On the Convergence of Message Passing Computation of Harmonic Influence in Social Networks

Abstract

International audienceThe harmonic influence is a measure of node influence in social networks that quantifies the ability of a leader node to alter the average opinion of the network, acting against an adversary field node. The definition of harmonic influence assumes linear interactions between the nodes described by an undirected weighted graph; its computation is equivalent to solve a discrete Dirichlet problem associated to a grounded Laplacian for every node. This measure has been recently studied, under slightly more restrictive assumptions, by Vassio et al., IEEE Trans. Control Netw. Syst., 2014, who proposed a distributed message passing algorithm that concurrently computes the harmonic influence of all nodes. In this paper, we provide a convergence analysis for this algorithm, which largely extends upon previous results: we prove that the algorithm converges asymptotically, under the only assumption of the interaction Laplacian being symmetric. However, the convergence value does not in general coincide with the harmonic influence: by simulations, we show that when the network has a larger number of cycles, the algorithm becomes slower and less accurate, but nevertheless provides a useful approximation. Simulations also indicate that the symmetry condition is not necessary for convergence and that performance scales very well in the number of nodes of the graph

Effects of Network Communities and Topology Changes in Message-Passing Computation of Harmonic Influence in Social Networks

“…It was introduced in Vassio et al (2014) and implicitly used in Acemoglu et al (2013); Yildiz et al (2013). We recall its equivalent definition given by Rossi and Frasca (2018). Consider a simple weighted graph 1 G = (I, E, C) with node set I = {f, 1, 2, .…”

“…We may interpret the dynamics that follows the change as an MPA running on the new network, but starting from with a different initialization. The result in Rossi and Frasca (2018) does not guarantee the convergence, because its proof uses a monotonicity property of the dynamics of W i→j (t) that is only valid for the standard initialization. Moreover, it is not clear whether the MPA would carry some memory of the initial network.…”

Control of Average and Deviation in Large-Scale Linear Networks

“…all of its individual nodes. A natural choice for such aggregate function is the average of the node states, which has indeed been defined as a control objective in some prior work that was motivated by opinion dynamics in social networks [12], [13]. More broadly, the control of a generic output of a largescale network was studied in [14], [15], [16], [17], [18]: the energy required to control aggregated outputs instead of all states is much less.…”

Graph structure based Heuristics for Optimal Targeting in Social Networks

“…Therefore, this kind of greedy approach typically results in O(M N ) cost, with the guarantee of a bounded error. Other heuristic approaches may achieve O(M ) cost, though without bounded-error guarantees [10,11].…”