link-pred

The Link Prediction Problem for Social Networks ∗ David Liben-Nowell† Laboratory for Computer Science Massachusetts Institute of Technology Cambridge, MA 02139 USA dln@theory.lcs.mit.edu Jon Kleinberg‡ Department of Computer Science Cornell University Ithaca, NY 14853 USA kleinber@cs.cornell.edu January 8, 2004 Abstract Given a snapshot of a social network, can we infer which new interactions among its members are likely to occur in the near future? We formalize this question as the link prediction problem, and develop approaches to link prediction based on measures for analyzing the “proximity” of nodes in a network. Experiments on large co-authorship networks suggest that information about future interactions can be extracted from network topology alone, and that fairly subtle measures for detecting node proximity can outperform more direct measures. 1 Introduction As part of the recent surge of research on large, complex networks and their properties, a consid- erable amount of attention has been devoted to the computational analysis of social networks— structures whose nodes represent people or other entities embedded in a social context, and whose edges represent interaction, collaboration, or influence between entities. Natural examples of social networks include the set of all scientists in a particular discipline, with edges joining pairs who have co-authored papers; the set of all employees in a large company, with edges joining pairs working on a common project; or a collection of business leaders, with edges joining pairs who have served together on a corporate board of directors. The availability of large, detailed datasets encoding such networks has stimulated extensive study of their basic properties, and the identi- fication of recurring structural features. (See, for example, the work of Watts and Strogatz [28], Watts [27], Grossman [9], Newman [19], and Adamic and Adar [1], or, for a thorough recent survey, Newman [20].) Social networks are highly dynamic objects; they grow and change quickly over time through the addition of new edges, signifying the appearance of new interactions in the underlying social structure. Understanding the mechanisms by which they evolve is a fundamental question that is still not well understood, and it forms the motivation for our work here. We define and study a basic computational problem underlying social network evolution, the link prediction problem: ∗An abbreviated version of this paper appears in Proceedings of the Twelfth Annual ACM International Conference on Information and Knowledge Management (CIKM’03), November 2003, pp. 556–559. †Supported in part by an NSF Graduate Research Fellowship. ‡Supported in part by a David and Lucile Packard Foundation Fellowship and NSF ITR Grant IIS-0081334. 1 Given a snapshot of a social network at time t, we seek to accurately predict the edges that will be added to the network during the interval from time t to a given future time t′. In effect, the link prediction problem asks: to what extent can the evolution of a social network be modeled using features intrinsic to the network itself? Consider a co-authorship network among scientists, for example. There are many reasons, exogenous to the network, why two scientists who have never written a paper together will do so in the next few years: for example, they may happen to become geographically close when one of them changes institutions. Such collaborations can be hard to predict. But one also senses that a large number of new collaborations are hinted at by the topology of the network: two scientists who are “close” in the network will have colleagues in common, and will travel in similar circles; this suggests that they themselves are more likely to collaborate in the near future. Our goal is to make this intuitive notion precise, and to understand which measures of “proximity” in a network lead to the most accurate link predictions. We find that a number of proximity measures lead to predictions that outperform chance by factors of 40 to 50, indicating that the network topology does indeed contain latent information from which to infer future interactions. Moreover, certain fairly subtle measures—involving infinite sums over paths in the network—often outperform more direct measures, such as shortest-path distances and numbers of shared neighbors. We believe that a primary contribution of the present paper is in the area of network evolution models. While there has been a proliferation of such models in recent years—see, for example, the work of Jin et al. [11], Barabasi et al. [2], and Davidsen et al. [5] for recent work on collaboration networks, or the survey of Newman [20]—they have generally been evaluated only by asking whether they reproduce certain global structural features observed in real networks. As a result, it has been difficult to evaluate and compare different approaches on a principled footing. Link prediction, on the other hand, offers a very natural basis for such evaluations: a network model is useful to the extent that it can support meaningful inferences from observed network data. One sees a related approach in recent work of Newman [17], who considers the correlation between certain network growth models and data on the appearance of edges of co-authorship networks. In addition to its role as a basic question in social network evolution, the link prediction problem could be relevant to a number of interesting current applications of social networks. Increasingly, for example, researchers in artificial intelligence and data mining have argued that a large organization, such as a company, can benefit from the interactions within the informal social network among its members; these serve to supplement the official hierarchy imposed by the organization itself [13, 23]. Effective methods for link prediction could be used to analyze such a social network, and suggest promising interactions or collaborations that have not yet been utilized within the organization. In a different vein, research in security has recently begun to emphasize the role of social network analysis, largely motivated by the problem of monitoring terrorist networks; link prediction in this context allows one to conjecture that particular individuals are working together even though their interaction has not been directly observed [14]. The link prediction problem is also related to the problem of inferring missing links from an observed network: in a number of domains, one constructs a network of interactions based on observable data and then tries to infer additional links that, while not directly visible, are likely to exist [8, 22, 26]. This line of work differs from our problem formulation in that it works with a static snapshot of a network, rather than considering network evolution; it also tends to take into account specific attributes of the nodes in the network, rather than evaluating the power of prediction methods based purely on the graph structure. We now turn to a description of our experimental setup, in Section 2. Our primary focus is on 2 training period Core authors papers edges authors |Eold | |Enew | astro-ph 5343 5816 41852 1561 6178 5751 cond-mat 5469 6700 19881 1253 1899 1150 gr-qc 2122 3287 5724 486 519 400 hep-ph 5414 10254 17806 1790 6654 3294 hep-th 5241 9498 15842 1438 2311 1576 Figure 1: The five sections of the arXiv from which co-authorship networks were constructed: astro-ph (astrophysics), cond-mat (condensed matter), gr-qc (general relativity and quantum cosmology), hep-ph (high energy physics—phenomenology), and hep-th (high energy physics— theory). The set Core is the subset of the authors who have written at least κtraining = 3 papers during the training period and κtest = 3 papers during the test period. The sets Eold and Enew denote edges between Core authors which first appear during the training and test periods, respec- tively. understanding the relative effectiveness of network proximity measures adapted from techniques in graph theory, computer science, and the social sciences, and we review a large number of such techniques in Section 3. Finally, we discuss the results of our experiments in Section 4. 2 Data and Experimental Setup Suppose we have a social network G = 〈V, E〉 in which each edge e = 〈u, v〉 ∈ E represents an interaction between u and v that took place at a particular time t(e). We record multiple interactions between u and v as parallel edges, with potentially different time-stamps. For two times t < t′, let G[t, t′] denote the subgraph of G consisting of all edges with a time-stamp between t and t′. Here, then, is a concrete formulation of the link prediction problem. We choose four times t0 < t ′ 0 < t1 < t ′ 1, and give an algorithm access to the network G[t0, t ′ 0]; it must then output a list of edges, not present in G[t0, t ′ 0], that are predicted to appear in the network G[t1, t ′ 1]. We refer to [t0, t ′ 0] as the training interval and [t1, t ′ 1] as the test interval. Of course, social networks grow through the addition of nodes as well as edges, and it is not sensible to seek predictions for edges whose endpoints are not present in the training interval. Thus, in evaluating link prediction methods, we will generally use two parameters κtraining and κtest (each set to 3 below), and define the set Core to be all nodes incident to at least κtraining edges in G[t0, t ′ 0] and at least κtest edges in G[t1, t ′ 1]. We will then evaluate how accurately the new edges between elements of Core can be predicted. We now describe our experimental setup more specifically. We work with five co-authorship networks G, obtained from the author lists of papers at five sections of the physics e-Print arXiv, www.arxiv.org. (See Figure 1 for statistics on the size of each of these five networks.) Some heuristics were used to deal with occasional syntactic anomalies; and authors were identified by first initial and last name, a process that introduces a small amount of noise due to multiple authors with the same identifier [18]. The errors introduced by this process appear to be minor. Now consider any one of these five graphs. We define the training interval to be the three years [1994, 1996], and the test interval to be [1997, 1999]. We denote the subgraph G[1994, 1996] on the training interval by Gcollab := 〈A, Eold〉, and use Enew to denote the set of edges 〈u, v〉 such that u, v ∈ A, and u, v co-author a paper during the test interval but not the training interval—these are the new interactions we are seeking to predict. 3 graph distance (negated) length of shortest path between x and y common neighbors |Γ(x)∩ Γ(y)| Jaccard’s coefficient |Γ(x)∩Γ(y)| |Γ(x)∪Γ(y)| Adamic/Adar ∑ z∈Γ(x)∩Γ(y) 1 log |Γ(z)| preferential attachment |Γ(x)| · |Γ(y)| Katzβ ∑∞ `=1 β ` · |paths 〈`〉 x,y | where paths 〈`〉 x,y := {paths of length exactly ` from x to y} weighted: paths 〈1〉 x,y := number of collaborations between x, y. unweighted: paths 〈1〉 x,y := 1 iff x and y collaborate. hitting time −Hx,y stationary-normed −Hx,y · piy commute time −(Hx,y + Hy,x) stationary-normed −(Hx,y · piy + Hy,x · pix) where Hx,y := expected time for random walk from x to reach y piy := stationary distribution weight of y (proportion of time the random walk is at node y) rooted PageRankα stationary distribution weight of y under the following random walk: with probability α, jump to x. with probability 1 − α, go to random neighbor of current node. SimRankγ { 1 if x = y γ · � a∈Γ(x) � b∈Γ(y) score(a,b) |Γ(x)|·|Γ(y)| otherwise Figure 2: Values for score(x, y) under various predictors; each predicts pairs 〈x, y〉 in descending order of score(x, y). The set Γ(x) consists of the neighbors of the node x in Gcollab . Evaluating a link predictor. Each link predictor p that we consider outputs a ranked list Lp of pairs in A×A−Eold ; these are predicted new collaborations, in decreasing order of confidence. For our evaluation, we focus on the set Core, so we define E∗new := Enew ∩(Core×Core) and n := |E ∗ new |. Our performance measure for predictor p is then determined as follows: from the ranked list Lp, we take the first n pairs in Core × Core, and determine the size of the intersection of this set of pairs with the set E∗new . 3 Methods for Link Prediction In this section, we survey an array of methods for link prediction. All the methods assign a connection weight score(x, y) to pairs of nodes 〈x, y〉, based on the input graph Gcollab , and then produce a ranked list in decreasing order of score(x, y). Thus, they can be viewed as computing a measure of proximity or “similarity” between nodes x and y, relative to the network topology. In general, the methods are adapted from techniques used in graph theory and social network analysis; in a number of cases, these techniques were not designed to measure node-to-node similarity, and hence need to be modified for this purpose. Figure 2 summarizes most of these measures; below we 4 discuss them in more detail. We note that some of these measures are designed only for connected graphs; since each graph Gcollab that we consider has a giant component—a single component containing most of the nodes—it is natural to restrict the predictions for these measures to this component. Perhaps the most basic approach is to rank pairs 〈x, y〉 by the length of their shortest path in Gcollab . Such a measure follows the notion that collaboration networks are “small worlds,” in which individuals are related through short chains [18]. (In keeping with the notion that we rank pairs in decreasing order of score(x, y), we define score(x, y) here to be the negative of the shortest path length.) Pairs with shortest-path distance equal to 1 are joined by an edge in Gcollab , and hence they belong to the training edge set Eold . For all of our graphs Gcollab , there are well more than n pairs at shortest-path distance two, so our shortest-path predictor simply selects a random subset of these distance-two pairs. Methods based on node neighborhoods. For a node x, let Γ(x) denote the set of neighbors of x in Gcollab . A number of approaches are based on the idea that two nodes x and y are more likely to form a link in the future if their sets of neighbors Γ(x) and Γ(y) have large overlap; this follows the natural intuition that such nodes x and y represent authors with many colleagues in common, and hence are more likely to come into contact themselves. Jin et al. [11] and Davidsen et al. [5] have defined abstract models for network growth using this principle, in which an edge 〈x, y〉 is more likely to form if edges 〈x, z〉 and 〈z, y〉 are already present for some z. • Common neighbors. The most direct implementation of this idea for link prediction is to define score(x, y) := |Γ(x) ∩ Γ(y)|, the number of neighbors that x and y have in common. Newman [17] has computed this quantity in the context of collaboration networks, verifying a correlation between the number of common neighbors of x and y at time t, and the probability that they will collaborate in the future. • Jaccard’s coefficient and Adamic/Adar. The Jaccard coefficient—a commonly used similarity metric in information retrieval [24]—measures the probability that both x and y have a feature f , for a randomly selected feature f that either x or y has. If we take “features” here to be neighbors in Gcollab , this leads to the measure score(x, y) := |Γ(x)∩Γ(y)|/|Γ(x)∪Γ(y)|. Adamic and Adar [1] consider a related measure, in the context of deciding when two personal home pages are strongly “related.” To do this, they compute features of the pages, and define the similarity between two pages to be ∑ z : feature shared by x, y 1 log(frequency(z)) . This refines the simple counting of common features by weighting rarer features more heavily. This suggests the measure score(x, y) := ∑ z∈Γ(x)∩Γ(y) 1 log |Γ(z)| . • Preferential attachment has received considerable attention as a model of the growth of net- works [16]. The basic premise is that the probability that a new edge involves node x is proportional to |Γ(x)|, the current number of neighbors of x. Newman [17] and Barabasi et al. [2] have further proposed, on the basis of empirical evidence, that the probability of co-authorship of x and y is correlated with the product of the number of collaborators of x and y. This corresponds to the measure score(x, y) := |Γ(x)| · |Γ(y)|. Methods based on the ensemble of all paths. A number of methods refine the notion of shortest-path distance by implicitly considering the ensemble of all paths between two nodes. 5 • Katz [12] defines a measure that directly sums over this collection of paths, exponentially damped by length to count short paths more heavily. This leads to the measure score(x, y) := ∞∑ `=1 β` · |paths〈`〉x,y|, where paths 〈`〉 x,y is the set of all length-` paths from x to y. (A very small β yields predictions much like common neighbors, since paths of length three or more contribute very little to the summation.) One can verify that the matrix of scores is given by (I − βM)−1 − I , where M is the adjacency matrix of the graph. We consider two variants of this Katz measure: (1) unweighted, in which paths 〈1〉 x,y = 1 if x and y have collaborated and 0 otherwise, and (2) weighted, in which paths 〈1〉 x,y is the number of times that x and y have collaborated. • Hitting time, PageRank, and variants. A random walk on Gcollab starts at a node x, and iteratively moves to a neighbor of x chosen uniformly at random. The hitting time Hx,y from x to y is the expected number of steps required for a random walk starting at x to reach y. Since the hitting time is not in general symmetric, it is also natural to consider the commute time Cx,y := Hx,y+Hy,x. Both of these measures serve as natural proximity measures, and hence (negated) can be used as score(x, y). One difficulty with hitting time as a measure of proximity is that Hx,y is quite small whenever y is a node with a large stationary probability piy , regardless of the identity of x. To counterbalance this phenomenon, we also consider normalized versions of the hitting and commute times, by defining score(x, y) := −Hx,y · piy or score(x, y) := −(Hx,y · piy + Hy,x · pix). Another difficulty with these measures is their sensitive dependence to parts of the graph far away from x and y, even when x and y are connected by very short paths. A way of counteracting this is to allow the random walk from x to y to periodically “reset,” returning to x with a fixed probability α at each step; in this way, distant parts of the graph will almost never be explored. Random resets form the basis of the PageRank measure for Web pages [3], and we can adapt it for link prediction as follows: Define score(x, y) under the rooted PageRank measure to be the stationary probability of y in a random walk that returns to x with probability α each step, moving to a random neighbor with probability 1 − α. • SimRank [10] is a fixed point of the following recursive definition: two nodes are similar to the extent that they are joined to similar neighbors. Numerically, this is specified by defining similarity(x, x) := 1 and similarity(x, y) := γ · ∑ a∈Γ(x) ∑ b∈Γ(y) similarity(a, b) |Γ(x)| · |Γ(y)| for some γ ∈ [0, 1]. We then define score(x, y) := similarity(x, y). SimRank can also be interpreted in terms of a random walk on the collaboration graph: it is the expected value of γ`, where ` is a random variable giving the time at which random walks started from x and y first meet. Higher-level approaches. We now discuss three “meta-approaches” that can be used in con- junction with any of the methods discussed above. • Low-rank approximation. Since the adjacency matrix M can be used to represent the graph Gcollab , all o