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(a) Public-private graph (b) Public graph G (c) In the view of v9: G ∪ Gv9

Figure 1. An example of undirected and simple public-private graph. In Figure 1(a), public edges are depicted in solid black lines and private edges are depicted in dashed edges. The red edges incident to vertex v9 are private to v9. The blue edges incident to vertex v3 and the edge (v1, v2) are private to v3. Figure 1(b) shows the public graph G consisting of all the solid black edges. Figure 1(c) shows the structure of public-private graph in the view of v9. Table I NETWORK STATISTICS. II. PUBLIC-PRIVATE NETWORKS Network |V | |E| |Vprivate| |Eprivate| δ(G) In this section, we first introduce a public-private graph PP-DBLP-2013 1,791,688 5,187,025 825,170 2,636,570 0.086 model for online social networks. Then, we generate real- PP-DBLP-2014 1,791,688 5,893,083 686,292 1,930,512 0.087 PP DBLP PP-DBLP-2015 1,791,688 6,605,428 515,549 1,218,167 0.087 world - datasets, and compare state-of-the-art of PP-DBLP-2016 1,791,688 7,378,090 263,937 445,505 0.083 public-private graph algorithms on PP-DBLP.

A. Public-Private Graph Model new approach to generate real-world datasets of public-private DBLP collaboration networks (PP-DBLP), according to the We present the model of a public-private graph G [2] as public and private information on DBLP records [10]. follows. Given a public graph G = (V, E), the vertex set V The intuition is that the information of one accepted pa- represents users, and the edge set E represents connections per available in public is usually later than the co-author between users. For each vertex u in the public graph G, u has collaboration really happened in private. In addition, such an associated private graph G = (V , E ), where V ⊆ V u u u u collaborations are always only known for authors themselves are the users from public graph and the edge set E satisfies u in person, and are not known to others. Thus, we take E ∩ E = ∅. The public graph G is visible to everyone, and u collaboration relationships in the published papers as public the private graph G is only visible to user u. Thus, in the u edges, and regard collaboration relationships in the ongoing view of user u, she/he can see and access the structure of works as private edges, which are only known by their authors. graph that is the union of public graph G and its own private Note that if two authors have had a pubic collaboration graph G as G ∪ G = (V, E ∪ E ). Let the private vertex set u u u relationship already, then their private ongoing collaboration of G as V = {u ∈ V : E 6= ∅} and the private edge set private u is not accounted as private. of as E v, w E u V . Note that G private = {( ) ∈ u : ∈ private} The public-private DBLP network is constructed as follows. E E , and each edge presented in different private private ∩ = ∅ We first obtain the DBLP raw data published in 2017 [10]. graphs only counts once in the private edge set E . private Next, we select one timestamp Y to distinguish the published Example 1: Consider a pubic-private graph with 11 vertices yet papers and on-going papers. For example, taking the cut- in Figure 1(a). The pubic-private graph consists of two kinds off timestamp Y as 2013/01/01, all collaborations happened of edges: public edges and private edges. The public edges before timestamp Y are regarded as public edges and the col- are depicted in solid black lines, e.g., v , v , indicating that ( 3 6) laborations happened on and after timestamp Y are regarded as v , v ( 3 6) is visible to everyone. The private edges are depicted private edges. We first construct the public graph. We sort all using dash lines in red and blue. Edges in red are private and papers in the increasing order of published dates in DBLP. For visible to v , e.g., v , v . Edges in blue are private and visible 9 ( 6 9) each paper p published before timestamp Y , we consider each v v , v G to 3, e.g., ( 1 2). Figure 1(b) shows the public graph author of this paper as a vertex, and add public edges between consisting of all public edges. Figure 1(c) shows the structure any pair of authors in this paper. Similarly, we construct all v of public-private graph in the view of 9, which is richer than private graphs as follows. For each paper p published on and G v the public graph in Figure 1(b). Because vertex 9 can after timestamp Y , if the two authors u, v do not have a public access all private relationships in private graph G . v9 edge, we add a private edge between u and v. This private edge B. Constructing Public-Private DBLP Networks can be accessed by all authors of this paper p. We generate four PP-DBLP datasets using four different timestamps Y In light of privacy concerns, there exist no publicly re- in {2013-01-01, 2014-01-01 2015-01-01, 2016-01-01}. The leased datasets of real-world public-private networks by now. network statistics of PP-DBLP are shown in Table I. Previous algorithms for public-private social networks, used public graphs to simulate public-private graphs, by randomly C. Evaluations on Real-world PP-DBLP datasets selecting some vertices and hiding their incident edges (in a We use PP-DBLP to evaluate two pubic-private graph star or a clique) as private edges from the public graph [2]. algorithms proposed by Chierichetti et al. [2]: shortest path ap- To verify competitive algorithms in a fair way, we propose a proximation and personalized PageRank. Sampling algorithms Table II 2 pp-dblp-2013 pp-dblp-2013 EFFICIENCYANDACCURACYOFPERSONALIZED PAGERANK USING 10000 pp-dblp-2014 1.8 pp-dblp-2014 HEURISTIC ON PP DBLP NETWORKS pp-dblp-2015 pp-dblp-2015 [2] - . pp-dblp-2016 pp-dblp-2016 1.6 Network Speed-up Ratio RMSE Cosine τ@50 1000 1.4 PP-DBLP-2013 6646 0.0036 0.9907 0.6007 PP DBLP Speed-up Ratio 1.2 - -2014 5746 0.0034 0.9908 0.6067 Approximation Ratio PP-DBLP-2015 5462 0.0041 0.9906 0.5715 100 1 4 8 16 32 64 4 8 16 32 64 PP-DBLP-2016 6319 0.0041 0.9890 0.5370 Multiplicative Factor Multiplicative Factor (a) Efficiency Improvement (b) Quality Approximation of the rankings, the score of τ@50 is still quite high falling PP DBLP Figure 2. Evaluation of shortest path approximation on - networks in [0.537, 0.6067]. Similar results on other datasets are also reported in [2]. are developed to precompute the public graph G offline, and III. ATTRIBUTED PUBLIC-PRIVATE NETWORKS then run the online update algorithm on private graph Gu with samples of G. We use the publicly available implementation In this section, we first define an attributed graph model of of sampling algorithms2 and set all the same parameters with public-private networks. We extend the approach of PP-DBLP [2] by default. We randomly select private graphs and report generation to produce real-world datasets of attributed public- the running time and accuracy of each task averaged over 50 private networks using title keywords in the papers. Finally, independent tests. we discuss promising research directions on attributed public- private graphs. Shortest Path Approximation. Given a vertex u in public- private graph G ∪ Gu, this task is to compute the shortest A. Attributed Public-Private Graph Model path from u to another arbitrary vertex in G ∪ Gu. We An attributed public-private graph of G is modeled as evaluate the performance of one sampling algorithm of shortest follows. Given an attributed public graph G = (V,E,A), path [2] with one baseline method of classical Dijkstra’s where the vertex set V represents users, the edge set E algorithm [11]. Figures 2(a) and 2(b) respectively show the represents connections between users, and the public attribute results of efficiency improvement and quality approximation set A(u) describes the public attributes of each user u ∈ V . varied by the multiplicative factor, which decides the number For each user u in the public graph, u has an attributed private of samples. The results verify that [2] achieves the great graph Gu = (Vu, Eu, Au), where Vu ⊆ V is a set of users improvement of efficiency (more than 300 times faster than from the public graph, private edge set Eu ∩ E = ∅, and the baseline method) and obtains a good balance of shortest Au(v) represents the private attributes of vertex v ∈ Vu that path approximations (no greater than 1.6 times of the optimal are visible to u. The public attributed graph G is visible to answer) on all PP-DBLP datasets. Figure 2(b) shows that the everyone, and the private attributed graph Gu is only visible to sampling algorithm [2] improves with smaller approximation user u. In terms of network structure, attributed public-private ratios as the multiplicative factor decreases, due to the more graphs have no difference with the public-private graphs in samples used in the algorithm. Section II. In terms of attributes, consider an attributed public- Personalized PageRank. Given a vertex u in public-private private graph G ∪ Gu, the vertex u can access both public and graph G ∪ Gu, the problem of personalized PageRank (PPR) private attributes of vertex v, i.e., Av ∪ Au(v). is to find node similarities to u for all vertices in G ∪ Gu. We Example 2: Figure 3(a) shows an example of attributed compare two methods. The first one is personalized PageRank public-private graph, which has the same graph structure as using heuristic [2]. The second one is a baseline method that Figure 1(a). Public attributes are in black, and private attributes directly applies the algorithm of Andersen et al. [12] on graph are in blue and red. Consider the vertex v3. The public attribute ′ G ∪ Gu. The results obtained by the baseline are used as of v3 is A(v3) = {‘SQL }. Attributes in blue (e.g., the the ground-truth PPR ranking of u. Table II reports the ef- attribute of ‘XML’ associated with vertices v1, v2 and v3) are

ficiency performance and accuracy of personalized PageRank private and visible to v3. Thus, Av3 (v1)= Av3 (v2)= Av3 (v3) ′ ′ using heuristic on PP-DBLP networks. In terms of efficiency = { XML }. Attributes in red (e.g., vertex v6’s attribute of comparison, the heuristic algorithm [2] is faster by three orders ‘Skyline’) are private and visible to v9. Figure 3(b) shows the of magnitude than the baseline [12], which are shown in the public attributed graph G consisting of all public edges and column of speed-up ratio in Table II. Table II also shows the public attributes that are visible to everyone. Figure 3(c) shows accuracy of PPR ranking computed by the heuristic algorithm the attributed graph G∪Gv3 in the view of v3. The attributes of ′ ′ w.r.t. the ground-truth PPR ranking, in terms of three metrics: v1 are {‘Skyline , ‘XML } as a result of the union of public the Root Mean Square Error (RMSE), the Cosine Similarity, attributes and private attributes, i.e., A(v1)∪Av3 (v1), showing and the Kendall-τ index. In terms of RMSE, the heuristic that v1 extends her/his research interests. algorithm produces ranking achieving the RMSE close to 0 B. Constructing Attributed Public-Private DBLP Networks on all datasets; In terms of cosine similarity, it obtains nearly 1; In terms of the Kendall-τ correlation of the first 50 positions To construct the attributed public-private DBLP networks, we add attributes into vertices on PP-DBLP in Section II-B as 2https://github.com/aepasto/public-private follows. For each author, we collect keywords in the title of all XML v11 XML XML SQL v11 v11 v 2 v2 v2 SQL,XML SQL, XML SQL,XML XML XML v XML XML XML 10 SQL v10 v10 Skyline XML XML v public v 6 XML v v v v Skyline 3 Skyline 3 6 XML Skyline 3 6 XML XML v v SQL, Skyline v v 1 SQL 9 1 SQL v9 XML 1 SQL, XML v9 XML v private 8 v8 v8 XML XML XML SQL v v v7 v v v v 4 5 XML 4 v5 7 XML 4 v5 7 XML SQL XML SQL SQL XML SQL XML

(a) Attributed public-private graph (b) Attributed public graph G (c) In the view of v3: G ∪ Gv3

Figure 3. An example of attributed public-private graph. In Figure 3(a), public attributes are in black. Private attributes are in blue and red, which respectively are visible to v3 and v9. Figure 3(b) shows the attributed public graph G consisting of all public edges and public attributes. Figure 3(c) shows the attributed graph G ∪ Gv3 in the view of v3. published articles and extract the most frequent keywords as Besides PP-DBLP, one future plan is to construct a real-world the public attributes. For the private attributes, let’s consider pubic-private Facebook social network by conducting a survey one author u and its attributed private graphs Gu. For each of Facebook users, who will be asked to manually identify all author v in Gu, the private attributes of v as Au(v) are the of private relationships that they hid. The survey can make use most frequent keywords from the title of all ongoing papers of Facebook built-in app3 to conduct investigations. involving authors v and u. To select representative keywords, ACKNOWLEDGMENT we set each number of public attributes and private attributes is no greater than a maximum threshold of 5, i.e., |A(v)|≤ 5 This work is supported by the Hong Kong General Re- search Fund (GRF) Project Nos. HKBU 12200917, 12232716, and |Au(v)| ≤ 5. The difference of public attributes Av and 12258116, 12632816, and National Natural Science Founda- private attributes Au(v) shows the evolving research interests tion of China (NSFC) Project Nos. 61702435, 61602395. of author v. Note that, Av ∩ Au(v) 6= ∅ may hold. We use θu(v) to quantify the overlapping ratio of public attributes REFERENCES and private attributes of vertex v in graph Gu, denoted by [1] R. Dey, Z. Jelveh, and K. Ross, “Facebook users have become much θ v |Av∩Au(v)| Pv∈Vu u( ) more private: A large-scale study,” in Pervasive Computing and Commu- θu(v) = . 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PP-DBLP datasets for attributed public-private networks, which are useful to further research of public-private graphs. 3https://apps.facebook.com/my-surveys