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Strategies for utility maximization in social groups with preferential exploration

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Abstract

Social Group is group of interconnected nodes interested in obtaining common content (Scott, in Social network analysis, 2012). Social groups are observed in many networks for example, cellular network assisted Device-to-Device network (Fodor et al., in IEEE Commun Mag 50:170–177, 2012, Lei et al., in Wirel Commun 19:96–104, 2012), hybrid Peer-to-Peer content distribution (Christos Gkantsidis and Miller, in 5th International Workshop on Peer-to-Peer Systems, 2006, Vakali and Pallis, in IEEE Internet Comput 7:68–74, 2003) etc. In this paper, we consider a “Social Group” of networked nodes, seeking a “universe” of data segments for maximizing their individual utilities. Each node in social group has a subset of the universe, and access to an expensive link for downloading data. Nodes can also acquire the universe by exchanging copies of data segments among themselves, at low cost, using inter-node links. While exchanges over inter-node links ensure minimum or negligible cost, some nodes in the group try to exploit the system by indulging in collusion, identity fraud etc. We term such nodes as ‘non-reciprocating nodes’ and prohibit such behavior by proposing the “Give-and-Take” criterion, where exchange is allowed iff each participating node provides at least one segment to the node which is unavailable with the node. While complying with this criterion, each node wants to maximize its utility, which depends on the node’s segment set available with the node. Link activation between pair of nodes requires mutual consent of the participating nodes. Each node tries to find a pairing partner by preferentially exploring nodes for link formation. Unpaired nodes download data segments using the expensive link with pre-defined probability (defined as segment aggressiveness probability). We present various linear complexity decentralized algorithms based on the Stable Roommates Problem that can be used by nodes for choosing the best strategy based on available information. We present a decentralized randomized algorithm that is asymptotically optimal in the number of nodes. We define Price of Choice for benchmarking the performance of social groups consisting of non-aggressive nodes (i.e. nodes not downloading data segments from the expensive link) only. We evaluate performances of various algorithms and characterize the behavioral regime that will yield best results for nodes and social groups, spending the least on the expensive link. The proposed algorithms are compared with the optimal. We find that the Link For Sure algorithm performs nearly optimally.

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Notes

  1. These algorithms are not incentive compatible  [24, 39].

  2. Price of choice (defined in Sect.  4) for link for sure algorithm is very close to 1. It is only possible when most of the nodes obtain their best possible utility values.

  3. Lemma 2 shows that at least one pair of nodes will be give mutual consent for activation of link. nodes unpaired. These unpaired nodes might choose to download new segments via the expensive link based on their “aggressiveness” for new segments.

  4. \(X^c\) denotes the complement of set X.

  5. We use the terminology “slot” as it suggests an interval of time over which a number of things can happen. In the beginning of a slot, decision negotiation for pairings is done. During the slot, segment exchange among nodes takes place. We have refrained from using other terminologies such as “tick” as it suggests an instant and it is difficult to visualize a situation where multiple events can happen.

  6. For the purpose of analysis in this paper, we have assumed \(u_i(r)\) to be strictly increasing w.r.t. to cardinality of segment set as outlined in A3.

  7. Node i may choose to ignore some nodes in \(l_i(r)\) based on value of its PEF in \(r\mathrm{th}\) slot, \(e_i(r)\).

  8. Assumed equality in Eq. (8) has been used in Eq. (7) to obtain Eq. (12).

  9. MPSM can be treated as a special case of LPSM, in which the PEF of node i, viz., \(e_i(r)\) is 1 for all nodes \(i\in \mathscr {M}\).

  10. \((a\mod b)\) denotes the remainder when a is divided by b, where \(a,b\in \mathbb {N}\) and \(a\ge b\).

  11. \(s_{(r,i)}\) is different from \(s_i(r)\) as defined in Sect. 2.

  12. PoC is only defined for cases where nodes are non-aggressive, i.e. \(a_i(r)=0\forall i\in \mathscr {M}, r>0\).

  13. Optimal ‘centralized’ solution refers to the system-wide state in a multi-agent system [24, 27, 28], which maximizes the social utility function.

  14. To generate a k element subset \(O_i\) for node i, we pick k elements uniformly at random without replacement from universe \(\mathscr {N}\). We choose subsets for all nodes in \(\mathscr {M}\). If \(\underset{i\in \mathscr {M}}{\bigcup } O_i\ne \mathscr {N}\), we repeat the process for all nodes.

  15. Complexity for computing Maximal Partial Stable Matching (MPSM) in each slot is same as that of computing Stable roommate matching i.e. \(O(m^2)\).

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Authors and Affiliations

  1. Department of Electronic Systems Engineering, Indian Institute of Science, Bangalore, India

    Saurabh Aggarwal & Joy Kuri

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Appendices

Appendix 1: Algorithm pseudo codes

1.1 Limited stable pairing algorithm (LSPA)

Algorithm 1 presents the pseudo code of Limited Stable Pairing Algorithm (LSPA). In LSPA, each node tries to find a pairing node (Steps 2-9)in every decision slot. To find a pairing node, node i considers the truncated preference list having only top \(\max (1,\lfloor e_i(r)\left| l_i(r)\right| \rfloor )\). If node i is unable to find a pairing node in the \(r\mathrm{th}\) decision slot, then node i downloads a new segment with probability \(a_i(r)\) (Steps 10-17).

figure a

1.2 Preferential exploration pairing algorithm (PEPA)

Algorithm 2 presents the pseudo code of Preferential Exploration Pairing Algorithm (PEPA). Setting Segment Aggressive Probability (SAP) to zero in LSPA, we will obtain PEPA. PEPA is called Link for Sure Algorithm (LSA), if \(e_i(r)=1\forall r>0\).

figure b

1.3 Decentralized randomized algorithm

Algorithm 3 presents the pseudo code for decentralized randomized algorithm. If social group does not want to bear the cost of overhead communication among nodes with the help of facilitator, then nodes can use decentralized randomized algorithm.

figure c

Appendix 2: Stable roommates problem and variants

The stable-roommate problem (SRP) is the problem of finding a stable matching among a pair of elements, such that, there is no pair of elements, each from a different matched set, where each member of the pair prefers the other to their match [43]. In a given instance of the SRP, each of m participants ranks the other participants in order of preference. A matching is a set of \(\frac{m}{2}\) disjoint pairs of participants. A matching M in an instance of SRP is stable if there are no two participants x and y, each of whom prefers the other to his partner in M. Such a pair is termed as stable pair.

Multiple variants of the SRP have been proposed and studied by researchers. We consider one such variant namely, Stable Roommates Problem with Ties and Incomplete lists (SRPTI), in which, the participants are allowed to have ties among participants and can chose to ignore certain participants [42]. Our problem is each slot can be mapped to an instance of SRPTI. Stable Matching as defined in [42, 43] exists iff all participants are able to find a pairing participant. But, there might exist some matchings in which only some participants are able to find stable pairing participants. We term such matchings as Partial Stable Matching.

We have defined Limited Preference Stable Matching and Maximal Partial Stable Matching in context of our problem.

Appendix 3

Lemma 2

(Rewriting Lemma 1 of [37]) For any order of link activations (resulting in a completely disconnected graph) and \(O_i\subsetneq \bigcup _{j\in \mathscr {M}} O_j \forall i\in \mathscr {M}\), at least two nodes will have \(\bigcup _{i\in \mathscr {M}} O_i\).

Proof

Please refer to proof for Lemma 1 of [37]. \(\square \)

Corollary 2

For any order of link activations (resulting in a completely disconnected graph) optimal aggregate cardinality is upper bounded by \(nm-\left( m\mod 2\right) \).

Proof

Trivial upper bound on aggregate cardinality is given by nm as system consists of m nodes and n segments. For odd number of nodes, it follows from the fact that number of nodes with universe can be even only. Hence, considering the best possible scenario where \(m-1\) nodes have got the universe. And node without universe will have at least 1 segment missing, therefore, optimal aggregate cardinality is upper bounded by \(nm-\left( m\mod 2\right) \). \(\square \)

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Aggarwal, S., Kuri, J. Strategies for utility maximization in social groups with preferential exploration. Auton Agent Multi-Agent Syst 31, 107–129 (2017). https://doi.org/10.1007/s10458-015-9315-3

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