Shifting Opinions in a Social Network Through Leader Selection
aa r X i v : . [ c s . S I] M a y Shifting Opinions in a Social Network ThroughLeader Selection
Yuhao Yi, Timothy Castiglia, Stacy Patterson
Abstract —We study the French-DeGroot opinion dynamics ina social network with two polarizing parties. We consider anetwork in which the leaders of one party are given, and we posethe problem of selecting the leader set of the opposing party so asto shift the average opinion to a desired value. When each partyhas only one leader, we express the average opinion in terms of thetransition matrix and the stationary distribution of random walksin the network. The analysis shows balance of influence betweenthe two leader nodes. We show that the problem of selectingat most k absolute leaders to shift the average opinion is NP -hard. Then, we reduce the problem to a problem of submodularmaximization with a submodular knapsack constraint and anadditional cardinality constraint and propose a greedy algorithmwith upper bound search to approximate the optimum solution.We also conduct experiments in random networks and real-worldnetworks to show the effectiveness of the algorithm. Index Terms —Social Network, French-DeGroot model, Bal-ance of Opinions, Optimization, Approximation Algorithm.
I. I
NTRODUCTION
Social networks have become increasingly influential inshaping public opinions. Within this field, the problem ofdesigning mechanisms to effectively shift opinions in a socialnetwork has received great interest in last two decades [1], [2],[3], [4], [5], [6], [7], [8]. Much of the existing work studies theproblem of choosing individuals in the network to be opinionleaders so as to maximize the influence of a particular opinion,for example, to shift the average opinion of the network to anextreme opinion. However, fine-grained optimization of theaverage opinion has not been well studied.In this paper, we study the problem of shifting the averageopinion of a network to a given value, which generalizesthe intensely studied influence maximization problem. Weconsider a continuous-time French-DeGroot opinion modelwith two polarizing parties. The French-DeGroot model [9],[10] is one of the most popular models for opinion dynamics.In the model, the social network is represented by a graph,with nodes corresponding to individuals. Each node has areal scalar-valued state that represents the individual’s opinion.Each node updates its state continuously by comparing its stateand the states of its neighbors. We consider a variation on thismodel where the nodes consists of leaders nodes, defined asthe nodes with external reference values, and follower nodes,defined as those without external information.We assume that there are two opposing sources of opinion, and . These could represent, for example, support for Y. Yi, T. Castiglia, and S. Patterson are with the Department of ComputerScience, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA (email:[email protected], [email protected], [email protected]).
Party A (1) or Party B (0), or these sources could representsupport and opposition to an event or outcome. In the modelwe adopt, all members of the social network take opinionvalues in the interval [0 , . A firm supporter of party A(or pro-event individual) has an opinion close to , and aunquestioned supporter of Party B (or anti-event individual)holds an opinion close to . Individuals with opinion areconsidered as completely neutral.Each party controls a set of nodes as their opinion leaders.The leader nodes can be fully or partially controlled by eachparty. If a leader node is fully controlled, its opinion is set tothe constant opinion value of that party and never changesover time. We call these leaders absolute leaders and callsuch a system an absolute leader system . If a leader nodeis partially controlled, it receives a constant input from thecorresponding party as a reference value, and it adjusts its stateaccording to the reference value and the states of its neighbors.One can think of these leaders as being influenced through arelationship with an individual that is a direct source of theopinion but is not part of the network, i.e., an external partyleader. We define such partially controlled nodes as influencedleaders , and we call this kind of system an influenced leadersystems .We consider two leader selection problems, one for eachtype of system. In both cases, we assume that the networkalready has a leader set for party 0, and our goal is to identifythe leader set for party 1 so as to shift the average opinionof the social network towards a target value. In the absoluteleader system , this translates to selecting individuals in thenetwork to act as absolute leaders , for example, by hiringthem into the party. In the influenced leader system , thisleader selection translates to forming relationships betweenthe identified set of influenced leaders within the network andthe external party leader with opinion 1.Our problem formulation is related to the well studiedproblem of influence maximization [2], [3], [4], [7], i.e., maxi-mizing the average opinion of the network by choosing leadersfor party 1, while the leaders for party 0 are fixed. However,there are cases where maximizing the average opinion is notbeneficial. It is well known that a large group of people tendsto have polarizing opinions, and the problem of depolarizingthe opinions in a group of interacting individuals has receivedinterest in social psychology [11]. In this case, one may seekto balance the network opinion around a target value of .Moreover, the opinion of an individual relates to his or herbehaviors [12], [13]. In particular, it can be viewed as theprobability that a user adopts a behavior. From this perspective,party 1 can achieve a desired level of participation in a voluntary activity in a large network by shifting the averageopinion to a certain target value.We begin by analyzing the two proposed models, and wepropose the concept of domination score to characterize thebalance of influence between leaders of two parties. Thisanalysis relates the models to properties of random walksin a network. We also identify the optimal solution to theleader selection problem for each model when a single leaderis chosen for each party. Next, we study the general problem ofchoosing a leader set for party 1 with a given cardinality, whenthe leader set for party 0 is already identified. For absoluteleader systems, we prove the NP -hardness of the problemby a reduction from the vertex cover problem on -regulargraphs. We also show the monotonicity and submodularity ofthe average steady-state opinion as a function of the leader setof party , for both absolute and influenced leader systems.Then, we propose an algorithm for the leader selection prob-lems with provable approximation guarantees. Our algorithmfinds an appropriate upper bound for a greedy routine thatapproximately solves a submodular cost submodular knapsack(SCSK) problem with an additional cardinality constraint. weare not aware of any previous work on SCSK problems withcardinality constraints. Related work:
In the last two decades, many works con-sidered the French-DeGroot model with leaders accessingthe same reference value [14], [15], [16], [17], [18]. Insuch systems, leader selection problems have been formulatedfor different objectives such as minimizing the convergenceerror [18] or minimizing the total deviation from the referencevalue of the system in the presence of additional noise onfollowers [15], [16], [17]. These combinatorial optimizationproblems are often intractable. For example, the leader selec-tion problem proposed in [15] has been proven to be NP -hard [19]. Various approaches have been proposed to addressthese problems, including convex relaxation heuristics [16]and greedy algorithms [18], [20] with constant approximationratios.Another line of works consider leaders with different ref-erence values, in particular two group of leaders with po-larizing opinions. In this case, the steady-state opinions ofall nodes fall into the interval of leader states [21], [22].In such systems, different leader selection problems havealso been studied. [3] investigated the problem of singleleader placement to maximize its influence. The work [7]studied a problem of choosing leaders to maximize influenceof the leader set in a French-DeGroot model where leadershave specified stubbornness, and [4] investigated a similarmaximization problem. Both works proved the monotonicityand submodularity of the average opinion in a French-DeGrootopinion network with influenced leader dynamics. [2] studiedthe influence maximization problem in the Friedkin-Johnsenmodel, which is related to a French-DeGroot opinion networkwith absolute leaders in special cases but not equivalent, ingeneral. This work proved the submodularity of the averageopinion in their model as a function of leader nodes and the NP -hardness of the average opinion maximization problem.Typical greedy algorithms were applied to these problems dueto submodularity of the objective functions. In contrast, our work studies the problem of shifting the average opinion of thenetwork to any specified value. Our problem thus includes theinfluence maximization problem as a special case. In addition,we show that our problem cannot be directly treated as sub-modular maximization problem with a cardinality constraint.Thus, a more sophisticated optimization algorithm is needed. Paper outline:
The reminder of the paper is organizedas follows. In Section II, we introduce basic notations andconcepts. In Section III, we present the system model andthe problem formulation. In Section IV, we give an explicitform of the steady-state opinion vector using the Laplacianof an augmented graph, and we show how this relates to thebalance of the leader nodes’ influence in a network. We alsoprove the hardness of the investigated problem in influencedleader systems. In Section V, we propose a greedy algorithmwith an upper bound search and provide provable bounds onthe approximation ratio of the algorithm. Section VI presentsexperimental results. Finally, we conclude in Section VII.II. P
RELIMINARIES
In this section, we introduce the notation of a graph and itsmatrix representations. Further, we review the concepts of hit-ting time, commute time, resistance distance, and informationcentrality, which are used as analytical tools in this paper.
Vectors and Matrices:
We use e u to denote the u -thcanonical basis vector of R n . The vector b u,v is defined as b u,v def = e u − e v . n represents the all-one vector with length n , and n ( p × q ) represents the all-zero vector (or matrix) withlegnth n (or size p × q ). We also use these notations withoutspecifying the sizes if they are implied in context. Apart fromthese exceptions ( e u , b u,v , n and n ), a vector or matrixwith subscripts denotes the vector or submatrix with indicesspecified by the subscripts. For example, given a vector x , x i is its i -th entry, and x I is a vector consisting of entries x i forall i ∈ I . For a matrix X , X i,j is the ( i, j ) -th entry of X and X I , J is the submatrix of X consisting of the entries of X whose rows are in I and columns are in J . In addition, weuse I to denote the identity matrix, and we use X † to denotethe Moore Penrose pseudoinverse of the matrix X . Graphs and their Matrix Representations:
We denote adirected graph as G = ( V , E , w ) , where V and E are the nodeset and edge set of the graph, respectively, with |V| = n and |E| = m . An undirected graph can be viewed as asymmetrically coupled bidirectional graph in this context. Welet e = ( u, v ) ∈ E represent an edge from nodes u to node v ,and w : E → R + is the edge weight function. We denote N ↓ v as the set of in-neighbors of v ( u ∈ N ↓ v iff ( u, v ) ∈ E ), and N ↑ v as the set of out-neighbors of v ( u ∈ N ↑ v iff ( v, u ) ∈ E ).In addition, for a graph G = ( V , E , w ) , and a subset ofnodes V ⊆ V , we denote the subgraph supported on V as G [ V ] = ( V, E, ω ) , where E = { e = ( u, v ) ∈ E : u, v ∈ V } and ω ( e ) = w ( e ) for all e ∈ E . Further, we define the plus op-eration on graphs as follows. For two graphs G = ( V , E , w ) and G = ( V , E , w ) , let H = ( U , M , ω ) = G + G be a new graph with U = V ∪ V , M = E ∪ E , and ω : M → R + the new edge weight function defined as ω ( e ) = w ( e ) if e ∈ ( E \E ) , ω ( e ) = w ( e ) if e ∈ ( E \E ) ,and ω ( e ) = w ( e ) + w ( e ) if e ∈ ( E ∩ E ) . I et al. : SHIFTING THE OPINIONS IN A SOCIAL NETWORK THROUGH LEADER SELECTION 3 The weighted Laplacian matrix of a graph is defined as L def = D − A , where A is the adjacency matrix with A u,v = w ( e ) for e = ( u, v ) ∈ E and A u,v = 0 for ( u, v ) / ∈ E , and D is the out-degree diagonal matrix, where D u,u = P v A u,v and D u,v = 0 if u = v . From the definition, it is clear that L = P ( u,v ) ∈E w ( u, v ) b u,v e ⊤ u .For a matrix (vector, scalar) associated with a graph, wesometimes use a superscript to explicitly show that it corre-sponds to the graph. For example, L G is the Laplacian matrixof graph G . Random walks on graphs:
We define W def = A ⊤ D − as the random walk transition matrix of graph G . A randomwalker has a probability W i,j = A j,i D j,j to transition from vertex j to vertex i . When the graph is strongly connected there existsa positive vector (unique up to scaling) such that π = W π .When the vector is scaled such that P v π v = 1 , π is calledthe stationary distribution of the random walk defined by W .We define Π as a diagonal matrix in which Π v,v def = π v forall vertex v ∈ V . We note that L = D ( I − W ⊤ ) and L = .In a connected graph G , the hitting time from vertex u to v is the expected number of steps that a random walker, startingfrom vertex v , takes until it hits u for the first time. We denoteby H u,v the hitting time from u to v . Lemma II.1 (Hitting time [23], [24]) . In a strongly connectedgraph G , H G u,v = b ⊤ u,v ( I − ( W ) ⊤ ) † Π − ( π − e v ) . If G is an undirected graph, H G u,v = 2 m · b ⊤ u,v L † ( π − e v ) . The commute time C u,v is defined as C u,v def = H u,v + H v,u . Lemma II.2 (Commute time [23], [24]) . In a strongly con-nected graph G , C G u,v = b ⊤ u,v ( I − ( W ) ⊤ ) † Π − b u,v . If G is an undirected graph, C G u,v = 2 m · b ⊤ u,v L † b u,v .Effective Resistance and Information Centrality: Givenan undirected graph G , we define an electrical network G . In G , every edge e of G is replaced by a resistor of resistance /w ( e ) , and the resistors are connected if the edges areincident. Then, the effective resistance between node u and v in graph G (or electrical graph G ) is defined as the voltagedifference between vertices u and v in G when unit current isinjected from u and extracted from v . We recall the followinglemma relating to effective resistance. Lemma II.3 (Effective Resistance [25]) . In a connectedundirected electrical network defined by G = ( V , E , w ) , theeffective resistance between nodes u and v is R G u,v = ( L † ) v,v − L † ) v,u + ( L † ) u,u . We further recall the related definition of information cen-trality.
Definition II.4 (Information Centrality [26]) . In a connectedundirected graph G = ( V , E , w ) , the information centrality ofa vertex u is defined by θ G ( u ) = n P v ∈V R G u,v . From Lemma II.3 we obtain X u ∈V R G u,v = n · (cid:16) L † (cid:17) v,v + Tr (cid:16) L † (cid:17) . III. P
ROBLEM F ORMULATION
We consider a directed strongly connected graph G = ( V , E , w ) . Nodes represent individuals in the socialnetwork, and an edge ( u, v ) ∈ E models a social link fromnode u to node v , indicating that node u follows node v , ornode v exerts influence on node u . Edge weights represent thestrengths of the social links. Each node v has a scalar-valuedstate x v ∈ R , which represents its opinion. The node set canbe divided into a leader set S and a follower set F . The set S can be further divided into two disjoint sets S and S ,which are sets of nodes controlled by two parties, namelyparty and party . All nodes in S have access to referencevalue , and all nodes in S have access to reference value .Nodes in F update their states according to a diffusion law. A. System Dynamics
We consider the French-DeGroot opinion model with ab-solute leaders and a variation of this model with influencedleaders that are connected to external absolute sources ofinformation. The two models differ in how the leaders usetheir reference values.In the absolute leader system, leaders initialize their stateswith (for v ∈ S ) or (for v ∈ S ), and their states remainunchanged over time. The dynamics of a leader node v ischaracterized by ˙ x v ( t ) = 0 . A follower node v begins withan arbitrary initial state x v (0) = x v , and it updates its stateby the dynamics ˙ x v ( t ) = − X u ∈N ↑ v w ( v, u ) ( x v ( t ) − x u ( t )) . We partition the state vector x as x = (cid:0) x ⊤ S x ⊤ F (cid:1) ⊤ where x S is associated with the leaders and x F is associatedwith the followers. Similarly, we partition the Laplacian matrix L G and adjacency matrix A into blocks as L G = (cid:18) L S,S L S,F L F,S L F,F (cid:19) . Then, the dynamics of the leaders and the followers can bewritten as ˙ x S ( t ) = (1) ˙ x F ( t ) = − L F,F x F − L F,S x S . (2)In the system described by (1) and (2), the steady-statevalues of the leader nodes are ˆ x S = x S (3) for v ∈ S . Since − L F,F is Hurwitz for a non-empty leaderset S [27], the system converges to a single stable steady-state [28]. Letting ˙ x F ( t ) = , we obtain the steady-state ofthe followers ˆ x F = − ( L F,F ) − L F,S ˆ x S . (4)We note that L F,S ˆ x S can be viewed as the sum of columnsof L F,S that correspond to -leaders (columns of -leaders areweighted by ).In the influenced leader system, disjoint subsets of nodes S and S are influenced by two external party leaders withopinions 0 and 1, respectively. These external nodes are notpart of the graph G , and further, they do not change theiropinions. Each of the influenced leaders in S ∪ S updates itsstate according to its current state, the states of its neighbors,and the reference value from its external leader, 0 for nodesin S and 1 for nodes in S .The system can start from any initial state and the dynamicsis given by ˙ x v = − X u ∈N ↑ v w ( v, u )( x v ( t ) − x u ( t )) + κ v (0 − x v ( t )) , v ∈ S , ˙ x v = − X u ∈N ↑ v w ( v, u )( x v ( t ) − x u ( t )) + κ v (1 − x v ( t )) , v ∈ S , ˙ x v = − X u ∈N ↑ v w ( v, u )( x v ( t ) − x u ( t )) , v ∈ F . where the value κ v is the weight that the influenced leader putson its reference value. We also refer to it as the stubbornness of the node. The dynamics can be expressed more compactlyas ˙ x = − (cid:16) L G + E S K (cid:17) x + E S K , (5)where E S is the diagonal matrix with E Sv,v = 1 for v ∈ S and E Sv,u = 0 otherwise; E S is defined similarly with non-zeroentries for v ∈ S . The matrix K is diagonal with K v,v = κ v ,the stubbornness of vertex v if chosen as an influenced node.For system (5), − ( L G + E S K ) is Hurwitz for a non-emptyleader set S , so the system converges to a single steady-state.We let ˙ x ( t ) = 0 and obtain the steady-state values of all nodes ˆ x = (cid:16) L G + E S K (cid:17) − E S K . (6)In this paper, we study the average opinion of all nodes in thenetwork. Definition III.1.
In both the absolute and influenced leadersystems, given the leader set S , the average opinion µ of anetwork as a function of leader set S is defined as µ ( S ) def = 1 n X v ∈V ˆ x v . (7)Besides the above definition, µ ( S ) has an interesting inter-pretation in an opinion-behavior model based on the French-DeGroot model. We can model the opinion-behavior linkagein the system by treating ˆ x v as the success probability ofa Bernoulli random variable X v of taking the value . Inthe social network, X v = 1 indicates the event that node (individual) v takes an action, and X v = 0 indicates theevent that v does not take an action. We recall that n = |V| for both the absolute and influenced leader systems. We thenassume X , X , . . . , X v , . . . , X n to be n mutually indepen-dent Bernoulli random variables associated with correspondingnodes in the network. In particular X v is defined byPr ( X v = 1) = ˆ x v , Pr ( X v = 0) = 1 − ˆ x v , for all v ∈ V , and therefore E [ X v ] = ˆ x v .We are interested in the fraction of nodes that take an action.We define the random variable X := n P v X v . Since X v areindependent bounded random variables, X concentrates at µ ( S ) = 1 n X v ˆ x v . (8)According to the Hoeffding’s inequality,Pr | X − µ ( S ) | ≥ r ln nn ! ≤ n , (9)which indicates that µ ( S ) determines the fraction of thepopulation that take an action in a large network, with adiminishing error bound and a diminishing probability that thisbound is violated. Therefore, a party can control the fractionof population that take part in an activity or event by shiftingthe average opinion of the network to a certain value. B. Leader Selection Problems
In a system where the set S is given, we define the problemof choosing at most k leaders for the set S , such that theaverage opinion of all nodes (including leaders and followers) µ ( S ) is closest to a given value α . Specifically, we areinterested in minimizing the following objective function, f ( P, α ) def = | µ ( P ) − α | . (10)We first formally define the problem for the absolute leadersystem. Problem 1 (Absolute Leader Selection) . In an absoluteleader system, given a strongly connected directed graph G = ( V , E , w ) , an opinion absolute leader set S = ∅ , aspecified value α ∈ [0 , , a candidate set Q ⊆ V\ S , | Q | = q ,and an integer ≤ k ≤ q , find the node set S ⊆ Q , | S | ≤ k such that S ∈ arg min P ⊆ Q, | P |≤ k f ( P, α ) . (11)We define a similar problem for the influenced leadersystem. Problem 2 (Influenced Leader Selection) . In an influencedleader system, given a strongly connected directed graph G = ( V , E , w ) , an opinion leader set S = ∅ , a stubbornnessfunction of leader nodes κ : S → R + , a specified value α ∈ [0 , , a candidate set Q ⊆ V\ S , | Q | = q , anotherstubbornness function κ : Q → R + , and a integer ≤ k ≤ q ,find the node set S ⊆ Q , | S | ≤ k such that S ∈ arg min P ⊆ Q, | P |≤ k f ( P, α ) . (12) I et al. : SHIFTING THE OPINIONS IN A SOCIAL NETWORK THROUGH LEADER SELECTION 5 We note that for both Problems 1 and 2, influence maxi-mization corresponds to the degenerate case of α = 1 .IV. A NALYSIS
In this section, we give analytical solutions for Problems 1and 2 for the case where k = 1 . We also present hardnessresults for the case where k > .Our analysis utilizes a leader-equivalent graph to giveanalytical expressions for the average opinion of the network.Furthermore, for a network with a single leader for each party,we express the average opinion using the transition matrix andthe stationary distribution of random walks in the network. A. Opinions in Leader-Equivalent Systems
We note that the dynamics of both the absolute leader sys-tem and the influenced leader system can be fully characterizedby a system defined in a leader-equivalent graph . For thesetwo different kinds of systems, we construct the correspondingleader-equivalent graphs in different ways. uv ijS S G ⇒ s ′ s ′ G ′ Fig. 1: An example of constructing a leader-equivalent graphfor an absolute leader system. Nodes u and v in G become theleader s ′ in G ′ , and nodes i and j in G become the leader s ′ in G ′ . Edges without labels are weighted ; otherwise, edgesare labeled with their weights.The system described by (1) and (2) is equivalent to asystem in which all nodes in S are identified as a singleabsolute leader s ′ , and all nodes in S are identified as asingle absolute leader node s ′ . We denote the contractedgraph by G ′ = ( V ′ , E ′ , w ′ ) , where V ′ = F ∪ { s ′ } ∪ { s ′ } , E ′ = { ( u, v ) : u, v ∈ F } ∪ { ( u, s ′ ) : ( N u ∩ S ) = ∅} ∪ { ( u, s ′ ) : ( N u ∩ S ) = ∅} , and w ′ ( u, v ) = w ( u, v ) if u, v ∈ F , w ′ ( u, s ′ ) = P v ∈ ( S ∩N u ) w ( u, v ) , and w ′ ( u, s ′ ) = P v ∈ ( S ∩N u ) w ( u, v ) . In addition, we define S ′ = { s ′ , s ′ } and F ′ = V ′ \ S ′ . Note that F ′ = F in this case. Figure 1 showsan example of constructing a leader-equivalent graph for anabsolute leader system.We denote the Laplacian matrix of G ′ as L G ′ . Then thedynamics of F ′ in the system defined on the leader-equivalentgraph is expressed by ˙ x F ′ ( t ) = − L G ′ F ′ ,F ′ x F ′ − L G ′ F ′ , { s ′ } . (13)The influenced leader system described by (5) is equivalentto a system in which two virtual absolute leaders s ′ and s ′ are added to the graph, and all nodes in the original network G are treated as followers. We define the augmented graphas G ′ = ( V ′ , E ′ , w ′ ) , where V ′ = V ∪ { s ′ } ∪ { s ′ } , and E ′ = E ∪ { ( u, s ′ ) : u ∈ S } ∪ { ( u, s ′ ) : u ∈ S } ∪ { ( s ′ , u ) : u ∈ S } ∪ { ( s ′ , u ) : u ∈ S } , w ′ ( u, v ) = w ( u, v ) if ( u, v ) ∈ E , w ′ ( u, s ′ ) = w ′ ( s ′ , u ) = κ u if u ∈ S , and uv ij κ u = 1 κ v = 2 κ i = 2 κ j = 1 S S G ⇒ uv ij s ′ s ′ S S G ′ Fig. 2: An example of constructing a leader-equivalent graphfrom an influenced leader system. w ′ ( u, s ′ ) = w ′ ( s ′ , u ) = κ u if u ∈ S . We again define S ′ = { s ′ , s ′ } and F ′ = V ′ \ S ′ ; in this case F ′ = V . Figure 2shows an example of constructing a leader-equivalent graphfor an influenced leader system. With this augmented graph,the dynamics of the influenced leader system is also describedby (13).By constructing the corresponding leader-equivalent graphs,we can study both absolute and influenced leader systemsusing a unified framework. We remark that this does notmean the systems are equivalent. Choosing leaders in differentsystem models leads to different leader-equivalent graphs andhence different steady-states, although system (5) approachessystem (2) as K v,v → + ∞ for all v ∈ ( S ∪ S ) .For both the absolute and influenced leader systems, thenodes s ′ and s ′ are the only nodes that directly use referencevalues as their states in the leader-equivalent graph. Theirsteady states are x s ′ = 0 and x s ′ = 1 . The steady statesof all remaining nodes satisfy L G ′ F ′ ,F ′ ˆ x F ′ + L G ′ F ′ , { s ′ } = . (14)We note that the edges from s ′ or s ′ to other nodes are notused according to the dynamics. We deliberately add theseedges to make the graph strongly connected, which facilitatesour analysis.Let ( A G ′ ) ⊤ ( D G ′ ) − be the random walk matrix of a leader-equivalent graph G ′ . Then, we define the following matricesfor G ′ : L G ′ def = Π ( I − ( W G ′ ) ⊤ ) , (15) R G ′ def = ( I − ( W G ′ ) ⊤ ) † Π − . (16)In general ( L G ′ ) † = R G ′ , but for any p ⊥ , q ⊥ , p ⊤ ( L G ′ ) † q = p ⊤ R G ′ q . For more details we refer the readersto the full version [24] of [23]. For an undirected graph, L G ′ = m L G ′ . Proposition IV.1.
For either an absolute leader system oran influenced leader system, we consider its leader-equivalentgraph G ′ . For any node v ∈ V ′ , the steady state value ˆ x v isgiven by ˆ x v = b ⊤ v,s ′ R G ′ b s ′ ,s ′ b ⊤ s ′ ,s ′ R G ′ b s ′ ,s ′ = b ⊤ v,s ′ ( L G ′ ) † b s ′ ,s ′ b ⊤ s ′ ,s ′ ( L G ′ ) † b s ′ ,s ′ . (17) When G is an undirected graph, the expression degenerates to ˆ x v = b ⊤ v,s ′ ( L G ′ ) † b s ′ ,s ′ b ⊤ s ′ ,s ′ ( L G ′ ) † b s ′ ,s ′ . (18) The correctness of the result in Proposition IV.1 can beverified by plugging (17) into (14), and the uniqueness isguaranteed by the fact that L G ′ F ′ ,F ′ is full rank and E S K ′ is non-zero. We leave the details to Appendix B.The value of ˆ x v is, in fact, the escape probability of node v , which is defined as the probability that a random walkerstarting from vertex v , reaches node s ′ before it reaches node s ′ , We note that the expression (18) was given in [22] ina different context. [22] studied an opinion dynamics modelwhere the sum of differences between the states of a node andits neighbors is divided by the out-degree of the node before itis applied as a negative feedback to the state of the node. If theleaders take the same values, the system studied in [22] has adifferent convergence rate than the absolute leader system butshares the same steady-state values. B. Single Leader for Each Party
For absolute leader systems, if | S | = | S | = 1 , the leader-equivalent graph G ′ is the same as the original graph G . We letthe leaders in G be denoted s and s for parties with opinion and , respectively. Then, by Proposition IV.1, µ ( S ) = ( L † ) s ,s − ( L † ) s ,s ( L † ) s ,s − ( L † ) s ,s +( L † ) s ,s − ( L † ) s ,s . (19)Intuitively, we can view this expression as the influenceof node s to node s , normalized by the sum of theirmutual influence. We quantify this influence with the followingdefinition. Definition IV.2.
In a strongly connected directed graph G , the domination score of node u over v is defined as D G u,v = ( L † ) v,v − ( L † ) v,u . (20)We provide two physical interpretations for D G u,v in specialcases. The first interpretation is that in a balanced regular(directed or undirected) graph, D G u,v is the hitting time H G u,v .A larger H G u,v indicates that a random walker, starting fromnode u , spends more time in the network before it reaches v , therefore, exerting greater influence in the network. Thesecond interpretation is that in an undirected graph G and itsinduced electrical network G , m D G u,v is the average voltagevalue of all nodes in the electrical network when unit currentis injected at u and extracted from v , and v is grounded ( s has voltage ).From the definition of domination score and the expressionof commute time in Lemma II.2, we immediately obtain µ ( S ) = D G s ,s C G s ,s = D G s ,s D G s ,s + D G s ,s . (21)As for the deviation of the average opinion from the givenvalue α , we give its expression the following theorem. Theorem IV.3.
For absolute leader systems, if | S | = | S | =1 , f ( S , α ) = (cid:12)(cid:12) (1 − α ) D G s ,s − αD G s ,s (cid:12)(cid:12) D G s ,s + D G s ,s . (22) The references [23], [24] discussed the escape probability of a node in adirected graph, although these papers did not include a correct expression.
The proof of Theorem IV.3 follows directly from (19)and Definition IV.2. The numerator is the absolute value of aweighted average of D G s ,s and − D G s ,s . Therefore, TheoremIV.3 shows a weighted balance between the domination scoreof s over s and the domination score of s over s , whichdecides the deviation of the average opinion from α . TheoremIV.3 indicates that for Problem 1, if | S | = | S | = 1 ,given the leader s , it suffices to find a node s such that (1 − α ) D G s ,s = αD G s ,s to shift the average opinion to α .For influenced leader systems, the vector ˆ x is given by (6).We do not apply the leader-equivalent graph analysis in thiscase because G ′ = G . We instead interpret ˆ x using propertiesof G . Fortunately, when we choose one leader for each party, E S is a rank- matrix, and E S = E S + E S is a rank- matrix. Applying the rank- update of matrices twice leads tothe following theorem. Theorem IV.4.
For influenced leader systems, if | S | = | S | =1 , we obtain f ( S , α ) = (cid:12)(cid:12)(cid:12) (1 − α )( d s κ π s + D G s ,s ) − α ( d s κ π s + D G s ,s ) (cid:12)(cid:12)(cid:12) ( d s κ π s + D G s ,s ) + ( d s κ π s + D G s ,s ) , where the entries of the vector d are defined as d v = D v,v , ∀ v ∈ V . We defer the proof of Theorem IV.4 to Appendix C.As observed in Theorem IV.3 for absolute leader systems,for influenced leader systems, Theorem IV.4 also shows thebalancing behavior of domination scores in the social network,which decides the deviation of the average opinion from α . Inaddition, Theorem IV.4 indicates that for Problem 2, if | S | = | S | = 1 , given the leader s , it suffices to find a node s suchthat (1 − α )( d s κ s π s + D G s ,s ) = α ( d s κ s π s + D G s ,s ) to shiftthe average opinion to α . Assuming κ = κ , and they bothapproach infinity, then the condition is the same as what wehave derived in the absolute leader system.The balancing behaviors shown in Theorem IV.3 and IV.4exhibit interesting results when G is undirected and α = 1 / .In particular, Theorems IV.3 and IV.4 imply the followingcorollaries. Corollary IV.5.
For absolute leader systems, when G isundirected, α = 1 / , and | S | = | S | = 1 , f ( S , /
2) = (cid:12)(cid:12) θ G ( s ) − − θ G ( s ) − (cid:12)(cid:12) R G s ,s . (23) Corollary IV.6.
For influenced leader systems, when G isundirected, α = 1 / , and | S | = | S | = 1 , f ( S , /
2) = (cid:12)(cid:12) θ G ( s ) − + 1 /κ − θ G ( s ) − − /κ (cid:12)(cid:12) R G s ,s + 1 /κ + 1 /κ ) . (24)These corollaries show the role of information centrality ofleader nodes in an undirected network when the objective isto balance the opinions in the network. If s has the sameinformation centrality as s (assuming κ = κ for influencedleader systems), then µ ( S ) = , and so the opinion networkis balanced. If there is no such an s , then it is beneficialto find a node s such that | θ G ( s ) − θ G ( s ) | is small while R G s ,s is relatively large. I et al. : SHIFTING THE OPINIONS IN A SOCIAL NETWORK THROUGH LEADER SELECTION 7 C. Hardness of Choosing Optimal k Leaders
Next we show that Problem 1 is NP -hard. The hardness ofProblem 2 remains an open question. Theorem IV.7.
The Absolute Leader Selection problem forshifting social opinion, described in Problem 1, is NP -hard. The proof of Theorem IV.7 is given in Appendix D.We note that in both Problems 1 and 2, µ ( S ) , as a functionof S , is monotone and submodular. Theorem IV.8.
For both absolute and influenced leader sys-tems, the set function µ ( S ) is monotone and submodular. The monotonicity and submodularity of µ ( S ) for influ-enced leader systems follows in a straightforward manner fromresults in [4], [7]. We are unaware of prior analogous resultsfor absolute leader systems. We give simple proofs for bothcases in Appendix G. Our proofs are based on analyzing theescape probabilities of random walks in the network.V. A LGORITHM
In this section, we present an algorithm for selecting a setof nodes to be leaders in S , given set of leaders S , to shiftthe average opinion as close as possible to a given value α . A. Algorithm Intuition
It is well known that greedy algorithms give a (1 − /e ) ap-proximation for monotone submodular maximization problemswith cardinality constraints [31]. According to Theorem IV.8,for either Problem 1 or 2, a greedy algorithm provides a (1 − /e ) approximation for the problem when α = 1 .However, for other values of α , the problems are not trivial tosolve. We observe that if µ ( S ) ≤ α always holds, we have asubmodular maximization problem with cardinality constraint;if µ ( S ) ≥ α always holds, the problem is a submodularminimization problem with the same cardinality constraint.However, we do not know the value of µ ( S ) beforehand.Therefore, we need to design a more sophisticated algorithmto approximately solve Problem 1 and 2.The intuition behind our algorithm is to consider theseproblems as submodular cost submodular knapsack (SCSK)constraint maximization problems [33], [34]. An SCSK con-strained maximization problem is defined asmaximize f ( X ) subject to g ( X ) ≤ b. for submodular functions f and g , and upper bound b ∈ R .Problems 1 and 2 can be interpreted as special cases of SCSKwith additional cardinality constraints:maximize µ ( S ) subject to: S ⊆ Q, µ ( S ) ≤ b, | S | ≤ k. (25)Our algorithm is motivated by an approach in [34] forthe general SCSK problem. We approximate the optimum µ ( S ) for Problem 1 or 2 by imposing an upper bound forthe submodular function µ and then applying a submodularmaximization algorithm to the bounded problem. Specifically,we find an appropriate upper bound constraint µ ( S ) ≤ b , such that a greedy algorithm for maximizing µ ( S ) with upperbound b leads to an approximation algorithm for optimumsolution S ∗ , where S ∗ ∈ arg min P ⊆ Q, | P |≤ k | µ ( P ) − α | is an optimal solution for Problem 1 or 2, respectively.We apply a greedy algorithm to problem (25). For an upperbound b , the algorithm Greedy returns a solution S b . Wecan compare different upper bounds by the solutions Greedy returns. The bound b is a better upper bound than b if | µ ( S b ) − α | < | µ ( S b ) − α | . We further define the best upperbound input for algorithm Greedy as b ∗ , or formally, b ∗ ∈ arg min b ∈ [ α, | µ ( S b ) − α | . (26)We use a modified binary search to converge to the bestupper bound b ∗ for Greedy . In the next subsection, wedescribe both the bound search algorithm and the routine
Greedy . B. Bounded Search Approximation Algorithm
We first define the algorithm in terms of Problem 1. Wedescribe the changes of the algorithm in order to solveProblem 2 in the end of the subsection.
Algorithm 1: P = BoundSearch ( G , Q, α, k, δ ) P ← ∅ b min ← α ; b max ← ˆ b ← // current upper bound ˇ b ← ˆ b // current best upper bound d min ← α // minimal | µ − α | so far t ← do t ← t + 1 ( S, µ ) =
Greedy ( G , Q, ˆ b, k ) ˆ d ← | µ − α | if ˆ d < d min then P ← S ; d min ← ˆ d ; ˇ b ← ˆ b if µ > α then while µ ≤ ( b min + b max ) / do b max ← ( b min + b max ) / else b min ← ˆ b if ( α + ˆ d ) < b max then b max ← ( α + ˆ d ) ˆ b ← b max ; continue ˆ b ← ( b min + b max ) / while b max b min > exp( δ ) return P Our algorithm, BoundSearch, is given in Algorithm 1. Thealgorithm takes as input a graph G , a candidate vertex set Q , an objective opinion α , a cardinality constraint k , and a precision parameter δ for binary search. It returns a set ofnodes P , which is a subset of Q satisfying | P | ≤ k .The bound ˆ b is initialized with value , and the algorithmsearches for b ∗ in the interval [ b min , b max ] that might include abetter upper bound than ˇ b , the current best bound found by thealgorithm that leads to the smallest | µ ( S b ) − α | . We update b min and b max until b min ≈ δ b max , and b ∗ , ˇ b, ˆ b ∈ [ b min , b max ] . Weobtain ˇ b ≈ δ b ∗ . Since ˇ b is the current best upper bound foundby the algorithm, for any b / ∈ [ b min , b max ] , | S ˇ b − α | ≤ | S b − α | .Before analyzing Algorithm 1, we recall the concept of ǫ -approximation [32]: Definition V.1.
Given two numbers a, b ∈ R , a, b ≥ , if exp( − ǫ ) a ≤ b ≤ exp( ǫ ) a , then a is an ǫ -approximation of b , denoted by a ≈ ǫ b . Note that a ≈ ǫ b if and only if b ≈ ǫ a .In Algorithm 2, we present the greedy routine P = Greedy ( G , Q, ˆ b, k ) for the constrained submodular maximiza-tion described in (25). The algorithm takes as input a graph G , a candidate set Q , an SCSK upper bound ˆ b , and an integer k for the cardinality constraint. It returns a set of nodes P ,which is a subset of Q satisfying | P | ≤ k and µ ( P ) ≤ ˆ b .The algorithm chooses the node that most increases µ ( P ) without violating the upper bound from the candidate set ineach iteration, deletes it from the candidate set, and adds it tothe current leader set. Algorithm 2: ( P, µ ) =
Greedy ( G , Q, ˆ b, k ) P ← ∅ while | Q | > and | P | < k do s ← arg max u ∈ Q µ ( P ∪ { u } ) if µ ( P ∪ { s } ) ≤ ˆ b then P ← ( P ∪ { s } ) Q ← ( Q \{ s } ) γ ← µ ( P ) return ( P, γ ) To analyze Algorithm 2, we introduce the concept of theminimum cover number.
Definition V.2.
The minimum cover number k µ,b for setfunction µ ( S ) , S ⊆ Q , and b ∈ R is defined as k µ,b def = min {| S | : µ ( S ) ≥ b } , if there exists S satisfying µ ( S ) ≥ b , otherwise k µ,b def = + ∞ . Then, we obtain the approximation ratio of
BoundSearch . Theorem V.3.
Consider a graph G , a candidate set Q ,an objective α , a cardinality constraint | P | ≤ k , and aprecision parameter δ > . Let S ∗ be an optimal solutionfor Problem 1 for these parameters. The algorithm P =BoundSearch( G , Q, α, k, δ ) returns a node set P such that µ ( P ) ≈ σ µ ( S ∗ ) , in which σ = − ln(1 − ζ ) + δ , and ζ def = max(1 /e, /k µ,α ) . α Optimum DS ER Random0.25 0.249830 0.250214 0.000083 0.2487510.50 0.499699 0.500495 0.000083 0.1228480.75 0.750014 0.750014 0.000083 0.0023911.00 0.999645 0.999645 0.000083 0.576522
TABLE I: Average opinion in an absolute leader system. Thegraph is the Twitter Retweet Network rt-higgs with a fixed s and a node s chosen via various methods. α Optimum DS&K ER Random0.25 0.250010 0.250010 0.000028 0.3270450.50 0.500010 0.500010 0.000028 0.3456620.75 0.750111 0.750111 0.000028 0.3059451.00 0.997124 0.997124 0.000028 0.000000
TABLE II: Average opinion in an influenced leader system.The graph is the Twitter Retweet Network rt-higgs with afixed s and a node s chosen via various methods.We defer the proof of Theorem V.3 to Appendix E.The guarantee given in Theorem V.3 can also be written as: (1 − ζ ) e − δ µ ( S ∗ ) ≤ µ ( P ) ≤ (1 − ζ ) − e δ µ ( S ∗ ) . The
BoundSearch algorithm can be applied to Problem 2with the same approximation guarantee with the only differ-ence that the stubbornness function κ is an input of the algo-rithm. The stubbornness function is also passed into Greedy to calculate the average opinion. Theorem V.3 holds for thecorresponding algorithm P = BoundSearch( G , Q, α, k, κ, δ ) ,which calls Greedy( G , Q, ˆ b, k, κ ) . C. Complexity Analysis
A naive implementation of the proposed algorithm runsin O ( kqn log δ ) time, which is expensive for large graphs.Using blockwise inversion and rank- update of matrices wecan improve the running time of BoundSearch to O ( n log δ ) . Theorem V.4.
There exists an implementation of Algorithm 1for a graph with n nodes that has running time O ( n log δ ) . The proof of Theorem V.4 is given in Appendix F.VI. E
XPERIMENTS
In this section, we present experiments to highlight theanalytical results and to show the effectiveness of the proposedalgorithm.We first study the properties of µ ( S ) when | S | = | S | = 1 in absolute and influenced leader systems for α = 0 . , . , . , and . The leader s is chosen uniformlyat random. We run experiments on a directed and weightedsocial network. We utilize the largest strongly connectedcomponent of the Twitter Retweet Network with the keyword“higgs”, which we refer to as rt-higgs [35]. The edges areweighted by the number of retweets to a user. The networkhas , nodes and , edges.For the absolute leader system, we find the average opinionof the network for the optimal solution to Problem 1, i.e.,the optimal s as given by Theorem IV.3. We also show theaverage opinion when s is chosen using heuristics motivatedby the theorem. The first heuristic, DS, is based on the dom-ination score; we find the s such that the resulting µ ( { s } ) I et al. : SHIFTING THE OPINIONS IN A SOCIAL NETWORK THROUGH LEADER SELECTION 9 minimizes the numerator of (22). We also use a heuristicbased on effective resistance (ER); here, s is chosen so asto maximize the denominator of (22). Finally, we computethe average opinion for a randomly chosen s . The results ofthis experiment are shown in Table I,We also conduct an experiment for an influenced leadersystem using the rt-higgs network. Influenced leaders haveuniform stubbornness κ = 1 , and the other parameters are thesame as the experiment for the absolute leader system. We findthe optimal s as well as the s chosen by heuristics motivatedby the numerator (DS&K) and denominator (ER) of the resultgiven in Theorem IV.4. We note that the s that minimizesdenominator of the result in Theorem IV.3 also minimizesdenominator of the result in Theorem IV.4. The results areshown in Table II.Table I and II show that when | S | = | S | = 1 , thedomination score well captures the behavior of µ ( { s } ) . Wehave observed similar results in various Erd˝osR´enyi graphswith different choices of a single leader s . μ ( S ) α=0.25 BoundSearchα=0.25 Optimalα=0.5 BoundSearchα=0.5 Optimalα=0.75 BoundSearchα=0.75 Optimal (a) Absolute leader system. μ ( S ) α=0.25 BoundSearchα=0.25 Optimalα=0.5 BoundSearchα=0.5 Optimalα=0.75 BoundSearchα=0.75 Optimal (b) Influenced leader system. Fig. 3: Average opinion of Optimum vs. average opinionof BoundSearch in an Erd˝osR´enyi graph with nodes andconnecting probability . . S leader sets are chosen randomly,and S leader sets are chosen by brute-force search andBoundSearch with different k and α values.Next, to show the effectiveness of our leader selectionalgorithm, we compare the result returned by our algorithmBoundSearch with the optimal value returned by brute-forcesearch. We use an unweighted undirected Erd˝osR´enyi graphwith nodes and connecting probability . . We choose an S leader set of size at random. We run the BoundSearchalgorithm for both absolute leader system and influencedleader systems with α ∈ { . , . , . } . Influenced leadersuse uniform stubbornness κ = 1 . The results are shown inFigure 3. In all cases, BoundSearch returns nearly optimalresults. α BoundSearch PDS Random0.25 0.250730 0.237610 0.4281060.50 0.500975 0.565233 0.2067650.75 0.750976 0.843537 0.4959801.00 1.000000 1.000000 0.466443
TABLE III: Average opinion in an internal leader system onthe rt-higgs network with | S | = 100 and k = 100 .We next run our leader selection algorithm on rt-higgs with | S | = 100 and k = 100 . We compare the resultproduced by our algorithm BoundSearch with a heuristic wecall the Propositional Domination Score and with a randomly α BoundSearch PDS&K Random0.25 0.250732 0.252612 0.6405160.50 0.500976 0.499114 0.4180730.75 0.750976 0.748867 0.4557671.00 0.975863 0.973205 0.533567
TABLE IV: Average opinion in an external leader system onthe rt-higgs network with | S | = 100 and k = 100 .select set. To calculate the Proposition Domination Score, foreach leader in S , we choose a leader for S according toTheorem IV.3 for absolute leaders and IV.4 for influencedleaders. For all influenced leaders, κ = 1 . Tables III and IVshows that our algorithm converges to the desired value andoutperforms the heuristic in all tested cases. −4 −3 −2 −1 δ0.10.20.30.40.50.60.7 μ ( S ) α=0.2α=0.35α=0.5 (a) Internal Leader System −4 −3 −2 −1 δ0.10.20.30.40.50.60.7 μ ( S ) α=0.2α=0.35α=0.5 (b) External Leader System Fig. 4: Effect of varying δ on BoundSearch at α ∈ { . , . , . } . Experiment uses the Haggle graph.Finally, we explore the effect of varying the δ parameterin BoundSearch . We run
BoundSearch on the Haggle [36]social contact graph. The Haggle graph is a multigraph, whichwe turn it into an undirected simple graph by deleting allduplicate edges. We use the largest connected component ofthe graph which has nodes and edges. All edges haveunit edge weight. We set k = 15 and α ∈ { . , . , . } . Wevary δ from . to . . For the absolute leader system,we have | S | = 80 , and for the influenced leader system,leaders we have | S | = 15 . Influenced leaders use uniformstubbornness κ = 1 . The results are shown in Figure 4. Weobserve that as δ decreases, the results from BoundSearchconverge to a value close to α .VII. C ONCLUSION
We have studied two French-DeGroot opinion dynamicsmodels where leaders have polarizing opinions. For bothmodels, we showed expressions for the steady-state opinionusing the Laplacian matrix of a leader-equivalent graph. Forthe single leader case, we gave an explicit expression for thesteady-state opinion vector and analyzed the average opinionbased on the expression. Then, we studied the problem ofshifting the average steady-state opinion to a given value byselecting an opposing leader set with a cardinality constraint.We gave both a hardness result for this problem and an al-gorithm with provable approximation ratio. We also presentedexperiments showing that our algorithm returns results close tooptimal in practice. Future work will focus on algorithms withbetter approximation ratios and running time and the hardnessof the influenced leader selection problem. R EFERENCES[1] M. E. Yildiz, A. E. Ozdaglar, D. Acemoglu, A. Saberi, andA. Scaglione, “Binary opinion dynamics with stubborn agents,”
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A. Some Useful Matrix Identities
We introduce some matrix identities.
Lemma A.1.
For any p ⊥ , r ⊥ , p ⊤ L † r = p ⊤ R r . The proof was given in [24, Appendix C.2].
Lemma A.2. I − LL † = 1 k D − π k D − ππ ⊤ D − , I − L † L = 1 n ⊤ . Proof: I − LL † = I − I Im ( L ) = I ker( L ⊤ ) , I − L † L = I − I Im ( L ⊤ ) = I ker( L ) , which completes the proof. Lemma A.3.
For any p ⊥ , r ⊥ , p ⊤ L † D Π − r = p ⊤ L † r . Proof:
From Lemma A.1 we know that p ⊤ L † r = p ⊤ R † r = p ⊤ ( I − W ⊤ ) † Π − r . Therefore it suffices toprove p ⊤ (( D ( I − W ⊤ )) † D Π − − ( I − W ⊤ ) † Π − ) r = 0 . (27) I et al. : SHIFTING THE OPINIONS IN A SOCIAL NETWORK THROUGH LEADER SELECTION 11 Since p ⊤ (cid:16) ( D ( I − W ⊤ )) † D − ( I − W ⊤ ) † (cid:17) ( I − W ⊤ )= p ⊤ (cid:0) I Im ( I − W ) D − I Im ( I − W ) (cid:1) and p ⊥ , ∈ ker( I − W ⊤ ) , ∈ ker( D ( I − W ⊤ )) , thenwe attain p ∈ Im ( I − W ) and p ∈ Im (( I − W ) D ) , whichleads to p ⊤ (cid:16) ( D ( I − W ⊤ )) † D − ( I − W ⊤ ) † (cid:17) ( I − W ⊤ ) = 0 . Therefore (cid:0) ( D ( I − W )) † D − ( I − W ) † (cid:1) p ∈ ker( I − W ) Then we know that p ⊤ (cid:16) ( D ( I − W ⊤ )) † D − ( I − W ⊤ ) † (cid:17) Π − r = v ⊤ Π − r = r ⊤ Π − v , where v ∈ ker( I − W ) . Therefore r ⊤ Π v = r ⊤ Π − π · β , β is a scaling factor. Since r ⊤ Π − π = r ⊤ = 0 , we attain(27), which proves the lemma. Lemma A.4.
For any y ⊥ , (cid:16) Π ( I − W ⊤ ) (cid:17) (cid:16) ( I − W ⊤ ) † Π − (cid:17) y = ( I − n ⊤ ) y . Proof:
It suffices to prove that (cid:16) Π ( I − W ⊤ ) (cid:17) (cid:16) ( I − W ⊤ ) † Π − (cid:17) y = y . Since Π − y ⊥ π and ( I − W ) π = 0 , therefore Π − y ∈ Im ( I − W ⊤ ) . Then Π ( I − W ⊤ )( I − W ⊤ ) † ( Π − y ) = Π I Im ( I − W ⊤ ) ( Π − y ) = ΠΠ − y = y . B. Proof of Proposition IV.1Proof:
We can express (1) and (2) in the following form (cid:18) ˙ x S ( t )˙ x F ( t ) (cid:19) = − (cid:18) L F,S L F,F (cid:19) (cid:18) x S ( t ) x F ( t ) (cid:19) . When the equilibrium is reached, (cid:18) L F,S L F,F (cid:19) (cid:18) ˆ x S ( t )ˆ x F ( t ) (cid:19) = , Since L = D ( I − W ⊤ ) , this is equivalent to solving (cid:18) [ I − W ⊤ ] F,S [ I − W ⊤ ] F,F (cid:19) (cid:18) ˆ x S ( t )ˆ x F ( t ) (cid:19) = , When S = { s ′ } and S = { s ′ } , x S ( t ) = (1 0) ⊤ ; it sufficesto solve [ I − W ⊤ ] F,s ′ [ I − W ⊤ ] F,s ′ [ I − W ⊤ ] F,F x F ( t ) = . (28) By solving ( I − W ⊤ )( z ⊤ S z ⊤ F ) ⊤ = ( − π − s ′ π − s ′ ⊤ ) ⊤ (29)we obtain a z F that satisfies the latter n − equations in (28).We note that (29) has solutions because ( − π − s ′ π − s ′ ⊤ ) ⊤ ∈ ker( I − W ⊤ ) . Since the rank of ( I − W ⊤ ) is n − and ( I − W ⊤ ) = , for any z satisfying the system of equations (29), y = z + γ also satisfies (29), where γ can be any realnumber. We observe that z = ( I − W ⊤ ) † Π − b s ′ ,s ′ (30)is a solution of (29). This can be verified by plugging it into(29): ( I − W ⊤ )( I − W ⊤ ) † Π − b s ′ ,s ′ = Π − (cid:16) Π ( I − W ⊤ ) (cid:17) (cid:16) ( I − W ⊤ ) † Π − (cid:17) b s ′ ,s ′ = Π − ( I − n ⊤ ) b s ′ ,s ′ = Π − b s ′ ,s ′ The second equality follows from Lemma A.4. Then, wefurther set z ′ = z − z s ′ to make z ′ s ′ = 0 . Now we havefound z ′ which satisfies (28) except for the second equation.We note that by multiplying a factor β to z ′ , the other n − equations are still satisfied. So we let y ′ = ( z u − z v ) − z ′ .Then ˆ x = y ′ is the solution of (28). C. Proof of Theorem IV.4Proof:
According to the Sherman-Morrison formula, ˆ x v = e ⊤ v ( L + E s κ ) − − κ ( L + E s κ ) − E s ( L + E s κ ) − κ e ⊤ s ( L + E s κ ) − e s ! e s κ . (31)Let us then consider ( L + E s κ ) − . Since L is a singularmatrix, the Sherman-Morrison formula cannot be applied inthis case. Instead we apply the rank- update given in [37].By further applying some matrix identities discussed in Ap-pendix A, we obtain ( L + E s κ ) − = L † − q s · ( L † e s ) q ⊤ − ( e ⊤ s L † )+ (1 /κ + e ⊤ s L † e s ) · q s · q ⊤ , (32)where q = D − π . Plugging (32) into (31), we arrive at ˆ x v = κ q s + b ⊤ v,s L ⊤ D Π − b s ,s κ q s + κ q s + b ⊤ s ,s L ⊤ D Π − b s ,s . (33)We further note that for any p ⊥ , r ⊥ , p ⊤ L † D Π − r = p ⊤ L † r (see Appendix A for details). Then we obtain µ ( S ) = κ q s + e ⊤ s L † b s ,s κ q s + κ q s + b ⊤ s ,s L † b s ,s , which directly leads to the desired result. D. Proof of Theorem IV.7
Problem 3 (Vertex Cover on Regular Graphs) . Given anundirected connected -regular graph G = ( V, E ) and aninteger k , decide whether there is a vertex set S ⊆ V , suchthat | S | ≤ k and | S | is a vertex cover of graph G . We give a decision version of Problem 1 as follows. Problem 4 (Absolute Leader Selection Decision Problem) . Inan absolute leader system, given a strongly connected directedgraph G = ( V , E , w ) , an opinion leader set S = ∅ , two realnumbers α, β ∈ [0 , , a candidate set Q ⊆ V\ S , | Q | = q ,and an integer ≤ k ≤ q , decide whether there is a leaderset S ⊆ Q with opinion with at most k nodes, such that theaverage opinion of all nodes (including leaders and followers) µ ( S ) = n P v ˆ x v satisfies f ( S , α ) = | µ ( S ) − α | ≤ β . Lemma A.5.
Given an instance of problem 4, it is NP -hard todecide if there is a set S , | S | ≤ k , such that | µ ( S ) − α | ≤ β Proof:
In this proof, we consider undirected graphs. Let F = ( V , E , w ) be a graph consisting of a star graph S n plusa -regular subgraph F [ V ] = ( V, E, ω ) supported on n − leaves of S n . Edges in S n are weighted and edges in F [ V ] are weighted . Then, we can construct an instance of Problem1 by letting S = { s } be the central node of S n , and thecandidate set Q be the node set V = V\{ s } . and k be anyinteger that satisfies ≤ k ≤ q . Completeness: If | S | = k and S is a vertex coverof the -regular graph G = F [ V ] , then we consider thesteady-state of the followers ˆ x F given by (4). In this case, ˆ x F = diag ([6 , . . . , − [3 , . . . , ⊤ = [ , . . . , ] ⊤ . Thereare n − − k follower nodes; thus, we have µ ( S ) = n (cid:0) · ( n − − k ) + k (cid:1) = n − k n . Soundness: If S is not a vertex cover of graph G , then thefollower node set is not an independent set. So, the matrix L F,F is a block diagonal matrix with each block associatedwith a connected component of graph G [ V \ S ] . Let T ⊆ V \ S , | V | ≥ be the node set of a connected component. Followingthe analysis given in the proof of [2, Theorem 4.1], we obtain ˆ x u < for any u ∈ T . Then µ ( S ) < n − k n . Next, we give a polynomial reduction form VC3 to ALSD: p : { G = ( V, E ) , k } → {G = ( V , E , w ) , Q, k, α, β } . For anygiven 3-regular graph G with n − nodes, we construct aweighted graph G = G + S n , with all edges in the originalgraph weighted and all edges in the star S n weighted . Let Q = V , k be the same integer, α be any constant t greater orequal to c = n − k n , and β = t − c . Then p ( G = ( V, E ) , k ) =( G + S n , V, k, t, t − c ) is a reduction from VC3 to ALSD.Lemma A.5 immediately implies Theorem IV.7. E. Proof of Theorem V.3Proof:
We let ˇ b be the best bound found by the algorithmwith the smallest | µ ( S ˇ b ) − α | . And, Greedy with the best upperbound b ∗ returns the result µ ( S b ∗ ) . b ∗ is given by (26).If µ ( P ∪ { s } ) ≤ α is always satisfied during the execution,then µ ( P ∪{ s } ) ≤ ˆ b is also always satisfied. Then the returned S ˇ b is the same as what we get from a greedy algorithm whichadds the element with largest marginal gain to the current setin each iteration until the cardinality constraint is violated. Wefurther define e S ∈ arg max T ⊆ Q, | T |≤ k µ ( T ) , therefore by the result in [31] we obtain µ ( e S ) ≥ µ ( S ˇ b ) ≥ (cid:18) − e (cid:19) µ ( e S ) . If α ≥ µ ( e S ) , then µ ( e S ) = µ ( S ∗ ) , we attain the guarantee µ ( S ˇ b ) ≈ γ µ ( e S ) , where e − γ = (1 − /e ) . If µ ( S ˇ b ) ≤ α ≤ µ ( e S ) , then µ ( S ∗ ) ∈ [ µ ( S ˇ b ) , µ ( e S )] , which implies µ ( S ˇ b ) ≈ γ µ ( S ∗ ) , where e − γ = (1 − /e ) .If µ ( P ∪ { s } ) ≤ α is first violated when we add the ( t + 1) th node, we define P t as the set of chosen nodesof size t in Greedy , therefore | P t | = t . We further define ρ ( s t +1 ) = µ ( P t ∪ { s t +1 } ) − µ ( P t ) . From the submodularityof µ ( S ) we know ρ ( s t +1 ) ≤ t +1 µ ( P t ∪ { s t +1 } ) holds for thegreedy algorithm. Then µ ( P t ) = µ ( P t ∪ { s t +1 } ) − ρ ( s t +1 ) ≥ tt +1 µ ( P t ∪ { s t +1 } ) ≥ k µ,α − k µ,α µ ( P t ∪ { s t +1 } ) . By letting ¯ b = µ ( P t ∪ { s t +1 } ) (then by definition ¯ b = µ ( S ¯ b ) = µ ( P t ∪ { s t +1 } ) ), we obtain µ ( S α ) ≥ (1 − k µ,α ) µ ( S ¯ b ) . Wefurther attain µ ( S ∗ ) , µ ( S b ∗ ) ∈ [ µ ( S α ) , µ ( S ¯ b )] , and b ∗ ∈ [ α, ¯ b ] .Since ˇ b, b ∗ ∈ [ b min , b max ] and b min ≈ δ b max , ˇ b ≈ δ b ∗ weobtain ˇ b ∈ [ α, e δ ¯ b ] and therefore µ ( S ˇ b ) ∈ [ µ ( S α ) , e δ µ ( S ¯ b )] ,so µ ( S ˇ b ) ≈ γ µ ( S ∗ ) , where e − γ = (1 − /k µ,α ) e − δ . F. Proof of Theorem V.4Proof:
We take the algorithm for the Absolute LeaderSelection problem as an example. In each execution of Line3 of Algorithm 2 , we need to calculate the sum of steadystates of followers given by − ⊤ ( L F,F ) − L F,S x S , for all P ∪ { u } , u ∈ Q . P and Q are the current leader set of opinion and the current candidate set. Calculating ( L F,F ) − when S = ∅ takes O ( n ) running time. L F F can be updated atiteration t + 1 by deleting the row and column associated withcandidate node u . From block matrix inversion, we obtain thatits inverse can be updated by (cid:0) L ( F ( t ) \{ u } ) , ( F ( t ) \{ u } ) (cid:1) − = (cid:0) L F ( t ) ,F ( t ) (cid:1) − − (cid:0) L F ( t ) ,F ( t ) (cid:1) − e u e ⊤ u (cid:0) L F ( t ) ,F ( t ) (cid:1) − e ⊤ u (cid:0) L F ( t ) ,F ( t ) (cid:1) − e u ! ( F ( t ) \{ u } ) , ( F ( t ) \{ u } ) . To calculate µ ( P t ) , we do not need to find (cid:0) L ( F ( t ) \{ u } ) , ( F ( t ) \{ u } ) (cid:1) − explicitly. It suffices to compare − ⊤ (cid:0) L ( F ( t ) \{ u } ) , ( F ( t ) \{ u } ) (cid:1) − L ( F ( t ) \{ u } ) , ( S ( t ) ∪{ u } ) x S ∪{ u } , for all u in the current candidate set. We note that e u ( e ⊤ u ) takes a column (row) of (cid:0) L F ( t ) ,F ( t ) (cid:1) − , and L ( F ( t ) \{ u } ) , ( S ( t ) ∪{ u } ) x S ∪{ u } is a column vector. By theassociative law, we compute the vector inner product first andfind the updated µ ( S ) for at most n candidates in O ( n ) total running time. The operations of taking the submatricesdo not change the complexity because for any candidate u ,these operations only take O ( |N ↑ u | + |N ↓ u | ) running time. So,in each execution of Line 3 of Algorithm 2, these operationscan be done in O ( m ) total running time, where m is thenumber of edges in the graph. After we find the best choice s t +1 in step t + 1 , we update ( L F,F ) − explicitly, whichtakes additional O ( n ) time. Therefore, execution of Line3 of Algorithm 2 takes O ( n ) time. By using this simple I et al. : SHIFTING THE OPINIONS IN A SOCIAL NETWORK THROUGH LEADER SELECTION 13 acceleration, the complexity of Algorithm 2 is improved to O ( n + kn ) = O ( n ) . Algorithm 1 calls Greedy O (log δ ) times until b max ≈ δ b min . Since b max − b min decreasesgeometrically in Algorithm 1, the total running time of BoundSearch is O ( n log δ ) .For the Influenced Leader Selection Problem, the the rank- update is obtained using the Sherman-Morrison formula. And,the running time of the Greedy routine is also O ( n ) by asimilar implementation. We omit the details of the analysis. G. Monotonicity and Submodularity of µ ( S ) We present simple proofs for the submodularity based onthe escape probability interpretation of ˆ x .
1) Steady-State Opinion Interpreted as Escape Probability:
The entries of ˆ x F can be interpreted as the escape probabilityof a random walker [38] in a Markov chain with absorbingstates define on graph G . Consider an absorbing Markov chain P with S ∪ S the set of absorbing states and F the set ofnon-absorbing states. Then the transition matrix has the form P ⊤ = (cid:18) I R Q (cid:19) . (34)where R = ( D F,F ) − A F,S and Q = ( D F,F ) − A F,F .Define a harmonic function y with boundary y B = ˆ x S .The interior y D is determined by (see [38], for a similarformulation for undirected graphs) y D = ( I − Q ) − Ry B . (35)Then we obtain ˆ x F = y D = − ( L F,F ) − L F,S x S . Combining with the boundary condition y B = ˆ x S , we obtain y = ˆ x . y defines the concept of escape probability explainedbelow.Let S and S be two sets of absorbing states in a Markovchain (34). We let τ G v ( S , − S ) represent the event that ina Markov chain defined by graph G , a random walker startsfrom node v , hits any state u ∈ S before it reaches anystate u ∈ S . Then ˆ x v is the probability that τ G v ( S , − S ) happens. We denote the escape probability as p G v ( S , − S ) def = Pr (cid:0) τ G v ( S , − S ) (cid:1) . This escape probability is given by theharmonic function y defined above (for example, see [38]).We have shown that y = ˆ x , so ˆ x v = p G v ( S , − S ) . Similarly,we define τ G v ( S , − S ) as the event that in the Markov chaindefined by graph G , a random walker starts from node v ,hits any state u ∈ S before it reaches any state u ∈ S ,and we also denote by p G v ( S , − S ) def = Pr (cid:0) τ G v ( S , − S ) (cid:1) the probability that event τ G v ( S , − S ) happens. Since arandom walker is either absorbed by u ∈ S or u ∈ S , p G v ( S , − S ) + p G v ( S , − S ) = 1 .
2) Internal Leader System:
In the considered leader-follower system with absolute leaders, given fixed S , µ ( S ) is defined as µ ( S ) def = 1 n X v ∈V p G v ( S , − S ) . (36) To prove that µ ( S ) is monotone and submodular, it sufficesto show that p G v ( S , − S ) is monotone and submodular for all v ∈ V . Lemma A.6.
For any S ⊆ T ⊆ V , S ⊆ V , and T ∩ S = ∅ ,for any v ∈ V p G v ( T , − S ) ≥ p G v ( S , − S ) Proof:
We first consider S (0)1 = S and S (1)1 = S ∪ { u } ,where u ∈ ( T \ S ) . For a random walker in graph G startingfrom node v , we observe that p G v ( S (1)1 , − S ) = p G v (( S ∪ { u } ) , − S )= p G v ( S , − ( S ∪ { u } )) + p G v ( { u } , − ( S ∪ S )) (37)and p G v ( S (0)1 , − S ) = p G v (( S , − S )= p G v ( S , − ( S ∪ { u } ))+ p G v ( { u } , − ( S ∪ S )) · p G u ( S , − S ) , (38)by the Markov property. Therefore p G v ( S (1)1 , − S ) − p G v ( S (0)1 , − S )= p G v ( { u } , − ( S ∪ S )) · (cid:0) − p G u ( S , − S ) (cid:1) = p G v ( { u } , − ( S ∪ S )) · p G u ( S , − S ) ≥ . (39)Similarly, by defining a sequence of S ( i )1 , i = 1 , . . . , t suchthat t = | T \ S | and S ( t )1 = T , we attain p G v ( S ( i )1 , − S ) ≥ p G v ( S ( i − , − S ) (40)holds for all i ∈ [ t ] . And this leads to the result in lemma A.6.Since p G v ( S , − S ) = 1 − p G v ( S , − S ) , we attain thefollowing corollary Corollary A.7.
For any S ⊆ V , S ⊆ T ⊆ V , and T ∩ S = ∅ p G v ( S , − T ) ≤ p G v ( S , − S ) . Lemma A.8.
For any S ⊆ T ⊆ V , S ⊆ V , T ∩ S = ∅ ,and u ∈ V\ ( T ∪ S ) , p G v ( T ∪ { u } , − S ) − p G v ( T , − S ) ≤ p G v ( S ∪ { u } , − S ) − p G v ( S , − S ) (41) Proof:
For any v ∈ V , p G v ( T ∪ { u } , − S ) − p G v ( T , − S )= p G v ( { u } , − ( T ∪ S )) · p G u ( S , − T ) (42)and p G v ( S ∪ { u } , − S ) − p G v ( S , − S )= p G v ( { u } , − ( S ∪ S )) · p G u ( S , − S ) (43)Using corollary A.7 we get the inequality in the lemma bycomparing (42) and (43).
3) External Leader System:
In the considered leader-follower system with influenced leaders, given fixed S , µ ( S ) is defined as µ ( S ) def = 1 n X v ∈V p G ′ v ( { s ′ } , −{ s ′ } ) , (44)in which p G ′ v ( { s ′ } , −{ s ′ } ) represents the probability that arandom walker in augmented graph G ′ starting from v reaches s ′ before it reaches s . To prove that µ ( S ) is monotoneand submodular, it suffices to show that p G ′ v ( { s } , −{ s } ) ismonotone and submodular for all v ∈ V . Lemma A.9.
For any S ⊆ T ⊆ V , S ⊆ V , and T ∩ S = ∅ ,we consider the augmented graph G ′ defined by G , S , and S ;and the augmented graph H ′ defined by G , S and T . Then H ′ has the same node set as G ′ , the edge set of H ′ consists ofall edges in the edge set of G ′ , and all ( u, s ′ ) , u ∈ ( T \ S ) .For any v ∈ V p H ′ v ( { s ′ } , −{ s ′ } ) ≥ p G ′ v ( { s ′ } , −{ s ′ } ) . Proof:
Let G + ( u, v ) be the graph attained by adding anedge ( u, v ) to the graph G . We start by considering G (0) = G ′ and G (1) = G ′ + ( u, s ′ ) , u ∈ ( T \ S ) . Let ξ G ′ v ( u, s ′ ) be theevent that a random walker in G ′ starting from node v passesthrough edge ( u, s ′ ) before it reaches any absorbing state,and ξ G ′ v ( u, s ′ ) be the event that a random walker does notpass through ( u, s ′ ) before reaching an absorbing state. p G (1) v ( { s ′ } , −{ s ′ } )= Pr (cid:16) τ G (1) v ( { s ′ } , −{ s ′ } ) (cid:12)(cid:12)(cid:12) ξ G (1) v ( u, s ′ ) (cid:17) Pr (cid:16) ξ G (1) v ( u, s ′ ) (cid:17) + Pr (cid:18) τ G (1) v ( { s ′ } , −{ s ′ } ) (cid:12)(cid:12)(cid:12) ξ G (1) v ( u, s ′ ) (cid:19) Pr (cid:18) ξ G (1) v ( u, s ′ ) (cid:19) = Pr (cid:16) ξ G (1) v ( u, s ′ ) (cid:17) + Pr (cid:18) τ G (1) v ( { s ′ } , −{ s ′ } ) (cid:12)(cid:12)(cid:12) ξ G (1) v ( u, s ′ ) (cid:19) · (cid:16) − Pr (cid:16) ξ G (1) v ( u, s ′ ) (cid:17)(cid:17) . We note thatPr (cid:18) τ G (1) v ( { s ′ } , −{ s ′ } ) (cid:12)(cid:12)(cid:12) ξ G (1) v ( u, s ′ ) (cid:19) = Pr (cid:16) τ G (0) v ( { s ′ } , −{ s ′ } ) (cid:17) , therefore p G (1) v ( { s ′ } , −{ s ′ } ) − p G (0) v ( { s ′ } , −{ s ′ } )= Pr (cid:16) ξ G (1) v ( u, s ′ ) (cid:17) · (cid:16) − p G (0) v ( { s ′ } , −{ s ′ } ) (cid:17) = Pr (cid:16) ξ G (1) v ( u, s ′ ) (cid:17) · p G (0) v ( { s ′ } , −{ s ′ } ) ≥ . (45)Similarly, by defining a sequence of G ( i ) , i = 1 , . . . , t suchthat t = | T \ S | , we attain G ( t ) = H ′ and p G ( i ) v ( { s ′ } , −{ s ′ } ) ≥ p G ( i − v ( { s } , −{ s } ) holds for all i ∈ [ t ] . This leads to the result in lemma A.9.Since P G ′ v ( { s ′ } , −{ s ′ } ) = 1 − P G ′ v ( { s ′ } , −{ s ′ } ) , weobtain the following corollary Corollary A.10.
For any S ⊆ V , S ⊆ T ⊆ V , and T ∩ S = ∅ , G ′ and H ′ have the same definitions as they aredefined in Lemma A.9. then p H ′ v ( { s ′ } , −{ s ′ } ) ≤ p G ′ v ( { s ′ } , −{ s ′ } ) . Lemma A.11.
For any S ⊆ T ⊆ V , S ⊆ V , T ∩ S = ∅ ,and u / ∈ ( S ∪ T ) , we consider the augmented graph G ′ defined by G , S , and S ; and the augmented graph H ′ definedby G , S and T . Then H ′ has the same node set as G ′ , theedge set of H ′ consists of all edges in the edge set of G ′ , andall ( l, s ′ ) , l ∈ ( T \ S ) . For any v ∈ V and u / ∈ ( S ∪ T ) , p H ′ +( u,s ′ ) v ( { s ′ } , −{ s ′ } ) − p H ′ v ( { s ′ } , −{ s ′ } ) ≤ p G ′ +( u,s ′ ) v ( { s ′ } , −{ s ′ } ) − p G ′ v ( { s ′ } , −{ s ′ } ) . (46) Proof:
Following similar analysis as the proof ofLemma A.9, we obtain p H ′ +( u,s ′ ) v ( { s ′ } , −{ s ′ } ) − p H ′ v ( { s ′ } , −{ s ′ } )= Pr (cid:16) ξ H ′ +( u,s ′ ) v ( u, s ′ ) (cid:17) · p H ′ v ( { s ′ } , −{ s ′ } ) (47) p G ′ +( u,s ′ ) v ( { s ′ } , −{ s ′ } ) − p G ′ v ( { s ′ } , −{ s ′ } )= Pr (cid:16) ξ G ′ +( u,s ′ ) v ( u, s ′ ) (cid:17) · p G ′ v ( { s ′ } , −{ s ′ } ) (48)Then we extend the definition of ξ G ′ v ( u, s ′ ) and denote ξ G ′ v ( U, s ′ ) as the event that a random walker in G ′ startingfrom v passes through any edge ( u, s ′ ) , u ∈ U before itreaches any absorbing state. Similarly we define ξ G ′ v ( U, s ′ ) asthe event that the random walker reaches an absorbing statewithout passing through any ( u, s ′ ) , u ∈ U .Pr (cid:16) ξ H ′ +( u,s ′ ) v ( u, s ′ ) (cid:17) = Pr (cid:16) ξ H ′ +( u,s ′ ) v ( u, s ′ ) (cid:12)(cid:12)(cid:12) ξ H ′ +( u,s ′ ) v (( T \ S ) , s ′ ) (cid:17) · Pr (cid:16) ξ H ′ +( u,s ′ ) v (( T \ S ) , s ′ ) (cid:17) + Pr (cid:16) ξ H ′ +( u,s ′ ) v ( u, s ′ ) (cid:12)(cid:12)(cid:12) ξ H ′ +( u,s ′ ) v (( T \ S ) , s ′ ) (cid:17) · Pr (cid:16) ξ H ′ +( u,s ′ ) v (( T \ S ) , s ′ ) (cid:17) (49)In addition,Pr (cid:16) ξ H ′ +( u,s ′ ) v ( u, s ′ ) (cid:12)(cid:12)(cid:12) ξ H ′ +( u,s ′ ) v (( T \ S ) , s ′ ) (cid:17) = 0 and Pr (cid:16) ξ H ′ +( u,s ′ ) v ( u, s ′ ) (cid:12)(cid:12)(cid:12) ξ H ′ +( u,s ′ ) v (( T \ S ) , s ′ ) (cid:17) = Pr (cid:16) ξ G ′ +( u,s ′ ) v ( u, s ′ ) (cid:17) , (50)then we obtainPr (cid:16) ξ H ′ +( u,s ′ ) v ( u, s ′ ) (cid:17) ≤ Pr (cid:16) ξ G ′ +( u,s ′ ) v ( u, s ′ ) (cid:17)(cid:17)