
In order to meet the needs of the human-computer interaction experience of health testing products and improve the decision-making efficiency of intelligent decision support systems, we visualized the design of health testing products. We summarized the design methods for the human-computer interaction experience of health testing products, analyzed health testing data visualization requirements in terms of thematic databases, data visualization diagrams, thematic dashboards and knowledge management systems, and introduced the general process of monitoring information visualization. The visual health testing product information display interface is designed to visualize the testing data in three aspects: information architecture, interaction mode and visual language presentation. The visual intelligent decision support system and the visual interface design are combined for the functional design of the visual intelligent decision support system. The experimental part of the study investigates the effectiveness of the visualized health testing product of the intelligent decision support system, using the questionnaire method and health data measurement method to collect results on the interactivity, convenience, health decision accuracy and product satisfaction of the health monitoring product, with the data presented as a percentage system. The experimental results show that the interactivity, convenience and health decision accuracy of the intelligent decision support visual health monitoring product are higher than those of traditional health monitoring products, with interactivity evaluation results above 85% and high satisfaction with product use, indicating that the product can provide new and innovative design ideas in home healthcare.
Citation: Yinhua Su. Visualization design of health detection products based on human-computer interaction experience in intelligent decision support systems[J]. Mathematical Biosciences and Engineering, 2023, 20(9): 16725-16743. doi: 10.3934/mbe.2023745
[1] | Clinton Innes, Razvan C. Fetecau, Ralf W. Wittenberg . Modelling heterogeneity and an open-mindedness social norm in opinion dynamics. Networks and Heterogeneous Media, 2017, 12(1): 59-92. doi: 10.3934/nhm.2017003 |
[2] | Sergei Yu. Pilyugin, Maria S. Tarasova, Aleksandr S. Tarasov, Grigorii V. Monakov . A model of voting dynamics under bounded confidence with nonstandard norming. Networks and Heterogeneous Media, 2022, 17(6): 917-931. doi: 10.3934/nhm.2022032 |
[3] | Sergei Yu. Pilyugin, M. C. Campi . Opinion formation in voting processes under bounded confidence. Networks and Heterogeneous Media, 2019, 14(3): 617-632. doi: 10.3934/nhm.2019024 |
[4] | Rainer Hegselmann, Ulrich Krause . Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model. Networks and Heterogeneous Media, 2015, 10(3): 477-509. doi: 10.3934/nhm.2015.10.477 |
[5] | Sabrina Bonandin, Mattia Zanella . Effects of heterogeneous opinion interactions in many-agent systems for epidemic dynamics. Networks and Heterogeneous Media, 2024, 19(1): 235-261. doi: 10.3934/nhm.2024011 |
[6] | Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil . Opinion Dynamics on a General Compact Riemannian Manifold. Networks and Heterogeneous Media, 2017, 12(3): 489-523. doi: 10.3934/nhm.2017021 |
[7] | Yuntian Zhang, Xiaoliang Chen, Zexia Huang, Xianyong Li, Yajun Du . Managing consensus based on community classification in opinion dynamics. Networks and Heterogeneous Media, 2023, 18(2): 813-841. doi: 10.3934/nhm.2023035 |
[8] | Marina Dolfin, Mirosław Lachowicz . Modeling opinion dynamics: How the network enhances consensus. Networks and Heterogeneous Media, 2015, 10(4): 877-896. doi: 10.3934/nhm.2015.10.877 |
[9] | Mayte Pérez-Llanos, Juan Pablo Pinasco, Nicolas Saintier . Opinion fitness and convergence to consensus in homogeneous and heterogeneous populations. Networks and Heterogeneous Media, 2021, 16(2): 257-281. doi: 10.3934/nhm.2021006 |
[10] | Michael Herty, Lorenzo Pareschi, Giuseppe Visconti . Mean field models for large data–clustering problems. Networks and Heterogeneous Media, 2020, 15(3): 463-487. doi: 10.3934/nhm.2020027 |
In order to meet the needs of the human-computer interaction experience of health testing products and improve the decision-making efficiency of intelligent decision support systems, we visualized the design of health testing products. We summarized the design methods for the human-computer interaction experience of health testing products, analyzed health testing data visualization requirements in terms of thematic databases, data visualization diagrams, thematic dashboards and knowledge management systems, and introduced the general process of monitoring information visualization. The visual health testing product information display interface is designed to visualize the testing data in three aspects: information architecture, interaction mode and visual language presentation. The visual intelligent decision support system and the visual interface design are combined for the functional design of the visual intelligent decision support system. The experimental part of the study investigates the effectiveness of the visualized health testing product of the intelligent decision support system, using the questionnaire method and health data measurement method to collect results on the interactivity, convenience, health decision accuracy and product satisfaction of the health monitoring product, with the data presented as a percentage system. The experimental results show that the interactivity, convenience and health decision accuracy of the intelligent decision support visual health monitoring product are higher than those of traditional health monitoring products, with interactivity evaluation results above 85% and high satisfaction with product use, indicating that the product can provide new and innovative design ideas in home healthcare.
The mathematical modelling of opinion formation within a group of interacting agents has recently been attracting considerable interest. Various models for opinion formation have been proposed, covering a diverse range of issues including consensus formation or the polarization of opinions [25, 36, 42], the emergence of extremism [13], the evolution of political organizations [5], and media influence [8]. Such models may be viewed as falling within the more general study of self-organized dynamics [10] and aggregation phenomena [43], which has also seen a surge of interest in the last decade: Due to inter-individual interactions among the members of a group, self-organization may occur in a physical space (insect swarms, fish schools, robots) or, more abstractly, in an opinion space.
The modelling of opinion dynamics predominantly seeks to represent different aspects of social influence [4]. This is manifested in the general tendency of most individuals to change their own opinion to align with those of their peers, and has been studied extensively since the classical work of Festinger on social comparison and cognitive dissonance [17, 18]. A standard mathematical framework for modelling opinion formation considers a number
A model for opinion formation is a set of rules for opinions
An individual's response to peer opinions is in general not homogeneous; instead, there is an enhanced tendency of individuals to interact with and be influenced by others with similar opinions (homophily). Mathematically, this can be modelled by permitting no social influence between individuals
The model considered in this paper is a time-continuous social influence model which generalizes the Motsch-Tadmor model [11, 41, 42] described in Section 2 below. The generalization is done in two stages. First, we focus on the heterogeneous response within a population [31, 33] and consider the psychological mechanism behind its occurrence. Second, we include media in the model and investigate its effects on the opinions of the population. The two generalizations are elaborated below.
We model heterogeneity by recognizing different approaches to communication, while acknowledging that in a social interaction with peers, an individual acts simultaneously as both communicator and communicatee. Among communicatees, we distinguish between open-and closed-minded individuals; while as communicators, agents are assumed to act either by encouraging open discussion or by avoiding it [51]. Open-minded communicatees who also act openly as communicators are simply referred in the paper as open-minded individuals; regardless of their opinion, they are assumed to interact with everyone else within their interaction range
The remainder of the population is assumed to interact in a cold, discussion-averse manner when acting as communicators, and also to be disposed to behave in a closed-minded manner as communicatees when their opinions become too extreme. Specifically, we introduce a critical threshold
A key focus of our work concerns the diversity and stability of equilibria obtained with the proposed heterogeneous opinion formation model. In previous homogeneous bounded confidence models [25, 12, 42], the dynamic evolution leads either to a consensus at an averaged opinion value or to several isolated opinion groups. Specifically, in a multi-cluster equilibrium, all opinions within each group asymptotically approach a common value [30], but the groups are disconnected from each other (the opinions of individuals in different clusters are separated by a distance larger than the bound of confidence
By contrast, we demonstrate that the heterogeneous model proposed in this paper, incorporating both open-and closed-minded agents, supports additional types of equilibria, in particular connected multi-cluster groups, extreme polarization and consensus at an extreme opinion. We illustrate numerically various qualitatively different equilibria that emerge from our model, and provide bifurcation studies showing their dependence on some important parameters. Most significantly, it turns out that the extreme consensus and polarization equilibria are asymptotically stable, as are the multi-cluster states provided they are suitably connected; we prove this result analytically using a graph-theoretical framework. Such stability is not often found in such aggregation or opinion dynamics models; in the present case it is due to the attraction experienced by (non-moderate) closed-minded individuals towards the extremes at
We note that various authors [20, 32, 34, 35] have recently considered heterogeneity in bounded confidence models using a different approach. The common strategy in these papers is to distinguish between open-and closed-minded agents by assigning different bounds of confidence to individuals, with the main goal of studying how heterogeneity affects the likelihood of consensus formation in society. In [35], for instance, it is noted that heterogeneity substantially enriches the complexity of the dynamics and in particular, perhaps surprisingly, enhances the chances of reaching a consensus. We point out that our approach of modelling heterogeneity is different, and also that our focus is on connected steady states and not solely on pure consensus. Note that while some partially connected equilibria have been obtained using other heterogeneous models [35], the discussion there does not dwell at all on this aspect.
The effect of attraction to (possibly) extreme views has also received attention. In one recent such study, Hegselmann and Krause [26] incorporated radical individuals or charismatic leaders into their original time-discrete model [25]. Such individuals, who by assumption maintain constant opinion values, send signals to the rest of the population (referred to as normals in [26]), signals which may or may not drive normals to their radical view. Among the differences between their model and ours is that we assume two radical opinions (placed at the extremes,
The time-discrete bounded confidence model of Deffuant et al. [12] has also been extended [13, 38] by incorporating extremists (at two extremes) as well as moderates; this generalization can also give rise to central multi-cluster and polarization states. In this model, which evolves by randomly-chosen pairwise interactions, when the bounds of confidence are fixed and sufficiently large, a statistically stationary state can arise in which the moderates' opinions fail to equilibrate [38]; this may be interpreted as an alternative form of ''connectedness'' maintained by ongoing opinion fluctuations.
Our second major extension to the Motsch-Tadmor model [42] is to incorporate the effect of media influences on the opinions in a population. It is important to take into account that, as for their reactions to peer opinions, individuals' responses to media exposure are heterogeneous. As noted in [31, 33], while some individuals become more moderate when exposed to cross-cutting media (which expresses a different partisan perspective from their own), a majority of individuals experiencing media influences with such contrary partisan biases tend either to not change their opinion, or to react by becoming more extreme in their own beliefs.
Existing mathematical models that consider the effects of media focus primarily on the interplay between consensus-seeking behaviour and the media [8, 39], and thus seem to overlook the effect of closed-mindedness or consensus-resisting behaviour in the presence of media. The model proposed in this work does consider these effects, systematically extending the above-mentioned ideas on open-and closed-mindedness to allow for media influences: We discuss the role of media in this framework and, in particular, the effect of social norms of open-mindedness in facilitating consensus or polarization. These questions are particularly intriguing since the perception of news media is often influenced by a social phenomenon known as the hostile media effect [52].
The outline of the paper is as follows. Section 2 introduces the heterogeneous model for opinion formation proposed in this paper. Section 3 focusses on the equilibria of this model, including a numerical illustration of various qualitatively different states accessible in our model, a bifurcation study with respect to several parameters, and a rigorous analysis of the stability of connected equilibria. In Section 4 we introduce media into our model, including both some numerical examples and some rigorous stability results for the extended system.
The modeling of group opinion dynamics using differential equations has grown in popularity over the last few years [42, 16, 8, 39, 43, 7, 2]. Our work is based is an ordinary differential equation (ODE) model recently proposed and investigated by Motsch and Tadmor [41, 42].
Consider
dxidt=α∑j≠iaij(xj−xi),i=1,...,N, | (1) |
where
aij=ϕ(|xj−xi|)/ϕi,with ϕi=∑j≠iϕ(|xj−xi|). | (2) |
Here,
∑j≠iaij=1,i=1,…,N, | (3) |
and that the adjacency matrix
One of the central ideas in [42] is that heterophily, the tendency to interact more strongly with those who have rather different opinions than with those holding more similar views (the opposite of homophily), may play a key role in the formation of opinion consensus and of group clusters in general. Mathematically, such tendencies in social interactions may be encapsulated in the choice of influence function
These observations concerning the effect of heterophily [42] foreshadow a general principle that consensus is more likely to occur when individuals are more open to being influenced by opinions somewhat different from their own. In the present paper we restrict our attention to non-increasing influence functions
In formulating our model, we consider different approaches to communication, recognizing that in each interaction, an individual acts simultaneously to transmit opinions and ideas as the communicator, and to receive them as the communicatee [6, 9]. As a communicatee, an individual can be open-minded, willing to listen to and be influenced by others' ideas; or on the contrary, be closed-minded and impervious to the opinions held by others. On the other hand, for communicators one can broadly distinguish between those who are warm and open in their communication, and thus more likely to be considered reliable and to influence others, versus those whose communication style is colder and more averse to open discussion [51].
In our model we consider two classes of agents. The first class consists in individuals who act warmly and openly as communicators and also behave open-mindedly as communicatees. We denote by
1With an abuse of notation we also let O and C be the sets of indices i∈{1,…,N} that correspond to such individuals. No confusion should arise from this dual interpretation.
However, when functioning as communicatees, the response of individuals in
Throughout this paper the term non-closed-minded refers both to all individuals in
The general form of the opinion model is
dxidt=fi,i=1,…,N, | (4) |
where
Model without open-mindedness social norm. In the absence of an open-mindedness social norm, we assume that closed-minded individuals are attracted to the nearest extremist views, regardless of with whom they are interacting [15, 27, 45]. The model reads:
fi={α2(N−1)ˆai(sgn(xi)−xi)if | xi | >Xc and i∈C,α1∑j≠iaij(xj−xi)otherwise. | (5) |
Here
The psychological forces shaping
One might typically expect that the rate at which a closed-minded individual
Model with open-mindedness social norm. Empirical studies [50, 49, 51, 40] have identified a strong link between open and warm behaviour and the enhancement of mutual understanding between individuals. It was found in these studies that an open and warm approach to communication induced curiosity, which in turn led to a search for more information. Participants confronted with opposing views in open, direct discussions started to doubt and feel uncertain about their own position. They showed more interest in learning, and felt motivated to ask questions and to search for more arguments [51]. The result was an enhancement of shared understanding on the issue in question. Throughout the paper we refer to this approach to communication using the notion of an open-mindedness social norm [40].
We modify our model (4)-(5) to incorporate an open-mindedness social norm by letting the extremist closed-minded individuals act in an open-minded manner, but only when interacting with open-minded individuals. In this case, the model is given by (4) with
fi={α2(N−m−1)ˆai(sgn(xi)−xi)+α3∑j∈Oˆaij(xj−xi)if|xi|>Xcandi∈C,α1∑aijj≠i(xj−xi)otherwise | (6) |
In (6), the interactions of non-closed-minded individuals with all others are as before, with interaction coefficients
The new features of this model, by comparison with (5), arise when we consider the behaviour of closed-minded agents. Thus suppose that the
If
If on the other hand
These interactions are represented schematically in Figure 1. Expressing the above description quantitatively, we thus model
ˆai=ˆϕ(|sgn(xi)−xi|)/ˆϕiandˆaij=ϕ(|xj−xi|)/ˆϕi. | (7) |
The difference between
(N−m−1)ˆai+∑j∈Oˆaij=1, | (8) |
so that the normalization factor
ˆϕi=(N−m−1)ˆϕ(|sgn(xi)−xi|)+∑j∈Oϕ(|xj−xi|). | (9) |
For simplicity we shall take
We end this section with some remarks:
Remark 2.1. Since
Remark 2.2. In the absence of open-minded individuals (
ˆai=ˆϕ(|sgn(xi)−xi|)/ˆϕiwithˆϕi=(N−1)ˆϕ(|sgn(xi)−xi|). | (10) |
This is simply
The introduction of distinct open-minded and closed-minded individuals in a Motsch-Tadmor-type opinion dynamics model considerably broadens the range of possible long-time behaviours. To illustrate the qualitatively different equilibria, we show the results of numerical simulations of model (4), with
In our computations, we have two ways of initializing the time evolution of the model. The first, referred below as a type NS (non-symmetric) initial condition, distributes the opinions
In all of our simulations we use a total population
Figure 2 shows the time evolution for various different initial data and parameters, demonstrating several qualitatively different equilibrium states.For all these plots, we have simulated the model (4) with (6), corresponding to a social norm of open-mindedness, with
As noted in the Introduction, in addition to consensus and isolated multi-cluster equilibria, our model captures connected multi-cluster states, which do not appear to have been observed with previous homogeneous bounded confidence models.
● If all individuals are open-minded (
Figure 2(a)shows one of our simulations for this case of purely open-minded individuals, confirming the emergence of neutrally stable isolated opinion clusters.
● As shown in Figure 2(b), for sufficiently large
● More interestingly, our model (4) with (6) also allows for the creation of an extremist consensus; such behaviour requires closed-minded agents and thus cannot occur in (1)-(2). As seen in Figure 2(c), convergence of all the opinions in a population to one of the extremes at
● Figure 2(d) demonstrates a common consequence of closed-mindedness, namely extreme polarization, in which the population splits into two groups each settling at an extreme opinion; this corresponding to situations containing two hostilely opposed groups who are unwilling to listen to each other. In the simulation shown in (d), there are no open-minded individuals (
● The versatility of our model becomes particularly apparent in the remaining two plots. In Figure 2(e), the equilibrium consists of seven opinion clusters distributed non-symmetrically in a connected configuration (none is isolated from the others); that is, the separation between adjacent clusters is less than
● Figure 2(f) shows another example of a connected, asymptotically stable equilibrium configuration; in this simulation, with Type S initial data, opinions are distributed symmetrically in five clusters, with the middle (
As seen in the examples shown in Figure 2, in all our simulations the initial opinion distribution converges to a stationary state; that is,
Neutrally stable clusters. For the original Motsch-Tadmor model [42], Jabin and Motsch [30] have recently made the above ideas precise; in particular, they rigorously proved such convergence to clusters with
Now it is readily seen that an attracting equilibrium consisting of such dynamically disconnected clusters is neutrally stable but cannot be asymptotically stable, due to the existence of non-decaying perturbations. Most straightforwardly, since the dynamics (1)-(2) are translation-invariant, a perturbation of each
Linearly stable clusters. By contrast, in our model the presence of closed-minded individuals (
The simplest types of such asymptotically stable equilibria are the extremist consensus and polarization states, seen in Figure 2(c) and (d), respectively. In each of these cases, all individuals in
In addition and possibly most interestingly, our model supports the existence of asymptotically stable, connected multi-cluster states, as shown in Figure 2(e) and (f). Such states can have a variety of properties: for instance, they may, but do not need to, include clusters at the extreme opinions at
The common feature of these asymptotically stable
Note that the main theorems from Section 3.4 guarantee only linear stability, that is, local stability to sufficiently small perturbations. We illustrate this numerically on the 5-cluster equilibrium solution of Figure 2(f) (which satisfies the assumptions of Theorem 3.7): We perform a time evolution of the initial data of Figure 2(f), allowing it to settle down to the 5-cluster state, which is then perturbed at time
We conclude this discussion on connected clusters by pointing out some similarities between our results and those in [35]. Unlike the approach in this paper, Lorenz modelled heterogeneity by assigning different bounds of confidence to individuals, to distinguish between open and closed-minded agents. We refer in particular to Figure 8 in [35]. In this Figure, the upper plot shows an extremist consensus (as also captured by our model; see Figure 2(c)), while the lower one displays a partially connected cluster equilibrium, in the following sense: the equilibrium configuration has three clusters, the one in the centre consisting of only open-minded agents and the two on the sides containing exclusively closed-minded individuals. The distance between the open-minded group's opinion and the two closed-minded clusters is less than the bound of confidence of the open-minded, meaning that the open-minded agents communicate with the closed-minded at equilibrium. On the other hand, neither of the two closed-minded clusters has the open-minded cluster within its own bound of confidence, so the closed-minded clusters are isolated in that sense. We call this configuration partially connected, to distinguish it from the fully connected equilibria that we find in our model. Note that Lorenz [35] does not elaborate further on this new type of equilibrium, aside from mentioning it briefly.
Explicit construction of a connected equilibrium opinion distribution. It is instructive to verify the existence of multi-cluster equilibria satisfying the hypotheses of Theorem 3.7 (with a social norm of open-mindedness) through explicit calculation. The steady state constructed below is similar to the long-time state depicted in Figure 2(f):
We postulate an equilibrium consisting of
The most extreme clusters at
Choosing
Due to symmetry, the equilibrium criteria for such a configuration reduce to just two equations, the balance conditions for the clusters at
(N−m−1)(1−z)+(m1(y−z)−m0z)=0,−m1(2y)−m0y−(N−m−q)y+q2(z−y)=0, |
which using
−m1y+(N−m1−1)z=(N−m−1), | (11) |
−(N−q2)y+q2z=0 | (12) |
It may readily be verified that for
The equilibrium configuration of opinion clusters in our model clearly depends on the various parameters as well as the initial opinion distribution. A full investigation of the parameter space and/or the dependence on initial data is well beyond the scope of the present study. However, to gain some insight into the global structure of the solutions of our model, we explore the influence of variations in the open-mindedness threshold
Parameter-dependence of equilibrium cluster count. The simplest configuration to consider is that in which all individuals are open-minded, that is,
Figure 4(b) represents the opposite extreme
An intermediate situation is shown in Figure 4(c), in which only 10\% of the population is open-minded (
As seen in the figure, there is a critical value of the interaction range (occurring at
Extremist consensus. A consensus of extremism has been observed in societies (such as Nazi Germany in the 1930s, for instance) and investigated with previous bounded confidence opinion models [13, 26, 35, 38]. In Figure 5 we explore the emergence of an extremist consensus within our model (4)-(6) for representative parameter values. Within each plot, the same non-symmetric (type NS) opinion configuration is used to initialize all the computations, and as before the long-time cluster locations are shown with their stability. Note that an unbalanced distribution of initial opinions is essential for reaching extremist consensus; as verified in the inserts of plots (a) and (c), symmetric (type S) arrangements of open-minded individuals do not lead to such equilibria.
In Figure 5(a) and(b) , extremist consensus is observed to emerge from 3-cluster equilibria through bifurcations in
Figure 5(c) shows extremist consensus emerging from a polarized equilibrium, through a bifurcation in the range
Sensitivity to the total population size
In a second set of runs, we tested the sensitivity to
Effect of a social norm of open-mindedness on consensus formation.Figure 6 demonstrates the relationship (for a specific set of parameters) between the proportion of open-minded individuals in a population and the extent to which an open-mindedness social norm enhances agreement within a population. We set
Figure 6(a) shows results for the case when no social norm of open-mindedness is present, while Figure 6(b) corresponds to the model incorporating an open-mindedness social norm. The most common equilibrium configuration in the two plots consist of three clusters, where the outside clusters contain exclusively closed-minded individuals, while the centre cluster has a mixed population. As
When no social norm of open-mindedness is present, the closed-minded adopt extreme opinions and the qualitative configuration of the three-cluster equilibrium changes very little with increasing
Figure 6(b) demonstrates how including an open-mindedness social norm has the potential to enhance agreement. We observe there that, by increasing the number of open-minded individuals, the two outside clusters drift monotonically toward the central opinion cluster, meaning that opinions in the population become less diverse. This trend is consistent for values of
In this section we provide the details and proofs for the stability results referred to in Section 3.2 above.
A key concept used in the stability analysis is the (graph) connectivity of a group of individuals
Definition 3.1.An individual
For instance, in model (6) a closed-minded individual
Remark 3.2. In graph-theoretic terminology, a population is connected according to Definition3.1 if the directed graph associated to its opinion state is strongly connected. This property can be equivalently characterized in terms of the interaction matrix. Indeed, it is a well known result [53] that a matrix is irreducible if and only if its associated directed graph is strongly connected. Consequently, a population is connected if and only if its interaction matrix is irreducible.
The following assumptions are made to prove the asymptotic stability results. First, the influence functions
Assumption 3.3 The equilibrium opinion state
1.
2.
The next two assumptions apply to model (6) only
3.
4.
Assumption 1 simply excludes the ambiguous case when an opinion of an individual in
Remark 3.4 We noted above that all coefficients
The stability analysis requires us to distinguish between the closed-minded individuals (individuals in
The stability proofs for the models without a social norm of open-mindedness, with interactions defined by (5), and with such a social norm, described by (6), need to be treated separately, due to the differences in the interaction coefficients corresponding to the
Stability for model without open-mindedness social norm. We start by investigating the linear stability of an equilibrium solution
Given that the
J1=(−α2I0BA), | (13) |
where
A=(−α1α1a∗ij[i<j]…α1a∗ij[i>j]⋱⋮⋮…−α1). | (14) |
The indices
The following classical result in linear algebra is an essential tool in the proof of our stability result:
Theorem 3.5. If a matrix is irreducible and weakly diagonally dominant, with strict diagonal dominance holding for at least one row, then it is non-singular [47, Theorem 2].
We now state and prove the stability theorem for model (5).
Theorem 3.6. Let
Proof. We have to show that the eigenvalues of
To conclude the proof we have to rule out the possibility that one (or more) of these eigenvalues is
Also by assumption, there exists at least one non-closed-minded individual connected to a closed-minded individual, which means that at least one of the entries
Stability for model with open-mindedness social norm. The study of the stability of equilibria for the model (4) with (6) follows along similar lines. Notation is as before; in particular, for an equilibrium opinion state
Now for linear stability to be possible, at least one of these
Recall that under Assumption 3.3, all coefficients
E=(−α2(N−m−1)D100E), | (15) |
with
The
E=(D2FBA), | (16) |
where
di=−α2(N−m−1)ˆa∗i−α2i∑j∈Oˆa∗ij,i=r+1,…,q. | (17) |
The matrix
F=(0|α3i(ˆa∗ij)j∈O). | (18) |
The zero block has
Finally, the matrices
The stability theorem for our model (4) with (6) now reads:
Theorem 3.7Let
Proof.We follow the strategy from the proof of Theorem 3.6. First note that the closed-minded individuals
Matrix
We rule out the possibility of
In the following we further extend the Motsch-Tadmor [42] model to incorporate the effects of media on opinion formation. Various attempts to model media-influenced opinion models have appeared in the recent literature [39, 48, 8, 2, 37, 44]. In particular, Pineda and Buendia [44] have recently considered media effects in the discrete-time bounded confidence models of Deffuant et al.[12] and Hegselmann and Krause [25]. Their approach is to assume that individuals interact with the media with a certain probability, or otherwise interact with other individuals in their confidence range; within this framework, the effectiveness of the media is studied in terms of the sizes of the bounds of confidence and the media intensity [44].
We introduce media via a natural extension of the basic model (4), modelling a media effect similarly to the inter-individual interaction term
dxidt=fi+g(xi,μ),i=1,…,N, | (19) |
where
In modelling
g(xi,μ)={α4ˆbi(sgn(xi)−xi)if|xi|>Xcandi∈C,α5bi(μ−xi)otherwise. | (20) |
Here
The analogies between (20) and (5) (or (6)) are immediate. We assume in (20) that the closed-minded individuals approach the extremes upon exposure to media, regardless of the actual bias of the media. This feature is similar to the closed-minded individuals' response in (5). Non-closed-minded individuals do respond to the media, however, and their opinions are steered toward the media bias
To model coefficients
bi=ϕ(|μ−xi|)andˆbi=ˆϕ(|sgn(xi)−xi|). | (21) |
More general media models. Model (19) can be generalized in various ways. First, one could consider more media sources, in which case the media term
g(xi,μ)={∑pα4pˆbip(|sgn(xi)−xi)if|xi|>Xcandi∈C,∑pα5pbip(μp−xi)otherwise, | (22) |
where
A second possible generalization is to allow the media bias(es) to vary in time, potentially driven by the opinion distribution, to take into account changing media perspectives due to economic and competitive pressures [24]. A general framework to describe the time evolution of the bias of a single media source in model(20) may be given by
dμdt=λ(t,μ,x), | (23) |
where
In the remainder of this paper we focus on the restricted case of a single media source of constant bias
In all simulations of opinion dynamics with one constant media source, we assume that both
Representative simulations with media. Figure 7 shows the time evolution of the model (19)-(20) with (6) in two cases: (a) media bias located at an extreme (
In Figure 7(a) all individuals approach the extreme media bias. This occurs both because the open-minded individuals are attracted to the media and drag the non-open-minded individuals along with them, and due to interactions among closed-minded individuals which cause them to approach the extremes. For more moderate media, as in Figure 7(b), open-minded agents again approach the media bias, but this time the non-open-minded agents do not follow entirely, but instead form two distinct opinion groups. Note that in this 3-cluster equilibrium configuration, with the media bias
As shown in Theorem 4.3 below, the equilibria in both (a) and (b) are asymptotically stable; indeed, the presence of media enhances the potential for stability by providing an additional external attractive force for the dynamics. As in the non-media model, the connectivity of the configuration plays a major role in stability; in particular, note that in (b) the two groups of non-open minded individuals are within the interaction range
The effect of the media bias on equilibria. We perform a bifurcation study, similar to that done for Figure 4 but now focussing on the effect of the parameter
Figure 8 shows the equilibrium cluster locations using three values for the number of open-minded individuals
Figure 8(a) corresponds to a population consisting only of closed-minded individuals; predictably, the equilibrium consists of two opinion clusters located at the extremes, regardless of the location of the media. On the other hand, Figure 8(b) and (c) demonstrate a stronger persuasive effect of the media as the number of open minded individuals increases. Note in particular figure (b), showing significantly different outcomes for the two types of initialization: The insert, which corresponds to a symmetric placement of the open-minded individuals, shows the formation of a group that deviates from the media bias; while a non-symmetric initial distribution of open-minded agents somehow seems to enhance the effect of the media (for
As seen in Figure 8(b) and (c), there are two typical configurations of such opinion dynamics models with media and a social norm of open-mindedness in the absence of symmetry. One is a 3-cluster configuration, occurring for relatively modest media, as discussed already in Figure 7(b). In this case the open-minded individuals form the intermediate cluster located at or near the media bias, experiencing a balance between an attraction to the closed-minded individuals at higher and lower opinion values, and to the media. Simultaneously, there are two clusters of closed-minded individuals: one at a positive opinion between
We conclude by noting that all equilibria illustrated in Figure 8 are asymptotically stable, as they satisfy the connectivity properties required by the analytical results in Section 4.2.
The stability of equilibria for the opinion formation model in the presence of media, (19) with (5) or (6) coupled to the media term (20), can be studied similarly to that for the model without media (Section 3.4). We present the analysis briefly, highlighting the changes from Section 3.4.
Consider an equilibrium opinion state
We assume, similarly, that the media-influence functions
Assumption 4.1.The equilibrium state
1.
2.
Assumption 4.1 guarantees that the coefficients
Extending the terminology introduced in Definition 3.1, we say that a non-closed-minded individual
As in Section 3.4, we study stability via the eigenvalues of the linear approximation. The terms in the Jacobian that correspond to the inter-individual interactions modelled by
Stability for model with media and no social norm of open-mindedness. We use the setup from Section 3.4, and label by
J1m=(Dm0BAm), | (24) |
where
Am=(−α1−α5b∗q+1α1a∗ij[i<j]...α1a∗ij[i>j]⋱⋮⋮…−α1−α5b∗N). | (25) |
Here the indices
The argument now proceeds as in the proof of Theorem 3.6: By the normalization condition (3),
Hence for linear stability to hold in this case, one of the connectivity assumptions in Theorem 3.6 can be replaced by the presence of a nontrivial media influence at equilibrium. The stability result is given by the following theorem:
Theorem 4.2. Let
Stability for model with media and a social norm of open-mindedness. As in Section 3.4, closed-minded individuals separate into two groups: a group consisting of the individuals
The Jacobian for (19) can be written as
J2m=(D1m00Em), | (26) |
where
−α2(N−m−1)ˆai∗−α4ˆb∗i,i=1,…,r, |
that corresponds to the first group of closed-minded individuals. The matrix
Em=(D2mFBAm), | (27) |
where the
−α2(N−m−1)ˆai∗−α3∑j∈Oˆaij∗−α4ˆbi∗,i=r+1,…,q. |
The matrices
Note that
The stability result in this case reads:
Theorem 4.3. Let
Theorems 4.2 and 4.3 suggest that media has the potential to increase stability. This is not surprising, given the tendency of the media to attract individuals who have been perturbed away from it. Empirical studies show indeed that the distance from a media source in opinion space is positively correlated with the tendency to move towards said media source upon exposure [33, 3].
We have generalized the continuous-time opinion dynamics model introduced by Motsch and Tadmor [42] to incorporate closed-minded, extreme-seeking individuals who may, however, be induced to act more open-mindedly when they interact with naturally open-minded individuals. Our model yields a richly diverse range of potential equilibrium states, which we have explored analytically and numerically: in addition to moderate consensus and mutually disconnected opinion clusters, we have shown the existence of extremist consensus and extreme polarization states, as well as of connected opinion clusters. Moreover, we have proved that these latter configurations are asymptotically, not just neutrally, stable. We have further extended the model to include media, which are assumed to have different influences on open-and closed-minded individuals, and confirm using analysis and simulation that media introduce an additional attractive and stabilizing effect on the long-time opinion distribution.
Our modelling framework suggests numerous natural extensions. For instance, at present our model contains sharp cutoffs both at the bound of confidence
Understanding the influence of media on societal opinion distributions is particularly important, and much more could be done within our modelling framework to explore this influence, by for instance adding multiple and potentially competing media sources with possibly time-varying strengths and/or intrinsic biases, accounting for the hostile media effect, or even as suggested in equation (23), incorporating a feedback loop by letting media bias be driven by the opinion distribution.
This research was funded in part by Canada's National Science and Engineering Research Council (NSERC) through grants to RF and RW.
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1. | GIULIA BRAGHINI, FRANCESCO SALVARANI, EFFECTS OF HIDDEN OPINION MANIPULATION IN MICROBLOGGING PLATFORMS, 2021, 24, 0219-5259, 10.1142/S0219525921500090 |