The Game-Theoretic Formation of Interconnections Between Networks

The Game-Theoretic Formation of Interconnections Between Networks

Ebrahim Moradi Shahrivar and Shreyas Sundaram Ebrahim Moradi Shahrivar is with the Department of Mechanical and Mechatronics Engineering at the University of Waterloo. E-mail: emoradis@uwaterloo.ca.Shreyas Sundaram is with the School of Electrical and Computer Engineering at Purdue University. E-mail: sundara2@purdue.edu.This material is based upon work supported by the Natural Sciences and Engineering Research Council of Canada.
Abstract

We introduce a network design game where the objective of the players is to design the interconnections between the nodes of two different networks and in order to maximize certain local utility functions. In this setting, each player is associated with a node in and has functional dependencies on certain nodes in . We use a distance-based utility for the players in which the goal of each player is to purchase a set of edges (incident to its associated node) such that the sum of the distances between its associated node and the nodes it depends on in is minimized. We consider a heterogeneous set of players (i.e., players have their own costs and benefits for constructing edges). We show that finding a best response of a player in this game is NP-hard. Despite this, we characterize some properties of the best response actions which are helpful in determining a Nash equilibrium for certain instances of this game. In particular, we prove existence of pure Nash equilibria in this game when contains a star subgraph, and provide an algorithm that outputs such an equilibrium for any set of players. Finally, we show that the price of anarchy in this game can be arbitrarily large.

Interconnected Networks, Network Design, NP-hardness, Nash Equilibria, Price of Anarchy, Hub-and-Spoke.

I Introduction

There is a growing realization that many large scale networks consist of interconnected subnetworks [1, 2, 3]. Examples include coupled energy infrastructure and communication networks [4], cyber-physical systems [5], and transportation networks (such as the flight networks of different airlines) [6]. Understanding the structure of large-scale networks and the implications of this structure for the effective functioning of the network has been the subject of many studies throughout the past decade [7, 8]. One approach to investigate this problem is through the framework of random graphs where each subnetwork is drawn from a certain probability distribution [9, 10]. For (single) random networks, properties such as connectivity, robustness against structural and dynamical failures, and edge expansion have been widely investigated in the literature [11, 12, 13, 14], with recent extensions to interconnected networks [15, 4, 16, 17, 18].

An alternative perspective on understanding the structure of networks is to view the edges as being optimally placed (either by a central designer, or by different decision makers) in order to maximize some given utility function(s) [19, 20, 21, 22]. The classical literature on optimal network design has predominantly focused on the construction of a single network [7, 23]. In [24], we proposed a multi-layer network formation setting in which, given a network , the network designer aims to find a network such that distances between nodes that are neighbors in are minimized in ; in this context, networks and represent different types of relationships between the set of nodes . We then exploited this setting to formulate a multi-layer network formation game where each layer corresponds to a player that is optimally choosing its edge set in response to the edge sets of the other players.

In this paper, we consider the game-theoretic formation of edges between two given networks and on two different sets of nodes and . We assume that there are dependencies between nodes in and , i.e., some of the nodes in require connections to (or information from) some of the nodes in in order to function. These dependencies are captured by a bipartite network where , and an edge indicates that is dependent on . We assume that each node in is a player and builds a set of edges between itself and nodes in in order to maximize a distance-based utility function. As a motivating abstraction for this problem, consider a cyber-physical system where is a power network (with the nodes representing substations) and is a sensor network. Suppose that the sensor nodes are responsible for gathering critical information (e.g., power usage, line failures, etc.) from different geographical regions and this information is required by the power stations that supply electricity to those regions. This setting has been investigated from different perspectives over the past few years [25, 26, 15, 5, 1, 4, 27] and our results add to this literature by studying the game-theoretic allocation of interconnecting edges between the power and sensor networks. We model dependencies between the substations and sensors (which correspond to nodes in and , respectively) by the network . Suppose that neighboring nodes in each network are capable of exchanging information with each other. The substation operators wish to construct connections to the sensor network in such a way that they minimize the number of hops required to gather data from their interdependent nodes (where the number of hops is measured with respect to the connections within and and the edges constructed between the networks). This leads to an interconnection network design game (INDG) with distance utilities where the utility of each player (operator) depends on its own set of edges as well as the set of edges constructed by other players.

The INDG setting also matches the framework studied in [28] for merging two social networks where the goal is to construct a set of edges between the networks such that the integrated network has diameter no more than a fixed value. Besides considering a cost for constructing edges and having a different utility function, the nodes are the decision makers in our setting, whereas [28] assumes a central network designer. Distance-based utilities have also been used to study computer networks (where nodes represent the computers and edges are the communication links) [29, 30]. In this case, network models the virtual dependencies among the computers in cluster and cluster , indicating the set of pairs of nodes that wish to exchange information. The designed interconnection network represents the physical communication network between the two clusters. Yet another application of the INDG with distance utilities arises in studying interconnections between the transportation networks of two countries. We will elaborate on this example in Section IV.

We start our investigation of the INDG by showing that it is NP-hard to find a best response for each player. Despite the NP-hardness of the problem, we characterize some useful properties of the best response which consequently enable us to determine a Nash equilibrium instance for certain cases of the game. Specifically, we study the existence of Nash equilibria in an INDG with distance utilities when network has a star subgraph (similar to the “hub-and-spoke” structure seen in various transportation networks [31, 32] or in sensor networks with fusion centers [33, 34]) and there is full interdependency between nodes in and . We show that this setting possesses a Nash equilibrium for any set of players with arbitrary benefit functions and edge costs. We partition the set of players into two sets consisting of high and low edge cost players and show that in any Nash equilibrium, all of the high-cost players that have a low-cost player in their vicinity “free ride” and choose not to construct any interconnections to . At the end, we provide some insights about our results via a simulation involving random network models that have been previously used to capture interdependencies between power and sensor/communication networks [26, 27, 4, 17]. Our simulations suggest that the social welfare of the constructed networks is higher when all of the players have equal cost of constructing edges, compared to the case where they have heterogeneous edge costs.

Ii Definitions

An undirected network (or graph) is denoted by where is the set of nodes (or vertices) and denotes the set of edges. If there is an edge between two nodes, they are said to be neighbors. The number of neighbors of a node in graph is called its degree and is denoted by . A path from node to in graph is a sequence of distinct nodes where there is an edge between each pair of consecutive nodes of the sequence. The length of a path is the number of edges in the sequence. We denote the shortest distance between nodes and in graph by . If there is no path from to , we take . The diameter of the graph is . A cycle is a path of length two or more from a node to itself. A graph is called a subgraph of , denoted as , if and . A graph is connected if there is a path from every node to every other node. A subgraph of is a component if is connected and there are no edges in between nodes in and nodes in . A graph is called bipartite if there exist two disjoint subsets such that and , i.e., does not have any edge with both endpoints in or . The set of all possible bipartite graphs with two partitions and is denoted by .

Iii Distance-Based Utility

Jackson and Wolinsky introduced a canonical problem in network formation which involves distance-based utilities [7]. In their formulation, each node is a decision maker, and chooses its connections to other nodes in the network. In any formed network, each node receives a benefit of from nodes that are hops away, where is a real-valued, nonincreasing, nonnegative function (i.e., nodes that are further away provide smaller benefits) and . Furthermore, constructing the edge incurs a cost of to both endpoints and . The total utility that node receives from the constructed network is

(1)

Thus, the nodes have to compromise between adding more links (which provides a larger benefit by reducing the distances between nodes) and decreasing the cost by using fewer edges. When is a strictly decreasing function, there are only a few different kinds of socially optimal (or efficient) networks, depending on the relative values of the link costs and connection benefits: the empty network (for high edge costs), star (for medium edge costs) and the complete network (for low edge costs) [23].

Iv Interconnection Network Design Game

Assume that we are given two arbitrary networks and . In this paper, we consider a setting in which each node in constructs a set of edges to nodes in such that some utility function is maximized. This leads to a game with the nodes of as the players.

Definition 1

Consider two arbitrary networks and with and . An instance of the interconnection network design game (INDG) has a set of players where player is associated with node for . The strategy space of player is , i.e., all possible subsets of edges from to nodes in . The action of player is an element of and is denoted by , i.e., is a set of edges from to a certain subset of . By an abuse of notation, we take to indicate the bipartite graph . The utility of player is given by a function , where the argument111The utility function is also a function of and which will be omitted from the argument list as long as it is clear from the context. is the action of the player for .

The characteristics of the game and the optimal strategies for each player will depend on the form of the utility functions . In this paper, we consider a modified version of the distance utility function in (1) as the payoff to the players. Specifically, we assume that there are dependencies between nodes in the graphs and which is represented by a bipartite network with two partitions and and . Let , denote the set of neighbors of in the network . Then the objective of player is to find the optimal set of edges to construct to such that distance between its associated node and the set of nodes in is minimized. In addition to the technological applications that we mentioned in Section I, the INDG can be utilized to model problems in transportation. For instance consider a modified version of the problem studied in [16] where we are given the traffic flow between cities of two different countries and . Each of these countries has a domestic transportation service which connects its cities and is modeled by networks and . A city in and a city in are said to be interdependent if the traffic flow between them is higher than some threshold, and this interdependency is represented by an edge between them in the network . The players of the game correspond to transportation service planners at each node in , who are faced with the problem of finding the optimal set of routes to establish from their associated city to cities of the country such that distance between the interdependent cities is minimized. It is clear that the structure of the interconnection between cities inside the countries and (modeled as and ) affects the optimal decisions made by the players.

Definition 2

An instance

of the game in Definition 1 is said to be an interconnection network design game with distance utilities if the utility function of player , , with action has the form

(2)

where .

As we can see in the utility function , only the distances between node and the set of nodes matter. Furthermore, each player has to pay only for his/her constructed edges. The benefit functions are nonnegative, nonincreasing and satisfy , and all costs are positive, and can be different across players.

We will use to denote the vector of actions of all players except player , and use to denote the utility of player with respect to the given vector . Based on the definition of the game, we say that a vector of actions is a Nash equilibrium if and only if for all . In this case, is said to be a best response action to with respect to the utility function . For the rest of this paper, whenever we say INDG, by default we mean an interconnection network design game with distance utilities.

Remark 1

The benefit function can be chosen to capture how quickly (in terms of number of hops) node needs to communicate with its interdependent nodes in . For example, consider again the cyber-physical system abstraction described in Section I, where the nodes in are power substations and nodes in are sensors that measure certain quantities of interest. If substation is able to tolerate a routing delay of up to hops from each of the sensors it depends on, but higher routing delays are useless, then the associated benefit function can be chosen as , and for . Alternatively, if node is able to tolerate any routing delay, but would prefer shorter delays, can be chosen to be an appropriate strictly decreasing function. Our formulation allows different nodes to have different benefit functions and edge costs, encoding heterogeneity in the players of the game.

V Characteristics of the Best Responses

In this section, we characterize some important properties of the best response actions for the players. We start by determining the complexity of finding a best response action for the players in the INDG.

V-a Complexity

In order to characterize the complexity of finding the best response actions for the papers, we first formulate the decision problem corresponding to optimizing the utility (2), as follows.222Decision problems are those with “yes” or “no” answers, and form the basis of the complexity classes P and NP. Since optimization problems can be solved by repeatedly solving a corresponding decision problem (e.g., by determining whether there is a solution that provides a utility larger than a certain threshold), showing that the decision problem is NP-hard is sufficient to show NP-hardness of the optimization problem. We refer to standard textbooks such as [35] for more details and background on complexity theory.

Definition 3

Best Response Interconnection (BRI).
INSTANCE: A given instance

of INDG, a player , a joint strategy by all other players and a threshold .

QUESTION: Does there exist an action for the player such that

where ?

We now provide the following theorem showing that finding a best response for the players, given arbitrary networks , and arbitrary non-increasing benefit functions and edge costs for the players, is impossible in polynomial-time (unless P = NP).

Theorem 1

The Best Response Interconnection problem is NP-hard.

To prove this theorem, we provide a reduction from the NP-complete Dominating Set Problem [35]. A dominating set of the network is a subset such that for all , has a neighbor in the set .

Definition 4

Dominating Set Problem.
INSTANCE: Network and positive integer .
QUESTION: Does the network have a dominating set with ?

We are now in place to prove Theorem 1.

Proof:

Given an instance of the dominating set problem with and , define an instance of the BRI problem with and as follows

(3)

For example and for all satisfies the above conditions. In the above instance of the BRI, there is only one node in (with associated player ), and this player is fully dependent on all nodes in (i.e., we have in (2)). Hence, the BRI problem is to determine whether has an action such that .

The above instance of the BRI problem can be constructed in polynomial time. In the rest of the proof, we show that the answer to the above instance of the BRI problem is “yes” if and only if the answer to the given instance of the Dominating Set Problem is “yes”.

Suppose that the graph has a dominating set with and thus the answer to the given instance of the Dominating set problem is “yes”. Then by defining , the distance between node and any node in is at most 2. Since ,

Therefore, the answer to the constructed instance of the BRI problem in (3) is “yes” as well.

Next suppose that the answer to the defined instance of BRI in (3) is “yes”, i.e., there exists a such that . If there is a node such that , we can add the edge to ; this would increase the benefit of the network by at least and incur a cost of . Since , this would increase the utility of . Thus without loss of generality we can take the distance between node and any node in to be at most 2 under the constructed edge set .

Consider the set of nodes that are incident to at least one edge in , i.e., . All of the nodes in are connected to at least one of the nodes in due to the assumption that the distance between any node in and node is at most 2. Thus is a dominating set of the network . On the other hand, the assumption that yields

Since , we must have that . Hence, . This means that network has a dominating set of size less than . Thus the answer to the given instance of the Dominating Set Problem is “yes”. \qed

Given that BRI is a NP-hard problem, finding best response actions in the INDG with distance utilities is nontrivial in general. In the next section, we provide some properties of the best response actions that will be helpful in characterizing the best responses of the players in certain cases.

V-B Properties of the Best Response

Lemma 1

Let be a best response to in the INDG

Then we have that

  1. .

  2. If , then if and only if .

Proof:

Let . We use contradiction to prove the first statement. Assume that , then

where . Thus is not a best response to which is a contradiction to the assumption of the lemma.

To prove the second statement, note that if , then . Thus we only have to show that when , if , then . Assume by way of contradiction that there exists such that . This means that and thus

where, again, . This is a contradiction and thus we must have . We also know that and therefore, have the required result. \qed

The next lemma characterizes a best response action of the players when the cost of constructing edges is less than a certain threshold. The proof follows the same reasoning as the proof in [23] for the formation of (single) networks under low edge costs.

Lemma 2

Let be a best response to in the INDG

If , then . Furthermore, if , then is a best response action for player .

Proof:

Suppose that and . Then where . Adding the edge to increases the utility of by at least which contradicts the assumption that is a best response and thus . Hence . By Lemma 1, we know that and therefore, .

For the case that , note that adding the edge to does not decrease the utility of and thus as in the above argument, is a best response action for . \qed

The next result gives an upper-bound on the maximum number of edges that a player with will form in a Nash equilibrium.

Lemma 3

Let be a best response to in the INDG

If , then , where denotes the smallest dominating set of the network .

Proof:

If , we have the result by the first statement of Lemma 1. Thus consider the case that . Assume by way of contradiction that . Let . Then

where . Thus produces more utility than for player which is a contradiction to the assumption that is a best response to . \qed

We will apply Lemma 3 later in Section VI to determine a Nash equilibrium instance of the INDG when has a star subgraph. The next lemma provides a threshold on the edge costs of the players in order for them to have nonempty actions.

Lemma 4

Let be a best response to in the INDG

If , then , i.e., it is not beneficial for the player to construct any edges incident to its associated node .

Proof:

Assume by way of contradiction that . Given , we have

where in the above, we are using the fact that by the assumption of the lemma. Therefore, we must have that which yields the required result. \qed

In the next result, we propose a condition under which a player disregards the network constructed by another player when considering the best response. We define the -radius of a player with as the minimum integer (or ) such that .

Lemma 5

Consider two players with R-radii and , respectively. For a given instance of INDG

assume that and are best response actions to and , respectively. If , then the actions of the players and are such that shortest paths from nodes and to the nodes that they depend on in are node disjoint in .

Proof:

The idea behind the proof stems from the fact that for any two nodes with , there does not exist any node that simultaneously has distance less than to and less than to . To formally prove the lemma, consider . By way of contradiction, assume that the shortest paths from to and to intersect at a node . Without loss of generality, let . This means that where . Now consider as a modified action of player . This new action will increase the utility of player by at least , which is a contradiction to the assumption that is a best response to . \qed

The following example illustrates the application of Lemma 5 in determining a Nash equilibrium of the INDG.

Example 1

Consider networks and depicted in Fig. 1 with the given dependency network between them (shown by dashed edges). Assume that for which yields . Nodes and correspond to the players and , respectively. Note that since all of the other nodes have , their associated players do not construct any edges in any Nash equilibrium by Lemma 1. Both and are dependent on all nodes in , as illustrated for in Fig. 1.

(a)

(b)

(c)

(d)
Fig. 1: (a) Networks and with interdependencies shown by dashed edges. (b) Interdependencies of player with nodes in . (c) Best response action of (d) A Nash equilibrium instance.

The distance between nodes and in is and thus the networks constructed by players and will be such that the shortest paths from to the nodes in are node disjoint (in ) from the shortest paths from to the nodes in , by Lemma 5. Fig. 1 demonstrates a best response for player . Using the optimal action of and Lemma 5, we can determine a Nash equilibrium as shown in Fig. 1.

Vi Nash Equilibrium of INDG for Networks Containing Star Subgraphs

With our results on best responses in hand, we now turn our attention to proving the existence of a Nash equilibrium. While it is challenging to show this for general and , we will prove that the INDG always has a Nash equilibrium when contains a star subgraph,333Such networks can be used to represent, for example, sensor networks that have a fusion center, or transportation networks that have a “hub-and-spoke” structure [33, 32, 31, 6]. and is the complete bipartite network, i.e., . We allow to be arbitrary. Without loss of generality, let be a hub node in , i.e., . As we illustrate later, the presence of heterogeneous players (captured by individual benefit functions and edge costs) along with the arbitrary structure of leads to non-trivial interconnection networks in equilibrium, even under the above assumptions on and .

To develop our results, we partition the set of players into two sets: high-cost players

(4)

and low-cost players

(5)

Recall that we assumed and . For the rest of this section, we denote the number of players by and the number of nodes in by .

We begin our analysis in this section with the following useful corollary of Lemma 2, which determines a best response action for the low-cost players.

Corollary 1

Assume that . Then