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Influencing Opinion Dynamics in Networks with Limited Interaction

Influencing Opinion Dynamics in Networks with Limited Interaction

Opinion Dynamics has been an increasingly popular field of study in recent years with the rising usage of social media - leading to people being more connected to each other, and simultaneously becoming more accessible to an external agency aiming to influence the public opinion on a particular topic. This has become a crucial topic in modern socio-political discourse, given that social media has enormous potential to shape public opinion, which in turn significantly affects the future of various communities.

A number of agencies - including, but not limited to - political parties and company advertising teams - are constantly evolving ways to reach out to more people, and optimize their budget to convince them on a specific topic, using various influencing strategies.

The focus of this work is on designing influencing strategies to shape the collective opinion of a network of individuals. We consider two modes of opinion evolution, namely, organic evolution like in the voter model due to interactions with peers, and evolution due to external influence which attempts to move the network towards the favored opinion. Here, external influence refers to the influence by any externality that is not part of the network and whose opinion is not considered for calculating the average opinion of the network.

To elaborate further on opinion evolution - in both the models, time can be considered to be divided into slots and an individual is chosen uniformly at random in each time-slot. In the traditional voter model, the chosen individual adopts the opinion of one of her neighbours; our model considers this adoption to be probabilistic wherein the probability changes with the fraction of population in the network with opposing opinions (conformist behaviour). Under external influence, the opinion of the chosen individual is more likely to move in the direction preferred by the external agency, and in this case, the interactions with neighbours do not matter.

We define the influencing strategies using the time-slots in which the agency chooses to exert its influence; since the agency has a limited influencing budget to use, the goal is determine when to influence so as to optimize the overall opinion of the network using the finite budget.

Conventional wisdom states that the influence needs to be exerted strongly towards the end of the campaigning period, which is manifested in real-life with political parties ramping up their campaigning as the voting day comes near. This work tries to analyse if that strategy is the optimal strategy in all settings, and if not, what could be a better influencing strategy.

In each time-slot, the chosen individual collects a set of opinions from the network, using one of these two ways:

  1. Random Sampling: Here, the selected individual collects K opinions, each chosen uniformly at random from the set of M opinions, where M indicates the total number of individuals in the network. Note that there can be repetitions in the collected opinions.
  2. Graph-based local interactions: In this case, we model the social network using a connection-graph. The nodes represent the individuals, and the edges between them represent a relationship/connection between the individuals. Hence, if the underlying graph is not completely connected, then the selected individual collects opinions only from her direct neighbours in the connection-graph.

Thereafter, depending on whether external influence is present or not, and based on transition probabilities (p and q), she updates her opinion in the next time-slot.

We have assumed a conformist model of opinion updation - hence, if the fraction of population holding the Yes opinion is represented by δ and that holding the No opinion is represented by β, then the probability of a No node changing into Yes will be given by q*δ, while the probability of a Yes node changing into No will be given by p*β, while. In the presence of external influence, these probabilities will respectively change to qand p.

The primary mathematical tool employed here to study the opinion evolution is stochastic approximation, through which we are able to model the changes happening in discrete time-slots using a continuous ODE. This enables us to analyze the various influence strategies that could be used. Using monotone arguments on pairs of time-slots, we have established that the optimal influence strategies will be ones in which either all the influence budget is utilized together in the beginning of the time horizon (SF), or at the end of the time-horizon (SL) - or all the strategies will perform equally well.

More specifically, we consider the results of our analysis separately for the two previously-defined opinion-collection methods as:

  1. In the first case of random sampling of opinions, the findings are as follows:
    1. If p < q, then optimal strategy is to influence in the beginning of the time-horizon (SF).
    2. If p > q, then the optimal strategy is to influence in the end of the time-horizon (SL).
    3. If p = q, then all the strategies are equally optimal.
  2. In the second case of graph-based local interactions, the nature of the optimal influencing strategies depends on the nature of the graph, as:
    1. For popular network models like d-regular graphs, Barabási Albert graphs and Erdős–Rényi graphs, the optimal influencing strategies are identical to those as in case of random sampling of opinions. (Note: The Barabási Albert model of the graph is considered to be a close approximation of real-life social networks, since it’s based on preferential attachment theory.)
    2. In case of hub and spoke graph - where a single node, called the hub, has links to all other nodes in the network while the remaining nodes are only connected to the hub - the results do not follow the same pattern as above, possibly owing to the overarching influence of the central hub. Nevertheless, the following can be said in this case:
      1. If the hub is never chosen in the time-frame, then - if the hub has opinion Yes at t = 1, all strategies perform equally well for all values of p and q, else if the hub has opinion No at t = 1, SF is strictly sub-optimal for all values of p and q.
      2. If the hub has initial opinion as 1, and is chosen only once in the time-frame for opinion update, then any strategy that influences the hub outperforms any strategy that does not influence the hub.

To conclude, we have seen that the strategy to influence at the end of the time-horizon is not always the most optimal strategy, and strongly depends on the nature of the underlying social graph, as well as the probability of changing individuals' opinion from Yes to No and vice versa, through local interactions.

In a broader sense, the results proven in earlier work for complete interaction hold true for random limited interaction in completely connected graphs. The results also hold true in neighbouring interactions in multiple graphical structures, like d-regular graphs, Barabási Albert graphs and Erdős–Rényi graphs. The influence strategies in Hub and Spoke Model, while not in line with previous results, have been analyzed to deduce the optimal strategies in certain conditions.


Link to full paper (Accepted in MTNS 2020): https://arxiv.org/abs/2002.00664

The plots indicate the performance of various influence strategies in Barabási Albert (BA) graphs and Erdős–Rényi (ER) graphs in the possible scenarios defined by p and q values. ‘b’ indicates the fraction of time T for which the agency can exert influence, while ‘β(T)’ indicates the fraction of Yes at the end of time T. As is evident, the performance follows the trends of the three scenarios established for p and q by our results.