Opinion dissenting-in-part filed by Circuit Judge Chen.
Per Curiam.
Software Rights Archive, LLC (“Software Rights”) appeals inter partes review (“IPR”) decisions of the Patent Trial and Appeal Board (“board”) of the United States Patent and Trademark Office (“PTO”) holding that claims 18 and 45 of U.S. Patent No. 5,832,494 (the “’494 patent”) and claims 12 and 22 of U.S. Patent No. 6,233,571 (the ‘"571 patent”) are un-patentable over the prior art.1 See Facebook, Inc. v. Software Rights Archive, LLC, IPR No. 2013-00479, 2015 WL 470598, at *7-13, *16-17 (PTAB Feb. 2, 2015) (“Board Decision I”); Facebook, Inc. v. Software Rights Archive, LLC, IPR No. 2013-00481, 2015 WL 429750, at *12-16, *18-20 (PTAB Jan. 29, 2015) (“Board Decision II”). Facebook, Inc., Linkedln Corporation, and Twitter, Inc. (collectively “Facebook”) cross-appeal, challenging the [631] board’s determinations that claims 1, 5,15, and 16 of the ’494 patent are not anticipated, see Facebook, Inc. v. Software Rights Archive, LLC, IPR No. 2013-00480, 2015 WL 456539, at *8-13 (PTAB Jan. 30, 2015) (“Board Decision III”), and that claim 21 of the ’571 patent is not obvious over the prior art, see Board Decision II, 2015 WL 429750, at *16-18. For the reasons discussed below, we affirm in part and reverse in part.
BACKGROUND
The ’494 and ’571 patents are eontinua-tions-in-part of U.S. Patent No. 5,544,352 (the “’352 patent”). We recently affirmed the board’s determination that claims 26, 28-30, 32, 34, and 39 of the ’352 patent are unpatentable as obvious. See Facebook, Inc. v. Software Rights Archive, LLC, IPR No. 2013-00478, 2015 WL 470597 (PTAB Feb. 2, 2015), affd without opinion, 640 Fed.Appx. 995 (Fed. Cir. 2016).
The ’494 and ’571 patents, which relate to computerized research on a database, are both entitled “Method and Apparatus for Indexing, Searching and Displaying Data.” Joint Appendix (“J.A.”) I 5057; J.A II 5058.2 The patents purport to improve upon traditional Boolean search methods by analyzing non-semantic relationships between documents. See J.A. I 5057-59; J.A. II 5058-60. They describe a process for organizing and searching for data using a technique called “proximity indexing.” ’494 patent, col. 3 1. 28; ’571 patent, col. 3 1. 33. Proximity indexing is used to search for data, including textual objects, by “generating] a quick-reference of the relations, patterns, and similarity found among the data in the database.” ’494 patent, col. 3 11. 30-31; ’571 patent, col. 3 11. 34-36. The claimed inventions are designed to provide a “user friendly computerized research tool” which “emulates human methods of research.” ’494 patent, col. 3 11. 11-14; ’571 patent, col. 3 11. 15-18.
I. The ’494 Patent
The ’494 patent describes using non-semantic relationships to search for objects in a database. J.A. I 5057-58, A citation relationship between two documents is non-semantic because it is not based on words (or “terms”) common to both documents, but is instead based on one document’s reference to the other document. See J.A. I 5058, 5063. Two documents have a direct citation relationship when one document cites to the other document. See J.A. I 5063. Two documents can also have an indirect citation relationship, such as when they both cite to a third document (bibliographic coupling (“6c”)) or when they are both cited by a third document (co-citation (“cc”)). J.A. I 5063-64.3 These relationships, as well as other relationships between documents, can be used to create clusters of similar documents, thereby enhancing search and retrieval. J.A, I 5058-64.
The ’494 patent explains that data in the database to be searched may be represent ed as a “node.” ’494 patent, col. 12 11. 34-[632]*63241. A node can be “an object in a database, a portion of an object in a database, a document, a section of a document[ ] [or] a World Wide Web page.” Id. col. 12 11. 36-39 (diagram numbering omitted). The ’494 patent further states that a cluster link generator can be used to generate candidate cluster links between nodes. Id. col. 21 11. 54-67. It explains that “[c]andidate cluster links are the set of all possible cluster links between a search node and a target node.” Id. col. 21 1. 66-col. 22 1. 1 (diagram numbering omitted). Actual cluster links, which are “a subset of the candidate cluster links ... which meet a certain criteria,” can be “used to locate nodes for display.” Id. col. 22 11. 1-4 (diagram numbering omitted).
Independent claim 1 of the ’494 patent recites:
A method of analyzing a database with indirect relationships, using links and nodes, comprising the steps of: selecting a node for analysis;
generating candidate cluster links for the selected node, wherein the step of generating comprises an analysis of one or more indirect relationships in the database;
deriving actual cluster links from the candidate cluster links;
identifying one or more nodes for display; and
displaying the identity of one or more nodes using the actual cluster links.
Id. col. 51 11. 38-49.
Claim 5, which depends from claim 1, provides that “the step of generating the candidate cluster links comprises the step of[] eliminating candidate cluster links, wherein the number of candidate cluster links are limited and the closest candidate cluster links are chosen over the remaining links.” Id. col. 511. 66-col. 52 1. 4. Claim 15 recites: “The method of claim 14 further comprising the step of deriving the actual cluster links wherein the actual cluster links are a subset of the candidate cluster links.” Id. col. 52 11. 65-67 (emphasis omitted). Similarly, claim 16 recites: “The method of claim 15 wherein the step of deriving comprises the step of choosing the top rated candidate cluster links.” Id. col. 5311.1-3 (emphasis omitted).
Claims 18 and 45 of the ’494 patent describe search methods which use numerical representations of relationships between documents. Independent claim 18 recites:
A method of analyzing a database having objects and a first numerical representation of direct relationships in the database, comprising the steps of:
generating a second numerical representation using the first numerical representation, wherein the second numerical representation accounts for indirect relationships in the database;
storing the second numerical representation;
identifying at least one object in the database, wherein the stored numerical representation is used to identify objects; and
displaying one or more identified objects from the database.
Id. col. 53 11. 28-40.
Claim 45 depends from claim 19, which in turn depends from claim 18. It recites:
The method of claim 19, wherein the direct relationships are hyperlink relationships between objects on the world wide web and the second numerical representation of direct and indirect relationships is a value that is generated by analyzing direct link weights in a set of paths between two indirectly related objects, and wherein the step of identifying uses at least the value to determine an object’s importance for ranking. ,
[633] J.A. I 5092 (reexamination certificate) (emphasis omitted).
The board held that claim 18 of the ’494 patent is unpatentable over three prior art publications by Dr. Edward A. Fox (collectively the “Fox Papers”). See Board Decision I, 2015 WL 470598, at *7-13. These publications were originally part of one document but were eventually split into three separate documents: (1) Edward A. Fox, Characterization of Two New Experimental Collections in Computer and Information Science Containing Textual and Bibliographic Concepts (Sept. 1983) (Ph.D. dissertation, Cornell University) (“Fox Collection”); (2) Edward A. Fox, Some Considerations for Implementing the SMART Information Retrieval System under UNIX (Sept. 1983) (Ph.D. dissertation, Cornell University) (“Fox SMART”); and (3) Edward A. Fox, Extending the Boolean and Vector Space Models of Information Retrieval with P-Norm Queries and Multiple Concept Types (Aug. 1983) (Ph.D. thesis, Cornell University) (“Fox Thesis”). The Fox Papers describe clustering documents based on concepts (which are referred to as “vectors” or “subvectors”) as well as terms. J.A. I 5629^0. In the Fox system, a query will not only “retrieve1 clusters containing documents whose terms match its terms,” but -will also retrieve “documents which have little in common with the query terms but are highly correlated through other components of the extended vectors.” J.A. I 5659. Fox Thesis explains that “bibliographic measures,” such as be and cc, are “useful in both retrieval and clustering applications.” J.A. I 5635.
Although Software Rights argued that the Fox Papers do not teach claim 18’s limitation requiring “a database having objects and a first numerical representation of direct relationships in the database,” ’494 patent, col. 53 11. 27-29, the board rejected this contention. See Board Decision I, 2015 WL 470598, at *10-11. According to the board, “it would have been obvious to modify the databases of the Fox Papers to contain full text documents.” Id. at *10. In support, the board noted that the Fox Papers specifically state that “some [information retrieval] systems store the full text of the various documents.” Id. (citations and internal quotation marks omitted).
The board further concluded that the Fox Papers, when combined with Edward A. Fox et al., Users, User Interfaces, and Objects: Envision, a Digital Library, 44 J. Am. Soc’y Info. Sci. 480 (1993) (“Fox Envision”), rendered claim 45 of the ’494 patent obvious. See Board Decision I, 2015 WL 470598, at *16-17. According to the board, Fox Envision teaches analyzing web-based links, as claim 45 requires, because it specifically describes “applying citation analysis to hypertext systems, including the World Wide Web.” Id. at *17.
Tfye board determined, however, that the prior art failed to teach the limitation of claims 1, 5, 15, and 16 that requires “deriving” actual cluster links from the set of candidate cluster links. See Board Decision III, 2015 WL 456539, at *8-13. The board noted that “Fox SMART describes the clustering process as initializing a new tree as empty, adding documents to the tree, and recursively splitting overly large nodes of the tree.” Id. at *9. In the board’s view,, however, because Fox SMART does not disclose “deleting clusters other than those that simply overlap, or duplicate, other clusters,” it does not teach “deriving a subset of the already generated candidate cluster links.” Id. at *10; see also id. at *13 (concluding that Fox Thesis does not anticipate claims 15 and 16 because it does not teach “deriving” a subset of actual cluster links).
[634] II. The ’571 Patent
The ’571 patent is focused on search techniques for use in hypertext networks. Claim 12 describes “cluster analyzing” universal resource locators (“URLs”):
A method for visually displaying data related to a web having identifiable web pages and Universal Resource Locators with pointers, comprising:
choosing an identifiable web page;
identifying Universal Resource Locators for the web pages, wherein the identified Universal Resource Locators either point to or point away from the chosen web page;
analyzing Universal Resource Locators, including the identified Universal Resource Locators, wherein Universal Resource Locators which have an indirect relationship to the chosen web page are located, wherein the step of analyzing further comprises cluster analyzing the Universal Resource Locators for indirect relationships; and
displaying identities of web pages, wherein the located Universal Resource Locators are used to identify web pages.
’571 patent, col. 52 11. 38-56.
Claim 22 describes retrieving hyperjump data which “has an indirect reference to [a] chosen node.” J.A. II 5090. It recites:
A method for displaying information about a network that has hyperjump data, comprising:
choosing a node;
accessing the hyperjump data;
identifying hyperjump data from within the accessed hyperjump data that has a direct reference to the chosen node;
determining hyperjump data from within the accessed hyperjump data that has an indirect reference to the chosen node by 'proximity indexing the identified hy-perjump data; and
displaying one or more determined hy-perjump data, wherein the nodes are nodes in the network that may be accessed, the hyperjwmp data includes hyperjump links between nodes in the network, and the step of displaying comprises:
generating a source map using one or more of the determined hyperjump data, wherein the source map represents hy-perjump links that -identify the chosen node as a destination of a link, and wherein the method further comprises activating a link represented on the source map, wherein a user may hyper-jump to a node represented as a node of the link.
J.A. II 5090-91 (reexamination certificate).
Claim 21 of the ’571 patent describes deriving actual cluster links from a set of candidate cluster links in a hypertext network:
A method of displaying information about a network that has hyperjump data, comprising:
choosing a node;
accessing the hyperjump data;
identifying hyperjump data from within the accessed hyperjump data that has a direct reference to the chosen node;
determining hyperjump data from within the accessed hyperjump data that has an indirect reference to the chosen node using the identified hyperjump data, wherein the step of determining comprises non-semantically generating a set of candidate cluster links for nodes indirectly related to the chosen node using the hyperjump data, assigning weights to the candidate cluster links and deriving actual cluster links from the set of candidate cluster links based on the assigned weights; and
[635] displaying one or more determined hy-perjump data.
J.A. II5090 (reexamination certifícate).
The board determined that claims 12 and 22 of the ’571 patent are obvious over a combination of Fox Thesis, Fox SMART, and Fox Envision. See Board Decision II, 2015 WL 429750, at *12-16, *18-20. Specifically, the board concluded that Fox SMART and Fox Thesis taught all of the elements of claims 12 and 22—other than the hypertext limitations—and that Fox Envision taught applying the teachings of Fox SMART and Fox Thesis to hypertext networks. See id. Furthermore, although Software Rights contended that the commercial success of Google’s PageRank algorithm provided objective evidence of non-obviousness, the board determined that Software Rights had failed to establish any nexus between the success of Google’s algorithm and the “features recited in the claims of the ’571 patent.” Id. at *15.
The board concluded, however, that Fa-cebook had not established that claim 21 of the ’571 patent is unpatentable as obvious. See id. at *16-18. In the board’s view, since Fox SMART’S clustering algorithm does not delete documents from the cluster tree, it does not teach “deriving” a subset of actual cluster links from the set of candidate cluster links. Id. at *18.
The parties then timely appealed. This court has jurisdiction under 28 U.S.C. § 1295(a)(4)(A) and 35 U.S.C. § 141(c).
DISCUSSION
I. Standard of Review
We review the board’s legal conclusions de novo, but review for substantial evidence any underlying factual determinations. See Teva Pharm. USA, Inc. v. Sandoz, Inc., — U.S. -, 135 S.Ct. 831, 836-38, — L.Ed.2d-(2015); see also Nike, Inc. v. Adidas AG, 812 F.3d 1326, 1332 (Fed. Cir. 2016); In re Giannelli, 739 F.3d 1375, 1378-79 (Fed. Cir. 2014). Substantial evidence is “such relevant evidence as a reasonable mind might accept as adequate to support a conclusion.” Consol. Edison Co. v. NLRB, 305 U.S. 197, 229, 59 S.Ct. 206, 83 L.Ed. 126 (1938); see In re Applied Materials, Inc., 692 F.3d 1289, 1294 (Fed. Cir. 2012).
II. Claim 18 of the ’494 Patent
Software Rights argues that the board erred in concluding that claim 18 of the ’494 patent is obvious in view of the Fox Papers. In support, it contends that claim 18 operates on a full-text database containing documents with textual citations to each other, whereas the test collections used by Fox contained only limited information—such as abstracts, authors, titles of articles, and bibliographic records— instead of the full text of documents.
This argument fails. Even assuming ar-guendo that claim 18 requires a database containing full-text documents, the Fox Papers explicitly suggest the use of such a database. Fox Thesis states that some information retrieval “systems store the full text of the various documents ... being manipulated,” and that this approach is advantageous because it allows users “to locate documents of interest” and “examine paragraphs, passages, sentences, or single word occurrences (in context).” J.A. I 5482.4 It explains, moreover, that storing the full text of documents is a “straightforward generalization[ ] of document retrieval methods.” J.A. I 5482. Given that Fox Thesis specifically states that storing the [636] full text of documents is both beneficial and “straightforward,” the board had ample support for its conclusion that “the Fox Papers suggested to one of ordinary' skill in the art at the time of the invention the modification of the Fox databases to include full text documents.” Boa/rd Decision I, 2015 WL 470598, at *11.
III. Claim 45 of the ’494 Patent
The board also had ample support for its determination that claim 45 of the ’494 patent is obvious over a combination of the Fox Papers and Fox Envision. See id. at *15-17. Claim 45 recites that “the direct relationships are hyperlink relationships between objects on the world wide web.” J.A. I 5092 (emphasis omitted). Although Software Rights argues that “Fox Envision does not teach the idea that one should analyze hyperlinks as opposed to bibliographic citations to enhance search,” the board properly rejected this contention, concluding that Fox Envision explicitly teaches the application of citation analysis in hypertext systems. See Board Decision I, 2015 WL 470598, at *17.
Fox Envision, which was published ten years after Fox Thesis and Fox SMART, was designed to “reconceptualize the idea of digital libraries” in order to “envision their next generation.” J.A. I 5844 (emphasis omitted). Its objective was “to harmonize and integrate concepts from a variety of interrelated fields,” including “hypertext^] hypermedia[,] ... [and] information storage and retrieval.” J.A. I 5844. Fqx Envision explicitly teaches analyzing the links between objects in a hypertext system:
Links should be recorded, preserved, organized, and generalized. As we integrate documents into very large collections covering an entire scientific domain or professional area, links among those documents become increasingly important to help with search and browsing. Groupings of those links into paths, threads, tours, and webs are essential for organizing, personalizing, sharing, and preserving the structural, interpre-tational, and evolutionary connections that develop. We are beginning to see the emergence of wide area hypertext systems (Yankelovich, 1990) like the WorldWideWeb ... that carry this concept forward into a distributed environment. Clearly, we must coordinate hypertext and hypermedia linking with the various approaches to search and retrieval (Fox et al, 1991b). One approach is the idea of information graphs (including hypergraphs), where objects of all types are interrelated by links or arcs that capture not only citation (reference) but also inheritance, inclusion, association, synchronization, sequencing, and other relationships.
J.A. I 5845.
As the board correctly determined, “[t]he approach taught in Fox Envision interrelates ‘objects of all types,’ including objects on the World Wide Web, so as to capture citation relationships.” Board Decision I, 2015 WL 470598, at *17 (quoting J.A. I 5845). Indeed, Fox Envision explains that as the overall size of a document collection increases, the “links among ... documents become increasingly important to help with search anfl browsing.” J.A. I 5845. It further explains that one approach to “coordinating] hypertext and hypermedia linking” with known search and retrieval methods is to create “information graphs (including hypergraphs), where objects of all types are interrelated by links or arcs that capture ... citation ... relationships.” J.A. I 5845 (emphasis added); see also J.A. I 7084-88. Thus, as the board correctly concluded, Fox Envision teaches analyzing citation relationships in hypertext systems. See Board Decision I, 2015 WL 470598, at *17.
[637] Software Rights contends that because the Fox Papers predated the widespread use of the web and hyperlinks, their teachings are limited to bibliographic citations between paper documents. We are unpersuaded. In assessing obviousness, references are not read in isolation. See, e.g., In re Merck & Co., Inc., 800 F.2d 1091, 1097 (Fed. Cir. 1986) (“Non-obviousness cannot be established by attacking references individually where the rejection is based upon the teachings of a combination of references.”). Although the Fox Papers do not describe analyzing hyperlink relationships, they must be read in view of Fox Envision, which, as discussed above, provides express motivation to apply citation analysis to the links between objects found on the web. See, e.g., Applied Materials, 692 F.3d at 1298 (emphasizing that “[o]ne of ordinary skill in,the art is not foreclosed from combining” relevant related references). As the board correctly determined, moreover, a skilled artisan would readily have combined the teachings of Fox Envision with those of Fox Thesis and Fox SMART given that Fox Envision is a follow-on work to Fox’s earlier publications. See Board Decision I, 2016 "WL 470698, at *16; see also Board Decision II, 2016 WL 429750, at *12 (“Dr. Fox states it would have been obvious to one of ordinary skill in the art to combine the techniques of [Fox] Envision,. Fox Thesis, and Fox SMART because [Fox] Envision was built on, and was itself[] a follow-on work to Fox Thesis and Fox SMART.”). In short, Fox Thesis and Fox SMART teach the use of citation analysis in databases storing information related to paper documents, while Fox Envision extends that teaching to hypertext networks. We conclude, therefore, that substantial evidence supports the board’s determination that claim 45 of the ’494 patent is obvious over a combination of Fox Envision and the Fox Papers.5 See In re Gartside, 203 F.3d 1305, 1316 (Fed. Cir. 2000) (“The presence or absence of a motivation to combine references in an obviousness determination is a pure question of fact.”); Winner Int’l Royalty Corp. v. Wang, 202 F.3d 1340, 1349 (Fed. Cir. 2000) (“What a reference teaches and whether it teaches toward or away from the claimed invention are questions of fact.” (citations and internal quotation marks omitted)).
IV. Alleged Teaching Away
“A reference may be said to teach away when a person of ordinary skill, upon [638] reading the reference, would be discouraged from following the path set out in the reference, or would be led in a direction divergent from the path that was taken by the applicant.” In re Gurley, 27 F.3d 551, 553 (Fed. Cir. 1994). According to Software Rights, the board erred in finding its claims obvious because the Fox Papers teach away from using indirect relationships, such as 6c and cc, to enhance search and retrieval. In support, it argues that “when all of the Fox Papers’ teachings are fairly considered, they overwhelmingly demonstrate that use of 6c and cc harms search results for the vast majority of queries.”
We disagree. One of the central teachings of the Fox Papers is that using indirect citation relationships, such as 6c and cc, can improve search results. For example, Fox Thesis, after describing the results of one experiment, concludes that using a combination of 6c and cc for clustering “seemed rather good” and that the “simpler” combination of 6c and cc pro-, duced better clusters than a “mixture” of more subvectors. J.A. I 5693. Describing another experiment, Fox Thesis states that “[ojverall, the best behavior seemed to come when [6c] and [cc] were combined with equal weighting.” J.A. I 5745. Discussing yet another experiment, Fox Thesis concludes that “[o]f all the subvectors, terms are best, though co-citations are not much worse.... Using regression or guessed at coefficients, the [term vector] and [cc] combination yields a 5-6% improvement over the performance when terms alone are used.” J.A. I 5723. Fox SMART echoes the conclusions reached in Fox Thesis. It states that “[t]he extended vector model was introduced whereby various types of bibliographic data were utilized to supplement the standard term vectors” and explains that “[o]ne motivation for [this] model was the success of clustering studies considering bibliographic data instead of or in addition to terms.” J.A. I 5406 (emphasis added).
Although the Fox Papers indicate that certain subvectors will perform worse than others under differing conditions and with the use of particular data sets, see, e.g., J.A. I 5733-34, those references unequivocally convey that analysis of indirect citation relationships—such as be and cc—can improve search and retrieval. See, e.g., Dome Patent L.P. v. Lee, 799 F.3d 1372, 1381 (Fed. Cir. 2015) (concluding that there was no teaching away even where a prior art reference “disclose[d] potential disadvantages associated” with a claimed compound). Accordingly, substantial evidence supports the board’s determination that the Fox Papers do not teach away from using indirect relationships to improve search and retrieval. See In re Mouttet, 686 F.3d 1322, 1334 (Fed. Cir. 2012) (emphasizing that the board’s determination as to whether a particular reference teaches away from a claimed invention must be upheld if supported by substantial evidence).
We reject, moreover, Software Rights’ assertion that the board failed to give proper consideration to objective indi-cia of non-obviousness. “Commercial success . is relevant to obviousness only if there is a nexus ... between the sales and the merits of the claimed invention.” Applied Materials, 692 F.3d at 1299 (citations and internal quotation marks omitted). Although Software Rights argues that the success of Google’s search engine using the PageRank algorithm provides objective evidence of non-obviousness, it fails to provide any credible evidence that Google’s search engihe practices any inventive elements recited in its claims. See Board Decision I, 2015 WL 470598, at *13 (noting that Software Rights “failed to provide the source code of PageRank, or any other [639] detailed information beyond publicly-available, generalized hearsay statements about Google’s search”). As we have previously made clear, “[w]here the offered secondary consideration actually results from something other than what is both claimed and novel in the claim, there is no nexus to the merits of the claimed invention.” In re Huai-Hung Kao, 639 F.3d 1057, 1068 (Fed. Cir. 2011); see also Tokai Corp. v. Easton Enters., Inc., 632 F.3d 1358, 1369-70 (Fed. Cir. 2011) (“If commercial success is due to an element in the prior art, no nexus exists.”).
V. The Deriving Actual Cluster Links Limitation
We conclude, however, that the board erred when it determined that Face-book failed to establish that claims 1 and 5 of the ’494 patent are unpatentable over Fox SMART. See Board Decision III, 2015 WL 456539, at *8-11. In determining that Fox SMART does not teach deriving actual cluster links from the set of candidate cluster links, the board failed to consider fully Fox SMART’S procedure for splitting overly large clusters. See J.A. I 10272-81; see also Smith & Nephew, Inc. v. Rea, 721 F.3d 1371, 1378 (Fed. Cir, 2013) (emphasizing that the board is obliged to “read[ ] the prior art for all that it teaches”).
As an initial matter, we note that the specification of the ’494 patent describes the step of deriving actual cluster links as “simple” or routine. ’494 patent, col. 241, 2. It explains that “[o]nee the candidate cluster link set has been generated, deriving the actual cluster links is a simple matter of selecting or choosing the ... top rated candidate links, and eliminating the rest.” Id. col, 24 11. 1-4 (diagram numbering omitted). Nothing in the specification suggests that deriving actual cluster links from the set of candidate cluster links is a novel or inventive aspect of the claimed invention.
Even more fundamentally, Fox SMART provides explicit instructions both on: (1) how to generate a set of candidate cluster links using indirect relationships such as be and cc; and (2) how to derive a subset of actual cluster links from that candidate cluster link set. See J.A. I 10270-81. As discussed previously, a key teaching of Fox SMART is that search and retrieval can be improved by clustering documents based upon indirect bibliographic relationships such as be and cc. J.A. I 10255-82. The Fox SMART algorithm analyzes direct relationships between documents and then identifies and assigns weights to indirect relationships between documents. See J.A. I 10257-60. Both the direct and indirect relationships can be represented as vectors in a table or matrix. See J.A. I 10257-60, 10274.
In the Fox SMART system, clustering proceeds by determining the “overall similarity between documents” which is calculated “based on available subvectors,” J.A. I 10274, such as be and cc, see J.A. I 10257-60. If a cluster exceeds a preset size limit, see J.A. I 10273, an algorithm splits the cluster and re-groups its constituent documents into smaller clusters, J.A. I 10275-79. Splitting overly large clusters is accomplished by creating a “complete similarity matrix” that compares the “pairwise combined similarity values” for each potential pair of documents in the original cluster. J.A. I 10277; see also J.A. I 10257-60, 10271-76. Because the similarity values calculated during cluster-splitting provide a set of possible cluster links between each document and every other document in the original cluster, they qualify as “candidate cluster links.” See Board Decision III, 2015 WL 456539, at *5 (explaining that “.candidate cluster links” are the “set of possible cluster links between a search [640] node and a target node” (citations and internal quotation marks omitted)); see also ’494 patent, col. 13 11. 25-26 (defining a “cluster link” as “a relationship between two nodes”).
Fox SMART then describes the process for deriving a subset of actual cluster links from the set of candidate links. Fox SMART’S cluster-splitting procedure identifies “highly correlated pairs” of documents from the original cluster by “comparing the similarity of a pair of children to the average off diagonal similarity value and seeing if it is a sufficient number of standard deviations away from the mean.” J.A. I 10278. Next, Fox SMART forms new, smaller clusters which contain “highly correlated pairs,” J.A. I 10278, and which also pass other tests for similarity and overlap, J.A. I 10277-79. Simply put, when splitting an oversized cluster, Fox SMART first generates candidate cluster links between all of the documents in the original cluster and then derives a subset of those links—the actual cluster links—in order to create new, smaller clusters.
Indeed, Fox SMART and the preferred embodiment of the ’494 patent perform the relatively straightforward step of deriving actual cluster links from the set of candidate links in exactly the same way. The preferred embodiment, like Fox SMART, generates a set of candidate cluster links based on direct and indirect relationships and then represents those candidate cluster links as vectors in a matrix. Compare ’494 patent, col. 21 11. 60-64 (“[T]he cluster link generator .. .■ generates a set of cluster links based upon both the direct links and on the indirect paths. The cluster links may be represented by a table or a series of vectors.” (diagram numbering omitted)), with J.A. I 10275-80 (describing Fox SMART’S procedure for creating a complete similarity matrix that compares the similarity values for each possible pair of documents in a cluster). Next, the preferred embodiment, like Fox SMART, selects a subset of the candidate cluster links to become actual cluster links. Compare ’494 patent, col. 21 1. 66-col. 22 1. 4 (“Candidate cluster links are the set of all possible cluster links between a search node and a target node.... [T]he actual cluster links which meet a certain criteria are used to locate nodes for display.” (diagram numbering omitted)), with J.A. I 10277-80 (describing Fox SMART’S splitting procedure which generates candidate cluster links between pairs of documents in an oversized cluster and then selects a subset of those links, based on the strength of their similarity, to become the actual cluster links in new, smaller clusters). The board erred, therefore, in concluding that Fox SMART does not teach deriving actual cluster links from the set of candidate links as required by claims 1 and 5. See Board Decision III, 2015 WL 456539, at *8-11.
The dissent contends that Fox SMART does not teach deriving actual cluster links because its cluster-splitting algorithm does not eliminate candidate cluster links, but instead “simply redistributes the candidate cluster links among two or more clusters.” Dissent at 644. This argument fails. When it splits oversize clusters, the Fox system calculates pairwise similarity values between each document and every other document in the original cluster. See J.A. I 10275-79. The algorithm thus calculates the similarity value between document one and document two, document one and document three, document one and document four, etc., until it has obtained similarity values for every pair of documents in the original, oversized cluster. See J.A. I 10277-79. These similarity values are the candidate cluster links. The Fox system then chooses the best candidate cluster links to become actual cluster links. If, for example, document one is much more simi[641] lar to document two than it is to document three, then documents one and two will likely be placed together in one cluster while document three will likely be placed in a different cluster. See J.A. I 10278 (explaining that when an oversize cluster is split, the new, smaller clusters will contain “highly correlated pairs” of documents). The candidate cluster link between documents one and three will be eliminated when document one is placed in a separate cluster from document three. There is no merit, therefore, to the dissent’s assertion that Fox SMART does not teach eliminating candidate cluster links.
On appeal, Software Rights contends that Facebook waived the right to argue that Fox SMART’S cluster-splitting procedure teaches the deriving actual cluster links limitation by failing to properly present that argument to the board. We disagree. Facebook’s original IPR petition identified Fox SMART’S cluster-splitting procedure as teaching the “deriving” step. See J.A. I 2022-23. In his declaration submitted with Facebook’s IPR petition, moreover, Fox specifically stated that the process of “recursively splitting overly large nodes” in both Fox SMART and Fox Thesis teaches the deriving actual cluster links limitation. J.A. I 10851 (citations and internal quotation marks omitted). Likewise, when Fox was cross-examined by counsel for Software Rights, he explained in detail how the cluster-splitting procedure satisfies the deriving actual cluster links limitation. See J.A. I 20758-59; see also J.A. I 20695, 20770, 20772-75. We reject, therefore, Software Rights’ contention that Facebook waived the right to rely on Fox SMART’S cluster-splitting procedure to meet the requirement of deriving a subset of actual cluster links.
We also reject Software Rights’ argument that Fox SMART does not teach claim l’s limitation reciting “displaying the identity of one or more nodes using the actual cluster links,” ’494 patent, col. 51 11. 47-48. Fox SMART specifically states that “clustered search allows retrieval of groups in response to query submission,” J.A. I 10270, and that “most of the documents in a retrieved cluster are presented to the user,” J.A. I 10282. Because Fox SMART retrieves and displays whole clusters of documents, it necessarily “us[es] the actual cluster links” that form those clusters when displaying documents or “nodes” as required by claim l.6
We are likewise unpersuaded by Software Rights’ argument that Fox SMART does not anticipate because it teaches retrieving all documents and ranking multiple clusters of documents for display. As the board correctly determined, ranking documents from multiple clusters is not the same as deriving a subset of those documents. See Board Decision III, 2015 WL 456539, at *10 (explaining that “a set of ranked documents provides an indication of an order of presentation, but is not a subset”). Significantly, however, claim 1 does not require deriving or displaying a subset of documents. Instead, it requires deriving a subset of cluster links. See ’494 patent, col. 51 11. 44-45 (reciting “deriving actual cluster links from the candidate cluster links”). Likewise, claim 5 does not require eliminating documents, but instead only requires eliminating candidate cluster links. See id. col. 52 11. 1^1 (reciting “eliminating candidate cluster links, wherein the number of candidate cluster links are limited and the closest candidate cluster links are chosen over the remaining links”). As discussed above, Fox SMART’S cluster-[642] splitting procedure teaches deriving a subset of actual cluster links from the set of candidate links when it describes the procedure for splitting overly large clusters. Software Rights points to nothing in claims 1 and 5 that additionally requires deriving a subset of documents for display. We conclude, therefore, that the board erred in confirming the patentability of claims 1 and 5 over Fox SMART. See Board Decision III, 2015 WL 456539, at *8-11.
VI. Claims 15 and 16 of the ’494 Patent
For similar reasons, we conclude that the board erred in confirming the patentability of claims 15 and 16 of the ’494 patent over Fox Thesis. See id. at *13. The only limitation in claims 15 and 16 that the board found missing from Fox Thesis was the step of deriving a subset of actual cluster links from the set of candidate cluster links. Id. Because the process for splitting overly large clusters in Fox Thesis is essentially the same as that described in Fox SMART, the board erred in concluding that Fox Thesis fails to teach the step of deriving actual cluster links from the set of candidate cluster links.
Fox Thesis and Fox SMART arose from the same thesis project and were originally part of one document. See Board Decision I, 2015 WL 470598, at *9.'It is unsurprising, therefore, that the clustering process described in Fox Thesis tracks that described in Fox SMART. Indeed, Fox SMART specifically refers to Fox Thesis when it describes its clustering algorithm. See J.A I 10271. Like Fox SMART, Fox Thesis describes a procedure in which overly large clusters are split and new clusters are formed based on high pairwise similarity values. See J.A. I 1Q607, 10613-14; see also J.A. I 10619 (explaining that a cluster will be split when “the splitting limit of 20 [documents] is reached”). As Fox Thesis explains, its algorithm will continually split and reform clusters so that “the tree stays relatively balanced and all documents are the same distance from the root.” J.A. I 10613. In Fox Thesis, each new, smaller cluster contains a derived set of actual cluster links. See J.A I 10614-21 (providing an example of the procedure for splitting an overly large cluster and forming smaller clusters of closely linked documents).
Software Rights contends that Fox Thesis does not teach clustering with be and cc because it does not require the use of be and cc when forming clusters. This argument is unavailing. As discussed previously, Fox Thesis and Fox SMART are both replete with statements encouraging the use of indirect relationships, such as be and cc, to cluster documents and search databases. See, e.g., J.A. I 10584-89, 10633-38 (Fox Thesis); J.A. I 10255-82 (Fox SMART). The fact that they do not in all cases mandate the use of indirect relationships during the clustering process does not mean that they do not teach the use of be and cc for clustering. See Hewlett-Packard Co. v. Mustek Sys., Inc., 340 F.3d 1314, 1326 (Fed. Cir. 2003) (explaining that “a prior art product that sometimes, but not always, embodies a claimed method nonetheless teaches that aspect of the invention”); see also Perricone v. Medicis Pharm. Corp., 432 F.3d 1368, 1376 (Fed. Cir. 2005) (concluding that a prior art reference anticipated notwithstanding the fact that it disclosed a claimed compound in a list with other suitable “skin benefit ingredients”).
Software Rights attempts to salvage the board’s decision upholding the patentability of claims 1, 5, 15, and 16 by seeking affirmance on grounds not raised before the board. As a general rule, however, arguments not raised before the board are waived on appeal. See Redline Detection, [643] LLC v. Star Envirotech, Inc., 811 F.3d 435, 450 (Fed. Cir. 2015); see also Camp v. Pitts, 411 U.S. 138, 142, 93 S.Ct. 1241, 36 L.Ed.2d 106 (1973) (per curiam) (emphasizing that “the focal point for judicial review should be the administrative record already in existence, not some new record made initially in the reviewing court”); In re Lee, 277 F.3d 1338, 1345 (Fed. Cir. 2002) (explaining that “review of an administrative decision must be made on the grounds relied on by the agency”).
VII. Claim 21 of the ’571 Patent
Finally, we conclude that the board erred when it determined that claim 21 of the ’571 patent is patentable over a combination of Fox Thesis, Fox SMART, and Fox Envision. See Board Decision II, 2015 WL 429750, at *16-18. The board found that the prior art failed to teach a specific feature of claim 21, the limitation requiring “deriving actual cluster links from the set of candidate cluster links.” J.A. II 5090 (emphasis omitted); see Board Decision II, 2015 WL 429750, at *17-18. As discussed above, however, the cluster-splitting procedure disclosed in both Fox SMART and Fox Thesis teaches deriving actual cluster links from a set of candidate cluster links. Furthermore, the actual cluster links in Fox SMART are selected “based on. the assigned weights,” as claim 21 requires, J.A. II 5090 (emphasis omitted), because the final clusters are defined by pairs of documents that have been assigned high similarity values. See J.A. II 5961-62.
Software Rights argues that the Fox Papers do not teach “determining ... hy-perjump data that has an indirect reference to the chosen node,” J.A. II 5090, because “[t]he clusters produced from Fox’s clustering process contain documents that do not have any indirect relationships between them,” This argument fails. Fox Thesis and Fox SMART teach clustering using be and cc, even if some clusters contain documents that are assigned to that cluster based upon relationships other than be and cc. See, e.g., Merck & Co. v. Biocraft Labs., Inc., 874 F.2d 804, 807 (Fed. Cir. 1989) (explaining that the fact that the prior art discloses a “multitude of effective combinations” does not mean that any particular combination is not obvious). We therefore reverse the board’s decision concluding that Facebook failed to establish that claim 21 of the ’571 patent is obvious over the prior art.
Conclusion
We have considered Software Rights’ remaining arguments but do not find them persuasive. Accordingly, we affirm the determinations of the Patent Trial and Appeal Board that claims 18-20, 45, 48-49, 51, and 54 of the ’494 patent and claims 12 and 22 of the ’571 patent are unpatentable on the ground of obviousness, but reverse its determinations that claims 1, 5, 15, and 16 of the ’494 patent and claim 21 of the ’571 patent are patentable over the prior art.
AFFIRMED-IN-PART AND REVERSED-IN-PART
Costs
The cross-appellants shall have their costs.