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Fig. also shows [MATH] that best matches Sydney-2011 Census data, which is lower than the critical value [MATH] but nevertheless is in the proximity of the phase transition, being located in the region where the Fisher information undergoes a rapid growth. |
These results show that changes in the social disposition, away from its current value [MATH] , would significantly and abruptly change the flow distribution of income within Greater Sydney. This has an immediate effect on the spatial distribution of the population, driving an urban transformation from the sprawling ph... |
For the low value of [MATH] (Fig. (a)), corresponding to the sprawling urban phase, the model shows a quite homogeneous distribution of the population, with the areas around the City of Sydney and other major urban areas being only slightly more populated than the other surrounding areas. As we move to [MATH] (Fig. (b)... |
Interestingly, Sydney-2011 profile, lying within the sprawling phase but near the phase transition, displays features of a polycentric metropolis, which accentuate beyond the critical point. However, the dynamics of the polycentric phase are not steady (cf. Fig. for [MATH] ), and so the transformations may suffer from ... |
, and multiple peaks detected by the Fisher information may relate to this phenomenon, given the clustered connectivity of urban aggregations. |
3.2 Deepening the thermodynamic analogy An important consideration in making a rigorous thermodynamic analogy is a choice of the protocol according to which the control parameters are varied, so that the corresponding changes in the required work, energy and configuration entropy, as well as symmetry breaking |
, can be traced. Specifically, we consider a quasi-static protocol varying [MATH] , at the expenditure of some required work, and driving changes from the sprawling urban phase to the polycentric phase, across the phase transition. For a quasi-static protocol the required work is minimal, i.e., the work matches the fre... |
It has been recently shown that for quasi-static processes the second derivative of the generalised work [MATH] with respect to a control parameter is proportional to the negative of the Fisher information |
We refer to generalised work in the sense of Jaynes (for more details about generalised quantities and their relationship with the Fisher information see Supplemental Material, Sec. 3). Given this relationship, we obtain the rate of change of the work with respect to [MATH] by numerically integrating the negative of th... |
Fig. (a) also shows the rate of change of the internal energy of the system [MATH] This quantity is obtained from the rates of change of the work [MATH] and the configuration entropy |
[MATH] —according to the first law of thermodynamics (in the case of quasi-static processes) a change in the internal energy corresponds to the sum of the changes in entropy and work: [MATH] , where the angle brackets represent average values over the ensemble. The rate of change of the internal energy decreases with [... |
The thermodynamic efficiency of urban transformations for Greater Sydney, [MATH] , is shown in Fig. (b). It can be seen that [MATH] is very low in the sprawling phase, increases towards the phase transition and then tends to slowly decrease, while also exhibiting the secondary local peak. Interestingly, this ratio is i... |
Discussion The transition of cities between different patterns of urban settlement (dispersed, monocentric, polycentric, etc.) has become a central problem in urban planning. In this study we investigated the urban dynamics from a statistical mechanical viewpoint, deriving a thermodynamic description and applying it to... |
The model has been calibrated to Census data and geospatial datasets and exhibits a clear phase transition between a dispersed configuration, in which the population settles homogeneously within Greater Sydney, and a polycentric configuration, in which the population aggregates in a few highly populated urban clusters.... |
A recent plan by the Greater Sydney Commission envisaged a tripartite Greater Sydney region, with a western parkland city, a central river city around greater Parramatta, and an eastern harbour city. As shown in our study, such a tripartite arrangement is possible only under a narrow set of constraints and importantly ... |
Appendix A Greater Sydney and data sources Greater Sydney is an urban area covering more that [MATH] square kilometres, delimited in all directions by either the Pacific Ocean or by the several surrounding national parks. It includes the City of Sydney as well as other urban agglomerations such as Parramatta, Penrith, ... |
The employment areas are defined by the standard Destination Zone (DZN), which was designed by the New South Wales transport authority in order to spatially classify employment places, with the purpose of analysing commuting data and developing transport policies. The standard Statistical Area Level 2 (SA2), as defined... |
The Census data for year 2011 was geographically classified by the Australian Bureau of Statistics in accordance with the both geographical areas DZN and SA2, and included the travel-to-work matrix [MATH] , the average weekly income [MATH] and the average weekly rent [MATH] , for all DZN areas [MATH] and SA2 areas [MAT... |
The cost of travelling [MATH] was estimated as the Euclidean distance between the centres of the employment and residence areas. An alternative approach would be to calculate the time of travelling using Google Maps or OpenStreetMap data (however, that requires access to high resolution data which is not immediately av... |
Appendix B Calibration of the model The model was calibrated by identifying the optimal values [MATH] and [MATH] for which the output [MATH] best matches the actual flow of income [MATH] of Sydney-2011. The difference between actual and predicted flow of income was estimated as the sum of (the absolute values of) the d... |
Appendix C Thermodynamic analysis Let us consider the state functions [MATH] that describe a physical system over its configurations [MATH] In a stationary state, the Gibbs measure defines the probability of the states of the system: |
[EQUATION] where [MATH] are thermodynamic variables, [MATH] is the inverse temperature [MATH] [MATH] is the Boltzmann constant), [MATH] is the Hamiltonian defining the total energy at state [MATH] , and [MATH] is the partition function |
The Gibbs free energy of such system is: [EQUATION] where [MATH] is the internal energy of the system, [MATH] is the configuration entropy and [MATH] is an order parameter. Let us also consider the generalised internal energy [MATH] in the sense of Jaynes |
, such that [EQUATION] where the angle brackets represent average values over the ensemble. The generalised first law holds [MATH] , where [MATH] and [MATH] are, respectively, the generalised heat and the generalised work. |
The Fisher information measures the amount of information that an observable random variable [MATH] carries about an unknown parameters [MATH] If [MATH] is the probability of the realisation [MATH] of [MATH] given the parameters [MATH] , the Fisher information matrix is defined as |
[EQUATION] where the function [MATH] is the expected value of [MATH] For a physical system described by the Gibbs measure in ( 10 ), the Fisher information has several physical interpretations, e.g., it is equivalent to the thermodynamic metric tensor [MATH] , is proportional to the second derivative of the free entrop... |
[EQUATION] Furthermore [EQUATION] Under a quasi-static protocol the total entropy production is zero, and therefore any change in the configuration entropy due to the driving process is matched by the flow of heat to the environment: |
[EQUATION] Thus, combining ( 15 ) and ( 16 ) with the first law of thermodynamics yields another important result for the generalised work [MATH] |
[EQUATION] Appendix D Entropy and a proxy of order parameter A higher entropy indicates a more homogeneous distribution of the income to all suburbs, while a lower entropy indicates a less balanced distribution of the income biased towards one or few suburbs. We observe that the entropy decreases with both parameters [... |
To formalise this intuition, one typically introduces and traces corresponding order parameters. This is however hindered by an incomplete statistical-mechanical description of the system, and we first illustrate a simpler approach which considers a proxy of an order parameter. Such a proxy characterises the equilibriu... |
Fig. and Fig. also show the values [MATH] and [MATH] which best matches Sydney-2011 Census data (the red dot on either the entropy or the [MATH] surfaces). This value is within a close proximity to the social disposition where the abrupt change is observed. |
Acknowledgements All authors were supported by The University of Sydney’s DVC Research Strategic Research Excellence Initiative (SREI-2020) project, “CRISIS: Crisis Response in Interdependent Social-Infrastructure Systems” (Grant No. IRMA 194163). E.C. was supported by the University of Sydney’s “Post-graduate Scholars... |
# Source: arxiv 1806.03781 # Title: Noise-based control of opinion dynamics # Sections: all # Downloaded: 2026-03-02T08:57:32.756133+00:00 |
Noise-based control of opinion dynamics* Abstract Designing feasible control strategies for opinion dynamics in complex social systems has never been an easy task. It requires a control protocol which 1) is not enforced on all individuals in the society, and 2) does not exclusively rely on specific opinion values share... |
Keywords : Social control, noise-based intervention, Hegselmann-Krause model, opinion dynamics Introduction In the past decade, research on opinion dynamics and consensus problems in complex networked systems has drawn an increasing attention from a variety of fields, including mathematics, physics, social science, inf... |
In the study of complex social dynamics, developing control strategies for opinion consensus has been a central issue . However, designing a practical and effective intervention strategy for a social opinion system that should attain global agreement has always been a grand challenge, largely due to two main obstacles.... |
. Considering these hindering issues, developing a new intervention strategy which can circumvent the reliance on system states and instead act upon only a fraction of individuals is indicated. |
In recent years, a few noise-based control strategies for coordination enhancement and consensus generation in social networks have been advanced |
. For example, Shirado and Christakis devised a network coordination experiment which took a local rule into consideration. After adding some noisy agents into the network, they observed a remarkable improvement of the coordination efficiency of the group. Su et al. |
designed a simple noise injection scheme to eliminate the disagreement in a divisive opinion system. By injecting random noise to only one agent of a divisive opinion system, |
These noise-based control strategies generally meet the requirements for a social system control; however, the underlying general theory for noise-induced opinion control is still lacking. For example, one limitation of the strategy of Su and colleagues |
is that only one agent was allowed to receive noise, and moreover, the initial opinion state of the system had to be assumed a priori while the effective noise strength relied on the group size and was vanishingly small as the group size was increasing. The noise scheme developed in |
was inspired by the previous seminal works on the noisy opinion dynamics , and was later theoretically verified for the HK opinion dynamics model |
. However, the noise component in most of these previous models was typically added to all agents, which is far from realistic for a control of large-scale social systems. Critically, the mathematical approach developed in |
was inadequate for modeling the scenario of noise application to only a fraction of agents. Driven by these limitations of earlier studies, we herein aim to establish a general theoretical framework for the noise-based control strategy of large social networks in which no prior assumptions about the initial opinion sta... |
, and the other is the so-called bounded confidence model which originates from the works of Deffuant et al. and Hegselmann and Krause |
. The topology-dependent model is typically efficient in illustrating the opinion evolution of relatively small groups. For large social systems, however, the confidence-based model is much more convenient because it captures local-level self-organization processes that are the hallmark of large complex systems. |
In this paper, we employ the confidence-based HK dynamics to study the noise-based control theory of a large social opinion system. We rigorously prove that given any initial system states and any percentage of agents affected by noise, the system will almost surely attain a quasi-consensus in finite time. Crucially, o... |
The rest of the present paper is organized as follows: In section , we first present some necessary preliminaries. Our main findings are then presented and discussed in section ; in section , we present our numerical simulation results to verify the main theoretical analyses and finally, some concluding remarks and fut... |
Definitions and model Denote [MATH] as the set of [MATH] agents, [MATH] the nonempty set of controlled agents. Let [MATH] be the state of agent [MATH] at time [MATH] and [MATH] be the noise control. The update rule for the noise-induced HK dynamics takes the form: |
[EQUATION] where [EQUATION] and [EQUATION] is the neighboring set of [MATH] at [MATH] with [MATH] representing the confidence threshold of agents. Here, [MATH] is the indicator function which takes the value 1 or 0 according to [MATH] or not, and [MATH] stands for the cardinal number of a set or the absolute value of a... |
To proceed further, we need to introduce some preliminary definitions as follows. Let [MATH] be the graph of [MATH] at time [MATH] , and [MATH] if and only if [MATH] . A graph [MATH] is called a complete graph if and only if [MATH] for any [MATH] ; and [MATH] is called a connected graph if and only if for any [MATH] , ... |
The definition of a quasi-consensus of the noisy model ( 2.1 )-( 2.3 ) is then as in Definition 2.1 Denote [EQUATION] (i) if [MATH] , we say the system ( 2.1 )-( 2.3 ) will reach a quasi-consensus. |
(ii) if [MATH] , we say almost surely (a.s.) the system ( 2.1 )-( 2.3 ) will attain a quasi-consensus. (iii) if [MATH] , we say a.s. the system ( 2.1 )-( 2.3 ) cannot reach quasi-consensus. |
(iv) let [MATH] If [MATH] , we say a.s. the system ( 2.1 )-( 2.3 ) attains a quasi-consensus in finite time. Main Results For simplicity, we first present the result of a quasi-consensus for independent and identically distributed (i.i.d.) noises, and we then generalize these results with independent noises by a suffic... |
Theorem 3.1 Suppose the noises [MATH] are i.i.d. random variables with [MATH] Let [MATH] and [MATH] be arbitrarily given, then (i) if [MATH] when [MATH] or [MATH] when [MATH] , then a.s. the system ( 2.1 )-( 2.3 ) will attain a quasi-consensus in finite time; |
(ii) if [MATH] and [MATH] when [MATH] , or [MATH] and [MATH] when [MATH] then a.s. the system ( 2.1 )-( 2.3 ) cannot reach a quasi-consensus. |
Conclusion (i) shows that if noise strength is no more than [MATH] when [MATH] or [MATH] when [MATH] a.s., the system will a.s. achieve a quasi-consensus in finite time; Conclusion (ii) states that when noise strength has a positive probability to exceed [MATH] when [MATH] or [MATH] when [MATH] , the system will not re... |
Theorem 3.2 Suppose [MATH] are independent and satisfy: i) [MATH] with [MATH] when [MATH] , or [MATH] when [MATH] ii) there exist constants [MATH] such that |
[MATH] and [MATH] Then, for any initial state [MATH] and [MATH] , the system ( 2.1 )-( 2.3 ) will a.s. attain a quasi-consensus in finite time and [MATH] a.s. when [MATH] , or [MATH] a.s. when [MATH] |
Before the proofs of Theorems 3.1 and 3.2 we need introduce some lemmas: Lemma 3.3 Suppose [MATH] is a nonnegative nondecreasing (nonincreasing) sequence. Then for any [MATH] , the sequence |
[MATH] [MATH] is monotonically nondecreasing (nonincreasing) for [MATH] Lemma 3.4 Suppose [MATH] , then for every [MATH] and [MATH] , there exist constant [MATH] such that [MATH] for [MATH] , and either [MATH] or [MATH] holds for any [MATH] |
In what follows, the ever appearing time symbols [MATH] (or [MATH] , etc.) will all refer to the random variables [MATH] (or [MATH] , etc.) on the probability space [MATH] , and for simplicity, they will be still written as [MATH] (or [MATH] ). |
In the rest process of the proofs of Theorems 3.1 and 3.2 , we only consider the case [MATH] , and the proof for the case [MATH] can be obtained in a similar fashion. |
Lemma 3.5 For the system ( 2.1 )-( 2.3 ) with conditions of Theorem 3.2 i), if there exists a finite time [MATH] such that [MATH] , then on [MATH] , we have |
[MATH] for all [MATH] Proof. Denote [MATH] [MATH] , and this notation remains valid for the rest of the context. If [MATH] by ( 2.3 ) we have |
[EQUATION] Since [MATH] a.s., we obtain a.s. [EQUATION] Repeating ( 3.1 ) and ( 3.2 ) yields the conclusion. Lemma 3.6 For system ( 2.1 )-( 2.3 ) with conditions of Theorem 3.2 , if at the initial moment there exist subsets [MATH] such that [MATH] [MATH] , and [MATH] [MATH] for [MATH] , then there exist constants [MATH... |
[MATH] and [MATH] for [MATH] Proof. This proof uses the idea that “transforming the analysis of a stochastic system into the design of control algorithms” first proposed by |
At the initial moment, the systems forms 2 separate subgroups [MATH] of which one is not neighboring with the other. By ( 2.1 ), [MATH] Before one subgroup enters the neighbor region of the other, for each [MATH] , we have |
[EQUATION] when [MATH] Suppose [MATH] , and consider the following noise protocol: for [MATH] [EQUATION] Here [MATH] is a constant defined in Theorem 3.2 Under the protocol ( 3.4 ), it is easy to check that before one subgroup enters the neighbor region of the other, and by ( 3.3 ), we will have |
[EQUATION] suggesting that after each step, the maximum difference of opinion values decreases at least [MATH] This implies that there must exist a constant [MATH] such that under the protocol ( 3.4 ), |
[EQUATION] From moment [MATH] , design the following protocol: for [MATH] [EQUATION] Since [MATH] by ( 3.6 ), we can check that for each [MATH] , under the protocol ( 3.7 ), either [MATH] or [MATH] or both hold, implying it is neighbor to agents with extreme opinion. By ( 2.1 ), we know that ( 3.5 ) also holds. This im... |
[EQUATION] By independence of [MATH] , we know that the probability of the protocol ( 3.4 ) occurring [MATH] times and protocol ( 3.7 ) occurring [MATH] times is no less than [MATH] . Moreover, under the protocols ( 3.4 ) and ( 3.7 ), by Lemma 3.3 , it holds [MATH] for [MATH] Let [MATH] and consider Lemma 3.5 , then we... |
Lemma 3.7 Suppose the noise satisfies the conditions of Theorem 3.2 i), then for any [MATH] , there exist a noise protocol and constants [MATH] such that [MATH] . Furthermore, if [MATH] , it has under the noise protocol that [MATH] for [MATH] |
Proof. If [MATH] , the conclusion holds directly from Lemma 3.5 . Now we consider the case of [MATH] , and use the method of induction for the group size [MATH] . Note that there exist constants [MATH] such that |
[MATH] and [MATH] . When [MATH] , consider the following noise protocol: for [MATH] , if [MATH] , take [MATH] ; otherwise, take [MATH] . It can be easily seen that under the protocol, the opinion difference of the two agents will decrease at least [MATH] for each step. If [MATH] , we obtain the conclusion by taking [MA... |
Let [MATH] , and consider the following noise protocol: For [MATH] [EQUATION] By Lemma 3.3 , it is easy to obtain that under the protocol ( 3.9 ), [MATH] is nondecreasing, and [MATH] is nonincreasing. Let [MATH] be the first moment when the graph of the system ( 2.1 )-( 2.3 ) is not connected under the protocol ( 3.9 )... |
First consider the case when there is no agent controlled by noise in one of the two subgroups, say [MATH] . By Lemma 3.4 , there is a constant [MATH] such that [MATH] converges in [MATH] By assumption, there is a constant [MATH] such that [MATH] reaches a quasi-consensus in [MATH] with a positive probability [MATH] . ... |
Now we consider the case when both [MATH] and [MATH] are intervened by noise. By assumption, there exist constants [MATH] and [MATH] , such that [MATH] reaches a quasi-consensus in [MATH] with probability [MATH] and [MATH] achieves a quasi-consensus in [MATH] with probability [MATH] . If [MATH] , the conclusion holds b... |
To sum up, the conclusion holds for [MATH] . This completes the proof. Proof of Theorem 3.2 Define [MATH] 2.1 )-( 2.3 ) does not reach quasi-consensus in period [MATH] |
and [MATH] 2.1 )-( 2.3 ) does not reach quasi-consensus in finite time [MATH] By Lemma 3.7 , there exist constants [MATH] such that |
[MATH] Since [MATH] is arbitrarily given in [MATH] , following the procedure of Lemma 3.7 , it has [EQUATION] Then [EQUATION] and hence, |
[EQUATION] This completes the proof. [MATH] Next, we will present the necessary part of the noise induced consensus, which shows that when the noise strength has a positive probability of exceeding [MATH] , the system a.s. cannot reach a quasi-consensus. |
Theorem 3.8 Let [MATH] [MATH] are arbitrarily given. Assume the zero-mean random noises [MATH] are i.i.d. with [MATH] or independent with [MATH] If there exists a lower bound [MATH] such that |
[MATH] and [MATH] then a.s. the system ( 2.1 )-( 2.3 ) cannot reach quasi-consensus. Proof. We only need to prove the independent case, while the i.i.d. case can be obtained similarly. For the independent case, we only need to prove that, for any constant [MATH] , there exists [MATH] a.s. such that [MATH] , i.e. |
[EQUATION] Given any [MATH] , by independence of [MATH] , it has [EQUATION] Hence, [MATH] . Similarly, [EQUATION] Thus [EQUATION] |
This completes the proof. Numerical Simulations In this section, we present the outcomes of simulation experiments to verify our main theoretical results in this paper. First, we present a fragmentation of noise-free HK model. We take [MATH] , and the initial opinion values are uniformly distributed on [MATH] . Fig. sh... |
Discussion and Conclusions Recently, HK opinion dynamics models with bounded confidence and its variants have attracted considerable attention across fields. Nevertheless, feasible control strategies for HK systems are still lacking. |
In this paper, we established a rigorous theoretical analysis for the noise-based opinion control strategy in large social networks where only a fraction of agents are noise-affected. The local rule based HK dynamics was taken as the underlying model of complex social systems. It was rigorously proved that, given any i... |
Indeed, the constructive role of noise has been revealed previously in a variety of physical, biological, and social systems (e.g. |
). However, its potential for the design of control strategies in dynamic social systems remained largely unexplored. While we have demonstrated that noise itself can serve as an efficient mechanism for control of opinion dynamics, its combined effects with other mechanisms are still unknown. For instance, recent studi... |
have evidenced nontrivial interactive effects of various combined mechanisms, including those with noise , that are not present when these mechanisms are considered in isolation. Future studies should therefore explore the effects of noise-related control strategies in social dynamics when they are employed in combinat... |
For example, while previous theoretical analyses showed that the bounded confidence structure of HK dynamics can be synchronized by noise, it is still unknown whether the noise-induced synchronization can occur with a topology-based mechanism. The theory of robust consensus in noisy multi-agent systems suggests that th... |
In studies of noise-induced phenomena, it is typically assumed that noise source is a Gaussian distributed variable . However, it has been shown that noise-induced transitions can be shifted significantly when using non-Gaussian noise sources |
characterized by nonextensive statistical properties that are often found in various biological and social processes . It thus remains a challenge for future research to also investigate noise-based mechanisms for the control of opinion dynamics when the noise source departs from the classical Gaussian behavior. Since ... |
, its applications could especially be interesting in systems in which only a temporary consensus is desired, after which the system can go back to its initial non-consensus state. |
In sum, our present study demonstrated a feasibility of noise-induced control strategy in HK dynamic systems that can alter the system behavior after being applied to only a fraction of individuals and without relying on the complete knowledge of system’s states. Our results thus represent the first step towards a more... |
Lemma .1 Suppose the graph of the system ( 2.1 )-( 2.3 ) remains connected under the protocol ( 3.9 ), then the system will reach a quasi-consensus before a constant moment [MATH] where [MATH] |
Proof. If [MATH] , the conclusion holds by Lemma 3.5 . Now we only consider the case of [MATH] . Denote [MATH] as the agent with the smallest opinion value at moment [MATH] and let [MATH] , then |
[EQUATION] If there exist [MATH] such that [MATH] , by ( .1 ) and Lemma 3.3 , we know that [MATH] increase at least [MATH] from [MATH] under the protocol ( 3.9 ). If an agent [MATH] is a neighbor of agent [MATH] at moment [MATH] and also at [MATH] , by ( .1 ) and Lemma 3.3 , we know that at [MATH] [MATH] will increase ... |
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