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Edited by: Carlos Gershenson, Universidad Nacional Autónoma de México, Mexico

Reviewed by: Genki Ichinose, Shizuoka University, Japan; Marcelo N. Kuperman, Bariloche Atomic Centre, Argentina; Rick Quax, University of Amsterdam, Netherlands

Specialty section: This article was submitted to Computational Intelligence, a section of the journal Frontiers in Robotics and AI

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this paper, we investigate influence maximization, or optimal opinion control, in a modified version of the two-state voter dynamics in which a native state and a controlled or influenced state are accounted for. We include agent predispositions to resist influence in the form of a probability

Processes of opinion formation play a role in a variety of real-world problems, ranging from political elections to marketing and product adoption, see Castellano et al. (

Starting with the seminal study of Kempe et al. (

Recognizing this limitation of the independent cascade model, recent work has also started to consider opinion control in dynamic models of binary opinion change, which appear more suitable to capture dynamic phenomena of opinion change if agents don’t have strong commitment to decisions. Research in this area so far has considered models based on the kinetic Ising model (Liu and Shakkottai,

In the context of opinion control in the voter model (Kuhlman et al.,

Studies like Masuda (

With the inclusion of predispositions, we aim to provide a framework that agrees with empirical evidence from recent work on social networks, in which it was observed that influence propagation follows a

Our study is organized as follows. In Section

In the following, we consider a variant of the voter model (Clifford and Sudbury, _{i}_{i}_{ij}_{ji}_{ij}_{ji}_{i}_{i}

In more detail, after random initialization of voters, the dynamics of opinions are updated as follows: (i) a focus agent _{x}_{y}

The above process allows for analytical solutions. Define _{i}_{i}

Equilibrium states can be obtained from
_{i}_{ii}_{i}

From equation (

In the following, we are interested in optimal control strategies (as quantified by ^{−α} according to the configuration model (Newman, _{x}_{y}_{x}_{y}_{x}_{y}_{x}_{y}^{4}, which makes sure no substantial improvements in control can be found any more. If not mentioned otherwise, we set

In this section, we present our main findings. We start by outlining numerical results in Section

In Figure

What are the best resource allocations? We proceed by investigating the dependence of optimal control strategies on the strength of predispositions

Examples of optimized control for a network of ^{−α} with

For a more systematic investigation, we define the average degree of a controlled node
_{k, controlled} of nodes to be controlled depending on their degrees.

The dependence of all three measures on

Dependence of optimal strategies for opinion control on the predisposition parameter

The low–high SD regime threshold depends on resource endowments. To evaluate this dependence, we have measured _{k, controlled}⟩ for various resource endowments _{k, controlled}⟩(

Dependence of the critical predisposition strength at which optimal control switches from hub control to low-degree control on _{k, controlled(q)} plots.

We also investigated dependencies of thresholds on the structure of the social network to be controlled as quantified by the degree exponent _{k, controlled}⟩(

All of the experiments conducted above have been carried out for networks with given degree heterogeneity, but without higher order correlations such as clustering or assortativity, which are typical for real-world networks (Newman,

Dependence of average controlled degree

To understand changes in optimal control strategies depending on predisposition strengths, we give an analytical argument for a star network and analyze two control scenarios: control of strength one focused at the central hub and control of strength one focused on a single peripheral node, cf. Figures

Illustration of a star network with control targeted at a central hub _{0} for the hub node, _{1} for uncontrolled periphery nodes, and _{2} for a controlled periphery node.

Our arguments below are based on applying equation (_{0} = 1 applied to the central hub, we obtain _{0} = (1 − _{1} = (1 − _{0} and thus
_{1} = (1–_{0}, _{0} = (1–^{2}(_{2} and _{2} = (1–^{2}/(^{2}(

Comparison of _{central(}_{q}_{)} with _{periphery(}_{q}_{)} reveals changes in the optimal strategy when _{central}(_{periphery}(_{periphery}/∂_{q=}_{0} = (1–2^{2})/(_{central}/∂_{q}_{=0} = (–1–4^{2})/(_{central} > _{periphery}. As for both control scenarios _{central}(_{periphery}(_{central} > _{periphery} for

Instead of a not very instructive exact calculation of the critical point _{crit} at which optimal control switches, we limit the analysis to the case of large _{crit} ≈ 1/2 for large _{central} > _{periphery} for _{central} < _{periphery} for q > 1/2 in the limit of

More importantly, calculations in this toy model illustrate why hub control weakens at large values of

Illustration of a chain network with control targeted at node 0 at the left end _{i}

To solve the above system of linear homogeneous difference equations, we use the ansatz _{i}^{i}

Solving for

We finally obtain

We observe that for _{i}

We thus see two opposing effects of hub control. On the one hand, hubs are the more difficult to control the larger their degree. On the other hand, because a hub node has more neighbors than an average node, control of hub nodes provides a controller with closer access to other nodes in the network, and this improved access can outweigh the enhanced difficulty of controlling high degree nodes for low predisposition strengths. In contrast, in high

In this paper, we have investigated the impact of predispositions to return to the uninfluenced state on opinion control in a variant of the voter model. Results have shown that predisposition strength has a strong influence on optimal control strategies, such that essentially two control regimes exist. For low predisposition strength, optimal control is found to be focused on hub nodes, whereas for large predisposition strength, optimal control should be focused on low-degree nodes. In the latter situation, controllers can only gain relatively little total influence over the system, but strategic allocation can still result in improvements of control gains of up to 40% relative to random allocation.

Through numerical simulations of the voting dynamics on scale-free networks and analytical calculations on star networks, we have established that both regimes tend to be separated by a transition, with details of the transition depending on resource endowments of the controller and the heterogeneity of the social network. Our numerical results suggest that more heterogeneous networks (i.e., scale-free networks with a smaller scaling exponent

Our main finding, i.e., the existence of regimes in which optimal control strategies should focus on low-degree nodes, differs markedly from previous investigations of the original voter model (Kuhlman et al.,

In the presented model, we have analyzed the case of binary scenarios in which nodes can either be controlled or not, but controllers cannot choose the strengths of control. An alternative scenario could be an allocation scheme in which controllers can distribute resource in such a way that some nodes are strongly influenced and others only experience a weak effect. It is thus possible that our choice of binary control could have affected the results. One could imagine that even in the low-degree regime in the binary model, optimal continuous schemes that allocate very strong control to hubs could outperform evenly balanced control that aims to influence many low-degree nodes. Investigations of the continuous scenario represent an interesting avenue for future work.

Another point worth emphasizing is that we have considered undirected networks in this study. Results in the voter dynamics may differ markedly on directed networks (Masuda,

On a more speculative note, we remark that predispositions to return to the uninfluenced state in the present model essentially introduce a degree-dependent resistance of nodes to align with the external control. A rewrite of equation (

MB designed the study, conducted and evaluated the experiments, and wrote the paper. All authors contributed to manuscript revision, read and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors acknowledge the use of the IRIDIS High Performance Computing Facility, and associated support services at the University of Southampton, in the completion of this work. This research was sponsored by the U.S. Army Research Laboratory and the U.K. Ministry of Defence under Agreement Number W911NF-16-3-0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Army Research Laboratory, the U.S. Government, the U.K. Ministry of Defence or the U.K. Government. The U.S. and U.K. Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon. VR is also partially supported by EPSRC (EP/P010164/1).

Here, we provide a more detailed derivation of equation (_{i}u_{j}_{i}

Re-arranging terms and recalling

In this section, we provide some additional detail for the calculation of equilibrium shares for star networks with central and peripheral control (cf. Section

From equation (_{1} = (1 − _{0} and inserting into equation (

We thus find the expression for _{0} given in Section _{0}.

We proceed with details of the calculation to obtain the controlled vote share for a periphery-controlled star. In this case, equation (

We immediately see _{1} = (1–_{0}. Inserting into equation (_{2} gives the expression in Section _{2}, which can be written in more convenient form using equation (

Finally, we have
_{2}, results in expression (9) in Section

In this section, we provide a more detailed argument how the voting model with predisposition can be mapped to the original voter model in the presence of two opposing zealots, where the passive zealot has a control strength, which is proportional to a node’s degree. In the following, we shall label the active controller as A and its control influence as _{i}

We can now rewrite equation (

Again rescaling time by a factor (1 +