Dijkstra agorithm is presented beow: MDA - Muti-aered Dijkstra Agorithm Input: MSN=<VE> V source node Output: the ist of shortest paths outgoing from to a vv and their engths SSPv P={} empt set T=V SSPv= for a vv SSP=0 pred=0 whie PV do begin v=argmin{sspv vt} P:=Pv T:=T\v if SSPv= then end whie for wmn out vα do if SSPw> SSPv+ dvw then begin SSPw:= SSPv+ dvw predw=v end end

17 Enccopedia of Socia Network Anasis and Mining Springer 2013 Now when it is possibe to cacuate shortest paths for MSN we can cacuate coseness centrait and betweenness centrait b utiizing reguar equations deveoped for SSN The compete evauation of described above soutions can be found in [Bródka 11c] Ke Appications Muti-aered socia networks can be used in an environment that contains more than one tpe of reationships In socioog various reationships can be etracted from separate human surroundings and interviews coected in famiies among acquaintances or at work Simutaneous most of us are aso users of compe IT sstems and onine services Various human traces activit tpes and achievements stored in these sstems can be anased in order to etract eisting reationships and each activit tpe or interaction ma be depicted b a separate aer in the muti-aered socia network The on prerequisite to estabish the muti-aered socia network is the common set of actors nodes identified in ever environment or sstem Muti-aered measures ma be used in an appication area of SNA together with reguar structura measures eg in coective cassification of nodes ink prediction socia communit etraction and anasis evoution modeing mutipurpose simuations on compe socia networks and man other domains Muti-aered socia networks ma be used in recommender sstems [Kazienko 11a] more comprehensive identification of ke actors and roe mining mutieve modeing coaborative Information Retrieva vira marketing spread of information as we as anasis of socia infuence and trust eve Future Directions There are severa interesting directions for further research in the domain of muti-aered socia network anasis In particuar the ma focus on visuaisation techniques deveopment of more compe structura measures that coud respect the importance weights of individua aers socia communities in the aer or even singe nodes whie their cacuation efficient agorithms for cacuation of more than one measure simutaneous and their incrementa update understandabe interpretation of combined measures appication of muti-aered measures in communit detection or prediction of network evoution Cross-References Centrait Measures Coective Cassification Coective Cassification Structura Features Current Metrics and Aternative Measures of Socia Media Effectiveness Eigenvaues Singuar Vaue Decomposition Etracting Socia Networks from Data Importance Assessment

18 Enccopedia of Socia Network Anasis and Mining Springer 2013 Independent Component Anasis Mapping Onine Networks Mutieve Modeing Network Representations of Compe Data Path-Based and Whoe Network Measures Principa Component Anasis Process of Socia Network Anasis Ranking Methods for Networks Recommender Sstems Retrieva Modes Roe Identification of Socia Networkers Roe Mining Simiarit Metrics on Socia Networks Socia Infuence Anasis Trust Metrics and Reputation Sstems Acknowedgements [optiona] The work was partia supported b the Poish Ministr of Science and Higher Education the research project References [Abraham 12] Abraham I Chechik S Kempe D Sivkins A ow-distortion Inference of atent Simiarities from a Mutipe Socia Network 2012 arxiv: v2 [Aeman-Meza 05] Aeman-Meza B Haaschek-Wiener C Arpinar I B Ramakrishnan C Sheth A P Ranking Compe Reationships on the Semantic Web IEEE Internet Computing vo 9 no pp [Bródka 09] Bródka P Musiał K Kazienko P A Performance of Centrait Cacuation in Socia Networks CASoN 2009 IEEE Computer Societ 2009 pp [Bródka 10] Bródka P Musiał K Kazienko P A Method for Group Etraction in Compe Socia Networks WSKS 2010 CCIS 111 Springer 2010 pp [Bródka 11a] Bródka P Fiipowski T Kazienko P An Introduction to Communit Detection in Muti-aered Socia Network WSKS 2011 Springer CCIS pp [Bródka 11b] Bródka P Skibicki K Kazienko P Musiał K A Degree Centrait in Muti-aered Socia Network CASoN 2011 IEEE Computer Societ 2011 pp [Bródka 11c] Bródka P Stawiak P Kazienko P Shortest Path Discover in the Muti-aered Socia Network ASONAM 2011 IEEE Computer Societ 2011 pp [Bródka 12] Bródka P Kazienko P Musiał K Skibicki K Anasis of Neighbourhoods in Muti-aered Dnamic Socia Networks Internationa Journa of Computationa Inteigence Sstems vo 5 no pp

19 Enccopedia of Socia Network Anasis and Mining Springer 2013 [Cai 05] Cai D Shao Z He X Yan X Han J Communit Mining from Muti-reationa Networks ECM PKDD 2005 [Cantador 06] Cantador C Castes P Mutiaered Semantic Socia Network Modeing b Ontoog-Based User Profies Custering: Appication to Coaborative Fitering EKAW 2006 NAI Springer 2006 pp [Domingos 03] Domingos P Prospects and Chaenges for Muti-Reationa Data Mining ACM SIGKDD Eporations Newsetter vo 5 no pp [Entwise 07] Entwise B Faust K Rindfuss RR Kaneda T Networks and Contets: Variation in the Structure of Socia Ties The American Journa of Socioog vo 112 no pp [Fienberg 85] Fienberg S E Meer M M Wasserman SS Statistica Anasis of Mutipe Sociometric Reations Journa of American Statistica Association vo 80 no pp [Garton 97] Garton Hathorntwaite C Weman B Studing Onine Socia Networks Journa of Computer-Mediated Communication vo 3 no [Geffre 09] Geffre J Deckro RF Knighton SA Determining Critica Members of aered Operationa Terrorist Networks The Journa of Defense Modeing and Simuation: Appications Methodoog Technoog vo 6 no pp [Gobeck 06] Gobeck J Hender J FimTrust: Movie Recommendations Using Trust in Web-based Socia Networks CCNC 2006 IEEE Conference Proceedings vo pp [Hami 06] Hami JT: Anasis of aered Socia Networks Department of the Air Force Air Force Institute of Technoog USA PhD Dissertation 2006 [Hami 07] Hami JT Deckro RF Wie VD Renfro RS Gaines osses and Threshods of Infuence in Socia Networks Internationa Journa of Operationa Research vo 2 no pp [Hathornthwaite 99] Hathornthwaite C A Socia Network Theor of Tie Strength and Media Use: A Framework For Evauating Muti-eve Impacts of New Media Technica Report UIUCIS 2002/1+DKRC Universit of Iinois at Urbana-Champaign 1999 [Kazienko 10] Kazienko P Bródka P Musiał K: Individua Neighbourhood Eporation in Compe Muti-aered Socia Network IEEE/WIC/ACM 2010 IEEE Computer Societ Press 2010 pp 5-8 [Kazienko 11a] Kazienko P Musia K Kajdanowicz T Mutidimensiona Socia Network and Its Appication to the Socia Recommender Sstem IEEE Transactions on Sstems Man and Cbernetics vo 41 no pp [Kazienko 11b] Kazienko P Musia K Kuka E Kajdanowicz T Bródka P: Mutidimensiona Socia Network: Mode and Anasis ICCCI 2011 NCS Springer 2011 pp [Kazienko 11c] Kazienko P Kuka E Musia K Kajdanowicz T Bródka P Gaworecki J: A Generic Mode for Mutidimensiona Tempora Socia Network ICeND2011 CCIS Springer 2011 pp 1-14 [Kenned 09] Kenned KT Snthesis Interdiction and Protection of aered Networks Department of The Air Force Air Force Institute of Technoog USA PhD Dissertation 2009

20 Enccopedia of Socia Network Anasis and Mining Springer 2013 [in 04] in S Interesting Instance Discover in Muti-reationa Data IAAI 2004 MIT Press 2004 pp [Magnani 11] Magnani M Rossi The M-mode for Muti aer Network Anasis ASONAM 2011 IEEE Computer Societ 2011 pp 5-12 [McPherson 01] McPherson M Smith-ovin Cook JM Birds of a Feather: Homophi in Socia Networks Annua Review of Socioog vo pp [Minor 83] Minor M J New Directions in Mutipeit Anasis Appied Network Anasis Sage Pubications 1983 pp [Monge 03] Monge P R Contractor NS Theories of Communication Networks Oford Universit Press Oford 2003 [Mucha 10] Mucha PJ Richardson T Macon K Porter MA Onnea J-P Communit Structure in Time-Dependent Mutiscae and Mutipe Networks Science vo 328 no pp [Newman 10] Newman MEJ Networks: An Introduction Oford Universit Press 2010 ISBN-10: ISBN-13: [Padgett 93] Padgett JF Anse CK Robust Action and the Rise of the Medici The American Journa of Socioog vo 98 no pp [Rodriguez 07] Rodriguez MA A Muti-reationa Network to Support the Schoar Communication Process Internationa Journa of Pubic Information Sstems vo 2007 no pp [Rodriguez 09] Rodriguez MA Shinavier J Eposing Muti-Reationa Networks to Singe-Reationa Network Anasis Agorithms Journa of Informetrics vo 4 no pp [Schneider 11] Schneider K Rainwater C Poh EA Assessing Muti-aered Socia Networks Using Reiabiit Modes RAMS 2011 pp [Sze 10] Sze M ambiotte R Thurner S Mutireationa Organization of arge-scae Socia Networks in an Onine Word Proceedings of the Nationa Academ of Science USA vo 107 no pp [Wasserman 94] Wasserman S Faust K Socia Network Anasis: Methods and Appications Cambridge Universit Press 1994 [Watts 98] Watts D J Strogatz S H Coective dnamics of 'sma-word' networks Nature pp [Weman 96] Weman B Saaff J Dimitrova D Garton Guia M Hathornthwaite C Computer Networks as Socia Networks: Coaborative Work Teework and Virtua Communit Annua Review of Socioog vo 22 no pp [Wong-Jiru 07] Wong-Jiru A Coombi J Suzuki Mis R Graph Theoretica Anasis of Network Centric Operations Using Muti-aer Modes CSER 2007 artice #55 [Zhuge 03] Zhuge H Zheng Ranking semantic-inked network WWW Recommended Reading [optiona]