Seeding with Costly Network Information

Abstract

The spread of behavior over social networks depends on the contact structure among individuals, and seeding the most influential agents can substantially enhance the extent of the spread. While the choice of the best seed set, known as influence maximization, is a computationally hard problem, many computationally efficient algorithms have been proposed to approximate the optimal seed sets with provable guarantees. Most of the previous work on influence maximization assumes the knowledge of the entire network graph. However, in practice, obtaining full knowledge of the network structure is very costly.In this work, we consider the choice of k initial seeds to maximize the expected number of adopters under the independent cascade model. We propose a "probe-and-seed" algorithm that provides almost tight approximation guarantees usingÕ(pn 2 + √ pn 1.5 ) edge queries inÕ(pn 2 + √ pn 1.5 ) time, where n is the network size and p is the probability of spreading through an edge. To the best of our knowledge, this is the first result to provide approximation guarantees for influence maximization, using a sub-quadratic number of queries for polynomially small p. We complement this result by showing that it is impossible to approximate the problem using o(n 2 ) edge queries for constant p. In the end, we consider a more advanced query model, where one seeds a node and observes the resultant adopters after running the spreading process. We provide an algorithm that uses onlyÕ(k 2 ) such queries and provides almost tight approximation guarantees.1 1 INTRODUCTION Decision-makers in marketing, public health, development, and other fields often have a limited budget for interventions, such that they can only target a small number of people for the intervention. Thus, in the presence of social or biological contagion, they strategize about where in a network to intervene -often where to seed a behavior (e.g., product adoption) by engaging in an intervention (e.g., giving a free product). The influence maximization literature is devoted to the study of algorithms for finding a set of k seeds so as to maximize the expected adoption, given a known network and a model of how individuals are affected by the intervention and others' adoptions [15]. However, finding the best k seeds for many models of social influence is NP-hard, as shown by Kempe, Kleinberg and Tardos [21]. Much of the subsequent influence maximization literature is concerned with developing efficient approximation algorithms with theoretical guarantees to make use of desirable properties of the influence function such as submodularity [10,34].With few exceptions [28,36], the seeding strategies that are studied in the influence maximization literature require explicit and complete knowledge of the network. However, collecting the entire network connection data can be difficult, costly, or impossible. For example, in development economics, public health, and education, data about network connections is often acquired through costly surveys (e.g., [4,7,8,30]). Indeed, t...

Influence maximization under limited network information: Seeding high-degree neighbors

“…The first category concerns approximation algorithms or heuristics developed for full network information to networks with partially disclosed structure [17][18][19][20][21][22][23]. The central idea is to efficiently disclose part of the network structure, i.e., the edge list of some nodes (or "subnetwork") so that the established algorithms or heuristics for the full network can be used and the seeding outcomes with full network information can be approximated [22,23].…”

“…Such a process is repeated a number of times and the highest degree nodes after all the probing is used as seeds. Similarly, Eckles et al [21] chose a set of random nodes and asked them to reveal their connections with a probability. The revealed connections will go through the same process again to reveal more of the network.…”

“…While letting go of the assumption of full network information, these studies still assume that there is large degree of control over what network information is collected, e.g. through snowball sampling [17,18] or probabilistic random walks [21], and pose nontrivial demands on available network information [21][22][23]. This renders it impossible to use them for seeding in the scenarios where limits on available network information are predetermined.…”

“…Strategies proposed in prior work either assume full network information [4,[10][11][12][13][14][15][16] or large degree of control over what network information is collected [17][18][19][20][21][22][23]. However, privacy settings on the Internet and high non-response in surveys often severely limit available connectivity information.…”

Targeted Intervention in Random Graphs

“…In this setting, for G(n, p) and power law random graphs, they derive bounds on the additional number of "seeds" needed to match optimal targeting when network information is limited. A version of this problem where network information can be purchased is studied in [19]. Another similar model was employed by Candogan, Bimpikis, and Ozdaglar [11] where they study optimal pricing strategies to maximize profit of a monopolist selling service to consumers in a social network where the consumer experiences a positive local network effect, where notions of centrality play a key role.…”