Communication on the y made possible by modern information and communication technology is a characteristic feature of modern society. This style of communication signi cantly changed social life and provided new ways of public opinion formation. New social practice poses new challenges for specialists studying social phenomena. One of these problems is the problem of the reliability of public opinion measurements, which are always based on indirect assessments. Unfortunately, indirect assessments depend essentially on the suggestions accepted during their development. The marked above changes raised by the technological progress require the new mathematical suggestions lied in the base of public opinion measurements. This paper is to draw attention to this situation and to begin the movement toward the rigorous theory of public opinion measurements basing on social phenomena mathematical models of adequate to features of modern communication processes. It seems that authors' rst results are consistent to hypotheses of a number of sociologists working in this area.
The modern information and communication technology (ICT) has essentially changed communication processes between communities and their members, generated new social phenomena. Social networks are of these phenomena. Their emergence and rapid development have markedly changed the character of communication streams.
In this context, an important point is the e ect of a combination of social networks and personal telecommunication microelectronic devices (tablets, smartphones). This e ect signi cantly speeds up communication, reduces in time and makes it discrete and as non-linear as possible. The cognitive and motivational results of such communication are still unknown to sociologists. Nevertheless, it can be argued that such discrete and nonlinear communication signi cantly diversi es the sources of potential in uence on the opinion formation of the of communication participants, and at the same time increases the frequency of such impact, potentially having a di erent result. Public opinion formation becomes the process di cult for predicting because we need to take into account a signi cant number of factors than before. At the same time, the e ect of these factors due to the speci city of the new segment of the communicative space (nonlinearity, discreteness and high communication rate) is characterised by more and more complex interconnections.
The reliable sociological information about public opinion is especially in demand in modern society. The reason for this is not to reinforce the need to e ectively predict changes in public opinion (this need has traditionally been high since the fties of the twentieth century). The reason is associated with an increase in such threats to social stability and security as information terrorism, manipulative technologies a ecting society, and the active use of fake-news in the political sphere. Today public opinion needs not only to predict but also to protect against the above threats. This, in turn, requires the development of the technique of sociological measurement based on the dependable quantitative theory. This necessity is also caused by the high cost of the evidential public opinion measurements. Improving public opinion measurements by the way of increasing the frequency of spot measurements as to record in time all possible deviations and shifts in public opinion seems to be of little prospect due to the high cost of such measurements and their technical complexity.
Based on the foregoing, it can be said that developing a theory of measuring public opinion to substantiate and improve polling tools aimed at making it possible to grasp the peculiarities of the formation of public opinion in modern conditions is a relevant challenge not only for social science but also for mathematics and computer science.
We believe that simulation of the public opinion formation within the modern information and communication environment is the origin point for this theory.
This paper is our attempt to attract the attention of researchers in the elds of ICT, mathematics and sociology to this challenge.
It is needed to stress that the idea to use mathematical modelling as a tool
for studying social processes is not novel. In the middle of the twentieth century,
the use of relational and statistical models for understanding social processes
was proposed by a number of scientists (N. Rashevsky [13], A. Rapoport [12]
and other). The idea was further developed with the inception of network science
(see [
Today there are a lot of works devoted to simulation of social network and studying social dynamics based on the corresponding methods. But we could not nd a paper that would present a simulation model for studying a public opinion measurement. This is what caused our research.
Model of Communication in Network In this paper, we consider the special class of homogeneous multicomponent discrete-time dynamical systems, whose components interact only pairwise via a network of channels. The components of such a system are entities for modelling members of the community being studied, and channels are entities for modelling stable pairwise communications between the members of this community. 2.1
The goal of this subsection is to formulate explicitly the modelling assumptions. Assumption 1. A network being simulated is a multicomponent dynamical system of the discrete time with a constant set of components called members. Some pairs of components communicate stable and, in this case, we say that there is an information channel between members of such a pair. Assumption 2. The property of an information channel to be active is a random Boolean variable. Moreover, for di erent channels, the corresponding variables are independent.
Assumption 3. Each system component estimates the claim being in the focus of community interests using an element of the set C = fRED; GREEN; BLUEg corresponding to either negative, or neutral, or positive estimation respectively. The mapping associating a claim estimation with each system components is below called a microstate of the system.
Assumption 4. The personal estimations of a member of a community (a system microstate) is not interest and is considered as not available for the direct observation. Only the occupacy measure of elements of C is available for the direct observation. The value of this measure for an element of C is a ratio between the number of community members that have the corresponding opinion and the total number of community members. This measure is considered below as a macrostate of the system.
Assumption 5. At each time-point, a member of the community participates at most in one communication. 2.2
Taking into account Assumption 1, an undirected simple nite graph G = (N; E) is the most natural mathematical structure for modelling pairwise communications in a community. The node set N of the graph models members of a community, and the edge set E of the graph models stable information channels between ones.
Assumption 2 can be ensured with associating a random Boolean variable activatede with each edge e 2 E. To do this it is su cient to specify a function activation rate : E ! [0; 1] and to think about its value activation rate(e) as about the activation probability of the channel corresponding the edge e. In other words, we set activation rate(e) = Pr(activatede = true). Thus, we come to the following de nition.
De nition 1. A network model is a triple hN; E; activation ratei where N and E are respectively the sets of nodes and edges of some undirected simple nite graph G = (N; E) and activation rate : E ! [0; 1] is the channel activation rate function.
Assumption 3 causes the following de nition of a microstate and microdynamics for the system class being studied.
De nition 2. A node colouring of the graph G in accordance with the colour set C is a microstate of the system.
Thereby, the system micro-dynamics is a discrete-time stochastic process explaining the observed sequences of system microstates.
Assumption 4 leads us toward the concepts of a macrostate and macrodynamics.
De nition 3. Let c : N ! C be a microstate of the system then a function c : C ! [0; 1] is the macrostate corresponding to c if for each x 2 C, it is de ned as follows3 c(x) = 1
X [c(n) = x] : jNj n2N Thereby, the system macro-dynamics is a discrete-time stochastic process explaining the observed sequences of system macrostates. 2.3
Based on the above assumptions and de nitions, a prototype framework has
developed for simulation of the community dynamics with various kinds of pairwise
communications. The general speci cation of a simulation process is presented
as a UML activity diagram in Fig. 1. For the realisation of this general speci
cation, the language Python 3 [
To construct a framework providing the presented simulation process we
propose the conceptual model shown as a UML class diagram in Fig. 2. This
model based on an undirected simple graph whose nodes are instances of the
class Node and the edges are instances of the class Edge. The attribute estimation
of the class Node is intended for saving the current value of a microstate for the
corresponding node. The association state gives access to the internal description
of a node state. This description is abstract on the framework level. Similarly,
the attribute activation rate of the class Edge is intended for saving the value
activation rate(e) for the Edge-instance that models edge e.
3 In the formula, the Iverson bracket is used (see, for example, [
Set nodes and edges parameters
Build and save initial microstate [otherwise]
Choose communicated pairs
Perform communication protocol
Renew microstate
Save macrostate
Node
Edge
2
incidence
1..*
activation_rate: Real
weight(colours: Colour[
Fig. 2. Conceptual model of the simulated net Algorithm 1: Method for forming communicating pairs
Data: a simple undirected graph G = (N; E) Result: the subset SELECTED of E representing communicating pairs /* initialise the target set and auxiliary sets 1 SELECTED := ?; AVAILABLE := ?; FORBIDDEN := ?;
/* activate communication channels 2 foreach e 2 E do 3 choose randomly True or False with probabilities a(e) and 1 respectively; if True is selected then
add e into AVAILABLE 4 5 6 else 7 add e into FORBIDDEN 8 end 9 end
/* form communicating pairs 10 while AVAILABLE 6= ? do 11 choose randomly an element e 2 AVAILABLE in accordance with uniform distribution on AVAILABLE; delete e from AVAILABLE; if for some e0 2 SELECTED, e and e0 are incindent then add e into FORBIDDEN a(e) */ */ */ 2.4
Above we were focused on modelling the structure of a network, and in this subsection, we pass to modelling the interaction (or the communication, in the case of social network) between nodes of the network. The association between instances of the class Edge and abstract entities classi ed as Protocol is foreseen for providing the speci cation of such interaction (see Fig. 2).
Taking into account that Assumption 5 is accepted we need some method to form the set of interacting pairs of nodes. We propose to use the method speci ed by Algorithm 1.
The following proposition establishes properties of the method.
Proposition 1. The method presented by Algorithm 1 has properties 1. a computation with respect to Algorithm 1 is halted for any input data after a nite number of steps; 2. after halting a computation with respect to Algorithm 1, sets SELECTED and
FORBIDDEN are disjoint; 3. after halting a computation with respect to Algorithm 1, set SELECTED does not contain incident edges; 4. adding to the set SELECTED an edge added to the set FORBIDDEN in loop 10{18 violates the property claimed in item 3.
Proof. The rst item of the proposition is true because of the set AVAILABLE decreases (see, line 12 of Algorithm 1) after each iteration of loop 10{18. The validity of the second item of the proposition is ensured by branching 13{17. The validity of the third item of the proposition is ensured by line 14. The validity of the fourth item of the proposition is ensured by branching 13{ 17. tu
We suggest that any communication protocol can be represented by the UML sequence diagram as in Fig. 3.
theEdge:Edge theEdge.incidence[0]:Node
theEdge.incidence[
Finally, the method estimate(state: State) of the abstract entity State (see Fig. 2) is intended to renew the current microstate. 3
In this section, we present and discuss the results of simulation for four kinds of systems: models A-IR and B-IR, which called below as models of components with an instant response, and models A-LR and B-LR, which called below as models of components with a lazy response. 3.1
The above classi cation of the models being studied is based on the general scheme of the interaction process modelled by the method communicate(. . . ) of the abstract entity Protocol.
We assume that the communication corresponding to edge e 2 E is modelled by the weight function we on C C and taking random value we(colour0; colour00) in the following outcome set fnobody; rst; secondg. The outcome is interpreted as follows { we(colour0; colour00) = nobody means that participants of the communication preserve their opinions; { we(colour0; colour00) = rst means that the rst participant of the communication preserves his opinion, but the second one does not preserve; { we(colour0; colour00) = second means that the second participant of the communication preserves his opinion, but the rst one does not preserve.
Based on this assumption, we propose to use the following abstraction speci ed by Algorithm 2.
Algorithm 2: The scheme of the method communicate(. . . )
Data: an edge e 2 E, the weight function we corresponding e
Result: the pair of new node states (newFirstState; newSecondState)
1 rstState := e:incidence[0];
2 secondState := e:incidence[
The model of a system of components with instant response (below IR-model) is based on the following model of a state called by SimpleState (see Fig. 4).
SimpleState colour: Colour estimate(): Colour Constrints: inv: self.estimate() = self.colour
State estimate(): Colour
The IR-model realises items 9 and 11 of Algorithm 2 as follows if outcome = nobody then if outcome = rst then if outcome = second then newFirstState = rstState newSecondState = secondState newFirstState = rstState newSecondState = rstState newFirstState = secondState newSecondState = secondState Simulation Experiment for the IR-model. The simulation experiment was carried out at the initial macrostate de ned as follows c0(RED) = 0:1 , c0(GREEN) = 0:8 , and c0(BLUE) = 0:1 . The typical simulation results are shown in Fig. 5.
1.0 0.8 0.6 0.4 0.2 Error Estimation for the IR-Model. We assume that the measurement of the system is performed sequentially by observing a xed number of system components. Thus, the measurement rate depends on the number of observed components in one step. More precisely, our assumption is that the measurement procedure under our study is sequential and represented by the Algorithm 3.
We estimate the measurement error by using Kullback-Leibler divergence [
Remind that Kullback-Leibler divergence D is computed by the formula D(c jj c ) =
c(c) X c(c) log2 c (c) c2C and estimates the minimal information quantity needed to correct an error.
As mentioned above, the measurement speed depends on the number of k nodes observed during one simulation cycle. A small value of k corresponds to a slow measurement and a big value of k corresponds to a fast one. In Fig. 6, the dynamics of error estimation for slow (the blue curve with k = 20) and fast (the green curve with k = 250) measurements are presented.
Algorithm 3: Measurement procedure
Data: a model of a system, a number k of nodes observed per one simulation cycle
Result: the measured macrostate c 1 N [RED] = N [GREEN] = N [BLUE] = 0; 2 foreach simulation cycle do 3 choose randomly k nodes from the nodes not chosen yet; 4 increase each N [RED], N [GREEN] and N [BLUE] by the number of nodes from the sample correspondingly coloured 5 end 6 N = N [RED] + N [GREEN] + N [BLUE]; 7 c (RED) = N [RED]=N ; 8 c (GREEN) = N [GREEN]=N ; 9 c (BLUE) = N [BLUE]=N ; 10 return c 0.8 0.6 0.4 20 100 250 0.00.0 2.5 5.0 7.5
10.0 Relative time The LR-model is based on the following model of a state called by LazyState (see Fig. 7).
LazyState colour: Colour balance: Integer setColour(colour: Colour) getColour() : Coulor estimate(): Colour Constrints: post: self.colour = self.estimate(balance) inv: self.getColour() = self.colour inv: self.estimate() = self.colour
State setColour(colour: Colour) getColour() : Colour estimate(): Colour
Unlike the previous model, the model considered in this subsection is more inertial. This is provided by the method estimate(), which uses the function Pm(x) , and the eld balance, which equals the di erence between BLUE-arguments and RED-arguments (see Fig. 7).
The function Pm(x) is de ned as
Pm(x) = 8 > < >: 12 + x3 m3 1
1 arctan x 2m (x m m) if 0
x < m if x m
This function provides model inertness. Its value equals the probability that the corresponding system node is not green. We assume that the current balance of the node determines this probability.
The LR-model realises items 9 and 11 of Algorithm 2 as follows if outcome = nobody then
newFirstState if rstState:colour = RED then
newSecondState:balance if rstState:colour = GREEN then
newSecondState:balance if rstState:colour = BLUE then newSecondState:balance = rstState = secondState:balance
1 = secondState:balance = secondState:balance + 1 if outcome = second then
newSecondState if secondState:colour = RED then
newFirstState:balance if secondState:colour = GREEN then
newFirstState:balance if secondState:colour = BLUE then newFirstState:balance = secondState = rstState:balance
1 = rstState:balance = rstState:balance + 1
The positive parameter m controls the system inertia and in a certain sense can be considered as a mass. This interpretation is illustrated by Fig 8.
We should mark that the character of the measurement error behaviour is similar to one for the IR-model. This is a reason to omit the corresponding illustrating gure. 4
Thus, the paper has proposed a framework for simulating pair-chatting in
communities. The simulation results show that our fears associated with a
fundamental change in social behaviour caused by the widespread use of modern
information and communication technologies are not groundless. Moreover, these
changes have led to a violation of the basic assumptions on which the
mathematics of sociological measurements is based. The main argument in favour of
such a conclusion is the observable fact, saying for the existence of a positive
lower bound for measurement errors. The mention of this e ect demonstrated
by simulation modelling was described in the works of sociologists devoted to
the survey method. Their reasoning is informal and far from mathematical ones.
In the context of this reasoning, sociologists noted the existence of distortion
e ects always present in such measurements. In the context of this reasoning,
sociologists noted the existence of distortion e ects always present in such
measurements. One can mention, for example, the book of Walter Lippmann [
b) m = 50 Fig. 8. Behaviours of LR Model 1000
In the case, if this hypothesis is con rmed, we will have to admit that the assumption of complete observability [4, p. 14] is wrong for intensively communicating communities. In other words, for studying such communities we need to use models similar to rather quantum than classical models of physical systems. Of course, this does not mean that mathematics of quantum theory is adequate for describing dynamics of intensively communicating communities. Hence, the challenge to nd the adequate mathematical language for studying this class of systems.
Summing up our discussion, we can formulate the following problems for the top-priority research 1. conduct a detailed study of the dependence of the behaviour of the LR-model on the parameter m; 2. establish the dependence of the measurement error on the rate of this measurement; 3. generalise the obtained results for more complicated than pairwise communications; 4. build a simulation model for communities exposed to external in uences; 5. establish the character of the dependencies between parameters of the external in uence and the system behaviour; 6. nd out whether the community exposed to external in uences is a system managed by these in uences.
If all these studies give a positive result then the problem to ensure certain community behaviour in the presence of limited resources that provide external in uence on the system can be set. 12. Rapoport, A.: Contributions to the theory of random and biased nets. Bulletin of
Mathematical Biophysics 19, 257{277 (1957) 13. Rashevsky, N.: Mathematical Theory of Human Relations: An Approach to Mathematical Biology of Social Phenomena. Principia Press, Bloomington, 2nd edn. (1949)