1 Message Passing Optimization of Harmonic Influence Centrality Luca Vassio, Fabio Fagnani, Paolo Frasca and Asuman Ozdaglar Abstract—This paper proposes a new measure of node central- and the rest are regular agents. We define the HIC of node ` ity in social networks, the Harmonic Influence Centrality, which as the asymptotic value of the average opinion when node ` emerges naturally in the study of social influence over networks. switches from a regular agent to a type one stubborn agent. Using an intuitive analogy between social and electrical networks, we introduce a distributed message passing algorithm to compute This measure hence captures the long run influence of node ` the Harmonic Influence Centrality of each node. Although its on the average opinion of the social network. design is based on theoretical results which assume the network The HIC measure, beside its natural descriptive value, is to have no cycle, the algorithm can also be successfully applied also the precise answer to the following network decision on general graphs. problem: suppose you would like to have the largest influence on long run average opinion in the network and you have the capacity to change one agent from regular to type one I. INTRODUCTION stubborn. Which agent should be picked for this conversion? A key issue in the study of networks is the identification This question has a precise answer in terms of HIC; the agent of their most important nodes: the definition of prominence with the highest HIC should be picked. is based on a suitable function of the nodes, called centrality Though the HIC measure is intuitive, its centralized compu- measure. The appropriate notion of centrality measure of a tation in a large network would be challenging in terms of its node depends on the nature of interactions among the agents informational requirements that involve the network topology situated in the network and the potential decision and control and the location of type zero stubborn agents. We propose objectives. In this paper, we define a new measure of centrality, here a decentralized algorithm whereby each agent computes which we call Harmonic Influence Centrality (HIC) and which its own HIC based on local information. The construction emerges naturally in the context of social influence. We explain of our algorithm uses a novel analogy between social and why in addition to being descriptively useful, this measure electrical networks by relating the Laplace equation resulting answers questions related to the optimal placement of different from social influence dynamics to the governing equations of agents or opinions in a network with the purpose of swaying electrical networks. Under this analogy, the asymptotic opinion average opinion. In large networks approximating real world of regular agent i can be interpreted as the voltage of node i social networks, computation of centrality measures of all when type zero stubborn agents are kept at voltage zero and nodes can be a challenging task. In this paper, we present type one agents are kept at voltage one. This interpretation a fully decentralized algorithm based on message passing for allows us to use properties of electrical circuits and provide a computing the HIC of all nodes, which converges to the correct recursive characterization of HIC in trees. Using this charac- values for trees (connected networks with no cycles), but can terization, we develop a so called message passing algorithm also be applied to general networks. for its solution, which converges after at most a number of Our model of social influence builds on recent work [1], steps equal to the diameter of the tree. The algorithm we which characterizes opinion dynamics in a network consisting propose runs in a distributed and parallel way among the of stubborn agents who hold a fixed opinion equal to zero nodes, which do not need to know the topology of the whole or one (i.e., type zero and type one stubborn agents) and network, and has a lower cost in terms of number of operations regular agents who hold an opinion xi 2 [0; 1] and update with respect to the centralized algorithm recently proposed it as a weighted average of their opinion and those of their in [15]. Although originally designed for trees, our algorithm neighbors. We consider a special case of this model where can be employed in general networks. For regular networks a fixed subset of the agents are type zero stubborn agents with unitary resistances (corresponding to all agents placing equal weights on opinions of other agents), we prove that this L. Vassio is with Dipartimento di Ingegneria Meccanica e Aerospaziale, algorithm converges (although not necessarily to the correct Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Tel: +39-339-2868999. [email protected]. HIC values). Moreover, we show through simulations that the F. Fagnani is with Dipartimento di Scienze Matematiche, Politecnico di algorithm performs well also on general networks. Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Tel: +39-011- 0907509. [email protected]. P. Frasca is with Department of Applied Mathematics, University of Twente, A. Related works Drienerlolaan 5, 7522 NB Enschede, the Netherlands. Tel: +31-53-4893406. [email protected]. In social science and network science there is a large A. Ozaglar is with LIDS, Massachusetts Institute of Technology, 77 Mas- sachusetts Avenue, Cambridge, 02139 Massachusetts. Tel: +1-617-3240058. literature on defining and computing centrality measures [6]. [email protected]. Among the most popular definitions, we mention degree 2 centrality, node betweenness, information centrality [11], and be the degree of node i. A graph in which all nodes have Bonacich centrality [3], which is related to the well-known degree d is said to be d-regular. A path in G is a sequence Google PageRank algorithm. These notions have proven useful of nodes γ = (j1; : : : js) such that fjt; jt+1g 2 E for every in a range of applications but are not universally the appropri- t = 1; : : : ; s − 1. The path γ is said to connect j1 and js. The ate concepts by any means. Our interest in opinion dynamics path γ is said to be simple if jh 6= jk for h 6= k. A graph thus motivates the choice to define a centrality measure for is connected if any pair of distinct nodes can be connected our purposes. by a path (which can be chosen to be simple). The length As opposed to centralized algorithms, the interest for dis- of the shortest path between two nodes i and j is said to tributed algorithms to compute centrality measures has risen be the distance between them, and is denoted as dst(i; j). more recently. In [7] a randomized algorithm is used to Consequently, the diameter of a connected graph is defined compute PageRank centrality. In [14] distributed algorithms to be diam(G) := maxi;j2I fdst(i; j)g: A tree is a connected are designed to compute node and edge betweenness on trees, graph such that for any pair of distinct nodes there is just but are not suitable for general graphs. one simple path connecting them. Finally, given a graph G = Message passing algorithms have been widely studied and (I;E) and a subset J ⊆ I, the subgraph induced by J is are commonly used in artificial intelligence and information defined as GjJ = (J; EjJ ) where EjJ = ffi; jg 2 E j i; j 2 theory: they have demonstrated empirical success in numerous Jg. applications including low-density parity-check codes, turbo codes, free energy approximation, and satisfiability problems II. OPINION DYNAMICS AND STUBBORN AGENT (see e.g. [10] [4] [13] [8]). In such algorithms, nodes are PLACEMENT thought as objects with computational ability which can send Consider a connected graph G = (I;E). Nodes in I will be and receive information from their neighbors. thought as agents who can exchange information through the Interesting connections between message passing algo- available edges fi; jg 2 E. Each agent i 2 I has an opinion rithms and electrical networks are discussed in [12]. Moreover, xi(t) 2 R possibly changing in time t 2 N. We assume a electrical networks have also been put in connection with splitting I = S [ R with the understanding that agents in S social networks: two notable cases are the notion of resis- are stubborn agents not changing their opinions while those in tance distance [5] and the characterization of random walk R are regular agents whose opinions modify in time according betweenness in [9]. to the consensus dynamics X x (t + 1) = Q x (t) ; 8i 2 R B. Paper outline i ij j j2I In Section II we review our model of opinion dynamics where Q ≥ 0 for all i 2 R and for all j 2 I and P Q =1 with stubborn agents and we define our problem of interest, ij j ij for all i 2 R. The scalar Q represents the relative weight that is the optimal placement of a stubborn agent. In Section III ij that agent i places on agent j’s opinion. We will assume that we review basic notions of electrical circuits and explain Q only uses the available edges in G, more precisely, our their connection with social networks. Section IV is devoted standing assumption will be that to apply this electrical analogy on tree graphs and to study the optimal placement problem in this important case. Next, Qij = 0 , fi; jg 62 E (1) in Section V we describe the message passing algorithm to compute the optimal solution on trees, while in Section VI A basic example is obtained by choosing for each regular agent i.e.