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Long-Run Integration in Social Networks∗ Sergio Currarini† Matthew O. Jackson‡ Paolo Pin§ This Draft: January 12, 2011 Abstract We study network formation where nodes are born sequentially and form links with pre- viously born nodes. Connections are formed through a combination of random meetings and through search, as in Jackson and Rogers (2007). A newborn’s random meetings of existing nodes are type-dependent and the newborn’s search is then by meeting the neighbors of the randomly met nodes. We study “long-run integration,” which requires that as a node ages suﬃciently, the type distribution of the nodes connected to it approaches the overall type– distribution of the population. We show that long-run integration occurs if and only if the search part of the network formation process is unbiased, and that eventually the search process dominates in terms of the new links that an older node obtains. Integration, however, only oc- curs for suﬃciently old nodes, and the aggregate type-distribution of connections in the network still reﬂects the bias of the random process. We illustrate the model with data on scientiﬁc citations in physics journals. 1 Introduction Homophily patterns in networks have important implications. For example, citation patterns across literatures can aﬀect whether important ideas developed in one literature eventually diﬀuse into another. Homophily also aﬀects a variety of behaviors and the welfare of individuals connected in social networks.1 In this paper we analyze a model that provides new insight into patterns and the emergence of homophily, and illustrate its ﬁndings with an application to a network of scientiﬁc citations. ∗ This supersedes “Overlapping Network Formation”, Currarini, Jackson and Pin (2006), which also appeared as a chapter in Pin’s dissertation in (2007). This version contains some new theoretical results and strengthening of the existing ones, and adds an empirical analysis of citations. † a Universit` di Venezia. Email: s.currarini@unive.it ‡ Department of Economics, Stanford University and the Santa Fe Institute. Email: jacksonm@stanford.edu, http://www.stanford.edu/∼jacksonm/ § a Dipartimento di Economia Politica, Universit´ degli Studi di Siena (Italy). Email: pin3@unisi.it 1 See McPherson, Smith-Lovin, Cook (2001) and Jackson (2007, 2008) for more background and discussion. 1 The primary issue that we investigate is how homophily patterns change over time. Do nodes become more integrated as they age? How does integration relate to the link formation process? For instance, does the network end up more integrated if new connections are found through the existing network or if nodes always meet anonymously? Intuition would suggest that the extent to which the existing connections inﬂuence new ones will seriously aﬀect the long run behavior of the system. To answer these questions, we study a stochastic model of network formation in which nodes come in diﬀerent types, and in which the formation of links is sensitive to such types. More speciﬁcally, we extend the model of Jackson and Rogers (2007) so that a new node is born at each time period and has a given “type”, and forms (directed) links with the nodes born in previous periods. A newborn node selects older nodes to connect to in two ways. First, a set of older nodes are met and linked to according to a random, but potentially type-biased process. As an example, a given scientiﬁc paper written in a given ﬁeld (its type) has relations to a set of existing papers, and possibly with greater frequency within its own ﬁeld. These form a set of citations, or directed links from the new paper to older papers. Second, the newborn node then meets and links to some neighbors of the nodes to which it has already formed links. This is referred to as the search process. Again, this part of the process might be type biased, but we also consider a case where it is not. As an example with regards to citations, an author ﬁnds some references by examining the reference lists of the papers that he or she has already located and cited. This search part of the process may have a diﬀerent bias than the random part of the process, since these papers were cited by the papers that he or she has already chosen to cite and thus are more likely to be related. We examine the limiting, long-run properties of this process. One possible interpretation for the biases in the process is as a reduced form for agents’ pref- erences over the types of their neighbors and/or of biased meeting opportunities that agents face in connecting to each other. So, in one direction we enrich a growing network model by allowing for types and biases in connections, and in another direction we still bypass explicit strategic con- siderations by studying a process with exogenous behavioral rules. Since search goes through the out-neighbors only, strategic considerations are to an extent already limited, since a node cannot increase the probability of being found by choosing its out-neighborhood. While this may not be a good assumption in certain instances of social networks, such as friendships or job contacts, where search presumably goes both ways on a link, it is appropriate in other contexts, such as scientiﬁc citations where the time order of publications strictly determines the direction of search. Our results concern the dynamics of link formation among diﬀerent types, and in the extent to which biases in the process of link formation translate into biases in the long-run patterns of connections. In particular, we are interested in the conditions under which the system tends to “integrate” in the long run. We consider two deﬁnitions of integration. The ﬁrst, weak integration, requires that older nodes have a higher probability than younger nodes of being linked to by 2 newborn nodes, independently of their types. For example, this requires that an older, established paper in one ﬁeld have a higher probability of being cited by a newborn paper than some very young paper, regardless of the ﬁelds of the older, younger, and newborn papers. In this weak sense, age overcomes the bias in the link formation process. Eﬀectively, this notion of integration requires that old enough nodes become suﬃciently “authoritative” to be found by a newborn node with a relatively large probability even if this is of a diﬀerent type. The second and more demanding deﬁnition of integration is what we call long-run integration. It requires that as a node ages, the distribution of types of nodes that have linked to that node eventually approaches the distribution of types in the population. This requires that as a node ages any bias in the distribution of nodes who have connected to it disappears. Our main theoretical results are as follows. Weak integration is satisﬁed whenever the probability that a given node is found increases with that node’s in-degree. This holds in any version of our model where at least some links are formed through the search part of the process and there is some possibility of connecting across types.2 In contrast, long-run integration is signiﬁcantly more diﬃcult to satisfy. It is satisﬁed if and only if the search part of the link formation process is unbiased (type-blind), so that every node that is linked to by one of the nodes located through the random attachment part of the process has an equal probability of being linked to under search. Thus, this requires that any bias in the network formation process occur only in the random part of the process. In addition, we discuss how under mild conditions on the biases, the process moves monotoni- cally towards the long run behaviour. In particular, the aging of nodes has the eﬀect of weakening the bias in their in-degree, and when search is unbiased, the in-degree composition tends to the frequencies of types in the population. We also discuss some subtle aspects of the relation between the biases in the random meeting process, the short run composition of in-degrees and the total number of connections agents receive. For the case of unbiased search with two types, we show that the more homophilous type3 ends up accumulating more total connections from both types, and that it attracts a more balanced mix of types in the short run. To understand the long-run integration result, note that if both parts of the process are biased then long-run integration cannot hold. Thus, let us examine why long-run integration holds when the random part of the process is biased, but search is unbiased. As nodes age, the relative fraction of in-links that they obtain through the search part of the process begins to dominate, since the number of neighbors through whom they can be found grows and also since the probability that any give node is found via the random process decreases because there are more nodes. 2 As will become clear, weak integration would also hold in a variety of other models that also exhibit the property of having linking probabilities increase suﬃciently with in-degree. 3 The term “homophily” refers to the probability of meeting same type agents in excess of this type’s population share. See also footnote 13 3 Note that even though search begins to dominate and is unbiased, it is still not obvious that long-run integration will hold. That is because the likelihood of ﬁnding various neighbors is still biased in the random part of the process. So let us examine this in more detail, and for simplicity with just two types, say purple and green, as the logic extends easily. A given purple node can be found by a newborn green node of a diﬀerent type via search in diﬀerent ways: one is that the green newborn ﬁnds a neighbor of the purple node that is green, and the other is that the newborn ﬁnds a neighbor of the purple node that is purple. It is relatively easier for the green node to ﬁnd other green nodes given the bias in the random part of the process, but then the purple node tends to have more purple neighbors early in the process. The critical fact that it can happen either way, means that this bias is lower than the current bias in the purple node’s neighborhood, and thus tends to lower the bias overall. As the purple node’s neighborhood becomes less and less bias over time, then that leads it to become even less biased, and the bias in the process vanishes over time. It must be noted that long-run integration coexists with a contrasting feature: the fraction of links formed between agents of diﬀerent types is never uniform across types. This persistent asymmetry across types reﬂects the fact that although each node will eventually end up attracting an unbiased spectrum of links, in the meantime younger nodes still experience biases, and so integrating across the full population links can still be biased. This has to be the case since we know that the links formed randomly are always biased, and so there is at least a given fraction of links that are formed that are biased. In addition to the theoretical analysis, we also illustrate the model using data on scientiﬁc citations in journals of the American Institute of Physics (AIP) published between 1977 and 2007. We ﬁnd that the proportion of citations that a paper obtains from other papers in its own ﬁeld decreases as the paper becomes more cited. An interpretation of the observed citation patterns suggests biases in both the random and search parts of the process, but with a smaller bias in the search part of the process. In using this speciﬁc application we are motivated by two factors. First, patterns of scientiﬁc citations have important welfare consequences as they can aﬀect the diﬀusion of knowledge, and the contamination of diﬀerent research ﬁelds.4 Previous research, such as that by Palacios–Huerta and Volij (2004) and Koczyy, Nichiforz and Strobel (2010), generalizing popular concepts as the recursive impact factor, stress that the importance of a citation relies on the paths that it allows in the network of citations. We extend this argument considering under which conditions citations are likely to bridge scientiﬁc production across diﬀerent communities.5 Second, scientiﬁc citations possess all the features of the network formation process that we study: nodes (papers) appear in chronological order and never die, they only link to previously born nodes, they have types (scientiﬁc classiﬁcations), and they ﬁnd citations both directly and though search among the citations of other 4 See, for instance, Breschi and Lissoni (2006) and Jaﬀe and Trajtenberg (1996). 5 Rinia et alii (2001) study cross-ﬁeld citations in the scientiﬁc production of the 90’s, for three diﬀerent datasets. 4 papers.6 e Our analysis is independent of work by Bramoull´ and Rogers (2009) who examine a similar model, but with some diﬀerences in the questions asked and application. The paper is organized as follows. Section 2 described the model. Section 3 contains our deﬁnitions of integration and a mean-ﬁeld analysis. Section 4 illustrates the model using citation data. Section 5 concludes the paper. Finally, an appendix contains some additional results on Markov matrices, the proofs of the propositions, a more detailed description of a possible matching process, and some examples. 2 The Model Time is indexed by t = 1, 2, ..... In each period a new node is born. We index nodes by their birth dates, so that node t was born in period t. Nodes have “types,” with a generic type denoted θ belonging to a ﬁnite set Θ (with cardinality H). A newborn’s type is random and drawn according to the time invariant probability distribution p (so that types are i.i.d., across time). A newborn node sends (directed) links to n > 1 nodes that were born in previous periods. Of these n nodes, a fraction mr is selected according to a type-dependent random process (where mr n is an integer in the true process, but allowed to be arbitrary in the mean-ﬁeld continuous-time approximation). In particular, p(θ, θ ) denotes the probability that a link sent by a node of type θ reaches a node of type θ . Among nodes of type θ , each node has an equal probability of getting one of the mr n links - so there is no further discrimination in this part of the process. The remaining fraction ms = 1 − mr of the n links are determined according to a search process: each new node looks at the neighbors of the nmr nodes found in the ﬁrst step and, among these, selects nms nodes at random.7 If the random meeting process were uniform, the probability p(θ, θ ) would equal the share p(θ ) of θ agents in the system. We consider more general meeting processes that can be biased. 6 These longitudinal aspects of citation networks have motivated the use of growing network models in previous o papers including the seminal work on citation networks by Price (1965, 1976). B¨rner, Maru and Goldstone (2004) and Simkin and Roychowdhury (2007), among others, ﬁnd that citations on the PNAS on a 20 years interval show some aspects of a bias towards recently published papers, while Redner (1998) and Newman (2009) correcting for cohort size and idiosyncratic popularity ﬁnd an age eﬀect (ﬁrst mover advantage) and a frequency distribution of in- citations that are consistent with a growing network model such as the one that we develop here. Finally, Shi, Tseng and Adamic (2009) ﬁnd a positive correlation between homophily of out-citations and the number of in–citations, but this eﬀect is valid only for low number of in-citations. 7 In the process, if some node is found to which the newborn is already connected, then the node is redrawn. If there are too few new nodes in the neighborhoods of the nodes found in the ﬁrst part of the process, then the random nodes redrawn. To ensure that the process is well-deﬁned, we begin with a set of n2 nodes in a sequence, each connected to all predecessors. 5 This can be interpreted in diﬀerent ways. One is that the bias is a reduced form for preferences that nodes have over the type of connections they form. The case of “homophilistic” preferences for type θ is then captured by a situation where p(θ, θ) > p(θ). Of course, the search part of the process can also be (directly) biased. We describe that more fully below. In the Appendix we formulate a detailed process of link formation, which generates biased probability of matchings, and which is based on the possibility of agents “rejecting” connections according to their type. For convenience, we work throughout the paper with the reduced form introduced here.8 The case in which mr = 1 is referred to as the “purely random” model, while the case of 0 < mr < 1 is referred to as the “random-search” model. Pjt (θt , θj ) is the probability that a node born in period j of type θj receives a link from a node of type θt born at time t > j. 2.1 Purely random model In the purely random model, the probability that node j gets linked from a θ-type node born at time t + 1 is simply given by the joint probability that node t + 1 is of type θ, p(θ), times the probability it ﬁnds j among all the other nodes of type θj who are in the network at time t + 1. Under a mean-ﬁeld approximation, the fraction of nodes of type θj at time t is tp(θj ). Thus, a mean-ﬁeld approximation is that p(θ, θj ) Pjt+1 (θ, θj ) = n p(θ) . (1) tp(θj ) In the formula above, the term in brackets is multiplied by n - the number of links formed by node t + 1. It is useful to express the terms of the above formula in a compact way. For all θ, θ we write p(θ, θ ) Br (θ, θ ) ≡ p(θ) . p(θ ) Note that the ratio p(θ,θ)) in the above expression is a measure of the bias that type θ applies p(θ to type θ , so that when this ratio is 1 there is no bias, while when it is greater (less) than 1 there is a positive (negative) bias of type θ towards type θ . In case of no bias, Br (θ, θ ) is simply the probability of birth of a type θ node, and Pjt+1 (θ, θj ) is n times the joint probability that the newborn node is of type θ and that node j is found by drawing uniformly at random from a population of t nodes. In the Appendix we discuss the properties of such a bias in more detail. 8 See Currarini, Jackson and Pin (2009, 2010) for more details on other such models that can justify this reduced form. 6 Let Br denote the |Θ| × |Θ| matrix containing the terms Br (θ, θ ): Br (1, 1) Br (1, 2) . Br (1, H) . Br (2, 2) . . Br ≡ . . . . . . . . Br (H, H) We can decompose the matrix Br as the product of two matrices A and Q, where A may be seen as a transition matrix of a Markov process (a Markov matrix),9 and Q is a diagonal matrix where the diagonal is a probability vector: Br = QAQ−1 , with p(1) ... 0 Aθθ = p(θ, θ ) and Q = ... ... ... . 0 ... p(H) We can now rewrite (1) in compact form, to express the probability that at type t + 1 a node of a generic θ-type node links to a generic θ -type node born at time t0 < t + 1 : n Pt+1 = t0 Br . (2) t 2.2 Random-Search Model with Unbiased Search We now introduce the search part of the network formation process. Before describing the full model with biases in both search and random link formation, we begin with the case where the random link formation may be biased but where the search process is unbiased. When nodes search among the neighbors found via the random links, the probability that node j is found by newborn node t + 1 depends on the shape of the network that has formed up to time t. In particular, it depends on the type-proﬁle of neighbors of j at time t, and on the bias of the newborn node to randomly link to such types. If search is not type-biased, each link that agent t + 1 forms through search is drawn from a uniform distribution over the set of all neighbors of all nodes that agent t + 1 has found at random. For this unbiased search case the following expression is a mean-ﬁeld approximation of the overall linking probability: t nmr λ=j Pjλ (θ , θj ) 1 Pjt+1 (θ, θj ) = Br (θ, θj ) + nms p(θ)p(θ, θ ) (3) t tp(θ ) n θ ∈Θ The ﬁrst term in the right–hand side of (3) diﬀers from (1) only because mr < 1. The second term is the probability of node j being found through search. It is given by the number of search 9 In Appendix A we derive some general results on Markov matrices that will be useful in Appendix B, where we prove our propositions. 7 links (nms ) formed by the node born at t + 1, times the sum, over all possible types θ , of the probabilities that j is found through a node of type θ . For each possible type θ , this probability is given by the joint probability of the following events (corresponding to the four terms in the ﬁrst summation over types): i) the newborn node is of type θ; ii) it forms a link with a θ -type node; iii) that the θ -type node has linked to j since j was born;10 iv) among the n neighbors of this θ -type node, that exactly j is found. We can again express (3) for all possible types in a compact way using the matrix Br : t nmr ms Pt+1 = t0 Br + Br Pλ0 , t (4) t t λ=t0 t where λ=j Pλ0 expresses the expected in-degree, type by type, at time t for a node born at time t t0 . 2.3 Random-Search Models with Type-Biased Search We now describe the full model in which search can also be type-biased. 2.3.1 Type-Bias in Search In this variant of the model, the bias in search aﬀects the choice of randomly found nodes that are to be used to ﬁnd the additional nms search connections. Here, each new node directs a fraction of the nms search links to each given type of node found through the nmr random links. The bias in how these nodes are selected can diﬀer form the bias in the random process. This bias is described via an H × H matrix where each element is positive and of the form Bs (θ, θ ).11 A value of 1 indicates no bias, while a value greater (less) than 1 indicates a positive (negative) bias of type θ towards type θ . This is the distortion in the relative probabilities that θ searches its randomly found neighbors. The mean-ﬁeld approximation of the process is described by H t Pjt+1 (θ, θj ) = nmr t Br (θ, θj ) + ms t θ =1 Br (θ, θ )Bs (θ, θ ) λ=j Pjλ (θ , θj ) . (5) The product Br (θ, θ )B s (θ, θ ) in (5) describes the probability applied by type θ to the selection of random nodes in search of type θj . Note that the bias is independent of both time variables j and t, and of the type θj of the target. We discuss this in more detail, for the case of two types, in the Appendix. 10 Note that this ratio has the total (expected) number of links received by agent j from θ agents up to time t as numerator, and the total number of θ nodes in the system at time t as denominator 11 There are constraints on this bias matrix to have the resulting output be well-deﬁned probabilities, but much can be deduced for general forms of the matrix, and so we only specify the (obvious) constraints as they become necessary. 8 In matrix form, the system becomes: t nmr ms Pt+1 t0 = Br + (Bs Br ) Pλ0 , t (6) t t λ=t0 where is the Hadamard product: (Bs Br )(ij) ≡ Bs (ij) · Br (ij) . Note that, from the decompo- sition Br = QAQ−1 , it follows that Bs Br = Bs QAQ−1 = Q (Bs A) Q−1 , where the biases are such that Bs A is still a Markov matrix. We refer to this variation of the model as RSB, for random-search with biased search. 2.3.2 Type-bias on targeted nodes While we focus on the above-speciﬁed formulation of search-bias in what follows, we remark that search could also be biased in other ways. In particular, a newborn node has linked to a set of nodes via the random part of the process and then searches those nodes’ neighborhoods to ﬁnd new nodes with whom to link. It can bias this process in two ways: it might be biased in terms of which of its randomly found neighbors neighborhoods it searches through, and it might be biased in terms of whether it evenly samples neighborhoods or biases the sampling process. Above we have focused on the ﬁrst form of bias. For completeness, we note the formulation for the other form of bias, which is a variation on that above. In this variant of the model, the bias in the search process aﬀects directly the probabilities that nodes of the various types are found out of the pool of out-neighbors of nodes found through search. The system of equation (3) is modiﬁed to allow for such bias as follows: H t nmr ms Pjt+1 (θ, θj ) = Br (θ, θj ) + Br (θ, θ ) Pjλ (θ , θj )Bs (θ, θj ) . (7) t t θ =1 λ=j The search-bias parameter Bs (θ, θj , θ , λ, t) of type θ towards type θj . Rewriting: t nmr ms Pt+1 = t0 Br + Bs (Br Pλ0 ), t (8) t t λ=t0 When Bs (θ, θj ) > 1 the expected number of links that a given θj node receives from a newborn node of type θ is larger than what it would receive if search was unbiased. Since this applies to all nodes of type θj , it implies that a newborn node of type θ will form a fraction of its search links with nodes of type θj that exceeds what is the share of type θj nodes in her distance 2 neighborhood after the random part of our process. Similar but opposite considerations apply to the case in which Bs (θ, θj ) < 1. The justiﬁcation for this formulation of search bias is discussed in more detail in the Appendix. We refer to this variation of the model as RSBT. 9 2.4 A Mean-Field Continuous Approximation We study a continuous time approximation of the model, using the techniques of mean-ﬁeld theory. This provides approximations and limiting expressions of the process that ignore starting conditions and other short-term ﬂuctuations that can be important in shaping ﬁnite versions of the model, and so the results must be viewed with the standard cautions that accompany such approximations and limit analyses. We consider the expected change in the discrete stochastic process as the deterministic diﬀerential of a continuous time process. We deﬁne t Πt0 t ≡ Pλ0 . t λ=t0 With a continuous approximation: ∂ t Π = Pt+1 . ∂t t0 t0 We study equations (2), (4) and (6) in terms of ordinary diﬀerential equations in matrix form: ∂ t n Π = Br ; (9) ∂t t0 t ∂ t nmr ms Π = Br + Br Πt0 , (10) ∂t t0 t t t ∂ t nmr ms Π = Br + (Bs Br )Πt0 (11) ∂t t0 t t t with the initial condition Πt0 = 0. t0 Summarizing, (9) refers to the purely random model (R), (10) refers to the random–search model with unbiased search (RSU ), and (11) refers to the random–search model with search biased on intermediaries (RSB ). If Br is invertible (so that the speciﬁcation of types is not redundant) the unique solutions to these diﬀerential equations are, respectively, the following: t Πt0 t = nBr log ; (12) t0 ms Br mr t Πt0 t = n −I , (13) ms t0 ms (Bs Br ) mr t Πt0 t = n (Bs Br )−1 Br −I . (14) ms t0 where a constant to the power of a matrix is deﬁned as follows: µ ∞ t t ms Br “ ms log “ t ” ” Br ms log t0 Br =e t0 = . (15) t0 µ! µ=0 10 Example 1 The 2–types symmetric random model (R). π 1−π Consider a symmetric setting in which n = 2, p(1) = p(2) = 1 , mr = 1, Br = 2 . 1−π π This is a purely random model R. We implement equation (12) to ﬁnd the expected in–degree of a node born at t0 : its in–links from same–type nodes follow equation 2π log tt0 , while its in–links from diﬀerent–type nodes follow equation 2(1 − π) log tt0 . Note that there is a rapid fall-oﬀ in the growth of the in-degree of a given node over time, as it only grows logarithmically due to competition from other nodes and ﬁxed out-degree of newborns. Example 2 The 2–types symmetric random–search model with unbiased search (RSU). Let us contrast previous example with a random–search model with unbiased search RSU, with mr = ms = 1 .12 In this case the outcome of equation (13) is that for a node born at time t0 the 2 in–links from same–type nodes follow equation π− 1 t t 2 + −2 , t0 t0 while those from diﬀerent–type nodes follow equation π− 1 t t 2 − . t0 t0 Here we see that the growth of in-degree falls oﬀ more slowly than in the purely random model, as in-degree grows proportionally to the square root of t rather than log of t. This reﬂects that fact that as a node acquires more in-links, it becomes easier to ﬁnd, which counter-acts some of the competition from younger nodes. Example 3 The 2–types symmetric random–search model with biased search (RSB). Consider a symmetric RSB model with the same parameters as in the previous examples, and 1−σπ σ matrix of biases: Bs = 1−σπ 1−π .13 In this case the explicit outcome of equation (14) is 1−π σ that for a node born at time t0 the in–links from same–type nodes follow equation (we write t instead of t/t0 , normalizing t0 to 1) √ 1 2π − 1 σπ + π − 1 t + tπ− 2 −2 , 2σπ − 1 2σπ − 1 while those from diﬀerent–type nodes follow equation √ 1 2π − 1 σπ − π t − tπ− 2 −2 . 2σπ − 1 2σπ − 1 The same diﬀerences in comparison with the purely random model hold also for this model. 12 A more general 2 × 2 case is treated in Appendix D.2. 13 In Appendix D.3 we discuss why Bs must depend on Br and a more general 2 × 2 case. 11 3 Integration In all the network formation processes described in the previous sections, links are formed with some degree of type-bias. As we said, this can be though of as a reduced form of agents’ preferences on potential partners and/or if the biases they face in their meeting opportunities. From a welfare perspective it is then important to understand how these biases in linking behavior translate into biases in the realized connections of agents. In particular, in this section we study under which conditions and to what extent the type-biases in the in-degree of agents tend to vanish in the long run. To this aim we propose various deﬁnitions of integration, that reﬂect varying degrees to which the dynamic process weakens the biases in the way agents form links. 3.1 Weak integration The ﬁrst notion that we consider requires that there exist a large enough “age” diﬀerence such that an older node receives a link from a newborn node with higher probability than a younger node, independently of the types of the nodes involved. Definition 1 The network formation process satisﬁes the weak integration property if for every t0 , there exists t > t0 such that, for all t ≥ t and for all θ ∈ Θ, the node born at time t has a lower probability than node t0 to receive a link from a node of type θ born at time t + 1. Under this condition an old enough node of type θ ends up receiving a link from a newborn node of type θ with a higher probability that a young enough node of the same type θ as the newborn. It is clear that the basic random model R does not satisfy this property. We show instead that the random-search models satisfy this property, even with a bias in search. Proposition 1 If mr < 1, both versions of the random-search model, with or without biased search (RSU, RSB), satisfy the weak integration property.14 The proof (which appears in the appendix) shows that Property 1 is not speciﬁc to the ho- mogeneous search model. Indeed, various models in which the in-degree of a node determines the probability of being found by a newborn node in a suﬃciently increasing manner would give the same result. This can be directly checked in both models of type-biased search of Section 2.3–2.4. Moreover, search is not needed for this type of dependence to take place. In the conclusions we discuss another mode with “type-biased” preferential attachment in which the probability of re- ceiving a link is positively correlated with a node’s in-degree, and which exhibits the same weak integration property. 14 In fact, this also holds for RSBT, as we show in the appendix. 12 3.2 Long-run Integration A stronger notion of integration requires that eventually nodes attract connections according to the population shares of types. Deﬁnition 2 introduces this requirement for nodes of suﬃciently old age. In other words, in the long run any surviving diﬀerence in the proportion of links received by old nodes from diﬀerent types is only due to the distribution of types in the population, and not on the biases in preferences or on the type of the receiving node. Definition 2 The network formation process satisﬁes the long-run integration property if for every node t0 the ratio of each type θ in the in-degree of t0 converges to θ’s populations share as node t0 ages. Long run integration is not satisﬁed by the basic random model R. The next proposition shows that this property is satisﬁed by the random-search model with unbiased search RSU. Proposition 2 If mr < 1, the random–search model with unbiased search (RSU) satisﬁes the long– run integration property, while the random–search model with nontrivially biased search (RSB) does not.15 The intuition behind this result is as follows. First, it is clear that if the search part of the process is biased then long-run integration cannot hold, since as nodes age, the relative fraction of in-links that they obtain through the search part of the process begins to dominate. This happens, since the number of neighbors through whom they can be found grows and so it is relatively easier to ﬁnd an older node via search compared to a younger node, while there is no advantage to age in the random part of the process.16 So let us examine why it is that as search begins to dominate and is unbiased, then long-run integration holds. A given node can be found by a newborn node of a diﬀerent type via search in diﬀerent ways: one is that the newborn ﬁnds a neighbor of the given node that is of the same type as the newborn, and the other is that the newborn ﬁnds a neighbor of the given node that is of the same type as the given node. The ﬁrst way is relatively more likely given the bias in the random part of the search, but the fact that this can also occur via the second route then leads this process to be less biased than the original random part of the search process. So, the neighborhood of an older node then tends to start being less biased than the original search process. Once it is less biased, this makes it even easier to be found by nodes of the opposite type, and so the neighborhood becomes even less biased, and this continues with the limit reﬂecting an 15 “Does not” means that for generic speciﬁcations of the bias process, long run integration fails. However, there can be speciﬁc (nongeneric) where integration holds, even with non-trivial bias. See the discussion of Examples 3 and 4. 16 A node faces increased competition from other nodes over time in both parts of the process and indeed the growth of in-degree slows down over time, but a given node gains a greater relative advantage in the search part of the process over time which then comes to dominate. 13 unbiased process. As mentioned in the introduction, as a node ages it becomes more of a “hub” node (very loosely speaking), with many in-links. As its incoming bias becomes lower, then it leads to even less of a bias as it becomes increasingly easy to ﬁnd by other type nodes, and eventually the bias in the process disappears. Let us analyze whether long-run integration occurs in Examples 1–3 from Section 2.4. In the purely random model of Example 1 the ratio of same type in-links to diﬀerent type in-links is π/(1 − π) and it remains constant over time, implying that long-run integration does not occur unless there is no bias (π = 1/2). In the RSU model of Example 2, the ratio between in–links from same-type nodes and in–links from diﬀerent type nodes converges to 1 as t grows relative to t0 , reﬂecting the ratio of population shares and indicating that long run integration occurs. In the RSB model of Example 3, long-run integration holds due to the symmetry of the matrix Br (and all of the other symmetry in the example). As next example makes clear, Example 3 is “non-generic” and more generally long-run integration fails if there is biased search that does not happen to exactly balance across types. Example 4 An asymmetric 2–types random–search model with biased search (RSB). Consider an RSB model with n = 2 types that are equally likely p(1) = p(2) = 1 . Suppose that 2 1/2 1/2 there is no bias in the random part: from the speciﬁcation in (1) we have that Br = . 1/2 1/2 1 1 1 Finally, let us assume that the search part is such that mr = ms = 2 and Bs = , 2−σ σ with 1 < σ < 2, which means that only the second type has a self–bias in the search part. In this case the outcome of equation (14) is that (we write t instead of t/t0 , normalizing t0 to 1) √ 2−σ σ−1 1 √ 1 σ−1 1 t 3−σ + t 4 3−σ − 1 t 3−σ − t 4 3−σ Πt = 1 √ 2−σ √ 1 σ−1 σ−1 2−σ 2−σ t 3−σ − t 4 3−σ t 3−σ + t 4 3−σ − 1 √ In the long–run the terms with t dominate: a type 1 node ends up receiving same-type links with a ratio 2 − σ compared to diﬀerent-type links; whereas a type 2 node ends up receiving same-type 1 links with a ratio 2−σ . As types are equally represented in the population, long–run integration would require that both those ratios be 1, which holds only in the case of no bias σ = 1. Note from Examples 3 and 4 that long-run integration holds in the RSB model when the Hadamard product of the matrices Bs and Br is either symmetric or equals the matrix Br . So, in the 2 types case, the bias in search must either equal the bias in the random part (that is, the RSU model), or it must be symmetric. In these two (non-generic) cases, the random bias disappears in the long run once the search part of the process takes over. For the 2 types case this is immediately implied by equation (i) in the Proof of Proposition 2, in which the proportions of in-connections from the various types are derived explicitly. 14 3.3 Partial Integration Even when full long-run integration does not occur, it is still of interest to understand whether time has the eﬀect of integrating the system at least to a partial extent. More precisely, we can ask whether the in-degree of an agent becomes more and more “homogeneous” as time elapses, leading, as we know, to full integration in the limit when search is unbiased. This type of ”monotonicity” property may hold even under biased search, although, as we have seen, long run integration does not occur in this case. Definition 3 The network formation process satisﬁes the partial integration property if for every node t0 the fraction of each type θ in the in-degree of t0 is weakly closer to θ’s population share at time t than at time t, for t > t, and strictly closer for some types. In particular we will consider the matrix A, as deﬁned in the beginning of Section 2, and its biased analogous Bs A for the general RSB model. These Markov matrices represent the biases of the random parts cleaned form the eﬀect of size of the diﬀerent populations. ¯ Consider a Markov matrix M. As formally stated in Appendix A, if we call M ≡ limµ→∞ Mµ , we say that M satisﬁes a monotone convergence property if, for every pair i, j ∈ {1, . . . , H}, and µ for every µ ∈ N, the element Mij satisﬁes: ¯ µ µ+1 ¯ 1. if Mij > Mij , then Mij ≥ Mij ≥ Mij ≥ Mij ; ¯ µ µ+1 ¯ 2. if Mij < Mij , then Mij ≤ Mij ≤ Mij ≤ Mij . The monotone convergence property captures the idea that transition probabilities are monotone over time. Even with a strictly positive transition matrix, this condition does impose additional restrictions.17 It is beyond the scope of this paper to ﬁnd general or even necessary conditions for monotone convergence of Markov matrices. This monotone convergence property is a suﬃcient condition for us to prove the following result. Proposition 3 In a random-search model RSB, with mr < 1 and such that Bs A satisﬁes the monotone convergence property, then the partial integration property holds. Note that the RSU model is a particular case of the RSB model where Bs is a matrix of all 1’s. In this way the hypothesis of the monotone convergence property that must hold for Bs A in the statement above can be applied directly to A in the RSU model. It is easy to check directly that the partial integration property holds for both Example 2 of the RSU model, and Examples 3 and 4 of the RSB model. 17 Just as a simple illustrating example, consider a Markov process with three states where it is mostly likely to go from state 1 to state 2, and 2 to 3, and 3 to 1, but with small probabilities of transitioning directly to other states too. Then in one period going from 1 to 2 is likely, but then it is unlikely to occur in two periods or three periods, but more likely in four periods, and so forth. Things eventually converge to equal likelihood on all states, but convergence is not monotone. One can also ﬁnd such examples that are more complicated where homophily is present. 15 3.4 Homophily bias and linking patterns in the RSU model Having established some general results on the various notions of integration, we now use the framework of unbiased search to study the patterns of homophily that stem from biases in favor of same-type links in the matrix Br . Working with a simple example with two types, we examine the eﬀect of homophily on both the number of in-connections and on their composition, in the short and long run. Let both types 1 and 2 come with equal population shares p(1) = p(2) = 1 . Let 2 type 1 have a strong homophilous bias, and let type 2 have no bias at all, with a particular matrix of linking probabilities p(θ, θ ): 5/6 1/6 . 1/2 1/2 Finally, let n = 10, mr = .5 and ms = .5. Figure 1 illustrates the evolution of the in–degree of a representative node born at time t0 = 1000, decomposed by the type of the originating node. It is clear from the ﬁgure that homophilous nodes (type 1) receive more links from both types than the non–homophilous nodes. Intuitively, type 1 nodes face a higher probability than type 2 nodes of being found at random by type 1 nodes, and equal probability of being found at random by type 2 nodes, and end up attracting more links in total. Note that this is still consistent with long-run integration. Even though nodes’ in-neighborhoods homogenize over time, which then means that they face relatively even probabilities of being found eventually by diﬀerent sorts of nodes, there is an initial advantage to the homophilous types. Homophilous types have a higher probability of being found at random, and thus accumulate in- links at a faster rate in the early part of their lives. Thus, they have a larger in-degree at an earlier stage in their lives than the unbiased types. The fact that the probability of gaining new links through search ends up being proportional to in-degree, means that they then maintain this gap. Thus they grow faster over time, since growth is proportional to in-degree and they gain an initial advantage which then persists over time, even though the composition of their in-degrees homogenizes. Note that this relation between homophily and total connections is consistent with the dis- tributions found by Shi, Tseng and Adamic (2009) in the context of scientiﬁc citations between diﬀerent ﬁelds in computer science.18 There, for an average scientiﬁc paper, being homophilous in out-citations is positively correlated with the total number of in–citations. This was explained by the authors in terms of the positive eﬀect of a homophilous attitude of a given paper, in citing other papers, on its success in terms of in-citations. In our model the same correlation is explained instead as the eﬀect of the positive correlation between the homophilous behavior of a paper of type θ and the behavior of the future papers of the same type. Being part of a homophilous group favors the process of accumulation of in-connections. 18 See Section 4 for an analysis of the application of our model to scientiﬁc citations. 16 700 250 600 200 500 400 150 300 100 200 50 100 104 105 106 107 104 105 106 107 1: In–links by type of receiver (left panel for type 1, right panel for type 2) and of sender (dotted for type 1) Turning now to the composition of in-connections, we note that nodes of the homophilous type display a relatively lower degree of homophily (measured as the share of same type in-connections on total connections) than nodes of the unbiased type. This is clear from Figure 2, in which it is also shown that the diﬀerence in the in-degree composition of the two types integrates in the long run, consistently with proposition 2. The reason for this somewhat subtle pattern has to do with the eﬀect of homophily bias on total connections. Homophilous nodes attract more links from both types, but with a larger proportion from the opposite type at earlier times, while unbiased nodes end up attracting fewer links, mostly from same-type nodes in the earlier periods. 0.75 2000 0.70 1500 0.65 1000 0.60 500 0.55 104 105 106 107 108 104 105 106 107 108 109 2: Left panel: total in–links for a type–1 node (dotted line) and for type–2 node. Right panel: share of in–links that are from own type (dotted line for type 1) In general, one can show that in the two type case the more homophilous type always receives more connections from both types in the long run. To see this, we need to use the proof of Proposi- tion 2 (contained in the appendix), where it is shown that the expected number of connections that a node of type i receives in the long run from a given type j is proportional to the i − th element 17 of the unit eigenvector of the matrix A containing the meeting probabilities p(θ, θ ). Letting i = 1 be the homophilous type with probability p(θ1 , θ1 ) = π > 1 , the ﬁrst element of this eigenvector 2 1 2−2π is equal to 3−2π , while the second element is equal to 3−2π . It can easily checked that the ﬁrst 1 element is larger than the second for π > 2 , and that the diﬀerence is increasing in π. 3.5 Aggregate long-run integration The long-run integration properties described in the previous sections apply to individual nodes, who eventually homogenize their in-degree. A diﬀerent question concerns the long-run overall relation among diﬀerent types: what is the long-run average in-degree of a given type of nodes from any other give type? For a formal answer we need an additional deﬁnition. Definition 4 The network formation process satisﬁes the aggregate long–run integration prop- erty if the average fraction of in-degree from nodes of various types of all the nodes of the network of any given type converges to the actual ratios of the overall population. Deﬁnition 4 applies when the overall populations of diﬀerent types integrates on average in the long–run. It is clear that in the simple random model R proportions are ﬁxed and are described by the matrix Br , so that the aggregate long–run integration property coincides with the (individual) long–run integration property, and they both hold only under a speciﬁc and non–generic case. The (individual) long–run integration (Deﬁnition 2) is diﬀerent from the aggregate long–run integration (Deﬁnition 4). Thus, the qeustion is how quickly the aggregation happens, since the aggregate property requires that long-run integration must occur for most nodes. We now show that the unbiased random–search model does not satisfy aggregate long–run integration. Proposition 4 The random-search model with a bias in the random part of the process but unbi- ased search ( RSU) does not satisfy the aggregate long–run integration property. The intuition behind this results is straightforward. Although in the long-run any given node eventually becomes integrated, there are many relatively young nodes in the system for which their in-degree is still mostly formed via the random part of the process. In fact, we can see this also from the out-degree which is always biased for at least the mr fraction of the links formed directly at random. Even if the search overcomes the other part of the bias, a given fraction of links are formed in a biased manner, and so integrating over all nodes, in-degree will still be biased over time. 3.6 On the Dynamics of Out-degrees So far we have focused on the dynamics of agents’ in-degree. It is of interest to also look at the composition of the out-degrees, and how this evolves in time. This for two reasons. First, out-links 18 may aﬀect welfare and their composition may therefore be relevant. Second, there is a relation between the evolution of the out-degree of nodes and the tendency of in-degrees to integrate (either partially or totally). We ﬁrst look at the steady state composition of the out-degree and we focus on the RSU model. Let us denote by dij,t the proportion of total links that originate from a node of type i born at time t that are directed towards nodes of type j. The evolution of these proportions in the RSU model is given by: H t τ =1 dhj,τ dij,t+1 = (1 − ms )Br (i, j) + ms Br (i, h) (16) t h=1 The out–degree depends on the random part (ﬁrst term) and on the search part (second term) through the average out–degree of existing nodes. In matrix form, the steady state relation is written as follows: t τ =1 Dt Dt+1 = (1 − ms )Br + ms Br . (17) t To get a feeling for the limit of this process, it is useful to examine the steady state Ds of this system. The steady-state is such that the out-degree of each type remains unchanged in time: Ds = (1 − ms )Br + ms Br Ds , (18) yielding Ds = (1 − ms ) (I − ms Br )−1 Br . (19) Using the algebraic identity ∞ −1 (I − ms Br ) = (ms Br )µ , µ=0 we obtain the following expression: 19 ∞ 1 − ms Ds = B (ms A)µ B−1 ms µ=1 . In the above expression, the matrix in brackets is such that, as ms → 1, the elements of each column homogenize (see Lemma B of Appendix A). However, full homogeneity only occurs at the limit ms → 1. 19 ¯ Note that the matrix we obtain coincides with the matrix D, deﬁned in the Proof of Proposition 4 dealing with the aggregate in-degree of types. 19 To obtain some insight on the time evolution of the out-degree, let us express equation (20) as a diﬀerential equation, and solve it explicitly (as we have done in (10) and (13) for the in-degree). The system is ∂ ∆t ∆t = (1 − ms )Br + ms Br . (20) ∂t t with solution: ¯ ∆t = Dt + Ctms Br , where C is a constant matrix. For a given initial condition D1 (that we can identify with the matrix A of biases) the solution for Dt can be written as: ∂ ¯ 1 ¯ Dt = ∆t = D + (D1 − D)tms Br , (21) ∂t t ¯ where D is a constant term. For ms < 1, the second term approaches the null matrix as t → ∞. As ¯ long as the matrix D1 is more biased than the steady state D (which is true for D1 =A), the bias in excess of the steady state decreases with time, vanishing out in the long run (see also Appendix A). This means that the biases in the out-links formed by agents decrease over time, consisten with the homogenization of the search process and the in-degree of older nodes which are dominating the search part of the process. However, unlike the case of the in-degree of old nodes, full homog- enization does not occur even in the limit, since the random part of the out-degree formation does not vanish as time grows. Example 5 Out–degree dynamics for the 2–types symmetric case. Consider the symmetric RSU case discussed in Example 2, with a self–bias 1 < π < 1. In that 2 1 1−π ¯ 3−2π 2 3−2π case, if we deﬁne the matrix D = 1−π 1 , we obtain that the matrix of out–degrees 2 3−2π 3−2π follow the following functional form in matrix notation: 1 1 3 π− −π Dt = D + t− 2 +π ¯ 1 3−2π 3−2π 1 . 3−2π −π π− 3−2π 1 As 1 < 3−2π < π for every 1 < π < 1, there are two things to notice: 2 2 ¯ (1) Matrix D is a self–biased stochastic matrix (i.e. it has greater elements on the diagonal), but it is less biased than Br . (2) Dt is also a stochastic matrix. The second matrix on the right has positive elements on the ¯ diagonal, while the other two are negative, so that Dt is more self–biased than D. However, this 3 additional bias vanishes to 0 with t− 2 +π . Finally, D1 = Br and Dt is less biased than Br for t > 1, 3 because t− 2 +π is 1 for t = 1 and then it decreases to 0. 20 4 An illustration using scientiﬁc citations In this section we use our random-search model to study the patterns of cross-ﬁeld scientiﬁc citations in physics. The use of scientiﬁc citation is motivated by several factors. First, there is a large body of literature that shows how key aspects of the time evolution of citations can be captured by models in which some sort of preferential attachment mechanism is at work. The existence of a cumulative eﬀect of time was found by Price (1976), and then by Radner (1998) for ISI papers and by Newman (2008), showing that older papers eﬀectively enjoy a ﬁrst mover advantage in receiving citations, independently of the intrinsic quality of the paper. Although some bias in favor of recent papers seem to allow for a better ﬁt of certain datasets (see Borner, Maru and Goldstone (2004) and Simkin and Roychowdhury (2007)), the evidence of a rich-gets-richer mechanism seems sound. In addition, Simkin and Roychowdhury (2005) have shown that this evidence is best accounted for when preferential attachment is generated by a random-search mechanism as the one we use in this paper, where in looking for a citation authors ﬁrst randomly select papers, and then look at these papers’ reference lists to randomly pick additional citations. There is less evidence on the patterns of citations across disciplines or across other types of categories in which research may be organized. Among these, several works have shown that geographical distance and countries boundaries is one important determinant of citations patterns, while Lehman, Lautrup, and Jackson (2003) have shown that citations patterns are quite uniform across sub-ﬁelds in the high energy physics dataset (SPIRES). Also, Shi, Tseng and Adamic (2009) ﬁnd a relation between the homophily in citing other papers and the total citations received by computer science papers (we have discussed this in Section 3.4). Summing up, the generative process of citations possesses all the basic aspects of the network formation process studied in this paper. First, it is a growing network process, since new papers are written in chronological order, and old papers do not vanish or die. Second, citations are directional, and only citations from newer to older nodes are possible. Third, citations never disappear, and accumulate over time. In addition, and speciﬁcally to our mode, nodes have “types”, that we identify with the scientiﬁc classiﬁcation of a paper (see below for details). Finally, a key element of our process is that links are formed both at random and by search through established links. In the case of citations, these two channels of search are present, since one can distinguish between citations that come from direct knowledge of a paper from citations that originate from the list of references of other papers that one has read. So, all the key elements of our formal analysis are present, and this illustration can be used to test our integration results, and to learn more about the generative process of citations in general. We use the American Institute of Physics (AIP) citations dataset, which reports all the papers published in journals of the AIP between 1977 and 2007. There is a total of 241749 papers and 1982689 citations (8 citations on average). Around 10 per cent of the papers are never cited, while 21 the most cited one receives around 3700 citations). Types are deﬁned by the ﬁrst digit of the PACS classiﬁcation code: 00: General; 10: The Physics of Elementary Particles and Fields; 20: Nuclear Physics; 30: Atomic and Molecular Physics; 40: Electromagnetism, Optics, Acoustics, Heat Transfer, Classical Mechanics, Fluid Dynamics; 50: Physics of Gases, Plasmas, and Electric Discharges; 60: Condensed Matter: Structural, Mechanical, and Thermal Properties; 70: Condensed Matter: Electronic Structure, Electrical, Magnetic, and Optical Properties; 80: Interdisciplinary Physics and Related Areas of Science and Technology; 90: Geophysics, Astronomy, and Astrophysics. We ﬁrst note that the time proﬁles of types’ population shares, measured, for each type and for each year, as the proportion of the total papers published during that year that are of that given type, is somewhat stationary during the whole period (see Figure 3).20 The approximate stationarity of most categories is roughly in line with our assumption in the theoretical model that probabilities of birth of various types are time invariant. In order to identify the various elements of our theoretical model, we need to distinguish citations that originate from a direct random draw from the pool of all existing papers (“random” citations) from those that originate from a search process that goes through the references contained in one’s random citations (“search” citations). To do this, we proceed as follows. We ﬁrst identify a citation from paper A to paper C as a “search” citation if there exists some paper B with the following properties: 1) B is published before C and after A, 2) A cites B, and 3) B cites C. This method obviously has some degree of arbitrariness and will not perfectly identify how the authors found the papers they cite. The bias of this simpliﬁcation is however not clear. At one side, it overstates the weight of “search” in the citation process, since A may well cite C because C is an important paper in the ﬁeld, reason for which also B cites C, without A having known about C though B. On the other side, however, it could be that authors of paper A know about paper C only because they came into paper B, which cites C: they could decide to cite only C because it contains an older version of the same idea. It could also be that some papers are found through the search process, without the authors ever citing the intermediate paper, and so some 20 The only two sharp changes in the time proﬁles are around 1990 for type 10 (Physics of Elementary Particles and Fields) and type 70 (Condensed Matter: Electronic Structure, Electrical, Magnetic, and Optical Properties). These changes are explained by looking at more detailed classiﬁcation of types. The increase of type 70 is driven by the sharp increase in the subcategory 74 “Superconductivity”, to be put in relation with the fast development of the computer industry; the sharp decrease of type 10 is mainly driven by a decrease in the subcategory 11 “General theory of ﬁelds and particles”. 22 .4 .3 .2 .1 0 1970 1980 1990 2000 2010 year type_0 type_1 type_2 type_3 type_4 type_5 type_6 type_7 type_8 type_9 3: Shares of types’ proportions in time citations are coded as random even though they were found through search. We stick with the strict interpretation of the model, given that we have no other way of identifying the actual process that the authors followed. Using this method we identify 59 percent of total citations as “search” citations. We then classify the remaining 41 percent of citations as “random” citations, being the complement of the “search” citations. 4.1 Homophily Bias in Random Out-Citations In order to identify the bias in the random part of the process, we compare the share of “random” out-citations that are of the same type of the citing paper with the population share of the type of the citing paper. The ﬁrst share (qout in table 1) is obtained by averaging the share of random same-type out-citations of all papers of a given type during the whole time period. The second share (w in table 1) is obtained as the share of papers of a given type over all papers in the sample for the whole time period. The diﬀerence between these two shares is positive for all types, with maximum value of about 0.8 for type 2, minimum value is 0.33 for type 80 (Interdisciplinary physics), and average value of 0.63. Normalizing, for each type, this diﬀerence by the the maximal potential diﬀerence given by one minus the population share of the type, we obtain the Coleman (1958) homophily index of each 23 00 10 20 30 40 50 60 70 80 90 qout 0.67 0.85 0.87 0.72 0.64 0.77 0.64 0.86 0.35 0.67 w 0.11 0.11 0.08 0.08 0.06 0.016 0.14 0.35 0.02 0.03 ih 0.63 0.83 0.86 0.70 0.62 0.76 0.58 0.79 0.33 0.66 1: Same-type bias in the overall citations. type (ih in table 1).21 This index turns out not to be correlated with types’ population shares. 4.2 Search Bias, Long-run integration, and Partial Integration One challenge with an empirical investigation of the various concepts of integration is that certain papers happen to be intrinsically more cited than others, simply because they are more fundamental or important than others for their discipline. This type of “ﬁtness” is independent of the age of the paper, and is not modeled in our analysis. More importantly, it could potentially outweigh the eﬀect of time, and of the large in-degree that older nodes accumulate in time, which is one of the forces behind the long-run integration property. We deal with this problem by looking at the type-composition of the τ –th citation of each paper, thereby replacing time with citation order. This allows us to normalize the time–scale of each single paper, as if they all had the same ﬁtness. In this new context, the hypothesis we are testing is whether the homophily of the in-degrees of a paper decreases with the order of its in-citations, getting close to the relative size of that paper’s type as this order gets large. This is meant to capture the main force that leads to partial integrations: the growth of nodes’ in-degree is to a large extent composed of in-citations of the “search” type, which are, in the case of unbiased search, less biased towards one’s own type than in-citations of the “random” kind. In Figure 4 we illustrate the share of same type in-citations ordered by types’ population shares. Each dot measures on the x–axis the population share of a given type (measured as the average over the whole time period), and on the y–axis the average value (taken over all papers of that given type) of the share of same type in-citations out of the ﬁrst τ in-citations. The key feature of Figure 4 is that shares of same-type in-citations uniformly decreases with the in-degree of nodes for all types in the sample. Since the absolute levels of these shares are well above the levels of population shares for small in-degrees, this suggests that the citation process becomes less and less biased towards own type as in-degrees become large. Thus, what we observe is consistent with partial integration. In particular, this trend is con- sistent with our theoretical analysis of the more prevalent role of search over time, provided the 21 This normalization has the purpose of allowing for meaningful comparison of groups of diﬀerent sizes, by taking into account the maximal potential amount of homophily that each group has. See Currarini, Jackson and Pin (2009) for more discussion. 24 .6 share of same−type in−citations first 5 in−citations .4 first 10 in−citations first 15 in−citations first 20 in−citations first 25 in−citations first 30 in−citations .2 45 degree line 0 0 .1 .2 .3 .4 population share 4: Shares of same-type in-citations by order of citation. .6 share of same−type in−citations .2 0 .4 0 .05 .1 .15 population share first 5 in−citations first 10 in−citations first 15 in−citations first 20 in−citations first 25 in−citations first 30 in−citations 45 degree line 5: Shares of same-type in-citations by order of citation: marginal. 25 search process is less biased than the random process. In the limit, if search were unbiased, we should observe long-run integration, that is, the share of same-type in-citations coinciding with the 45 degree line. This trend is not found in Figure 4, where same-type shares are signiﬁcantly ﬂatter than the 45 degree line, and become ﬂatter for larger degrees. Interestingly, this behavior seems however driven by a single observation (type 20: “Nuclear Physics”), which refers to the largest group in the sample. If we omit this single type, we obtain the trend in Figure 5, where the regressed patterns of same-type shares uniformly approaches the 45 degree line for larger and larger in-degrees. 5 Concluding Remarks Our interest in this paper has been the extent to which biases in the way agents link to each other (that is, biases in the process of network formation) translate into biases in the patterns of the actual network (that is, in the outcome of the process). Our analysis provides one basic insight: when some of the connections are formed through a network-based search process (friends of friends), the type-composition of agents’ neighborhoods homogenizes in the long run, and in particular, full integration of types occurs when search part of the formation process is unbiased. As we have pointed out, the mechanism at work is intuitive: as nodes age, they accumulate more and more links through the rich-gets-richer dynamics of the search part of the process, ending up attracting links from all types (even from those types by which they are discriminated in the random part). Through these connections, they are found through search by even more nodes of all types, and the mechanism reinforces itself becoming less biased over time. As time elapses, old nodes are found by all types at rates that mirror population shares. Two things account for the integration in the RSU model: over time the probability of being found at random vanishes compared to the probability of being found through search; and the in- degree of old nodes becomes less biased, which then further reduces the biases in the probabilities that older nodes are found by newborn nodes of various types. Both conditions are made possible by the passing of time, which increases the total population and the in-degree of old nodes on one hand, and homogenizes in- (and out-) degree by mixing the meeting biases through the cumulative mechanism described in the proofs of the main propositions. We remark that it is not enough simply for the search part of the process to dominate, but one also needs the gradual homogenization of that process over time, as the more an old-node gets found by other types, the easier it becomes for it to be found by other types in the future. The distinction between these is clear if we examine the limit as mr → 0 in the RSU model; looking at the proof of Proposition 2, we note that the low powers of the matrix A of biases, which are not homogeneous, still maintain weight in the average deﬁning the in-degree as long as the age of the node, given by the ratio tt0 stays “small”. If we examine a diﬀerent model, then one could obtain the immediate integration of all nodes 26 as mr → 0. The diﬀerence would be that instead of having biased-random and unbiased-search, one could have biased-random and unbiased-preferential attachment (as a variant of Albert and Barabasi (1999)). This would parallel our model, but uncouple the search part of the process from the bias in the randomly selected nodes whose neighborhoods are searched. To parallel the RSU model, one can also assume that the preferential attachment part is unbiased, in the sense that the probability of a node being found is directly proportional to its relative in-degree in the whole population, irrespectively of its type. Using the same notation of the previous sections, we can express the probability of link j to obtain a link from a θ node at t + 1 as follows: t nmr θ ∈Θ Πj (θ , θj ) Pjt+1 (θ, θj ) =Br (θ, θj ) + nmp p(θ) , (22) t nt Using a mean-ﬁeld approximation we express the change in the in-degree of node j as: ∂ t nmr mp p(θ) Π (θ, θj ) = Br (θ, θj ) + Πt (θ , θj ), (23) ∂t j t t j θ ∈Θ When t grows large, the random part of the process vanishes and long-run integration occurs. Also, as mr → 0 nodes are found almost only via a type-blind manner, and then all nodes integrate. Note also that for large enough values of time, this would happen for nodes that have a large in- degree irrespective of their age (due, for instance, to some “ﬁtness” node-speciﬁc parameter). More research is needed to incorporate strategic elements into the link formation process. As it is, the model represents situations in which the meeting biases come from exogenous constraints (institutional, geographical, organizational barriers or underlying preferences), and agents cannot aﬀect the induced probabilities. Interesting considerations are likely to arise when such options are allowed, and when agents anticipate the outcome of link formation on the type mix of their in- e degree and on their welfare (this is also suggested by an example in Bramoull´ and Rogers (2009)). We believe that these issues, and more general analyses of the homophily and dynamic integration of network formation processes, lie at the heart of the research agenda in the ﬁeld, and will be the object of future research. References a [1] Albert R. and A.–L. Barab´si (1999), “Emerging of Scaling in Random Networks,” Science 286, 509–512. o [2] B¨rner K., J.T. Maru and R.L. Goldstone (2004) “The simultaneous evolution of author and paper networks,” PNAS 101, 5266–5273. e [3] Bramoull` P. and B. Rogers (2009) “Diversity and Popularity in Social Networks”, mimeo. 27 [4] Breschi, S. and F. Lissoni (2006): “Mobility of inventors and the geography of knowledge spillovers. New evidence on US data,” CESPRI WP n. 184. [5] Coleman, J. (1958) “Relational analysis: the study of social organizations with survey meth- ods,” Human Organization, 17, 28–36. [6] Currarini, S., M.O. Jackson, and P. Pin (2006) ““Overlapping Network Formation,” mimeo: Stanford University. [7] Currarini, S., M.O. Jackson, and P. Pin (2009) “An Economic Model of Friendship: Homophily, Minorities and Segregation,” Econometrica 77 (4), 1003–1045. [8] Currarini, S., M.O. Jackson, and P. Pin (2010) “Identifying the Roles of Choice and Chance in Network Formation: Racial Biases in High School Friendships”, Proceedings of the National Academy of Sciences, 107, 4857–4861. [9] Koczyy, L. S., A. Nichiforz and M. Strobel (2010) “Intellectual Inﬂuence: Quality versus Quantity”, Mimeo. [10] Jaﬀe, A. B. and M. Trajtenberg (1996): “Flows of knowledge from universities and federal laboratories: Modeling the ﬂow of patent citations over time and across institutional and geographic boundaries,” PNAS 93: 12671–12677. [11] Jackson, M.O., (2007) “Social Structure, Segregation, and Economic Behavior,” pre- sented as the Nancy Schwartz Memorial Lecture, 2007; SSRN working paper 1530885, http://ssrn.com/abstract=1530885. [12] Jackson, M.O. (2008) Social and Economic Networks, Princeton University Press. [13] Jackson, M.O. and B. Rogers (2007): “Meeting strangers and friends of friends : How random are social networks?” American Economic Review 97 (3), 890–915. [14] Lehmann, S., B. Lautrup and A.D. Jackson (2004): “Citation networks in high energy physics” Phys. Rev. E 68, 026113. [15] McPherson, M., L. Smith-Lovin and J. M. Cook (2001): “Birds of a Feather: Homophily in Social Networks,” Annual Review Sociology 27, 415–44. [16] Newman, M. E. J. (2009): “First-mover advantage in scientiﬁc publication,” Europhys. Lett. 86, 68001. [17] Palacios–Huerta, I., and O. Volij (2004) “The Measurement of Intellectual Inﬂuence,” Econo- metrica 72 (3), 963–977. 28 [18] Pin, P. (2007) Four multi-agents economic models: From evolutionary competition to social interaction, PhD Thesis, University of Venice. [19] Price, D.J.S., (1965) “Networks of scientiﬁc papers.” Science 149, 510 - 515. [20] Price, D.J.S., (1976) “A general theory of bibliometric and other cumulative advantage pro- cesses.” J. Am. Soc. Inf. Sci 27, 292 - 306. [21] Redner, S. (1998) “How popular is your paper? An empirical study of the citation distribu- tion,” Eur. Phys. J. B 4: 131–134. [22] Rinia, E. J., T. N. van Leeuwen, E. E. W. Bruins, H: G. van Vuren, and A. F. J. van Raan (2001) “Citation delay in interdisciplinary knowledge exchange,” Scientometrics 51 (1), 293– 309. [23] Shi, X., B. Tseng and L. Adamic (2009) “Information Diﬀusion in Computer Science Citation Networks,” Proceedings of the Third International ICWSM Conference. [24] Simkin M.V. and V.P. Roychowdhury(2007) “A mathematical theory of citing,” Journal of the American Society for Information Science and Technology, 58(11): 1661–1673. Appendix A Some results on Markov Matrices This ﬁrst Section of the Appendix provides some results that are necessary for the proofs of our results. Take an H × H Markov matrix M with all positive elements, i.e. a positive Markov matrix. Lemma A For every x > 0 the H × H matrix ∞ xµ µ exp (Mx) − I M(x) ≡ (ex − 1)−1 M = µ! exp (x) − 1 µ=1 is a Markov matrix. Proof: for every µ ∈ N, Mµ is a Markov matrix. To show that M(x) is a Markov matrix, we need to prove that for every i, j ∈ {1, . . . , H} we have that 0 < M (x)ij < 1, and that H M (x)kj = 1. k=1 The ﬁrst condition comes from the fact that M (x)ij is a convex combination of (an inﬁnite number of) probabilities. The second condition comes from the fact that H ∞ H ∞ xµ xµ [M (x)]kj = (ex − 1)−1 [M µ ]kj = (ex − 1)−1 =1 . µ! µ! k=1 µ=1 k=1 µ=1 29 M(x) can be seen as a weighted average of the inﬁnite elements of {Mµ }µ∈N . We know that v(M) . lim Mµ = . . , (a) µ→∞ v(M) where the row–vector v(M) is the unique eigenvector associated with eigenvalue 1 of matrix M (up to a normalization that it’s elements sum to one, by the Perron-Frobenius Theorem). We deﬁne ¯ this matrix at the limit, with all equal elements on each column, as M. Now we prove a relation between the limit of M(x) and M. ¯ Lemma B For every positive Markov matrix A, and for every couple i, j ∈ {1, . . . , H}, we have that ¯ lim [M (x)]ij = lim [M µ ]ij = [M ]ij . x→∞ µ→∞ ¯ ¯ Proof: By deﬁnition of M, for every > 0 there is a number k ∈ N, such that for every µ > k, ¯ ν we have [M µ ]ij − [M ]ij < . By driving x → ∞ we can impose to 0 the weight (ex x −1)ν! of every ¯ In this way [M (x)]ij becomes a weighted average of almost only elements from {M µ }µ∈N , ν < k. ¯ ¯ with µ > k. As for all of them we have [M µ ]ij − [M ]ij < , we have the result. Definition 5 M satisﬁes the monotone convergence property if, for every couple i, j ∈ {1, . . . , H}, µ and for every µ ∈ N, the element Mij has the following properties: ¯ µ µ+1 ¯ 1. if Mij > Mij , then Mij ≥ Mij ≥ Mij ≥ Mij ; ¯ µ µ+1 ¯ 2. if Mij < Mij , then Mij ≤ Mij ≤ Mij ≤ Mij . ¯ What comes out directly from the deﬁnition is that, if Mij > Mij , then there is at least one µ µ µ+1 for which the inequality is strict, i.e. Mij > Mij . Lemma C For every couple i, j ∈ {1, . . . , H}, and for every x > 0 If M satisﬁes the monotone convergence property, then ¯ 1. if Mij > Mij , then ∂ ∂x [M (x)]ij < 0; ¯ 2. if Mij < Mij , then ∂ ∂x [M (x)]ij > 0. Proof: We focus on case 1, as the other is proven by reversing inequalities. First, note that the function µ x (e − 1) − ex x 30 is negative if and only if xex µ< . ex − 1 xex xex Let us call ν(x) the minimum integer strictly above ex −1 , i.e. ν(x) ≡ ex −1 . Now we can show that ∞ ∂ 1 xµ µ x [M (x)]ij = (e − 1) − ex [M µ ]ij ∂x (e x − 1)2 µ! x µ=1 ∞ 1 xµ µ x < (e − 1) − ex M ν(x) (b) (e x − 1)2 µ! x ij µ=1 It is a matter of calculus to check that ∞ xµ µ x (e − 1) − ex = 0 , µ! x µ=1 and then the derivative in (b) is strictly negative. Appendix B Proofs Proof of Proposition 1: We provide the details of the proof for the RSU case, which easily extends to the other cases. Note ﬁrst that the node born at time t in deﬁnition 1 has, at the beginning of time t + 1 (before node t + 1 sends its links) an in-degree of 0. This directly implies that the probability of t to receive a link at time t + 1 from a node of type θ , given that such a node is born, is equal to the probability of being found at random among the t nodes in the network. This probability is equal to: nmr p(θ , θ(t )) . (c) t On the other hand, the probability that node t0 is be found at time t + 1 is the sum of the probability of being found at random and through search. In the model with homogeneous search, this is: nmr Πt (θ, θ(t0 )) 1 p(θ , θ(t0 )) + nms p(θ , θ) t0 . (d) t t p(θ(t0 )) n θ∈Θ Note that in (d) the terms in the vector Πt0 (θ, θ(t0 )) grow without bound as t tends to inﬁnity, t while the ﬁrst terms in (d) and in (c) are constant once t is eliminated from the denominator of both expressions. It follows that we can always choose a t large enough for (d) to be larger than (c). Proof of Proposition 2: We want to see how the matrix Πt0 of type–by–type links for a node t born at time t0 develops. To do this we compare its behavior with the behavior of the type–blind 31 process, where the in–links evolve according to22 ms mr t πt0 (t) = n −1 . ms t0 To make this comparison in the long run we study Πt0 t lim . (e) t→∞ πt0 (t) Consider the solution to the RSU model, as described by equation (13), with the decomposition Br = QAQ−1 . We rewrite (13) as: −1 t mr t ms QAQ Πt0 = −I . ms t0 By (2.1), and the facts that I = QIQ−1 and An = QAn Q−1 , we obtain: µ ∞ t mr ms log t0 A Πt0 t = n Q − I Q−1 ms µ! µ=0 µ −1 ∞ t mr t ms t ms ms log t0 = n − 1 Q −1 Aµ Q−1 . (f) ms t0 t0 µ! µ=1 Limit (e) implies that (we use Lemma B from Appendix A) µ −1 ∞ t Πt t ms ms log t0 lim t0 = lim Q −1 Aµ Q−1 t→∞ π(t) t→∞ t0 µ! µ=1 = Q lim Aµ Q−1 µ→∞ v(A) . . −1 = Q . Q , (g) v(A) where the row–vector v(A) is the unique eigenvector associated with eigenvalue 1 of matrix A (normalized to sum to 1). In this way, in the long run a node of type i born at time t0 receives a fraction of in–links from nodes of type j which is approximately a ratio v(A)i p(j) p(i) 22 This process reduces to the 1–type case studied in Jackson and Rogers (2007). 32 of the overall nodes that it would receive in a type–blind process. This proportion is the product of p(j) times a term that is constant for type i. Consider now the solution to the RSB model, as described by equation (15). Following the same procedure as above, since Bs A is still a Markov Matrix (see Appendix C). We obtain v(Bs A) Πt0 . t = (Bs Br )−1 Br Q −1 lim . . Q . (h) t→∞ πt0 (t) v(Bs A) In the long run a node of type i born at time t0 will receive a number of in–links from nodes of type j which is a fraction H Πt0 v(Bs A)i lim t = (Bs Br )−1 Br p(h) t→∞ πt0 (t) jh p(i) ij h=1 H H v(Bs A)i = (Bs Br )−1 [Br ]kh p(h) jk p(i) h=1 k=1 H H p(k, h) v(Bs A)i = (Bs Br )−1 p(k) p(h) jk p(h) p(i) h=1 k=1 H H v(Bs A)i = (Bs Br )−1 p(k)p(k, h) jk p(i) h=1 k=1 H v(Bs A)i = (Bs Br )−1 p(k) p(j) . (i) jk p(i) k=1 of the overall links that it would receive in a type–blind process, where the last line comes from the fact that H p(k, h) = 1. The second term is still a constant for type i, but the ﬁrst term is h=1 generically not proportional to p(j). Proof of Proposition 3: Let us start from the RSU model (which is a particular case of the RSB model such that Bs is a matrix of all 1’s). The result comes from the expression of matrix Πt0 as t deﬁned in equation (f), in the Proof of Proposition 2: µ −1 ∞ t Πt0 t ms ms log t0 t = Q −1 Aµ Q−1 . πt0 (t) t0 µ! µ=1 ms −1 t Here t0 −1 is just a rescaling term so that the matrix in brackets is again a Markov matrix (see Lemma A in Appendix A). From the Proof of Proposition 2 we know that it converges 33 to the distribution of the population shares. As A satisﬁes the monotone convergence property, we can apply Lemma C from Appendix A to prove that this convergence is monotonic. The general RSB case is analogous, with the same distinctions discussed above in the Proof of Proposition 2. In this case µ Πt0 t ms −1 ∞ ms log tt0 t = (Bs Br )−1 Br Q −1 (Bs A)µ Q−1 . πt0 (t) t0 µ! µ=1 As Bs A satisﬁes the monotone convergence property, we can apply Lemma C from Appendix A and the same reasoning applies. Proof of Proposition 4: From (13) we can compute the aggregate in–degree of all the nodes in the network. To do this we simply integrate the expression over time: t t ∞ t µ mr log τ ms Br Πt dτ = τ n dτ 0 0 ms µ! µ=1 µ t mr ∞ log τt dτ (ms Br )µ = n 0 ms µ! µ=1 ∞ mr t · µ! · (ms Br )µ = n ms µ! µ=1 ∞ mr = t·n (ms Br )µ . (j) ms µ=1 Column j of this matrix represents the aggregate in–degrees received by the t · p(θj ) nodes of type j born up to time t. To get the average in–degree for each type we can divide every column by t·p(θj ), that is right–multiply the whole matrix by (tB)−1 , obtaining (remember that mr = 1−ms ): ∞ 1 − ms nDQ−1 ≡ n ¯ (ms Br )µ Q−1 . (k) ms µ=1 We have deﬁned the matrix ∞ ∞ 1 − ms 1 − ms ¯ D= (ms Br )µ = Q (ms A)µ Q−1 . ms ms µ=1 µ=1 It is a constant quantity of the system. The term in brackets 1−ms ∞ (ms A)µ is a Markov ms µ=1 matrix (Lemma A in Appendix A). It approaches A ¯ as ms → 1 (Lemma B in Appendix A). This means that the more the search process dominates the random one, the more the steady- state distribution among types approaches homogeneity. Nevertheless, ﬁxing one positive value of 34 ms < 1, these proportions are constant and unequal, so that the complete homogeneity among types never happens. Appendix C Introducing formal biases in a discrete stochastic process In this appendix we discuss the constraints that should be satisﬁed by a linear distortion of a probability distribution over a ﬁnite sample so that the composition is still a probability distribution. We do this by proposing a simple stochastic mechanism, and by showing under which circumstances a bias can be identiﬁed to be equivalent to this mechanism. Start with some probability distribution over a ﬁnite discrete set Θ with cardinality H, denoted by p = (p1 , p2 , . . . , pH ). A bias is a multiplicative factor for each probability, such that the new probability distribution maintains unity measure, i.e. a vector of biases b = (b1 , b2 , . . . , bH ), such that H b·p= bi pi = 1 . (l) i=1 An implication of this is that if the original probability pi is 0 for some event i, there is no bias that can make i happen with positive probability. An intuitive way to imagine and even implement such a bias is the following. Suppose that we are extracting the elements of Θ from an urn, with probabilities given by p. Imagine that whenever we extract an element of type i, after observing it, we disregard it with a probability ri speciﬁc to i, and make a new extraction with replacement. More formally we have a new vector r that gives probabilities of refusing an extraction. In this way the probability to extract and accept an element i is given by the probability of doing it in the ﬁrst extraction, plus the probability to do it in the second extraction, and so on. . . In formulas, if we denote this new biased probability bi pi , it follows that H H 2 bi pi = pi (1 − ri ) + pi ri pi (1 − ri ) + pi ri pi (1 − ri ) + . . . i=1 i=1 ∞ = pi (1 − ri ) p·r t=0 1 − ri = pi . (m) 1−p·r This formulation is well deﬁned if there is at least one element ri of r, such that ri < 1. It is 1−ri immediate to call bi = 1−p·r the bias for event i, and to check that property (l) holds. 35 It is also easy to see that whenever two vectors p and b are speciﬁed, we can always ﬁnd a vector r that satisﬁes condition (m). The solution solves a system of H linear equations and it may not be unique. As an example it is possible to check that the vector r = (k, k, . . . , k), with all elements equal to k ∈ [0, 1), gives always the same (neutral) bias b = (1, 1, . . . , 1). Appendix D Examples Let us consider how the mechanism described in Appendix C can be applied to the growing networks that we deﬁne in Section 2. We restrict attention to the case where there are only two types in the populations, type 1 forming a fraction of the population of p(1) = p, and type 2 with p(2) = 1 − p. We call (r1,1 , r2,1 ) a vector of refusal probabilities of type 1, and (r1,2 , r2,2 ) a vector of type 2. In particular, let types be homophilous with refusal probabilities r1 = (0, r1 ), with r1 ∈ [0, 1), and similarly r2 = (r2 , 0), with r2 ∈ [0, 1). D.1 Purely random model (R) The system of equations that characterizes this system is P t+1 (1, 1) = n p j 1 t 1−(1−p)r1 t+1 P (1, 2) = n (1 − p) 1−r1 j t 1−(1−p)r1 , (n) P t+1 (2, 1) = j n 1−r p 1−pr22 t t+1 n 1 P (2, 2) = (1 − p) 1−pr2 j t so that we obtain a result with an explicit matrix of biases 1 1−r1 p 1−(1−p)r1 (1 − p) 1−(1−p)r1 Br = 1−r 1 . p 1−pr22 (1 − p) 1−pr2 D.2 Random Unbiased Search model (RSU) e This is the case analyzed by Bramoull´ and Rogers (2010). The system of equations that charac- terize our system is now Pt Pt t+1 P λ (1,1) 1−r1 P λ (2,1) P (1, 1) = nmr p j 1 1 + nms p 1−(1−p)r1 λ=j tpj + (1 − p) 1−(1−p)r1 λ=j j 1 t 1−(1−p)r1 t(1−p) n Pt Pt λ P λ (2,2) λ=j Pj (1,2) t+1 P (1, 2) = nmr (1 − p) 1−r1 + nms p j 1 1−r1 + (1 − p) 1−(1−p)r1 λ=j j 1 t 1−(1−p)r1 1−(1−p)r1 tp t(1−p) n Pt λ Pt . t+1 nmr 1−r2 1−r λ=jPj (1,1) 1 P λ (2,1) 1 P (2, 1) = j t p 1−pr2 + nms p 1−pr22 tp + (1 − p) 1−pr2 λ=j j t(1−p) n Pt Pt P λ (1,2) P λ (2,2) t+1 nmr 1 1−r 1 1 P (2, 2) = t (1 − p) 1−pr2 + nms p 1−pr22 λ=j tpj + (1 − p) 1−pr2 λ=j j j t(1−p) n (o) 36 Br is the same as above, and we again have an explicit solution from Section 2.4, that we can make explicit as the matrix is just 2 × 2: 1 1−r1 µ p 1−(1−p)r1 (1 − p) 1−(1−p)r1 ms log tt0 ∞ 1−r 1 mr p 1−pr22 (1 − p) 1−pr2 Πt0 = n − I . (p) t ms µ! µ=0 The explicit solution is „ « „ « r2 −1 r2 −1 8 00 1 1 «m p + 1 +m «−m p + 1 s s „ „ pr2 −1 −pr1 +r1 −1 pr2 −1 −pr1 +r1 −1 > > t t B @p(r2 −1)((p−1)r1 +1) t0 +(p−1)(r1 −1)(pr2 −1)A > BB C C t0 > > C > > t n mr B > B C > Πt0 (1, 1) > > > = ms B p(r2 (2(p−1)r1 −p+2)−pr1 )+r1 −1 − 1C C > > B C > > @ A > > > > > > „ « „ « «m p r2 −1 + «−m p r2 −1 + 0 0 1 1 1 1 > +m > s s > „ „ B (p−1)(r1 −1)(pr2 −1)B t pr2 −1 −pr1 +r1 −1 pr2 −1 −pr1 +r1 −1 −1A t > > C > @ t C > > > B 0 t0 C > Πt (1, 2) n mr B > B C = > C > > > t0 ms B p(r2 (2(p−1)r1 −p+2)−pr1 )+r1 −1 C > > B C > > @ A > < „ r2 −1 « „ « . «−m p r2 −1 + 0 0 1 1 «m p + 1 +m 1 s s > „ „ pr2 −1 −pr1 +r1 −1 pr2 −1 −pr1 +r1 −1 > > > B p(r2 −1)((p−1)r1 +1)B t @ t −1A t C C t0 > > > B 0 C n mr B > > Πt (2, 1) > = B C C > > > t0 ms B p(r2 (2(p−1)r1 −p+2)−pr1 )+r1 −1 C > > B C > > @ A > > > > > „ « „ « r2 −1 r2 −1 > 0 0 1 1 > > «m p + 1 +ms «−m p + 1 s s „ „ pr2 −1 −pr1 +r1 −1 pr2 −1 −pr1 +r1 −1 > B p(r2 −1)((p−1)r1 +1)B t t > −1A > > C C @ t t0 0 > > B C > > Πt (2, 2) > mr B C > t0 = nm B p(r2 (2(p−1)r1 −p+2)−pr1 )+r1 −1 − 1C s B > > B C > > C > > @ A > : (q) If we assume that the parameters of the system are p = 1/2, n = 10, mr = .5, ms = .5, r1 = .8, r2 = 0 and t0 = 1000, then we obtain exactly the example discussed in Section 3.4. D.3 Random-Search with Search bias (RSB) b1,1 b1,2 Now we have to consider a new matrix of bias B = (that will be the Bs deﬁned b2,1 b2,2 in the model), that can be derived from a homophilous matrix of additional refusal probabilities 0 s1 S= . s2 0 The system of equations that characterize our system is now Pt λ Pt λ ! λ=j Pj (1,1) λ=j Pj (2,1) 8 > P t+1 (1, 1) nmr 1 1 1−r1 1 > > j = t p 1−(1−p)r + nms b1,1 p 1−(1−p)rtp + b1,2 (1 − p) 1−(1−p)r t(1−p) n > > > 1 1 1 > > Pt λ Pt λ ! > nmr 1−r1 Pj (1,2) 1−r1 Pj (2,2) > P t+1 (1, 2) 1 λ=j λ=j 1 > = (1 − p) 1−(1−p)r + nms b1,1 p 1−(1−p)r + b1,2 (1 − p) 1−(1−p)r > > < j t 1 1 tp 1 t(1−p) n Pt λ (1,1) Pt λ (2,1) ! . (r) > P t+1 (2, 1) nmr 1−r2 1−r2 λ=j Pj 1 λ=j Pj 1 > = p 1−pr + nms b2,1 p 1−pr + b2,2 (1 − p) 1−pr > > > > j t 2 2 tp 2 t(1−p) n > > > Pt λ Pt λ ! > nmr 1−r2 Pj (1,2) Pj (2,2) > P t+1 (2, 2) λ=j λ=j > > = 1 (1 − p) 1−pr + nms b2,1 p 1−pr 1 + b2,2 (1 − p) 1−pr 1 j t tp t(1−p) n : 2 2 2 Biases are on the (already biased) probabilities of matrix Br . Essentially, we have now a new matrix of bias in the search part, that we deﬁned as Bs Br in Section 2.3. This matrix has the 37 form 1 1−r 1 p 1−(1−p)r (1−p) 1−(1−p)r (1−s2 ) 1 1 1−r1 1−r1 p (1−p) (1−s2 ) 1−s1 (1−p) 1−(1−p)r 1−s1 (1−p) 1−(1−p)r 1−s1 (1−p)(1−r1 ) 1−s1 (1−p) 1−r2 1 1 = . (s) p 1−pr (1−s2 ) 1 (1−p) 1−pr p(1−s2 ) (1−p) 2 2 1−s2 p 1−s2 p(1−r2 ) 1−r2 1−r2 1−s2 p 1−pr 1−s2 p 1−pr 2 2 We can replace Bs Br with (s) in the solution (14). It is possible to obtain an explicit solution analogously to the one obtained in (r) for the RSU case. D.4 Type bias on search bias on targeted nodes (RSBT) In this case, we still have a bias derived from a homophilous matrix S. The system of equations that characterize this system is similar to that in the case of RSB. However this leads to two matrices of biases, because biases are on the target: Pt λ Pt λ ! λ=j Pj (1,1) λ=j Pj (2,1) 8 > P t+1 (1, 1) nmr 1−r1 > > j = t 1 p 1−(1−p)r + nms b1 p 1−(1−p)r 1,1 1 tp + b2 (1 − p) 1−(1−p)r 1,1 t(1−p) 1 n > > > 1 1 1 > > Pt λ (1,2) Pt λ (2,2) ! λ=j Pj λ=j Pj > > P t+1 (1, 2) nmr 1−r1 1−r1 (1 − p) 1−(1−p)r + nms b1 p 1−(1−p)r 1 + b2 (1 − p) 1−(1−p)r 1 > = > > < j t 1 1,2 1 tp 1,2 1 t(1−p) n Pt λ Pt λ ! . (t) > P t+1 (2, 1) nmr 1−r2 1−r2 λ=j Pj (1,1) λ=j Pj (2,1) p 1−pr + nms b1 p 1−pr + b2 (1 − p) 1−pr 1 1 > = > > > > j t 2 2,1 2 tp 2,1 2 t(1−p) n > > > Pt λ (1,2) Pt λ (2,2) ! λ=j Pj λ=j Pj > > P t+1 (2, 2) > nmr 1−r2 (1 − p) 1−pr + nms b1 p 1−pr 1 + b2 (1 − p) 1−pr1 1 > : j = t 2,2 tp 2,2 t(1−p) n 2 2 2 The biases are on the probabilities of ﬁnding a target of that particular type, and these probabilities diﬀer according to the intermediary (superscript on the b’s). We obtain 0 tp (1−s1 )tp 1 0 t(1−p) (1−s1 )t(1−p) 1 Pt P λ (1,2) Pt P λ (1,2) Pt P λ (2,2) Pt tp−s1 tp−s1 t(1−p)−s1 t(1−p)−s1 P λ (2,2) B λ=j j λ=j j C B λ=j j λ=j j C 1 B C 2 B C B =B B C C and B =B B C C . @ (1−s2 )tp tp A @ (1−s2 )t(1−p) t(1−p) A Pt P λ (1,1) Pt P λ (1,1) Pt P λ (2,1) Pt tp−s2 tp−s2 t(1−p)−s2 t(1−p)−s2 P λ (2,1) λ=j j λ=j j λ=j j λ=j j This makes the biases depend on every element inside the brackets that characterize the search part of system (t). They can be taken out, as a rough approximation, only if at the limit of t j we have B ¯ 1 and B2 converging to a unique matrix B of biases. ¯ Taking out Bs as a single constant B, as we do in Section 2.3, is a big simpliﬁcation. Even so, that case is not so easily solvable as it has an additional bias compared to the RSU model. This is the case of the RSBT model analyzed here. 38