Seeding with Costly Network Information
Dean Eckles, Hossein Esfandiari, Elchanan Mossel, M. Amin Rahimian
SSeeding with Costly Network Information
Dean Eckles , ∗ , Hossein Esfandiari † , Elchanan Mossel , ∗ , and M. Amin Rahimian , ∗ We study the choice of k nodes in a social network to seed a diffusion with maximum expected spread size.Most of the previous work on this problem (known as influence maximization) focuses on efficient algorithmsto approximate the optimal seed sets with provable guarantees, while assuming knowledge of the entirenetwork. However, in practice, obtaining full knowledge of the network is very costly. To address this gap,we propose algorithms that make a bounded number of queries to the graph structure and provide almosttight approximation guarantees. We test our algorithms on empirical network data to quantify the trade-offbetween the cost of obtaining more refined network information and the benefit of the added information forguiding improved seeding policies.Keywords: Viral marketing, influence maximization, social networks, submodular maximization, query oracle Decision-makers in marketing, public health, development, and other fields often have a limitedbudget for interventions, such that they can only target a small number of people for the inter-vention. Thus, in the presence of social or biological contagion, they strategize about where ina network to intervene — often where to seed a behavior (e.g., product adoption) by engagingin an intervention (e.g., giving a free product) [9, 22, 24, 28, 33, 43]. The influence maximizationliterature is devoted to the study of algorithms for finding a set of k seeds so as to maximizeexpected adoption, given a known network and a model of how individuals are affected by theintervention and others’ adoptions [22]. However, finding the best k seeds for many models ofsocial influence is NP-hard, as shown by Kempe, Kleinberg and Tardos [33]. Much of the subsequentinfluence maximization literature is concerned with developing efficient approximation algorithmswith theoretical guarantees to make use of desirable properties of the influence function such assubmodularity [17, 57].With few exceptions [46, 60], the seeding strategies that are studied in the influence maximizationliterature require explicit and complete knowledge of the network. However, collecting the entirenetwork connection data can be difficult, costly, or impossible. For example, in developmenteconomics, public health, and education, data about network connections is often acquired throughcostly surveys (e.g., [8, 13, 14, 50]). Indeed, to reduce the cost of such surveys a few seeding strategieshave been proposed to avoid collecting the entire network information by relying on stochasticingredients, such as one-hop targeting, whereby one targets random network neighbors of randomindividuals [14, 18, 36]. Moreover, such methods have the advantage of scalability, since they can beimplemented without mapping the entire network. This is particularly important in online socialnetworks with billions of edges, where working with the entire contact lists might be impractical.Although the importance of influence maximization with partial network information has beennoted and there are a few papers considering this problem, none of these previous work comeswith provable quality guarantees for general graphs.In this work, we address the problem of influence maximization using partial information ofthe network which has attracted attention recently [45, 46, 54, 60]. We enhance the theoretical ∗1 Sloan School of Management, Institute for Data, Systems and Society, and Department of Mathematics, MassachusettsInstitute of Technology, {eckles,elmos,rahimian}@mit.edu. † Google Research, [email protected] are listed in alphabetical order.The authors gratefully acknowledge the research assistantship of Md Sanzeed Anwar through the MIT IDSS UROP. ElchananMossel is supported by ONR grant N00014-16-1-2227, NSF grant CCF-1665252 and ARO MURI grant W911NF-19-0217.
Working paper. February 17, 2020. Code for simulations is available at: https://github.com/aminrahimian/seeding-queries
An extended abstract appeared in ACM Economics & Computation 2019 (EC ’19). doi:10.1145/3328526.3329651 a r X i v : . [ c s . S I] F e b ckles et al. Seeding with Costly Network Information p , which is the same as the independentcascade probability along each edge. Accessing the graph information by performing edge queriesis a common technique in sublinear time algorithms that provide an output after inspecting a smallportion of their input [2, 3, 15, 23, 30]. In the second model (Section 4), each query consists of aspread whereby a signal node is seeded and the identities of the resultant adopters is observed. Withboth models, we propose algorithms that provide almost tight approximation guarantees for seedingwith upper-bounds on the number of queries. In Section 5, we explain how our bounded-queryframework can be applied to quantify the value of network information for seeding with costlynetwork information. We provide concluding remarks, and a discussion of the open problems andfuture directions in Section 6. The detailed proofs are provided in Appendix A. We present all ofour results for the case of symmetric influences (in an undirected graph with the same cascadeprobability p on all the edges). In Appendix B, we discuss the extension of our results to asymmetricinfluences with weighted, directed graphs. In Appendix C, we discuss the extension of our resultsto other diffusion models, beyond independent cascade. Motivated by the difficulties of acquiring complete network data, we are interested in influencemaximization methods that do not make explicit use of the full graph. Such methods have roots inapplied work — for vaccination [21] and sensing [19], in addition to seeding. One approach thathas received substantial attention is a “one-hop” strategy (sometimes called “nomination” [36] or“acquaintance” targeting [14, 21]) that selects as seeds the neighbors of random nodes; this exploitsa version of the friendship paradox that whereby the friend of a random individual is expected tohave more friends than a random individual [40]. For example, Kim et al. [36] report on the resultsof field experiments that target individuals for delivery of public health interventions. They arguethat that one-hop targeting (whereby a random individual nominates a friend to be targeted) leadsto increased adoption rates, compared with random or in-degree targeting. Some other empiricalwork has been less encouraging ([18], cf. [39]). While there are results about how these shortrandom walks affect the degree distribution of selected nodes [38], one-hop seeding currently lacksany theoretical guarantees under models of contagion. Furthermore, given the collection of dataabout the network neighborhoods of k nodes, it is natural to consider whether this data can bemore effectively used than just locally taking a random step, ignoring data collected from the other k − ckles et al. Seeding with Costly Network Information
We consider the independent cascade model of social contagion [33] that is fairly well-studied.In this model, each active agent has a single chance to activate each of its neighboring agentswith probability p , independently at random. Motivated by applications to product and technologyadoption, we refer to activated nodes as adopters. Starting from a set of initial adopters, followingthe independent cascade model, the adoption propagates through the network and the processterminates after a finite number of steps. The k -Influence Maximization, or k -IM problem, refersto the choice of k initial adopters to maximize the expected number of adoptions. Let L be theoptimum value for this problem. A µ -approximation algorithm outputs a set of k initial adopters to ckles et al. Seeding with Costly Network Information µL . In this work we assume a queryoracle to the graph structure. We study the k -Influence Maximization problem with limited numberof queries.In the first model, we consider edge queries. In particular, a query is of the form ( v , i ) and returnsthe i ’th edge of node v , where the edges are ordered arbitrarily. One natural question that ariseshere is to study the relation between the required number of queries and the cascade probability.In particular, is it possible to find an approximately optimal seed set using sub-quadratic number ofqueries when p is desirably small? We resolve this question by developing a non-trivial, dependentsampling of the edges of the network that approximately preserves the solution of the k -InfluenceMaximization problem.Theorem (Approximation Guarantee with bounded edge qeries). For any arbitrary < ϵ ≤ , there exist a polynomial-time algorithm for influence maximization that covers ( − / e ) L − ϵn nodes in expectation, using ˜ O ϵ ( pn + √ pn . ) queries, where L is the expected number of nodes coveredby the optimum seed set. To achieve this result, we apply some subsampling and stopping constraints that enable us to approximately simulate ˜ O ( k ) independent realizations of the cascade over the network, using only˜ O ( pn + √ pn . ) queries. Notice that a single simulation of cascade over the entire network (withoutusing our subsampling and stopping constraints) requires Ω ( pn ) queries. In fact, we show that inthe worst case one needs to query Ω ( n ) edges to guarantee that the expected number of coverednodes is at least a constant fraction of that of the optimum solution. The following theorem statesour claim formally.Theorem (Lower bound on reqired edge qeries). Let µ be any constant. There is no µ -approximation algorithm for influence maximization using sub-quadratic number of queries. In the next step, we present a complementary model of querying the spreading process. In thismodel, we assume that we can pay a cost to learn the outcome of a spreading process when anode is seeded. Accordingly, we learn the identity of the final adopters, but we do not observethe network edges through which the influence spread. In practical terms, one can seed a randomindividual with cheap traceable coupons that she can distribute to the people under her influence,and so forth. We can then observe the adoption status of the entire network by observing the useof the coupons. Repeating this process r times we observe r independent outcomes from seeding r (potentially different) nodes. We refer to this setup as spread query. Note that such queries arejust used to learn the network, i.e., to find a seeding strategy of size k . Interestingly, we show thatrunning the spreading process ˜ O ( k ) times is enough to provide a k -IM solution with almost tightapproximation guarantees. Notice that the number of queries depends poly-logarithmically on n , and hence in the case that k is poly-logarithmic, our algorithm only requires poly-logarithmicnumber of spread queries to find an almost efficient seeding strategy for the entire network.Theorem (Approximation Guarantee with bounded spread qeries). For any arbitrary < ϵ ≤ , there exist a polynomial-time algorithm for k -influence maximization that covers ( − / e ) L − ϵn nodes in expectation using no more than ˜ O ϵ ( k ) spread queries. For example on a star, with high probability all of our spread queries are leaves of the star.However, based on the results of the queries our algorithm finds and seeds the center of the star.To complement this, we show that an additive loss (e.g., ϵn ) is necessary, given o ( n ) queries.Theorem (Lower bound on reqired spread qeries). Let µ be an any constant. There is no µ -approximation algorithm for influence maximization using o ( n ) spread queries. ckles et al. Seeding with Costly Network Information
Consider graph G = (V , E) with the set of nodes V and the set of edges E . Consider a seed set S .Starting from the seeded agents in S , adoption spreads in graph G according to the independentcascade model. In this model each agent, upon adoption, has a “one time” chance of convertingeach of her non-adopter neighbors. Each conversion is successful with probability p , independentlyof others. Given S , for v ∈ V , let ϕ ( v , S) be the probability that v adopts when the nodes in S areseeded. Let Γ (S) = (cid:205) v ∈V ϕ ( v , S) be the value (influence) of the seed set S , which is the expectednumber of nodes that adopt if the nodes in set S are seeded.Definition 1 ( k -IM). Given graph G , the k -Influence Maximization ( k -IM) problem is to choosea seed set S ⊂ V , card (S) = k to maximize Γ (S) . We use Λ = argmax S , card (S) = k Γ (S) to denoteany such solution. Moreover, we use L = Γ ( Λ ) to denote the maximum expected spread size, using anoptimal seed set of size k . Definition 2 (Approximations).
Given graph G , any set of size k , Λ α ⊂ V , card ( Λ α ) = k ,satisfying Γ ( Λ α ) ≥ αL is an α -approximate solution to k -IM. Given the knowledge of graph structure G , submodular maximization algorithms (such as thesimple greedy) guarantee a ( − / e )− approximate solution to k -IM. Here we achieve roughly thesame guarantee using only partial information about the graph G .In our approach, rather than optimizing the influence function on the original graph we do soon a subgraph that is properly sampled from the original graph. As its main property, we showthat for the appropriate choice of α and ϵ , an α -approximate solution to k -IM on this subgraph, hasan influence on the original graph that is lower-bounded by αL − ϵn , where L is the optimal valueof k -IM. We can thus achieve similar worst-case guarantees using only partial information aboutthe network. In this section we present our edge query model, as well as the algorithm to output an approximateseed set given the outcome of the edge queries. The main idea is to simulate multiple independentspreads by querying each node about their edges. The way the queries are organized is by exploringthe extended neighborhoods of nodes. We start from a random set of initial nodes that are fixedfor subsequent steps (Subsection 3.1). We then proceed to probe their neighborhoods by askingthem to reveal their neighbors with probability p , and repeating the same process for the revealedneighborhoods, etc. (Subsection 3.2). This way we obtain multiple sub-sampled copies of the graph,each of which corresponds to a cascade starting from the same set of initial nodes (see Figure 1). InSubsection 3.3, we argue that one does not need to continue probing the extended neighborhood ofan initial node if the size of its connected component is large enough. This is a key observationbecause it allows us to upper-bound the total number of edge queries. In Subsection 3.4, we proposean algorithm to choose k seeds (approximately optimally), based on the outcome of the edge queries.The crux of the argument is in using the number of initial nodes in the connected components ofeach node to infer its value as a potential seed. We also prove our guarantees on the number ofedge queries (Subsection 3.4.1) and the run-time (Subsection 3.4.2).In Figure 1, we depict an example of three cascades that are obtained through edge queries.Consider the node that is marked in black. The connected component of the marked node has ckles et al. Seeding with Costly Network Information (A) (B) (C)Fig. 1. Three cascades are depicted in (A) red, (B) orange, and (C) blue. All cascades start from the samerandom initial nodes which are marked in the same color as the cascades. The node that is marked inblack scores as high as or higher than other nodes across the three cascades. The dotted sections consist ofunrealized influences and nodes that are not influenced in each cascade. three initial nodes in 1(A), two initial nodes in 1(B), and one initial node in 1(C). The value of eachconnected component is the number of initial nodes in that component. The total value of themarked node is 1 + + =
6, which is computed by adding the values of all of the componentscontaining that node. To prevent overlap with already chosen seeds, we set the value of a connectedcomponent to zero if one of its nodes is chosen as a seed. Using these valuations, we can approximatethe greedy algorithm by adding the most valuable node to the seed set.The hallmark of our analysis is in identifying an auxiliary submodular function Γ δ : 2 V → R to approximate our submodular function of interest Γ : 2 V → R . The approximation is suchthat | Γ δ (S) − Γ (S)| ≤ ϵn for all seed sets S of size k , with high probability. Here ϵ is the qualityof approximation and it depends on δ , which parameterizes the approximator Γ δ . Following thenotation introduced in Definitions 1 and 2, let us use Λ δ and Λ to denote the maximizers of Γ δ and Γ with constrained size k . The following Lemma (proved in Appendix A.1) is true for any setfunction Γ and its approximator Γ δ . It allows us to bound the loss that is incurred from optimizing Γ δ in place of Γ .Lemma 3. Consider set functions Γ δ and Γ that map subsets of V to R with k -IM optimal values L δ and L . Assume that for all seed sets S of size k we have | Γ δ (S) − Γ (S)| ≤ ϵn . Then any approximatesolution Λ ′ δ of k -IM for Γ δ , satisfying Γ δ ( Λ ′ δ ) ≥ αL δ − βn , also satisfies Γ ( Λ ′ δ ) ≥ αL − ( β + ( α + ) ϵ ) n . Recall our goal is to choose a seed set that (approximately) maximizes the influence function Γ . Inthis subsection, we show that we can estimate the value of Γ by choosing a large enough set ofnodes uniformly at random. To begin, fix 0 < ρ < nρ nodes uniformly at random. Wecall these the initial nodes and denoted them by V ρ . Given V ρ , for any set S ⊂ V we estimate thevalue of Γ (S) = (cid:205) v ∈V ϕ ( v , S) by: Γ ρ (S) : = ρ (cid:213) v ∈V ρ ϕ ( v , S)· (1)To proceed, also define ρ n , kϵ , δ : = ( + ϵ )( kδ log n + log 2 ) ϵ n · In the next Lemma, we bound the difference between Γ and Γ ρ for ρ ≥ ρ n , kϵ , δ . The proof is inAppendix A.2. In the proof, we use a standard concentration argument to control the deviation of Γ ρ (S) from Γ (S) for a fixed S , and then a union bound to make the inequality true for any S . ckles et al. Seeding with Costly Network Information
Let ρ n , kϵ , δ ≤ ρ ≤ . With probability at least − e − δ , forall seed sets S of size k we have | Γ ρ (S) − Γ (S)| ≤ ϵn . Note that our definition of Γ ρ in (1) is in terms of ϕ ( v , S) . The latter can only be computed giventhe knowledge of the entire graph. However, with only partial network information we need toreplace ϕ ( v , S) by a proper estimate. To this end, we sample the graph edges through a probingprocedure. Consider the nρ initial nodes in V ρ . For each initial node, we probe its neighborhood,keeping the edges with probability p . We then proceed to probe the neighborhoods of the revealednodes, etc. We never probe a node more than once and each edge receives at most one chance ofbeing sampled. The probing stops after a finite number of steps (bounded by n ). We repeat thisprobing procedure T times and obtain T subsampled graphs that we denote by G ( ) ρ , . . . , G ( T ) ρ .We can now estimate ϕ ( v , S) using the T copies {G ( ) ρ , . . . , G ( T ) ρ } as follows. For i = , . . . , T , set Y ( i ) ( v , S) = v has a path to S in G ( i ) ρ , otherwise set Y ( i ) ( v , S) =
0. Our estimate for ϕ ( v , S) is ϕ ( T ) ( v , S) : = T T (cid:213) i = Y ( i ) ( v , S)· (2)We can similarly construct an estimate for the influence function that we want to optimize: Γ ( T ) ρ (S) : = ρ (cid:213) v ∈V ρ ϕ ( T ) ( v , S)· (3)To proceed, define T n , kϵ , δ : = ( δ + log 2 )( k + ) log nϵ · Our next result bounds the difference between Γ ( T ) ρ and Γ ρ for T ≥ T n , kϵ , δ . In the proof, we useconcentration and union bound to ensure that ϕ ( T ) ( v , S) remains close to ϕ ( v , S) for all v and S .The proof details are in Appendix A.3.Lemma 5 (Bounding the probing loss). Let T ≥ T n , kϵ , δ . With probability at least − e − δ , for allsets S of size k we have | Γ ( T ) ρ (S) − Γ ρ (S)| ≤ ϵn . Here we consider a variation of the probing procedure described in the previous subsection wherebywe stop probing once there are more than a threshold τ nodes in a connected component. Notethat the probing may stop even before hitting τ nodes if no new edges are activated. Limiting theprobed neighborhoods in this manner helps us bound the total number of edges that are used inour sketch (see Subsection 3.4.1 and Theorem 8). In fact, we show that it is safe to stop probingwhen there are τ = τ n , kϵ nodes in a connected component, where τ n , kϵ : = n log ( / ϵ ) ϵk · Let us denote the T subsampled graphs obtained through limited probing by G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ . More-over, let Γ ( T ) ρ , ϵ be our estimate of the influence function that is constructed based on G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ in the exact same way as in (2) and (3). This new estimator is, itself, a submodular function since itcan be expressed as sum of coverage functions. Our following result ensures that by optimizing Γ ( T ) ρ , ϵ ckles et al. Seeding with Costly Network Information Γ ( T ) ρ , we do not loose more than ( − ϵ ) in our approximation factor. The proof follows aprobabilistic argument similar to [10, Lemma 2.4]. The crux of the argument is in constructing arandom set whose expected value on Γ ( T ) ρ , ϵ is no less than ( − ϵ ) of the optimum on Γ ( T ) ρ . We do soby starting from the optimum set on Γ ( T ) ρ and replacing ϵk of its nodes at random. Taking τ largeenough allows us to show that those nodes whose connections are affected by limiting the probedneighborhoods belong to a large component — of size τ = τ n , kϵ — that is likely to be covered byone of the ϵk random nodes. The complete proof details are in Appendix A.4.Lemma 6 (Bounding the loss from limited probing). For ≤ ρ ≤ and < ϵ ≤ , considerthe limited probing procedure with the probing threshold set at τ = τ n , kϵ . Then any α -approximatesolution to k -IM for Γ ( T ) ρ , ϵ is an α ( − ϵ ) -approximate solution to k -IM for Γ ( T ) ρ . Algorithm 1, below, summarizes the limited probing approach for performing edge queries on theinput graph. The output is a sketch comprised of the T independent copies G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ that fullydetermine the estimator Γ ( T ) ρ , ϵ . Algorithm 1: PROBE ( ρ , T , τ ) Input:
Query access to graph G Output: G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ (1) SAMPLE: Choose nρ nodes uniformly at random and call them V ρ .(2) PROBE: For i form 1 to T , do(a) Probe every node in V ρ by asking them to reveal their neighbors with probability p .(b) For any revealed neighbor that is not probed before, add the corresponding edge to G ( i ) ρ , ϵ and proceed to probe them.(c) Stop probing if there are no more new nodes to probe or if the size of the revealed componentexceeds τ .Lemmas 4, 5, and 6 provide the following appropriate choices of the algorithm parameters forPROBE ( ρ , T , τ ) : ρ = ρ n , kϵ , δ = ( + ϵ )( kδ log n + log 2 ) ϵ n , T = T n , kϵ , δ = ( δ + log 2 )( k + ) log nϵ , τ = τ n , kϵ = n log ( / ϵ ) ϵk · The following theorem (proved in Appendix A.5) combines our results so far (Lemmas 3 to 6)to show that any α -approximate solution Λ ⋆ that we obtain for the k -IM problem on Γ ( T ) ρ , ϵ , indeed,satisfies Γ ( Λ ⋆ ) ≥ α ′ L − ϵ ′ n for appropriate choices of α ′ and ϵ ′ , providing an approximate solutionto the original k -IM problem on Γ .Theorem 7 (Bounding the total approximation loss). Consider any < ϵ , α < , and fix ρ = ρ n , kϵ , δ and T = T n , kϵ , δ . Moreover, let α ′ = α ( − ϵ ) and ϵ ′ = ( α ( − ϵ ) + ) ϵ . With probability atleast − e − δ , any α -approximate solution to k -IM problem on Γ ( T ) ρ , ϵ , has value at least α ′ L − ϵ ′ n onthe original problem. ckles et al. Seeding with Costly Network Information ( ρ , T , τ ) .The output of the PROBE algorithm is the set of T copies G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ . From these T copies,we construct the estimate Γ ( T ) ρ , ϵ and can, then, use a submodular maximization algorithm to finda ( − / e − ϵ ′ ) -approximate solution for any ϵ ′ >
0. In Subsection 3.4.2, we describe a fastimplementation of submodular maximization on the sketch (the output of PROBE) that runs in˜ O ( pn + √ pn . + nk ) time. To proceed define E n , kϵ , p : = pτ n , kϵ ( τ n , kϵ − )/
2, (4) C n , kϵ , δ : = nρ n , kϵ , δ T n , kϵ , δ (cid:18) E n , kϵ , p + (cid:113) δ ( τ n , kϵ log ( n ) + log T n , kϵ , δ ) E n , kϵ , p (cid:19) . In Theorem 8, we bound the total number of edge queries, denoted by q , in terms of E n , kϵ , p and C n , kϵ , δ .The proof in Appendix A.6 relies critically on how we limit the probed neighborhoods (Subsection3.3). Roughly speaking, the total number of connected components in the output of PROBE ( ρ , T , τ ) is at most nρT , and each component contains at most τ nodes. Moreover, since each edges isrevealed with probability p , the expected number of edges in each of these components is at most pτ ( τ − )/
2. Subsequently, concentration allows us to give a high probability upper-bound on thetotal number of edges that appear in the output of PROBE ( ρ , T , τ ) . We can, similarly, also boundthe total number of edges that are queried but discarded since they have been pointing to alreadyprobed nodes — see step (2-a) of the PROBE algorithm.Theorem 8 (Bounding the edge qeries). For ρ = ρ n , kϵ , δ , T = T n , kϵ , δ and τ = τ n , kϵ , with probabilityat least − e − δ the total number of edge queries ( q ) can be bounded as follows: q ≤ C n , kϵ , δ + (cid:16) + √ (cid:17) T n , kϵ , δ n (cid:113) δ + log T n , kϵ , δ ∈ ˜ O ϵ , δ ( pn + √ pn . ) . In Appendix A.7, we prove a matching hardness (or rather, impossibility) result to show that, inthe worst case, it is impossible to approximate the problem using o ( n ) edge queries for constant p .To show that there are no µ -approximation algorithms making o ( n ) edge queries, we consider anarbitrary algorithm that makes less that C µ n edge queries, for some constant C µ that is specifiedin Appendix A.7. Our hard example consists of a collection of 9 / µ cliques of size nµ / / µ of these cliques at random and connect them as in Figure 2. With k = p =
1, an optimalalgorithm will seed one of the nodes in the connected cliques and achieves ( / µ )( nµ / ) = nµ / C µ n queries cannot detect the connectedclique with probability more than µ /
3. In Appendix A.7, we show that the expected spread sizefrom seeding the output of any such algorithm is less than nµ / µ of the optimum.Theorem 9. Let µ be any constant. There is no µ -approximation algorithm for influence maximiza-tion using sub-quadratic number of edge queries. This result extends the previous result of Wilder et al. [60] that lower-bounds the number ofnodes needed to be queried. Moreover, as opposed to the previous work of Wilder et al. the spreadsize in our hard example is linear in the total number of nodes (see Figure 2). This means thatour impossibility result holds even if, for some ϵ ≪ µ , an ϵn additive loss is tolerated. Indeed, thehard example used in our impossibility result is adversarially tuned to require Ω ( n ) queries. Inparticular, we set the cascade probability p to 1. Although one can adjust the hard example to workfor smaller constant cascade probabilities, the example fails when p is sub-constant. In fact, inmany realistic applications of the k -IM, the cascade probability is relatively small. For example, if ckles et al. Seeding with Costly Network Information Fig. 2. To lower-bound the required number of edge queries, our hard example consists of / µ cliques, / µ of which are connected in a circle. An algorithm that makes o ( n ) edge queries may detect the connectedcliques with probability at most µ / . The expected performance of any such algorithm is worse than a factor µ of the optimum. someone tweets about a product, not all of her tweeter followers are subjected to this influence, orconvinced to buy, or retweet about it. In this subsection, we provide a fast implementation of ouralgorithm for influence maximization on the sampled graph. In fact, we can achieve a runningtime that is linear in the number of queried edges. First note that Γ ( T ) ρ , ϵ is, by definition, a coveragefunction, ergo a submodular function. Hence, we can use the randomized greedy algorithm of [47]to provide a ( − / e − ϵ ′ ) approximation guarantee. We start with S = ∅ and as in any greedyalgorithm, we only use two types of operations: • We query the marginal increase of a node v on the current set S , denoted by: ∆ ( v |S) : = Γ ( T ) ρ , ϵ (S ∪ { v }) − Γ ( T ) ρ , ϵ (S) . • We choose a node v ⋆ with maximal marginal increase and add it to the seed set: S ← S ∪ { v ⋆ } .The only difference is that the search for the node v ⋆ is restricted to a subset R of size ( n / k ) log ( / ϵ ′ ) that is drawn uniformly at random from V \ S . Algorithm 2: SEED ( ϵ ′ ) Input:
The T copies: G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ Output: Λ ⋆ , ( − / e − ϵ ′ ) -approximate solution to k -IM for Γ Tρ , ϵ (1) Find the connected components of G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ .(2) For every connected component in each of the T copies initialize the current value ofthe component equal to the number of sampled initial nodes (belonging to V ρ ) in thatcomponent.(3) Initialize Λ ⋆ = ∅ .(4) For i form 1 to k , do(a) Choose a random subset, R ⊂ V \ Λ ⋆ , card (R) = ( n / k ) log ( / ϵ ′ ) .(b) For each v ∈ R compute ∆ ( v | Λ ⋆ ) by adding the current values of the connectedcomponents containing v and set v ⋆ = argmax v ∈R ∆ ( v |S) (c) Add Λ ⋆ ← Λ ⋆ ∪ { v ⋆ } and set the current value of the connected components con-taining v ⋆ to zero. ckles et al. Seeding with Costly Network Information ( ϵ ′ ) , we provide efficient implementations for the above operations. Ourimplementations are based on the structure of Γ ( T ) ρ , ϵ , as determined by the T copies, G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ .First using a graph search (e.g., DFS) we find the connected components of each of the T subsampledgraphs and count the number of initial nodes (belonging to V ρ ) in each connected component. Werefer to this count for each connected component as the "value" of that component. The main idea isthat maximizing Γ ( T ) ρ , ϵ is equivalent to finding a seed set S , such that the total value of all connectedcomponents containing at least one seed is maximized. If a connected component already contains(i.e., is covered by) some nodes in S , then the marginal increase due to that component should bezero. This is achieved by setting the value of a component to zero after adding a node from thatcomponent to the seed set S .Our next result combines our conclusions from Theorems 7 and 8, as well as the analysis ofthe performance of fast submodular maximization (randomized greedy) in [47]. The proof is inAppendix A.8.Theorem 10. For any arbitrary < ϵ ≤ , there exist an algorithm for influence maximizationthat covers ( − / e ) L − ϵn nodes in expectation in ˜ O ϵ ( pn + √ pn . + nk ) time, using no more than O ϵ ( pn + √ pn . ) edge queries. Here we present a complementary setup to the edge query model of Section 3. In this section, weassume that we can pay a cost to learn the outcome of a spreading process when a single node isseeded. Repeating this process gives us independent outcomes from randomly seeding a node. Werefer to this type of query as spread query, and ask how many times we should run the spreadingprocess with a randomly seeded node to be able to provide a k -IM approximate solution whosevalue is at least ( − / e ) L − ϵn . Our following algorithm outputs one such solution with the desiredguarantee for ρ = ρ n , kϵ = k log nkϵ ϵ ,where ρ is the number of spread queries we make to add one seed. Hence the total number ofspread queries of our algorithm is r n , kϵ = kρ n , kϵ = k log nkϵ ϵ · The main idea is to use the identity of adopters from kρ = ˜ O ( k ) independent cascades to seed k nodes with optimal approximation guarantees. Nodes that appear most in different cascades are thebest candidates for seeding. For example, in Figure 3 the black node is the only node that appearsin all three cascades, each starting from a random initial node. To prevent overlap with alreadychosen seeds, we discard those cascades that intersect with the chosen seeds. ckles et al. Seeding with Costly Network Information (A) (B) (C)Fig. 3. Three cascades are depicted in (A) red, (B) orange, and (C) blue. In each case, the initial node is markedin the same color as the cascade. The node that appears most across different cascades is marked in black.The dotted sections consist of unrealized influences and nodes that do not adopt in each cascade. Algorithm 3: SPREAD ( ρ ) Input:
Spread query access to graph G with independent cascade probability p Output: Λ ⋆ , approximate seed set of size k with value at least ( − / e ) L − ϵn For i form 1 to k , do(1) SAMPLE: Choose ρ nodes uniformly at random and call them u i , . . . , u iρ .(2) SPREAD: Run the spreading process ρ times, each time with one of the sampled nodes ( u ij )seeded. Call the resultant set of adopters A i , . . . , A iρ . For any j = , . . . , ρ , if A ij ∩ Λ ⋆ (cid:44) ∅ ,set A ij = ∅ .(3) SEED: For each node u ∈ V \ Λ ⋆ and j = , . . . , ρ , set X iu , j = u ∈ A ij and X iu , j = X iu = (cid:205) ρj = X iu , j . Choose v ⋆ = argmax u ∈V\ Λ ⋆ X iu . Add Λ ⋆ ← Λ ⋆ ∪ { v ⋆ } .Algorithm 3: SPREAD ( ρ ) follows [12, Algorithm 1] but is adapted to undirected networks. Towork with directed graphs, the spread queries in [12, Algorithm 1] are performed on the transposedgraph where the direction of influences is reversed. Such queries serve their purpose by summarizingthe graph information for fast influence maximization. However, our interest is in queries as acostly method of acquiring network information. For our purpose, running spread queries in atransposed graph is hard to motivate since one needs to conceive a mechanism to implementcascades in reverse (see Appendix B for a discussion of extensions to directed graphs).By focusing on undirected graphs, we can propose an algorithm for influence maximizationusing spread queries on the original graph (as opposed to its transpose). Moreover, by allowingfor an ϵn additive loss in our approximation guarantee, we only need O ϵ ( k log ( n )) = o ( n ) spreadqueries, whereas [12, Algorithm 1] needs Θ ϵ ( kn log n ) = ω ( n ) spreads to achieve the nearly optimalapproximation factor ( − / e − ϵ ) . Achieving an approximation guarantee with the fewer than o ( n ) queries is important in our setup where acquiring network information is costly. More recently,Sadeh et al. [51] provide a significantly improved sample complexity bound of O ( kτ log n ) if thediffusion is stopped after τ steps (e.g., in time-constrained applications or small-world networks).In the next theorem, we formalize our approximation guarantee and bound on the query com-plexity for Algorithm 3: SPREAD ( ρ ) . The proof is in Appendix A.9. They crux of the argument isin realizing that with X iu defined in step (3), ( n / ρ ) E [ X iu ] is the expected marginal gain form addingnode u to Λ ⋆ i − — the seed set in step i . Therefore, we can approximate one step of the greedyalgorithm by choosing ν ⋆ ∈ V \ Λ ⋆ i − to maximize X iu while using concentration to control thedeviation of X iu from E [ X iu ] . ckles et al. Seeding with Costly Network Information Fig. 4. Using f ( n ) ∈ o ( n ) spread queries on a graph comprised of a clique of size д ( n ) = (cid:112) n / f ( n ) and n − д ( n ) isolated nodes, one cannot achieve an approximation factor that is better than o ( ) . Theorem 11.
For any arbitrary < ϵ ≤ , there exist a polynomial-time algorithm for influencemaximization that covers ( − / e ) L − ϵn nodes in expectation in ˜ O ϵ ( nk ) time, using no more than k log ( nk / ϵ )/ ϵ ∈ ˜ O ( k ) spread queries. Next we show that an additive loss on the quality of the solution is inevitable with o ( n ) spreadqueries in the worst case. The proof details are in Appendix A.10. Our hard example consist of agraph with a small clique and many isolated nodes (see Figure 4). In such a structure, using o ( n ) spread queries one cannot achieve better than an o ( ) approximation factor. The hard example usedin this impossibility result is similar to that of Wilder et al. [60, Theorem 1]; however, with a morecareful analysis, we improve their O ( n − ϵ ) bound to o ( n ) .Theorem 12. Let µ be any constant. There is no µ -approximation algorithm for influence maxi-mization using o ( n ) spread queries. We can study the value of network information by considering how the expected spread sizefrom seeding the output of our algorithm increases with increasing number of input queries.Our simulations of spread sizes with increased queries on real networks indicate the existence ofan inflection point, whereby the first few queries improve the performance significantly beforehitting a plateau — Figures 6C and 7A. Therefore, we can extract the benefits of the network (A) (B)Fig. 5. (A) The degree distribution of the University of Pennsylvania Facebook social network with , nodes, average degree . , and a total of , , edges (B) The connections between the one hundredhighest degree nodes in this network ckles et al. Seeding with Costly Network Information l l l l l l l l l l l l l l
20 40 60481216 0 10 20 30 nodes queried (mean %)T s p r ead s i z e ( m ean % ) k l (A) nρ = l l l l l l l l l l l l l l
20 40 60481216 0 20 40 60 nodes queried (mean %)edges queried (mean %) s p r ead s i z e ( m ean % ) k l (B) nρ = l l l l l l l l l l l l l l
20 40 60 80481216 0 10 20 30 nodes queried (mean %)T s p r ead s i z e ( m ean % ) k l (C) nρ = l l l l l l l l l l l l l l
20 40 60 80481216 0 20 40 60 80 nodes queried (mean %)edges queried (mean %) s p r ead s i z e ( m ean % ) k l (D) nρ = Fig. 6. The spread sizes from seeding the output of Algorithm 2 applied to the University of PennsylvaniaFacebook social network. To estimate the influence of each output seed set, we average the spread sizesover independent cascades with p = . . To generate the T copies G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ that are input toAlgorithm 2, we run Algorithm 1 (PROBE) starting from nρ initial nodes (randomly chosen). We test theperformance for two values of nρ = , (A and B), and nρ = , (C and D). We vary T over a logarithmic scale: T = , , , , , , , , , , , , , . Note that the T = case corresponds to random seeding (using nonetwork information at all). For each T , we run the PROBE Algorithm times to generate random inputsfor Algorithm . The vertical axes show the mean spread sizes and confidence intervals that are computedover the outputs of Algorithm 2 for each T . Figures (A) and (C) show the performance with increasing T for and initial nodes, respectively. Their top axes show the average number of revealed nodes that iscomputed over the random inputs for each T . Figures (B) and (D) show the mean spread sizes versus theaverage number of nodes (top axis) and edges (bottom axes) that are revealed in the input at each T . information using just a few spread queries. We present our simulation results on the Universityof Pennsylvania Facebook social network (Figure 5) that is the largest network in a dataset ofFacebook social networks in 100 U.S. universities and colleges [55].In Figure 6, we show the performance of Algorithms 1 and 2 (PROBE & SEED) on the Universityof Pennsylvania Facebook social network. Running the PROBE algorithm with higher values of T ckles et al. Seeding with Costly Network Information lllllll l l l l l l l l query cost s p r ead s i z e ( m ean % ) k l (A) lllllll l l l l l l l l query cost m ean p r o f i t k l (B)Fig. 7. (A) The spread sizes over the University of Pennsylvania Facebook social network ( p = . ) fromseeding the output of Algorithm 3 with different number of spread queries in the input (the query cost). Themean spread sizes and their confidence intervals are computed over executions of the algorithm at eachquery cost. For each seed set that the algorithm outputs, we run the spread times to estimate its influence(expected spread size). In (B) we assume a cost c s = per seed and another cost c q = per query, as well asa revenue r = . per each adopter. The vertical axis shows the mean profits and confidence intervals that arecomputed from executions of the algorithm at each query cost. Note that the maximum expected profit isachieved at k = with spread queries. leads to discovery of more nodes and edges from the social network. The top and bottom horizontalaxes in Figures 6B and 6D show the average number of nodes and edges that are revealed in theinput for different values of T , with 10 and 100 initial nodes respectively. The vertical axes show themean spread sizes from seeding the output of Algorithm 2 for each T using 50 random inputs. Recallthat each input is a set of T probed samples G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ that is obtained through Algorithm1. The output performance improves with increasing T , since with more nodes and edges thatare revealed from the social network, the output seed set can be better optimized; nevertheless,there are diminishing returns to the increasing network information. It is worth noting that therandomness in the algorithm output also decreases with the increasing network information inthe input. There are two sources of randomness in the algorithm’s output: T < ∞ and ρ <
1. Theoutput variance for large T remains non-vanishing in Figure 6A; however, increasing the numberof initial nodes, i.e. the size of the sample set V ρ in Subsection 3.1, from 10 nodes in Figure 6A to100 nodes in Figure 6C allows us to remove the remnant randomness from the algorithm output atlarge T .In Figure 7, we show the mean spread sizes from seeding the output of Algorithm 3 with increasingnumber of input queries. Figure 7A shows that there are diminishing returns to increasing queries,and we can extract the benefits of complete network information using just 200 spread queries:With enough information, the mean spread size from seeding the output of the algorithm saturatesat the complete-information (deterministic) greedy algorithm output, and acquiring more networkinformation does not improve the performance beyond that. The number of queries before eachgraph hits its inflection point is higher for larger seeds sets; however, in general, one can extractthe benefits of complete network information with just a few queries . Subsequently, if we assume acost for each seeded node and another cost for running each spread query, as well as a revenue pereach unit spread, then there is a number of queries that is profit maximizing for a given number of ckles et al. Seeding with Costly Network Information k = k = k = k + x nodes at random (using no information about thenetwork) for some x = ω ( ) is enough to outperform the optimum spread size with k nodes, as thenetwork size increases ( n → ∞ ). They conclude that the benefits of acquiring network informationto identify the optimal k seeds may be offset by seeding a few more nodes at random (withoutusing any network information). Comparing the k =
10 and k = o ( n ) . However, our algorithms have the guaranteed performance over arbitrary graph inputswith any cascade size.In another related work, Manshadi et al. [44] study a model of spread where individuals contacttheir neighbors independently at random, and each contact leads to an adoption with some fixedprobability. The contacts occur repeatedly; therefore, every cascade eventually spreads to the entirepopulation. They characterize the time to reach a fraction of adopters as well as the contact cost(number of contacts made), in a random graph with a given degree distribution. They also proposeoptimal seeding strategies that only use the degree information. However, this model is not directlycomparable to the influence maximization setup that we study. In our model, the realization of theinfluences is random and adoption spreads only through the realized edges. For us, the objective isto maximize the expected spread size and the incurred cost is in acquiring information about theinfluence structure (who influences whom). We consider the problem of choosing the k most influential nodes under the independent cascademodel and using partial network information. We propose two natural ways of querying the graphstructure: (i) by accessing an unordered adjacency list of the input graph — edge queries, (ii) byobserving the identity of the adopter nodes when a single node is seeded — spread queries. Ineach case, we provide polynomial-time algorithms with almost tight approximation guaranteesusing a bounded number of queries to the graph structure (Theorems 10 and 11). We also provideimpossibility results to lower bound the query complexity and show tightness of our guarantees(Theorems 9 and 12). Finally, we show the utility of the bounded-query framework for studyingthe trade-off between the cost of acquiring more network information and the benefit of increasingthe spread size. Results that address the problem of seeding with partial network information arenascent and we foresee many directions for future research in this area.Theorem 12 implies that with o ( n ) edge queries an additive loss is inevitable in the edge querymodel. An interesting open problem is to provide a better result when edge queries exceed o ( n ) ;either showing that the ϵn additive loss is unavoidable, even with the increased number of edgequeries, or an algorithm that avoids it. Another interesting open problem is to provide tight lowerbounds on the query complexity of the spread query model. An Ω ( k ) lower bound is easy to show —for a hard example consider a collection of k stars of size n / k each. We speculate that a lower-boundcloser to k is plausible but the proof would involve significant new ideas. We presented all of ourresults for the independent cascade model over undirected graphs with the same cascade probability( p ) on every edge. In Appendix B, we discuss the extension of our results to the case of asymmetricinfluences (directed, weighted graphs). In Appendix C, we explain how our techniques can be ckles et al. Seeding with Costly Network Information
For any arbitrary < ϵ ≤ , there exist a polynomial-time algorithm for influencemaximization under the linear threshold model that covers ( − / e ) L − ϵn nodes in expectation in ˜ O ϵ ( nk ) time, using no more than nk log ( nk / ϵ )/ ϵ ∈ ˜ O ( nk ) edge queries. It is possible to provide tighter approximation guarantees or better query bounds by assumingthat the inputs follow a known distribution. For example, Wilder et al. [60] propose an algorithm forstochastic block model inputs, and it consists of taking a random sample of T nodes and exploringtheir extended neighborhoods in R steps of a random walk. The outcome of the random walks isused to estimate the block sizes of each of the T nodes, and this is achieved by revealing no morethan T R = O ( log n ) nodes. The k seeds are then selected from the initial T samples, such that the k largest blocks are seeded uniformly at random.In our queries we make use of the spread process, over which we optimize. Specifically, ourqueries depend on the independent cascade probability p . In practice, the spread process may notbe available for querying the influence structure. For example, if a person is asked to reveal herinfluencers (during an edge query), she may reveal them with a probability p ′ that is differentfrom the cascade probability p . Similarly, coupons (during a spread query) may have a differentcascade probability p ′ compared to the seeds, whose locations we want to optimize. Therefore, itis interesting to know how the discrepancy between the parameter of the query method p ′ andthe independent cascade probability p affects the influence maximization performance. One mayoffer new approximation guarantees that either depend explicitly on | p − p ′ | , or hold true whenthis difference is bounded ( | p − p ′ | < ϵ ). Such results can complement the prior literature on stableand robust influence maximization (cf. [27, 31]).Another venue for future work is to explore other ways of querying the graph structure. Wecan draw inspiration from the graph sampling literature (cf., e.g., [42, 49, 62]) to devise new querymethods. Accordingly, one would like to obtain subsampled graphs that preserve enough networkinformation for performing influence maximization in a satisfactory manner. We are particularlyinterested in queries that reveal the realization of influences over time. One can use the temporaldata to complement the influence information. This is especially relevant in practice, where a slowreaction time can compromise the influence of an otherwise well-positioned node and decision-makers have preferences for earlier, rather than later, adoption [43]. A PROOFSA.1 Proof of Lemma 3
Starting with an approximate solution satisfying Γ δ ( Λ ′ δ ) ≥ αL δ − βn on the one hand, we have Γ ( Λ ′ δ ) + ϵn ≥ Γ δ ( Λ ′ δ ) , (5)since | Γ δ ( Λ ′ δ ) − Γ ( Λ ′ δ )| ≤ ϵn .On the other hand, consider Λ δ and Λ , which are the optimum seed sets for Γ δ and Γ , respectively.By assumption we have Γ δ ( Λ ′ δ ) ≥ α Γ δ ( Λ δ ) − βn , and since, by optimality of Λ δ for Γ δ , Γ δ ( Λ δ ) ≥ Γ δ ( Λ ) , we get Γ δ ( Λ ′ δ ) ≥ α Γ δ ( Λ ) − βn ≥ α Γ ( Λ ) − ( β + αϵ ) n (6)where, in the last inequality, we have again invoked the | Γ δ ( Λ ) − Γ ( Λ )| ≤ ϵn property. The proof iscomplete upon combining (5) and (6) to get that Γ ( Λ ′ δ ) ≥ α Γ ( Λ ) − ( β + ( α + ) ϵ ) n . ckles et al. Seeding with Costly Network Information A.2 Proof of Lemma 4
Fix a seed set S and let ( + ϵ )( δ ′ + log 2 )( nϵ ) ≤ ρ ≤ . We use a Hoeffding-Bernstein bound to claim that with probability at least 1 − e − δ ′ we have (cid:12)(cid:12) Γ ρ (S) − Γ (S) (cid:12)(cid:12) ≤ ϵn . (7)Let X v be the random variable that is zero if v is not in V ρ and ϕ ( v , S) otherwise. Consider theirsummation and note that (cid:213) v ∈V X v = (cid:213) v ∈V ρ ϕ ( v , S) = ρ Γ ρ (S) . Hoeffding-Bernstein inequality [56, Lemma 2.14.19] provides that P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) nρ (cid:213) v ∈V ρ ϕ ( v , S) − n Γ (S) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ ϵ = P (cid:2)(cid:12)(cid:12) Γ ρ (S) − Γ (S) (cid:12)(cid:12) ≥ ϵn (cid:3) ≤ (cid:18) − nρϵ σ n + ϵ ∆ n (cid:19) , (8)where ∆ n = max v ∈V ϕ ( v , S) − min v ∈V ϕ ( v , S) ≤ σ n = n (cid:213) v ∈V (cid:18) ϕ ( v , S) − n Γ (S) (cid:19) = n (cid:213) v ∈V ϕ ( v , S) − (cid:18) n Γ (S) (cid:19) ≤ n (cid:213) v ∈V ϕ ( v , S) − (cid:18) n Γ (S) (cid:19) = ℓ − ℓ ≤ ℓ ≤ . In the last equality, we used the notation ℓ : = ( / n ) (cid:205) v ∈V ϕ ( v , S) = ( / n ) Γ (S) . The bound in (8)subsequently simplifies P (cid:2)(cid:12)(cid:12) Γ ρ (S) − Γ (S) (cid:12)(cid:12) ≥ ϵn (cid:3) ≤ (cid:18) − nρϵ + ϵ (cid:19) . Using nρ ≥ ( + ϵ )( δ ′ + log 2 )/ ϵ , we get that for all δ ′ > P (cid:2)(cid:12)(cid:12) Γ ρ (S) − Γ (S) (cid:12)(cid:12) ≥ ϵn (cid:3) ≤ (−( δ ′ + log 2 )) = e − δ ′ . (9)To complete the proof we use a union bound to claim that (7) holds for all choices of the seedset S simultaneously. To claim a union bound over all (cid:0) nk (cid:1) choices of the seed sets S , it suffices tochoose δ ′ = kδ log n in (9). ckles et al. Seeding with Costly Network Information A.3 Proof of Lemma 5
Consider Γ ( T ) ρ (S) = ( / ρ ) (cid:205) v ∈V ρ ϕ ( T ) ( v , S) for a fixed S ⊂ V . By Chernoff bound to ϕ ( T ) ( v , S) = / T (cid:205) Ti = Y ( i ) ( v , S) , we get that: P (cid:104)(cid:12)(cid:12)(cid:12) ϕ ( T ) ( v , S) − ϕ ( v , S) (cid:12)(cid:12)(cid:12) > ϵ (cid:105) ≤ (− ϵ T / ) . Using T = T n , kϵ , δ , by union bound over the choice of (cid:0) nk (cid:1) seed sets S ⊂ V , card (S) = k , and n nodes v ∈ V , we obtain that: P (cid:104)(cid:12)(cid:12)(cid:12) ϕ ( T ) ( v , S) − ϕ ( v , S) (cid:12)(cid:12)(cid:12) > ϵ , for all S and v (cid:105) ≤ (− δ − log 2 ) = e − δ . The proof is complete upon considering the summation over v ∈ V ρ : P (cid:104)(cid:12)(cid:12)(cid:12) Γ ( T ) ρ (S) − Γ ρ (S) (cid:12)(cid:12)(cid:12) ≤ ϵn , for all S (cid:105) = P (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:213) v ∈V ρ ϕ ( T ) ( v , S) − (cid:213) v ∈V ρ ϕ ( v , S) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ϵnρ , for all S ≥ P (cid:104)(cid:12)(cid:12)(cid:12) ϕ ( T ) ( v , S) − ϕ ( v , S) (cid:12)(cid:12)(cid:12) ≤ ϵ , for all S and v (cid:105) ≥ − e − δ . A.4 Proof of Lemma 6
Following the notation in Definition 1, let us use Λ ( T ) ρ and L ( T ) ρ to denote the maximizer of Γ ( T ) ρ and its maximal value subject to the size constraint (card ( Λ ( T ) ρ ) = k ). Similarly, let us denote theoptimal solution to k -IM on Γ ( T ) ρ , ϵ and its value by Λ ( T ) ρ , ϵ and L ( T ) ρ , ϵ , respectively. Moreover, followingDefinition 2, let us use Λ α , ( T ) ρ and Λ α , ( T ) ρ , ϵ to denote the α -approximate solutions to k -IM on Γ ( T ) ρ and Γ ( T ) ρ , ϵ , respectively. Our goal is to show that any Λ α , ( T ) ρ , ϵ is also Λ α ( − ϵ ) , ( T ) ρ .It is useful to think of G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ as subgraphs of G ( ) ρ , . . . , G ( T ) ρ . An immediate consequenceof this observation is that for any set of nodes S , we have Γ ( T ) ρ (S) ≥ Γ ( T ) ρ , ϵ (S) . We call the imaginaryprocess whereby G ( i ) ρ , ϵ is obtained after removing some nodes and edges from G ( i ) ρ an ϵ -cutting, andsubsequently, we refer to G ( i ) ρ , ϵ and G ( i ) ρ as the cut and uncut copies, respectively. Finally, it is alsouseful to define ϕ ( T ) ϵ ( v , S) in the exact same way as (2) but using the ϵ -cut copies G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ .The proof follows [10, Lemma 2.4] closely. In particular, we first note that it suffices to show theexistence of a set L , card (L) = k satisfying Γ ( T ) ρ , ϵ (L) ≥ ( − ϵ ) L ( T ) ρ . Because if there exists such a set L , then for any α -approximate solution Λ α , ( T ) ρ , ϵ we can write (recall ∀ S Γ ( T ) ρ (S) ≥ Γ ( T ) ρ , ϵ (S) ): Γ ( T ) ρ (cid:16) Λ α , ( T ) ρ , ϵ (cid:17) ≥ Γ ( T ) ρ , ϵ (cid:16) Λ α , ( T ) ρ , ϵ (cid:17) ≥ α Γ ( T ) ρ , ϵ (cid:16) Λ ( T ) ρ , ϵ (cid:17) ≥ α Γ ( T ) ρ , ϵ (L) ≥ ( − ϵ ) αL ( T ) ρ , implying that Λ α , ( T ) ρ , ϵ is also Λ α ( − ϵ ) , ( T ) ρ . To show the existence of such a set L we use a probabilisticargument by constructing a random set L , satisfying E (cid:110) Γ ( T ) ρ , ϵ ( L ) (cid:111) ≥ ( − ϵ ) L ( T ) ρ . The set L isconstructed is as follows: Starting from Λ ( T ) ρ , remove ϵk nodes randomly, and replace them with ϵk ckles et al. Seeding with Costly Network Information V . To see why E (cid:110) Γ ( T ) ρ , ϵ ( L ) (cid:111) ≥ ( − ϵ ) L ( T ) ρ , consider L ( T ) ρ = ρ (cid:213) v ∈V ρ ϕ ( T ) (cid:16) v , Λ ( T ) ρ (cid:17) , and E (cid:110) Γ ( T ) ρ , ϵ ( L ) (cid:111) = (cid:213) v ∈V ρ E (cid:110) ϕ ( T ) ϵ ( v , L ) (cid:111) . The inequality, E (cid:110) Γ ( T ) ρ , ϵ ( L ) (cid:111) ≥ ( − ϵ ) L ( T ) ρ , would follow if for any node v ∈ V we have, E (cid:110) ϕ ( T ) ϵ ( v , L ) (cid:111) ≥ ( − ϵ ) ϕ ( T ) ( v , Λ ( T ) ρ ) + ϵ ≥ ϕ ( T ) ( v , Λ ( T ) ρ ) . It only remains to verify the truth of the former inequality, E (cid:110) ϕ ( T ) ϵ ( v , L ) (cid:111) ≥ ( − ϵ ) ϕ ( T ) ( v , Λ ( T ) ρ ) + ϵ .First note that E (cid:110) ϕ ( T ) ϵ ( v , L ) (cid:111) represents the probability of node v being connected to one of thenodes in the random set L averaged over the T copies G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ . Consider each of the T copiesin our uncut sketch, G ( ) ρ , . . . , G ( T ) ρ , and the connections between node v and the optimal set Λ ( T ) ρ inthese uncut copies. If these connections remain unchanged in the ϵ -cut copies G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ , thenwith probability at least ( − ϵ ) they remain unchanged after ϵk nodes in Λ ( T ) ρ are randomly replaced.If, however, any of these connections are affected by the ϵ -cutting, then this is an indication that v belongs to a connected component of size τ n , kϵ . This connected component is large enough tocontain one of the ϵk random nodes of L with probability at least ϵ . Indeed, the probability thatnone of the τ n , kϵ nodes is chosen is at most ϵ : (cid:32) − τ n , kϵ n (cid:33) ϵk = (cid:18) − log ( / ϵ ) ϵk (cid:19) ϵk ≤ e − log ( / ϵ ) = ϵ . A.5 Proof of Theorem A.5
Following the notation in the proof of Lemma 6 (Appendix A.4), consider any Λ α , ( T ) ρ , ϵ . Lemma 6implies that Λ α , ( T ) ρ , ϵ is also Λ ( − ϵ ) α , ( T ) ρ , as the loss in approximation factor form limited probingis at most ( − ϵ ) . Next note that Lemma 5, together with Lemma 3, implies that for T = T n , kϵ , δ with probability at least 1 − e − δ , the value of Λ ( − ϵ ) α , ( T ) ρ for Γ ρ can be lower bounded as follows: Γ ρ ( Λ ( − ϵ ) α , ( T ) ρ ) ≥ ( − ϵ ) αL ρ − (( − ϵ ) α + ) ϵn . Finally, another application of Lemma 3 with Lemma4 yields that with at least 1 − e − δ probability, Γ ( Λ ( − ϵ ) α , ( T ) ρ ) ≥ ( − ϵ ) αL − (( − ϵ ) α + ) ϵn . Theproof is complete upon combining the preceding statements to geth that, with total probability atleast 1 − e − δ , Γ ( Λ α , ( T ) ρ , ϵ ) ≥ ( − ϵ ) αL − (( − ϵ ) α + ) ϵn . A.6 Proof of Theorem 8
In the first step, we bound the total number of edges used in our sketch, i.e. the T copies G ( ) ρ , ϵ , . . . , G ( T ) ρ , ϵ . Let us also denote the set of all edges that appear in our sketch by E T . Fix a choice of τ nodesin one of the copies. Let X be the number of edges between these τ nodes. Note that X is a randomvariable and its distribution is fixed by PROBE ( ρ , T , τ ) . Using the Chernoff upper-tail and the factthat E [ X ] ≤ p (cid:0) τ (cid:1) , we can upper-bound X as follows: P (cid:104) X ≥ pτ ( τ − )/ + δ ′ (cid:112) pτ ( τ − )/ (cid:105) ≤ P (cid:104) X ≥ p E [ X ] + δ ′ (cid:112) p E [ X ] (cid:105) ≤ e − δ ′ / · (10) ckles et al. Seeding with Costly Network Information E n , kϵ , p = pτ n , kϵ ( τ n , kϵ − )/
2. Setting τ = τ n , kϵ and δ ′ = (cid:113) δ ( τ n , kϵ log ( n ) + log T ) ≥ (cid:115) δ log ( T (cid:18) nτ n , kϵ (cid:19) ) ,in (10) is enough to ensure that, by union bound, with probability at least 1 − e − δ for any subset ofsize τ n , kϵ in all of the T copies, we have X ≤ X , where X : = E n , kϵ , p + (cid:113) δ ( τ n , kϵ log ( n ) + log T ) E n , kϵ , p · (11)Next note that starting from any of the nρ nodes in V ρ we never hit more that τ nodes followingthe limited probing procedure — see step (2-c) of the PROBE ( ρ , T , τ ) algorithm. Hence, the totalnumber of components with size τ in our sketch is always less than nρT , and given (11), we canbound the total number of edges in the T copies by nρT X . More precisely, with probability at least1 − e − δ , we have: card (E T ) ≤ nρT X = C n , kϵ , δ , (12)where ρ = ρ n , kϵ , δ , T = T n , kϵ , δ , τ = τ n , kϵ , and C n , kϵ , δ is defined in (4).Next note that some edges may have been queried (reported to the surveyor), but not appearin E T . This can happen for an edge e in a copy G ( i ) ρ , ϵ as follows: Such an edge would have beenreported by a newly probed node ν to an already probed node u . Since this edge has already got itsone chance of appearing in G ( i ) ρ , ϵ when u was being probed, it is discarded after being reported asan edge by ν — see step (2-b) of the PROBE algorithm. We can bound the number of such edges ineach copy as follows. Let A ( i ) e be the indicator variable for the event that both nodes incident toedge e are probed; let B ( i ) e be the indicator that edge e is reported (i.e., queried) on its second chance,i.e. when the second of the two nodes incident to e is probed. Finally, let C ( i ) e be the indicator thatedge e is reported when the second of its two incident nodes is probed, conditioned on both of itsincident nodes being probed (i.e., B ( i ) e conditioned on A ( i ) e = e for which B ( i ) e =
1, arethose which are queried but do not appear in G ( i ) ρ , ϵ . In (12) we bound the total number of edgesbelonging to E T , i.e. the edges that are queried and appear in one or more of the T copies. Our nextgoal is to provide a complementary bound on (cid:205) i (cid:205) e B ( i ) e , thus controlling the total number of edgequeries.We begin by noting that B ( i ) e = (cid:205) e A ( i ) e C ( i ) e . The indicator variables C ( i ) e , e ∈ E are i.i.d. Bernoullivariables with success probability p . Using the Chernoff upper-tail bound, conditioned on therealizations of A ( i ) e for all e ∈ E , we have: P (cid:34)(cid:213) e A ( i ) e C ( i ) e ≥ p (cid:213) e A ( i ) e + n (cid:112) δ + log T (cid:35) ≤ exp (cid:169)(cid:173)(cid:173)(cid:171) − n ( δ + log T ) (cid:16)(cid:205) e A ( i ) e + ( n / ) (cid:112) δ + log T (cid:17) (cid:170)(cid:174)(cid:174)(cid:172) ≤ exp (− δ − log T ) = T e − δ , (13) ckles et al. Seeding with Costly Network Information (cid:205) e A ( i ) e ≤ n , and (cid:112) δ + log T ≤ n for T = T n , kϵ , δ and n large enough. Union bound over i = , . . . , T provides that with probability at least 1 − e − δ , for all i : (cid:213) e B ( i ) e = (cid:213) e A ( i ) e C ( i ) e ≤ p (cid:213) e A ( i ) e + n (cid:112) δ + log T . (14)To proceed, for any edge e , let D ( i ) e be the indicator of the event that edge e gets at least onechance to appear in G ( i ) ρ , ϵ , i.e. at least one of the nodes incident to e are probed. Note that, bydefinition, A ( i ) e ≤ D ( i ) e for all i and e ; hence, replacing in (14) yields: (cid:213) e B ( i ) e ≤ p (cid:213) e D ( i ) e + n (cid:112) δ + log T , (15)with probability at least 1 − e − δ , for all i . In the next step, let E ( i ) e be the indicator of the event thatedge e is reported on its first chance — i.e., the first time that one of its incident nodes is probed.Note that E ( i ) e = e ∈ E , are those edges which are queried and appear in G ( i ) ρ , ϵ . Hence, from (12)we have: (cid:213) i (cid:213) e E ( i ) e = card (E T ) ≤ C n , kϵ , δ , (16)with probability at least 1 − e − δ . Finally, let F ( i ) e be the indicator of the event that edge e is reported onits first chance, conditioned on at least one of its incident nodes being probed (i.e., E ( i ) e conditioned on D ( i ) e = F ( i ) e are i.i.d. Bernoulli variables with success probability p , and E ( i ) e = D ( i ) e F ( i ) e .Similarly to (13), using a Chernoff lower-tail bound we can guarantee that, with high probability, (cid:205) e E ( i ) e = (cid:205) e D ( i ) e F ( i ) e is not much smaller than p (cid:205) e D ( i ) e . Subsequently, we can upper-bound (cid:205) e B ( i ) e in (15) in terms of (cid:205) e E ( i ) e . These details are spelled out next.Application of the Chernoff lower-tail bound to (cid:205) e E ( i ) e = (cid:205) e D ( i ) e F ( i ) e , yields: P (cid:34)(cid:213) e E ( i ) e = (cid:213) e D ( i ) e F ( i ) e ≤ p (cid:213) e D ( i ) e − n (cid:112) ( δ + log T ) (cid:35) ≤ exp (cid:32) − n ( δ + log T ) (cid:205) e D ( i ) e (cid:33) ≤ exp (− δ − log T ) = T e − δ , where in the second inequality we use (cid:205) e D ( i ) e ≤ n . Union bound over i = , . . . , T provides thatwith probability at least 1 − e − δ , for all i : p (cid:213) e D ( i ) e ≤ (cid:213) e E ( i ) e + n (cid:112) ( δ + log T ) . (17)Combing (15) and (17) and taking the summation over i = , . . . , T gives that with probability atleast 1 − e − δ : (cid:213) i (cid:213) e B ( i ) e ≤ (cid:213) i (cid:213) e E ( i ) e + (cid:16) + √ (cid:17) Tn (cid:112) δ + log T . (18)To complete the proof, we combine (16) and (18) to get the claimed upper-bound on the totalnumber of edge queries: q = (cid:213) i (cid:213) e B ( i ) e + card (E T ) ≤ C n , kϵ , δ + (cid:16) + √ (cid:17) T n , kϵ , δ n (cid:113) δ + log T n , kϵ , δ , with probability at least 1 − e − δ . ckles et al. Seeding with Costly Network Information A.7 Proof of Theorem 9
Here, similar to the entire paper, we assume that the algorithm has access to the input graph’s(unordered) adjacency list, and a query ( v , i ) asks for the i -th neighbor of node v . In this proof weset k = p =
1. Moreover, for simplicity of presentations we assume that 3 / µ , 1 / µ , and µ n / • G : This graph consists of 9 / µ cliques, each of size µ n / • G ′ : This graph is constructed from G via the following random process. We select µ clus-ters uniformly at random. Then we select one edge from each selected cluster uniformlyat random. Let ( v , u ) , ( v , u ) , ... ( v / µ , u / µ ) be the list of the selected edges. we remove ( v , u ) , ( v , u ) , ... ( v / µ , u / µ ) and replace them by ( u , v ) , ( u , v ) , . . . , ( u / µ − , v / µ ) , ( u / µ , v ) .Note that this process connects all of the selected clusters while preserving the degree distri-bution (see Figure 2).Let Alg be an arbitrary (potentially randomized) algorithm for influence maximization that queriesless than ( µ / ) (cid:0) µ n / (cid:1) edges. Note that with k = G ′ spreads to µn / Alg on G ′ is less than µ n /
3, which means that
Alg is not an µ -approximation algorithm. This implies that there is no µ -approximation algorithm that queries less than ( µ / ) (cid:0) µ n / (cid:1) ∈ O µ ( n ) edges as claimed.We use the run of Alg on G to analyze the run of Alg on G ′ . Note that due to symmetricconstruction of G we can assume that Alg seeds one of the nodes of G uniformly at random.Observe that the expected spread size of a random seed in G ′ is ( − / µ / µ ) · µ n + / µ / µ · µ · µ n ≤ µ n Alg on G and G ′ are the same unless Alg queries one ofthe positions (i.e., edges) that we change G to construct G ′ . Next we upper-bound the probabilitythat Alg queries one of the changed edges by µ /
3. This implies that the expected spread size is atmost ( − µ ) · µ n + µ · µ · µ n ≤ µ n , as claimed.Now we bound the probability that Alg queries one of the changed edges. Let A i be the randomvariable that indicates the number of edges Alg queries from the i ’th clique. Recall that by assump-tion, Alg queries less than ( µ / ) (cid:0) µ n / (cid:1) edges. Hence, we have A i < ( µ / ) (cid:0) µ n / (cid:1) . Therefore, theprobability that Alg queries a changed position in the i ’th clique is less than µ (cid:0) µ n / (cid:1)(cid:0) µ n / (cid:1) = µ . By a union bound over all µ cliques we can bound the probability that Alg queries a changedposition by 9 µ · µ = µ , as claimed. This completes the proof of the theorem. ckles et al. Seeding with Costly Network Information A.8 Proof of Theorem 10
We first discuss the running time of SEED ( ϵ ′ ) . Using a typical graph traversal algorithm (such asDFS or BFS), we can identify the connected components of all the T copies, in time that is in theorder of the size of E T , i.e. ˜ O ( pn + √ pn . ) , by Theorem 8. To compute ∆ ( v |S) , we go over theconnected component of v in each of the T subsampled graphs and add up their values. Thesevalues are pre-computed for each connected component, and hence this operation takes O ( T ) time.Recall that the value of a connected component is initially set equal to the number of initial nodes,belonging to V ρ , in that component. To add a node v to S , S ← S ∪ { v } , we reset the value ofthe connected component of v in each of the T copies to zero. This ensures that if we later pickanother node from these components we do not double count these initial nodes that are alreadycovered by v . This process can be done in O ( T ) = ˜ O ( k ) rounds as well. Recall that the submodularmaximization algorithm that we are using makes at most n log ( / ϵ ′ ) queries to the function Γ ( T ) ρ , ϵ .Therefore the total running time of this algorithm is ˜ O ϵ ′ , ϵ , δ ( pn + √ pn . + nk ) as claimed.Next we consider the approximation guarantee of SEED ( ϵ ′ ) . Fix ϵ ′ = ϵ / δ ′ = log 5 + log ( / ϵ ′ ) , ρ ′ = ρ n , kϵ ′ , δ ′ , T ′ = T n , kϵ ′ , δ ′ , and τ ′ = τ n , kϵ ′ , δ ′ . Running the Algorithm PROBE ( ρ ′ , T ′ , τ ′ ) , provides accessto the submodular function Γ ( T ) ρ ′ , ϵ ′ which has the k -IM optimal solution Λ ( T ′ ) ρ ′ , ϵ ′ . Using SEED ( ϵ ′ ) , weobtain an approximate solution of k -IM for Γ ( T ′ ) ρ ′ , ϵ ′ in ˜ O ( pn + √ pn . + nk ) time. Call this solution Λ ⋆ ( T ′ ) ρ ′ , ϵ ′ . The analysis of [47, Theorem 1] implies that E (cid:104) Γ ( T ′ ) ρ ′ , ϵ ′ ( Λ ⋆ ( T ′ ) ρ ′ , ϵ ′ ) (cid:105) ≥ ( − / e − ϵ ′ ) Γ ( T ′ ) ρ ′ , ϵ ′ ( Λ ( T ′ ) ρ ′ , ϵ ′ ) = ( − / e − ϵ ′ ) L ( T ′ ) ρ ′ , ϵ ′ , where the expectation is with respect to the randomness of the SEED algorithm.Combing the claims of Theorems 7 and 8 guarantees that with probability at least 1 − e − δ ′ = − ϵ ′ , Γ ( Λ ( T ′ ) ρ ′ , ϵ ′ ) = L ( T ′ ) ρ ′ , ϵ ′ ≥ ( − ϵ ′ ) L − ( − ϵ ′ ) ϵ ′ n , and we probe no more than ˜ O ( pn + √ pn . ) edges.Thus the expected number of nodes that are covered by the output of SEED algorithm, Λ ⋆ ( T ′ ) ρ ′ , ϵ ′ , canbe lower bounded as follows: E (cid:104) Γ ( Λ ⋆ ( T ′ ) ρ ′ , ϵ ′ ) (cid:105) ≥ ( − / e − ϵ ′ ) E (cid:104) L ( T ′ ) ρ ′ , ϵ ′ (cid:105) ≥ ( − / e − ϵ ′ )( − ϵ ′ )(( − ϵ ′ ) L − ( − ϵ ′ ) ϵ ′ n )≥ ( − / e − ϵ ′ )( − ϵ ′ ) L − ϵ ′ n ≥ ( − / e )( − ϵ ′ ) L − ( ϵ ′ )( − ϵ ′ ) L − ϵ ′ n ≥ ( − / e ) L − ϵ ′ n = ( − / e ) L − ϵn , where the first expectation is with respect to the randomness of both the SEED and the PROBEalgorithms, and the second expectation is with respect to the randomness of the PROBE algorithm. A.9 Proof of Theorem 11
Recall our notation in the SPREAD algorithm. The output of the algorithm Λ ⋆ is a set of k nodesthat are chosen, one by one, in k steps. Let us use Λ ⋆ i to denote the first i nodes that are selectedin steps 1 to i . In step i , we choose ρ initial nodes at random to run spread queries. Let us denotethe random initial nodes in step i , by u i , . . . , u iρ . We use A ij to denote the random subset of nodesthat are observed to adopt when u ij is seeded. We reset A ij to ∅ if it contains any of the i − u ∈ V \ Λ ⋆ i − , and choose the i -th seed to be the one that appears in the most subsets. To put this in mathematical notation, let ckles et al. Seeding with Costly Network Information X iu , j = { u ∈ A ij } be the indicator that u belongs to A ij , and set X iu = (cid:205) ρj = X iu , j to count the numberof times that u appears in any of the subsets A i , . . . , A iρ . Subsequently, in step i , we choose v ⋆ = argmax u ∈V\ Λ ⋆ i − X iu , and add it to Λ ⋆ .We analyze the steps of Algorithm 3 and show that for ϵ ′ = ϵ / ρ = ρ n , kϵ = k log ( nk / ϵ ′ )/ ϵ ′ ,the output of SPREAD satisfies the desired approximation guarantee. Let us define random variable N iu to be the expected number of nodes that are covered by Λ ⋆ i − ∪ { u } but not by Λ ⋆ i − . Notethat the probability that X iu , j = N iu / n . Therefore, we have E [( n / ρ ) X iu ] = E [ N iu ] .Moreover, notice that choosing ν ⋆ ∈ V \ Λ ⋆ i − to maximize E [ N iu ] is equivalent to one step ofthe greedy algorithm. This is equivalent to choosing ν ⋆ ∈ V \ Λ ⋆ i − to maximize E [ X iu ] , since E [( n / ρ ) X iu ] = E [ N iu ] .Next note that due to submodularity, the marginal values only decrease as we add more elements.Hence, if we stop the algorithm when ∀ u E [ N iu ] < ϵ ′ n / k , in total we do not loose more than k ( ϵn / k ) = ϵn . For the sake of analysis, let us assume that the algorithm stops if ∀ u E [ N iu ] < ϵ ′ n / k :This means that the algorithm stops if it selects k seeds or ∀ u E [ N iu ] < ϵ ′ n / k whichever comes first.Henceforth, without loss of generality, we assume that max u E [ N iu ] ≥ ϵ ′ n / k which means that wehave E [ X iu ] ≥ ϵ ′ ρ / k .Recall that X iu is the sum of i.i.d. binary random variables X iu , j . Hence, by the Chernoff boundwe have P (cid:2) | X iu − E [ X iu ]| ≤ ϵ ′ E [ X iu ] (cid:3) ≤ (cid:18) − ϵ ′ E [ X iu ] (cid:19) Chernoff Bound ≤ (cid:18) − ϵ ′ ρ k (cid:19) E [ X iu ] ≥ ϵ ′ ρ / k = ϵ ′ nk . ρ = k log nkϵ ϵ = k log nkϵ ′ ϵ ′ Union bound over all u ∈ V and 1 ≤ i ≤ k provides that with probability at least 1 − ϵ ′ , ( − ϵ ′ ) E [ X iu ] ≤ X iu ≤ ( + ϵ ′ ) E [ X iu ] for all u and i . This implies that the seed that our algorithmselects has marginal increase at least − ϵ ′ + ϵ ′ ≥ − ϵ ′ times that of the greedy algorithm. Suchalgorithm is called ( − ϵ ′ ) -approximate greedy in [25] and it is proven to return a ( − / e − ϵ ′ ) -approximate solution [4, 25, 37]. Therefore, we can bound the expected value of the output solutionof SPREAD ( ρ ) as follows. E [ Λ ⋆ ] ≥ ( − ϵ ′ )[( − / e − ϵ ′ ) L − ϵ ′ n ]≥ ( − ϵ ′ )[( − / e ) L − ϵ ′ n ]≥ ( − / e ) L − ϵ ′ n = ( − / e ) L − ϵn . A.10 Proof of Theorem 12
Pick an arbitrary function f ( n ) ∈ o ( n ) , and let д ( n ) = (cid:112) n / f ( n ) . Note that д ( n ) ∈ ω ( ) . We showthat an algorithm Alg that makes f ( n ) spread queries is not µ -approximation. Consider the theexample depicted in Figure 4. We have a clique of size д ( n ) (on д ( n ) nodes chosen uniformly atrandom) and n − д ( n ) isolated nodes and we aim to seed one node. One can bound the probability ckles et al. Seeding with Costly Network Information (A) (B) (C)Fig. 8. Three reverse cascades are depicted in (A) red, (B) orange, and (C) blue. All cascades start from thesame random initial nodes which are marked in the same color as the cascades. As the cascades diffusein reverse, each node reveals its incoming edges at random. To score the nodes, we count the number ofreachable initial nodes in each cascade and add them up. For example, the node that is marked in black scoresthree in the red cascade, two in the orange cascade, and one in the blue cascade. In total, it scores as high asor higher than other nodes across the three cascades. The dotted sections consist of unrealized influences ineach cascade. that Alg queries a node from the clique by1 − (cid:18) − д ( n ) n (cid:19) f ( n ) = − (cid:18) − д ( n ) n (cid:19) nд ( n ) ≤ − (cid:18) − e (cid:19) д ( n ) . Moreover, since д ( n ) ∈ ω ( ) we have 1 − (cid:0) − e (cid:1) д ( n ) ∈ o ( ) . If Alg does not query a node viaspread queries, it seeds one of the nodes of the clique with probability at most д ( n ) n − f ( n ) = o ( ) .Therefore, the expected number of nodes covered by Alg is at most o ( ) д ( n ) +
1, which means thatthe approximation factor of
Alg is o ( ) д ( n ) + д ( n ) = o ( ) as claimed. B EXTENSION TO ASYMMETRIC INFLUENCES
We presented our results for undirected graphs with a homogeneous cascade probability ( p ). Ingeneral, the influence graph may be directed and cascade probabilities may differ in each directionand across the edges. Consider a node v and let N v be the set of all its incoming neighbors (i.e.,its influencers). For a directed edge u → v , let p uv be the probability of u influencing v . We caneasily extend our results to undirected graphs with heterogeneous cascade probabilities. Note that inundirected graphs ∀ uv p uv = p vu . With heterogeneous cascade probabilities, the spread queries areperformed as before and the adopters are observed. When performing edge queries, the probednodes reveal each of their neighbors, with the probability associated that edge. Our bound on theedge queries would include p max = max uv p uv instead of p .We can also extend our results to directed graphs (or even graphs with both directed andundirected edges). In the case of edge queries, we first modify step (2-a) of the PROBE algorithm,such that a probed node v reveals each of her incoming neighbors u ∈ N v , with their associatedprobability p uv . Two directed edges that are between the same pair of nodes in opposite directions( u → v and v → u ) are distinguished. Therefore, as long as we do not probe a node more than once,each directed edge will get at most one chance of appearing in G ( i ) ρ , ϵ . Accordingly, we modify step(2-b) of the PROBE algorithm as follows: For any revealed neighbor, add the corresponding directededge to G ( i ) ρ , ϵ and proceed to probe the revealed neighbors that are not probed before. In fact, we canslightly improve our edge query upper-bound in the directed case since every directed edge that isrevealed in step (2-a) of PROBE is added to G ( i ) ρ , ϵ in step (2-b) of the PROBE algorithm. Subsequently, ckles et al. Seeding with Costly Network Information − e δ no more than C n , kϵ , δ ∈ ˜ O ϵ , δ ( pn + √ pn . ) edges are queried.When limiting the probed neighborhoods in step (2-c) of the PROBE algorithm, instead ofconsidering the size of the connected component of an initial node, we count the number of nodesthat are reachable from it (i.e., the size of its realized cone of influence ). For example in Figure 8(A),the reachable set for all of the initial nodes is empty, therefore we proceed to probe the incomingneighbors until there are no new nodes to probe. In fact, of all the initial nodes in all three cascadesin Figure 8, only the leftmost initial node in Figure 8(B) has a non-empty reachable set that is asingleton.Similarly, when scoring the candidates for seeding in steps (2) and (4-b) of the SEED algorithm,we count the number of reachable initial nodes (belonging to V ρ ) rather than the number ofinitial nodes in the candidate’s connected component (see Figure 8). As before, if an initial node isreachable from any of the chosen seeds, then in step (4-c) of SEED we nullify its value for scoringthe subsequent candidates. Fig. 9. With o ( n ) spread query, one is unlikely to discover the center of the directed star network. In such acase it is impossible to guarantee a µ > approximation factor with fewer than Ω ( n ) queries. In contrast, finding an approximately optimal seed set by performing spread queries in a directedgraph is very hard. For example, consider a star graph with n leafs where all of the edges are directedaway from the center toward the leafs (see Figure 9). Assume that the cascade probability on eachedge is 1. In this case, if we query a leaf we only observe an isolated node and hence we need Ω ( n ) spread queries to find the center of the star and seed it. In a situation where running reverse spreadsis a plausible way of acquiring network information (e.g., by querying the transposed graph as in[12]), our algorithm and proofs continue to hold exactly the same. In particular, we can estimatethe marginal increase of a candidate node on the current seed set by counting the number of timesthat it has appeared in the output of the queries without any of the currently chosen seeds (i.e., thenumber of times that the random initial nodes are reachable from the candidate node but not fromany of the currently chosen seeds). By running enough such queries and choosing the best node atevery step, we can approximate greedy with enough precision as in Algorithm 3 — see also Figure10 for a relevant illustration. C EXTENSIONS BEYOND THE INDEPENDENT CASCADE MODEL
Here we explain how a triggering set technique that is proposed in [35] helps us devise queriesin a large class of models, including the linear threshold model . Recall that the influence function Γ maps a seed set to a positive real number that is the (expected) number of adopters under a(randomized) model of diffusion. Kempe, Kleinberg, and Tardos [33–35] — through a conjecturethat is positively resolved by Mossel and Roch [48] — identify a broad class of threshold models forwhich the influence function is non-negative, monotone, and submodular. Influence maximizationin such models can be solved to within a ( − / e ) approximation guarantee, following a naturalgreedy node selection algorithm. ckles et al. Seeding with Costly Network Information (A) (B) (C)Fig. 10. Some diffusion processes can be reduced to a triggering set model; whereby, after randomly drawinga triggering set from among her neighbors, a node becomes active if any of the nodes in her triggering set areactive. In such models, we can devise queries by having a probed node reveal a random realization of hertriggering set, and then proceeding to probe the revealed nodes. In the case of the linear threshold model, thetriggering sets are either empty or a singleton chosen randomly according to the edge weights. The reversecascades in (A), (B) and (C) show three queries where the nodes sequentially reveal their triggering sets,starting form a random initial node. The black node appears in the output of all three “reversed cascades”,and therefore scores the highest as a seed candidate. In a general threshold model, each node v has an activation function f v and a threshold θ v ∈ [ , ] .The activation function maps subsets of neighbors of v to a real number between zero and one.Node v becomes an adopter at time t if f v (A v , t − ) ≥ θ v , where A v , t − ⊂ N v is the set of all active(adopter) neighbors of node v . Approximate influence maximization with deterministic thresholdsis known to be very hard (cf., e.g., [35, Section 3.2] and [52]). To avoid the intractable settings,Kempe et al. consider a randomized model where thresholds are i.i.d. uniform [ , ] variables.Mossel and Roch show that if the “local” activation functions f v are submodular, then the “global”influence function Γ is also submodular and influence maximization can be achieved with strongapproximation guarantees [48, Theorem 1.6 and Corollary 1.7]. In the special case of the linear threshold model, each node v is influenced by its incoming neighbours u ∈ N v according to theiredge weights b uv . Node v becomes an adopter at time t if the total weight of her adopting neighborsexceeds her threshold, i.e. if f v (A v , t − ) = (cid:205) u ∈A v , t − b uv ≥ θ v .At the heart of the proofs of Kempe et al. [33, 35] lie a triggering set technique. Accordingly, eachnode v chooses a random subset of its incoming neighbors, which we call its triggering set anddenote it by T v ⊂ N v . Node v becomes an adopter at time t if any of the nodes in T v is an adopterat time t −
1. The distribution according to which the triggering sets are drawn is determined bythe diffusion model. However, not all diffusion processes can be reduced to a triggering set model.For those that do, their influence function is guaranteed to be submodular [35, Lemma 4.4].For example, in the independent cascade model, the triggering set T v includes each of theneighbors u ∈ N v with probability p uv , independently at random. In the case of the linear thresholdmodel, Kempe at al. [35] devise the following construction, assuming (cid:205) u ∈N v b uv ≤
1: The triggeringset T v is comprised of a single node or no nodes at all. For u ∈ N v , the probability that T v = { u } isequal to b uv , and T v = ∅ with probability 1 − (cid:205) u ∈N v b uv .For diffusion processes that can be cast as a triggering set model, we can implement queries byhaving the probed nodes reveal their triggering sets, and proceeding to probe the revealed nodes.Starting form random initial nodes, each triggering set corresponds to a batch of directed edges thatare incoming to the probed node. We can use the number of reachable initial nodes to implementan approximate greedy heuristic as described in Appendix B — see Figure 10. One needs to analyzethe specific diffusion process and the triggering distributions to provide approximation guaranteesusing a bounded number of queries. ckles et al. Seeding with Costly Network Information k batches of ρ reversedcascades, we can provide an approximate k -IM solution for the linear threshold model over directedgraphs. At each stage we choose ρ initial nodes at random and implement ρ cascades in reverse. Ineach reversed cascade, we start from the initial node, reveal her triggering set, and then proceedingto probe the node that is revealed in the triggering set, etc. After each batch of ρ reversed cascades,we choose the node that appears the most number of times, discarding those cascades that includethe already chosen seeds. Following Theorem 11, we can provide a ( − / e ) L − ϵn approximationguarantee, by running kρ = k log ( nkϵ )/ ϵ reversed cascades.To bound the number of edge queries, recall that each triggering set in the linear thresholdmodel consists of at most a single node. Hence, each reversed cascade corresponds to a path oflength at most n — see Figure 10. Therefore, we can bound the total number of queried edges by nkρ = nk log ( nkϵ )/ ϵ . In Theorem 13, we state how our results translate to the case of k -IMwith the linear threshold model over directed graphs. REFERENCES [1] Mohammad Akbarpour, Suraj Malladi, and Amin Saberi. 2018. Just a Few Seeds More: Value of Network Informationfor Diffusion. Available at SSRN 3062830.[2] Noga Alon, Eldar Fischer, Michael Krivelevich, and Mario Szegedy. 2000. Efficient Testing of Large Graphs.
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