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Information about Socialmatchmaking

Published on May 31, 2008

Author: epokh



The problem of matchmaking in electronic social networks is formulated as an optimization problem.
In particular, a function measuring the matching degree of fields of interest of a search profile with
those of an advertising profile is proposed.

Mathematical model of interest matchmaking in electronic social networks Andreas de Vries∗ (Dated: March 31, 2007) The problem of matchmaking in electronic social networks is formulated as an optimization problem. In particular, a function measuring the matching degree of fields of interest of a search profile with those of an advertising profile is proposed. ACM Classification: I.2.4: Semantic networks, H.3.5: Web-based services, J.4: Sociology, G.2.3: Applications of discrete mathematics, C.2.4: Distributed applications Keywords: matchmaking algorithm, matching degree, electronic social network, matching fields of interest arXiv:cs/0703118v2 [cs.CY] 24 Mar 2007 I. INTRODUCTION a mathematical model of the matchmaking problem as an optimization problem is proposed, especially the data Social activities in electronic networks play an increas- structure of search and advertising profiles, the search ingly important role in our every-day lives. We are space, and the matching degree as the objective function. exchanging important information via electronic mails, A short discussion concludes the paper. wikis, web-based forums, or blogs, and meet new friends or business contacts in Internet communities and social network services. Parallel to this growing socialization of II. ELECTRONIC SOCIAL NETWORKS the World Wide Web, the requirements on the electronic services become more ambitious. Huge data quantities have to be processed, user-friendly interfaces are to be A social network consists of a finite set of actors and designed, and more and more sophisticated computations the direct relations defined on them. An actor here may must be implemented to offer complex solutions. be an individual, a group, or an organization, and the direct relation between two actors indicates that they di- This paper studies a special aspect of social network rectly interact with each other, have immediate contact, services, the matchmaking problem. In essence it asks, or are connected through social familiarities, such as ca- given a search profile, for advertising profiles matching sual acquaintance or familial bonds [2, 12]. Thus a social it best. This problem is in principle well-known in Grid network is naturally represented by a graph in which each computing, where computational tasks are seeking for node represents an actor and each edge a direct relation. appropriate resources such as CPU time and memory Empirically, the mean number of direct relationships of space on different computers. In electronic social net- an individual in a biological social network depends on works, however, the problem is more general because not the size of the neocortex of its individuals; the maximum only specified attribute ranges are to be compared but size of such relationships in human social networks tends more or less vaguely describable interests. to be around 150 people (“Dunbar’s number”) and the The aim of this paper is to formulate a mathemati- average size around 124 people [6]. cal model for the problem of matchmaking of attribute ranges and fields of interests in electronic social net- Since the popularization of the World Wide Web works. It tackles the following fundamental questions. in the middle of the 1990’s, there emerged several How can an appropriate system and its data structure Internet social networks, maintained by social net- be designed? How is the mathematical formulation of a work services such as “circle of friends” like friendster matching problem as an optimization problem? In par- ( ), MySpace ( ), or ticular, what is its search space, what is its objective orkut (, as platforms for business profes- function? Whereas the idea to use a fuzzy function to sionals like XING ( ), or virtual worlds like calculate the matching degree of two numerical ranges Second Life ( ). Internet social networks may suggest itself, how could a function calculating a are instances of electronic social networks. matching degree of two fields of interest look like? One In this paper, an electronic social network is defined of the central results of this paper is the proposal of a pre- as a network of at least three human individuals or or- cise definition of such a function computing this matching ganizations which use essentially, albeit not exclusively, degree and the presentation of a concrete example. electronic devices and media to get in contact and ac- The paper is organized as follows. After a definition quaintance, to meet new partners, to communicate, and of electronic social networks is given in the next section, to exchange information. Examples of electronic social networks are Internet social networks, as well as video- conference sessions and conference calls, especially if they serve to meet new people as in party lines, or as long ∗ South Westphalia University of Applied Sciences, Haldener Straße as they admit spontaneous communication between each 182, D-58095 Hagen, Germany, e-Mail: member of the network.

2 III. THE MATCHMAKING PROBLEM 1. Definitions In computer science, the term matching in general A profile consists of its owner, being an actor of the refers to the process of evaluating the degree of similarity electronic social network, a list of attributes of a given or of agreement of two objects. Each object is character- set A together with their values, a list of attribute sten- ized by a set of properties or attributes, which in many cils where each stencil represents a pair of an attribute systems are given by name-value pairs [3]. Matching name and its value range, and a list of fields of interest plays a vital role in many areas of computer science and specifying their respective levels of interest. Attributes communication systems. For instance, it is studied for re- are properties of the profile owner such as age, height, source discovery and resource allocation in grid comput- weight, eye color, or hair color. and we therefore sub- ing where matchmaking services are needed to interme- sume them under the class “Owner” (Fig. 1). In princi- diate between resource requesters and resource providers [1]. Other examples are given by the problem of match- Interest ing demands and supply of business or personal profiles Owner 1 ∗ field: String in online auctions, e-commerce, recruitment agencies, or id: String Profile —— level: [−1, 1] 1 ∗ dating services. height: integer ——— 1 ∗ eye color: String —— Stencil ... attribute: String range: T A. Profiles FIG. 1: UML diagram of the data structure of a profile and its relationship to the owner’s attributes, the attribute stencils and In most matching problems, the objects under consid- the fields of interest. An attribute stencil consists of an owner’s eration take asymmetric roles, viz., some try to search attribute name and its (searched or advertised) range. for information or request for a service, others try to ad- vertise information or provide a service. A single object ple, there are two different types of attributes, subsumed may naturally do both activities at a time, in electronic in the two disjoint sets N and D such that the set A of social networks this even is the usual case. In the sequel attributes separates as we will therefore more accurately consider the matching of a search profile, containing the requested information, A = N ∪ D. (1) and an advertising profile presenting the provided infor- mation. The set N consists of the numerical attributes of the owner which take integer or real numbers as values, the Given a specific search profile, the matchmaking prob- set D consists of discrete non-numerical values. (The lem then is to find those advertising profiles which match difference between numerical and non-numerical is not it best, in a sense to be specified in the sequel. Gener- sharp, for instance a string could well be considered as alizations of this problem ask for best global matchings, numerical via a symbol code, as well as non-numerical given a whole set of search profiles and a set of adver- since it seldom makes sense to multiply or divide strings; tising profiles. For instance, the global pairwise match- in most cases, strings are better considered as non- making problem seeks pairs of search/advertising-profiles numerical.) such that the entity of the pairs matches the best under Correspondingly, the stencil of an attribute is deter- the constraint that any profile is member of at most one mined by the attribute’s name and its range, being of a pair, the global multiple matchmaking problem searches certain set called Type T , for possibly multiple combinations of search and adver- tising profiles which as a whole match the best. The T = N ∪ D. (2) pairwise version of the problem typically occurs for dat- ing services or classical marriage matchmaking tasks, where N denotes the set of ranges for the numerical whereas the multiple version appears in grid computing attributes, or in brokering interest groups. In this paper we will focus on the local version of the N ⊂ {[a, b] : −∞ ≦ a, b ≦ ∞}, (3) matchmaking problem, i.e., finding an optimum advertis- i.e., N is a set of closed intervals [a, b] ⊂ R, and D ing profile to a specified search profile. Thus the match- denotes the set of ranges for the discrete non-numerical making problem is an optimization problem, and to for- attributes, mulate it precisely we have to specify the search space and the objective function. The search space will turn D ⊂ {E : E is a finite set or enum}, (4) out to be the set of pairs of the fixed search profile and the advertising profiles, and the objective function will i.e., D is a finite set or enum, specified by the respec- be a function measuring the “matching degree.” We will tive owner attributes determined by the system model. work out these notions in the next sections. We allow the empty set ∅ as null element in N and D.

3 If a given range R ∈ T contains only one element, say is the set of searched fields of interest with their desired R = {x}, then the stencil is often shortly written “p = x” levels, with the given mapping ls : Is → [−1, 1]). Note instead of “p ∈ R.” If, on the other hand, R = [x, ∞] that each of the pairs (p, Rs ), (p, Es ), (p, ls ) can be easily then we may write “p > x” instead of p ∈ R. For in- implemented as a table or a hash map. Analogously, an stance, “height = 180” means “height ∈ [180, 180],” or advertising profile is given by “height ¿ 180” means “height ∈ [180, ∞].” On the other hand, a field of interest is a name-value a = na ∪ da ∪ ia , (9) pair specifying the field itself as well as its level ranging on a scale from −1 to 1, coded by the interpolation of where the three sets are defined the same way as in the the following table, search case, with the index ‘s’ (for “search”) replaced by ‘a’ (for “advertising”). Level Meaning −1 aversion Example 1. In grid computing, a main matching prob- (5) lem is resource discovery and resource allocation [4, 8]. 0 indifference Assume a toy grid consisting of two resource providers 1 enthusiasm Haegar and Bond, and two resource requests by some The set of fields of interests is denoted by I and is a computational process. In our terminology, Bond and subset of words of a specified alphabet Σ, Haegar each offer an advertising profile, whereas the requests are represented by search profiles. Moreover, I ⊆ Σ∗ (6) in the widely used matchmaking framework Condor-G Usually, Σ is the set of ASCII or Unicode symbols. The [5, 7, 9, 10, 11], the profiles are called ClassAds (classified set I determines the set of all fields of interests available advertisements). Let us assume the profiles according to to the system. Depending on the system design, it may the following tables. be a fixed set of words, or an arbitrary word over the alphabet Σ. Search Profiles owner = owner = B. The search space CPU ≥ 1.6 GHz memory ≥ 2 GB memory ≥ 1 GB Given a set S of search profiles s and a set A of adver- tising profiles a as input, the search space S of a global Advertising Profiles matchmaking problem is given by all pairs of search and owner = owner = advertising profiles, i.e., S g = S × A. In this paper, how- CPU ≤ 3.6 GHz CPU = 2.5 GHz ever, we are considering the local matchmaking problem, memory ≤ 4.0 GB memory = 1.0 GB given a single search profile s, i.e., S = {s}, and the search space In each column of a profile there is listed its owner and S = {s} × A(s). (7) some attributes and their values. where A(s) = {a ∈ A : owner(a) = owner(s)}. A search Example 2. Assume a small social network for pooling profile s itself is a set of the given attribute stencils ns , interest groups, consisting of three persons, Alice, Bob, ds , and of fields of interest is , and Carl, who provide search and advertising profiles ac- cording to the following tables. s = ns ∪ ds ∪ is , (8) where Search Profiles ns = {(p, Rs (p)) : p ∈ Ns } owner = Alice owner = Carl is the set of attribute-range pairs, with the given map- age ∈ [20,40] age ∈ [20,30] ping Rs : Ns → N from the set Ns of the searched nu- height > 180 merical attributes to their associated desired ranges (Rs tennis = 1.0 basketball = 1.0 associates to each numerical attribute p in Ns an interval chess = 0.5 Rs (p) = [a, b]), Advertising Profiles ds = {(p, Es (p)) : p ∈ Ds } owner = Alice owner = Bob owner = Carl is the set of attribute-set pairs, with the mapping Es : age = 25 age = 26 age = 31 Ds → D from the given set Ds of searched discrete at- height = 165 height = 182 height = 195 tributes to their desired sets or enums, (Es associates tennis = 1.0 tennis = 0.5 basketball = 1.0 discrete nonnumerical attribute p a set Es (p)), and chess = 0.5 basketball = −1.0 is = {(p, ls (p)) : p ∈ Is } basketball = 0.5

4 In each column of a profile there is listed its owner, some with f = 0 meaning “total mismatch” and f = 1 mean- attributes and their values, and the fields of interests ing “perfect match.” In general, the matching degree will with their levels. For instance, Alice looks for someone be the weighted sum of several partial matching degrees, between 20 and 40 years of age being enthusiastic in ten- one for each property separately. Moreover, the match- nis and having some penchant to chess, whereas Carl ing degree of an attribute is calculated in a different way seeks a tall person in the 20’s with highest preference for than the matching degree of a field of interest. Proposals basketball. Looking at the advertising profiles in this so- for these different kinds of matchings are introduced in cial network, one sees that Alice may contact Bob, but the following paragraphs. Carl cannot find an ideal partner in this community. On the other hand, Alice would be a “better” partner for Carl than Bob, since she is partly interested in basket- 1. Matching degree of an attribute range ball. Formally Alice’s search profile, for instance, is given as follows. The sets for the searched attributes and fields A function measuring the matching degree of ranges of interest are of an attribute has to quantify how a given advertised attribute stencil [aa , ba ] fits into the stencil pattern given Ns = {age}, Ds = ∅, Is = {tennis, chess}, (10) by the corresponding range in the search profile. In case of a numerical attribute, the stencil is given by a closed the mapping Rs is given by interval [a, b] ∈ D in case of a discrete-value attribute it is a set or enum E. [20, 40] if p = “age,” Rs (p) = (11) a. Numerical attribute intervals matching. To de- ∅ otherwise. termine the matching degree of a searched value range and the mapping ls is given by the table [as , bs ] with a given advwertised value range x ∈ R, we define the fuzzy step function he (x) = he (x; a, b) with p tennis chess a ≦ b and 0 < e < 1, as (12) ls (p) 1.0 0.5   ae − 1−e if (1 − e)a < x ≦ a,  x e  1 if a < x ≦ b,  The mapping Es does not exist since Ds = ∅. To sum he (x; a, b) = (18) up, Alice’s search profile is given by  1+e − be if b < x ≦ (1 + e)b,  e x  0 otherwise.  sAlice = {(age, [20, 40])} (Fig. 2). The parameter e is called the fuzzy level . It ∪ {(tennis, 1.), (chess, .5), (basketball, .5)}, (13) denotes the relative length of the fuzzy transition region. The smaller e, the narrower this region, and the more Note in particular that ns,Alice = ∅. On the other hand, accurate an advertised attribute value must fit into the the advertising profiles read searched interval. In the limit e → 0, the function he is A(sAlice ) = {aBob , aCarl } (14) ae be 1 where aBob = {(age, [26, 26]), (height, [182, 182])} a b x ∪ {(tennis, .5), (basketball, −1.)}. (15) FIG. 2: The fuzzy step function he (x) of Eq. (18). aCarl = {(age, [31, 31], (height, [195, 195])} the step function, and for a → −∞ or b → ∞, it tends ∪ {(basketball, 1.)}. (16) to one of the Heaviside step functions Hb (−x) or Ha (x), respectively. With the definitions If, for instance, the searched attribute is “height > sB = (sAlice , aBob ), sC = (sAlice , aCarl ), (17) 180” and an advertised attribute is “height = 165” then for a fuzzy level of e = 10%, we have the search space S = {sAlice } × {aBob , aCarl } = {sB , sC } 165 .9 consists of two feasible solutions. h0.1 (165; 180, ∞) = − = 0.16, (19) 18 .1 i.e., the matching degree is 16.7%. Then the matching degree of two numerical ranges [as , bs ] as search range C. A matching degree function and [aa , ba ] as advertised range are given by The matching degree of a search profile and an adver- mn ([as , bs ], [aa , ba ]; e) = tising profile is a real number f , typically 0 ≦ f ≦ 1, max [he (ba ; as , bs ), he (bs ; aa , ba )] . (20)

5 b. Features matching finite sets or enums. If the val- ues of specific attribute are constrained to be of a finite mi set, or an enum, say E then the matching degree is deter- 1 mined by the Boolean characteristic function χE defined 0.75 0.5 1 0.25 0.5 by 0 -1 0 -0.5 la 0 -0.5 1 if x ∈ E, 0.5 χE (x) = (21) ls 1 -1 0 otherwise. FIG. 3: The matching degree function m = m(ls , la ) in Eq. (24). If the searched attribute, for instance, is “name ∈ {‘Smith’, ‘Taylor’}” and the advertised attribute is “name = ‘Tailor’” then E = {‘Smith’, ‘Taylor’} and 3. The total matching degree function χE (‘Tailor’) = 0, i.e., the matching degree is zero. Since the owner of an advertising profile can advertise at most Putting together all partial matching degrees consid- one value for an attribute, we have ered above, we have to construct a function f : S → [0, 1] md (E, {x}) = χE (x). (22) as a weighted sum of them. We notice that any s ∈ S rep- resents a feasible solution of the matchmaking problem and has the form s = (s, a) where s is the given search 2. Matching degree of a field of interest profile (8) and a is one of the given advertising profiles (9) in the network. Then f defined for each s ∈ S by First we notice that the matching degree as a function mn (Rs (p), Ra (p)) of the levels of interest ls for the search profile and la for f (s) = the advertising profile must be asymmetric. For instance, |Ns | + |Ds | + |Is | p∈Ns if ls = 0 and la = 1, i.e., the search is indifferent with md (E(p), Ta (p)) respect to the field of interest, and the advertising profile + |Ns | + |Ds | + |Is | has la = 1, then the matching degree should be greater p∈Ds than 0, but if the search requires ls = 1 and la = 0 then mi (ls (p), la (p)) the matching degree should be zero. In the first case, the + (27) |Ns | + |Ds | + |Is | searcher is indifferent about the field of interest, in the p∈Is second case he demands high interest. where Ra (p) and E(p) denote the attribute ranges of the Definition 3. An interest matching degree function is attribute p, la (p) is the advertised interest level of the a function m : [−1, 1]2 → [0, 1] such that the following field of interest p, and the vertical bars | · | embracing a conditions are satisfied. set denote the number of its elements. Thus for the computation of the matching degree, the (ls , la ) (x, x) (0, ±1) (±1, 0) attributes and fields of interest of the search search pro- 1 (∀x ∈ [−1, 1]) (23) m(ls , la ) 1 2 0 file s are leading, i.e., it is s which determines what is tried to be matched. The, if an attribute p of the search The first condition expresses the perfect matching of the profile does not occur in the advertising profile, then the diagonal, the second the search indifference, and the last matching degree functions mn (p) and md (p) vanish by the search necessity. definition. If, however, a searched field of interest p ∈ is A possible matching degree function is given by does not occur in the advertised profile, then it is the level la (p) which vanishes by definition. Note the crucial dif- mi (ls , la ) = max[ϕ(ls , la ), 0] (24) ference between null values of attributes and null values where of fields of interest in the advertising profile: searched at- tributes are mandatory, and at least with respect to this (c2 − 1)(x − y)2 attribute there is a complete mismatch; if a field of in- ϕ(x, y) = 1 − (25) c2 − 2 + (x − cy)2 terest, however, does not occur in the advertised profile, it is indifferent to its owner, but depending on the level with √ of interest in the search profile, the matching degree may 1+ 7 be positive nonetheless. c= ≈ 1.823. (26) 2 Example 2 (revisited). For Alice’s search space we By construction, m(ls , la ) satisfies the conditions in (23) have the two solutions (17), i.e., and therefore is an interest matching degree function. It is asymmetric with respect to its arguments, since we mn ([20, 40], [26, 26]) mi (1, .5) + mi (.5, 0) 2 2 have m(ls , la ) = m(la , ls ) if and only if ls = la . On f (sB ) = + 3 3 the other hand, it is an even function, i.e., m(ls , la ) = 1 .5636 + .6308 m(−ls , −la ). = + = 0.7315 (28) 3 3

6 and of all weights wn : N → R+ , wd : N → R+ , wi : N → R+ . mn ([20, 40], [31, 31]) mi (1., 0) + mi (.5, 0) f (sC ) = + 3 3 IV. DISCUSSION 1 0 + .6308 = + = 0.5436 (29) 3 3 In this paper, a mathematical model of the match- Hence Bob’s advertising profile has a matching degree making of search and advertising profiles in an electronic of 73.15% with Alice’s search profile, whereas Carl’s social network is proposed. Basing on the data structure matches it only by 54.36%. described by Figure 1 and distinguishing between match- making of attribute ranges via stencils and matchmaking Notice that the objective function (27) is constructed of fields of interests via comparison, the matchmaking in such a way that each searched item p of a search profile problem is formulated as an optimization problem, with has equal weight. If, however, each item should have the search space consisting of a fixed search profile and its own weight w(p), then the objective function (30) is several advertising profiles as in Eq. (7) and the match- easily be modified to ing degree as its objective function in Eq. (27). The main difficulty is to define a function measuring adequately the wn (p) matching degree of two fields of interest and obeying the f (s) = mn (Rs (p), Ra (p)) necessary conditions listed in Definition 3. A proposed wtot p∈Ns solution is the function given in Eq. (24) and depicted in wd (p) Figure 3. The implementation of a matchmaking service + md (E(p), Ta (p)) wtot in an electronic social network basing on this matching p∈Ds optimization is straightforward. wn (p) + mi (ls (p), la (p)) (30) wtot p∈Is Acknowledgments where wtot is the total sum I am indebted to Thomas Kowalski, Valerie Rinke, and wtot = wn (p) + wd (p) + wi (p) (31) Volker Weiß for valuable discussions. p∈Ns p∈Ds p∈Is [1] X. Bai, H. Yu, , Y. Ji, and D. C. Mari- [7] O. Lodygensky, G. Fedak, F. Cappello, V. N´ri, M. Livny, e nescu. ‘Resource matching and a matchmaking ser- and D. Thain. Xtremweb & condor sharing resources vice for an intelligent grid’. International Jour- between internet connected condor pools. In CCGRID, nal of Computational Intelligence, 1(3):197–205, 2004. pages 382–389. IEEE Computer Society, 2003. xbai/ICCI04.pdf . [8] R. Prodan and T. Fahringer. Grid Computing: Ex- [2] J. A. Barnes. ‘Class and Committees in a Norwegian periment Management, Tool Integration, and Scientific Island Parish’. Human Relations, 7(1), 1954. Workflows. Springer-Verlag, Heidelberg Berlin, 1st edi- [3] A. de Vries. ‘XML framework for concept de- tion, 2006. scription and knowledge representation’, 2004. [9] R. Raman, M. Livny, and M. H. Solomon. Match- making: Distributed resource management for high [4] I. Foster and C. Kesselman, editors. The Grid 2. throughput computing. In HPDC, pages 140–, 1998. Blueprint for a New Computing Infrastructure, Amster- . dam, 2004. Elsevier. [10] D. Thain and M. Livny. Building reliable clients and [5] F. J. Gonz´lez-Casta˜o, J. Vales-Alonso, M. Livny, a n services. In Foster and Kesselman [4], pages 285–318. E. Costa-Montenegro, and L. E. Anido-Rif´n. Condor o [11] D. Thain, T. Tannenbaum, and M. Livny. Distributed grid computing from mobile handheld devices. Mobile computing in practice: the condor experience. Concur- Computing and Communications Review, 7(1):117–126, rency - Practice and Experience, 17(2-4):323–356, 2005. 2003. . [12] S. Wasserman and K. Faust. Social Network Analysis. [6] R. A. Hill and R. I. M. Dunbar. ‘Social net- Cambridge University Press, Cambridge, 1994. work size in humans’. Human Nature, 14(1), 2002. Dunbar networks.pdf .

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