Social mood, the aggregated mood of a society, emerges from complex system of individual moods and their in uences on each other. The real social networks consist of millions or even billions nodes constantly interacting with each other. Can such a complex system be modeled by a graph consisting of a small number of agents with simple interactions between them? Pro le of Mood States, known and well-vetted psychometric instrument, distinguishes seven mood dimensions (Tension, Happiness, Calmness, Vigor, Fatigue, Confusion and Friendliness). If we apply them to a society at large, i.e. to social mood, is it possible to measure in uences of one mood dimension on another? In addition to this, is it possible to both maintain good approximations of social mood changes and be able to observe such interactions at the same time? In this work we investigate these questions and propose a framework which can approximate or even, in some circumstances, be predictive of future social mood states. The framework consists of a model of social in uence and an evolutionary algorithm learning proper network topology and model parameters.
From the psychological research it is known that the emotional state, as well as
the amount of information, play the main role in human decision-making [
However, emotions can profoundly a ect human decision-making process as
well, in many cases driving an individual to make a choice that seems "irrational"
in the framework of said theory. For example, behavioral nance has provided
proofs stating that nancial decisions are signi cantly a ected also by emotion
and mood and not only by rational utility maximization [
Thus, emotions a ect individual choices and decisions. Does this also apply to
larger groups of people, i.e. can societies experience mood states that a ect their
collective decision making? Prechter's socionomic hypothesis suggests that the
social mood drives various types of social action in the areas of cultural, political
and nancial behaviour [
The rst problem with such investigations is to actually nd a way to
measure the social mood. Large surveys of public mood are generally expensive and
di cult to undertake. That is why there were proposed some ways to assess
the social mood indirectly. For example: from the results of football games [
In this paper we propose a framework that can translate huge, highly complicated social network (of individual moods and their in uences on each other) to a fairly simple and an order of magnitude smaller graph of agents. It behaves in a similar manner to the real network concerning dynamics and interactions of the mood dimensions. The translated network can be then more easily analyzed.
Such a framework enhances the state of the art of social sciences, o ering a
tool to measure and to interpret social mood and the interactions between the
mood dimensions. Its novelty is provided by its data-driven approach. Most of
the social in uence models employ bottom-up methodology: begin with simple
rules (of agents interactions), then observe the emergent behaviour of the system
[
The paper is structured as follows. In Section 2 we describe our framework. Firstly, we explain how we assess the social mood. Then, we describe our model of social mood. The description involves a model of social in uence and an evolutionary algorithm aiming to nd the best network topology and model parameters. Section 3 describes the empirical experiments that were conducted. In Section 4 we discuss the results of the experiments. We draw nal conclusions in Section 5. 2
In a real world people a ect each others individual emotional states during communication. If we sum those individual emotional states up, we will receive a global measure called social mood. The question is, if we can replace the real social network of emotional in uences with its model, say with a number of nodes two times smaller? And at the same time be able to maintain similar dynamics of mood in uences and good approximation of a global mood state? Then, could we create a model four times smaller? How small could that model be? It is clear that the smaller it is, the easier it would be to analyze it and to understand its dynamics.
In this paper we propose a framework, which is able to translate huge complex social network of individuals to a simple graph with xed, small number of nodes (not more than 50 nodes). In this graph each node is an agent which is a representation of a class of individuals in the original network. Every agent apart from its mood state, has its own level of impressionability and in uence. Respectively, these are the measures of how much an agent is sensitive to in uence of others and how in uential it is. The edge between nodes denotes their ability to a ect each other. The values of parameters and the topology of the graph is determined by data-driven evolutionary algorithm which approximates the social mood time series.
The next two subsections will describe the framework in detail. First, we will describe how we measure social mood and then, how model of its dynamics is constructed. 2.1
We measure social mood by analyzing Twitter feed in terms of 7 mood dimensions. We list them here (with explanation of what does, respectively, the low and high score of each dimension mean): 1. Tension - relaxed or anxious, 2. Happiness - happy or depressed, 3. Calmness - calm or angry, 4. Vigor - apathetic or vital, 5. Fatigue - rested or tired, 6. Confusion - sure or confused, 7. Friendliness - aloof or kind.
We use similar mood dimensions and methodology of assessing the public mood
to the one used in [
Data We recorded a collection of public tweets which were posted during 14 days from July 7th to July 20th, 2014. We were interested only in tweets expressing author's mood state, thus we only tracked tweets containing words: "feel" and "feeling" (20,110,489 tweets). For each post, we obtained its date and time of submission, as well as the content of the message (which is a text limited to 140 characters).
One can have an impression that the dataset is particularly small (14 days), concerning the fact that in other papers datasets can span over several months. The di erence, though, is in temporal resolution of the datasets. Whereas researchers usually measure social mood in terms of days, in this work we measure it every 5 minutes. We do it to be able to observe the intraday dynamics of collective emotions and to be able to track the microchanges in social mood. We believe that such investigations may be helpful, for instance, for nancial intraday traders, for trading algorithms or for people responsible for communication and public relations. If we compare the sizes of the datasets, we will obtain 14d 24h 12 = 4032 time intervals for our dataset. In this paper experiments were conducted for time periods between 09:30 and 16:00 EST from Monday till Friday, as these are the times when New York Stock Exchange is opened. This gives us 10d 6:5h 12 = 780 time slots. On the contrary, if we take a daily resolution into consideration and, say, we will have a dataset of 9 months, it gives us around 9m 31d = 279 time intervals.
Another fact is that the volume of tweets posted nowadays is much greater
than it used to be in the past. In this paper we collected 20,110,489 tweets during
14 days and in [
To create a computational version of the test, we expand the basic lexicon of 65 adjectives from POMS with similar words, which we obtain by analysing word co-occurrences in big collections of texts. The expanded lexicon consists of 965 associated terms which are collected in the following procedure. We use Bing search engine to query for phrases "is [adj] and" and "was [adj] and", where [adj] denotes a particular adjective from the original lexicon which we want to nd similar words to1. For each of the queries, we download rst 200 results. For each result, we extract the word after conjunction and. Then we sort extracted words by most frequent occurrences. From the most frequent words we choose similar adjectives by hand. The advantage of querying search engines is that they are a relatively simple way of searching over a large collection of documents. Moreover, it also enable us to retrieve similar words which actually are in use.
Having the lexicon of emotional words, the social mood of Twitter feed is measured in the following way. Tokenization is performed on each tweet and then each word from a tweet is compared with each adjective from the lexicon. If there is a match, the adjective from the lexicon is mapped back to its original POMS term and via the POMS scoring table to its respective POMS dimension. Then, a counter of corresponding dimension is incremented by one.
To obtain a social mood time series we split our collection of tweets into groups of messages sent in 5 minutes long time periods. For each hour H, we distinguish time intervals: [H:00; H:05), [H:05; H:10), ..., [H:55; H + 1:00). Then for tweets from each of such time intervals, we employ our mood measuring procedure. At the end, we obtain times series: where t corresponds to successive time intervals and
M = fMt : t 2 T g
Mt0 = [d0t;1; d0t;2; :::; d0t;7] where, d0t;1; d0t;2; :::; d0t;7 2 N are values of respective mood dimensions: Tension, Happiness, Calmness, Vigor, Fatigue, Confusion and Friendliness.
For our social mood time series not to be dependent on the volume of tweets in a given period of time, we then normalize the values of mood dimensions in the following way. For each mood dimension dt;i, i 2 1; 2; :::; 7: dt;i =
d0t;i P7 j=1 d0t;j 1 Bing search engine distributes a dedicated http://www.bing.com/dev/en-us/dev-center.
API. For the details visit (1) (2) (3) Obtaining nal elements of social mood time series:
Mt = [dt;1; dt;2; :::; dt;7] (4) All mood times series in the rest of the paper are normalized in the same manner. 2.2
Real-world social mood networks consist of big number of people, each of them having their own mood state. These people can interact with their acquaintances, a ecting their moods, as well as being a ected by them.
Therefore, if we want to translate such a network to a smaller graph, we need to nd a way to model 3 things: 1. Collective mood state of individuals - mostly, we already have it done. We model it with 7-dimensional vector like in equation (4). 2. Topology of the social network - we need to nd a way to translate the connections between nodes in a big social network to analogous connections in a small graph. 3. Social in uence - we need to build a model of how agents are a ecting each others mood states in a small graph.
We will start with approaching the topology issue, then we will describe our model of social in uence and nally we will present the evolutionary algorithm which aims to nd the best topology and in uence parameters. All these components will, in the end, describe our framework.
Topology of the network We use evolutionary approach to nd the best network topology (as well as other parameters of the model). This choice is made, because we want the algorithm: { to be population-based - in order to be able to compare obtained solutions at any time of the algorithm run, { to be anytime - meaning that it can return a valid solution, even if it is interrupted before it ends, { not to make any assumptions about the topology and the parametrs of the model.
However, evolutionary algorithm needs initial population in which topologies are somehow constructed. To model the social network in the beginning stage of the evolution, we decided to use two classes of graphs.
First class are random graphs. We believe that they are the simplest and the most natural way to initialize network topologies, concerning the fact that we take advantage of an evolutionary approach. To construct the particular graph, rst we draw p 2 (0; 1) from the uniform distribution. Then, every possible edge occurs independently with the probability p.
Second class consists of scale-free graphs, which are graphs whose degree
distribution follows the power law. The motivation behind this choice is that
many real-world social networks, as well as cyberspace networks, are conjectured
to be scale-free [
Model of social in uence There exists a multitude of social in uence models in the sociophysics literature. They can be classi ed into discrete (including binary) models and continuous models depending on the representation of opinions that are being in uenced.
The typical discrete models include Ising model [
This fact brings our attention to the continuous models, that mainly include
Hegelsmann-Krause model [
Therefore, we propose our own model of social in uence, which is similar to Hegelsmann-Krause model, but also introduces some major di erences. They enable us to model the characteristics of mood dimensions dynamics, which we just mentioned. It is de ned as follows: 1. A = f1; 2; :::; ng is the set of agents. 2. Each agent i, at discrete moment in time t, has its own mood state:
Mi;t = [di;t;1; di;t;2; :::; di;t;D] where di;t;k 2 R and D is a constant denoting the number of mood dimensions2. 3. Each agent i, at discrete moment in time t, knows if each of its mood dimensions increased or decreased during the last time step: i;t = [ i;t;1; i;t;2; :::; i;t;D] i;0 = [ai;0;1; ai;0;2; :::; ai;0;D] where i;t;k 2 R and i;t;k = di;t;k di;t 1;k, for t > 0. For each agent i rst element of the sequence i is speci ed at the beginning: where ai;0;1; ai;0;2; :::; ai;0;D are speci ed initial values. 4. Each agent i has its level of: { inf luence 'i 2 [0; 1], which denotes how much it is a ecting others, { impressionability i 2 [0; 1], which denotes how much it is being a ected by others. 5. Agents are organized in the network N = (A; E), where E is a set of connections or edges, which are 2-element subsets of the set A. 6. Sequence of agent's mood states is speci ed as follows. For each agent i: { First element of the sequence is speci ed at the beginning:
Mi;0 = [bi;0;1; bi;0;2; :::; bi;0;D] (9) where bi;0;1; bi;0;2; :::; bi;0;D are initial values. { Elements of the next time steps t > 1 are de ned using a recursive rule.
For each mood dimension di;t;k, k 2 1; 2; :::; D: di;t;k = di;t 1;k + idi;t 1;k jsgn( j;t 1;k)'j (10) where the sum is made over all agents j connected to the agent i (indicated by the set E). 7. The global social mood state, at each discrete moment in time t, is de ned as a sum of agents' mood states:
Mt = i2AMi;t (11)
Thus, in the model in every discrete time step t part i of agent's i mood can be a ected by its neighbours. If their particular mood dimension went up in the previous time step, the neighbours will try, taking their in uence parameters ' into consideration, to increase it. In other case, analogously, they will try to decrease it. 2 In this paper D = 7. (6) (7) (8) How to construct the model from Twitter data? We only described aggregated social mood M = fMt : t 2 T g acquired from Twitter data so far. However, to be able to use it in our model of social in uence, we need split data. To achieve this, the idea is to split tweets into some kind of equivalence classes associated with mood dimensions. The agents then are not representatives of individuals, but representatives of mood dimensions. The easiest way to achieve this is to employ the mood measuring procedure for each tweet, identify its dominant mood dimension (the dimension with the highest score) and then classify the tweet as Tension, Happiness, Calmness, Vigor, Fatigue, Confusion or Friendliness representative. If there is more than one dimension with the highest score, classify the tweet randomly as a representative of one of its dominant dimensions.
Using this procedure, we can obtain decomposition of tweets into seven different groups associated with mood dimensions. We can then couple each group i 2 f1; 2; :::; 7g with di erent agent, obtaining corresponding mood time series:
Mi = fMi;t : t 2 T g
Having agents as representatives of mood dimensions, we can then apply our social in uence model to observe how di erent dimensions are in uencing each other. This way, we obtain a graphical representation of in uence dynamics. In this approach the in uence of one mood dimension on the other is not based on the actual Twitter social graph or other kind of individuals topology. The in uence is measured on the macro level, the same way that in a society optimists have an in uence on pessimists or electorate of one political party has an in uence on the other electorate. The in uence is measured as the change in aggregated sum of micro-interactions among the individuals.
On the other hand, conducting such a simulation (running the model), we are able to calculate mood estimators Mdi;t - set of vectors of mood scores of every agent i, in every time step t. Those estimators can be then summed to obtain global social mood estimator Mct. It is then easy to assess how good is our estimation (and all in all - simulation) calculating the mean absolute percentage error (MAPE):
M AP E = 1 t2T
X jMct jT j t
Mt
Mtj (12) (13)
We choose MAPE as a measure of performance of our simulations, because we need to compensate for two things: 1. scores of some mood dimensions are usually much greater than scores of other mood dimensions - therefore, we need percentage error to measure performance of all dimensions approximations, 2. the values of estimators may be greater or less than actual mood score therefore, we need to measure error in absolute values.
Another thing is that, as the experiments showed, only seven agents in the
model may not be a su cient number to approximate the global social mood M
well. We may therefore want to have more than just one representative of each
mood dimension. To achieve this, we introduce splitting parameter S. We then
employ the same grouping procedure to tweets set as before, but after obtaining
seven groups for seven di erent mood dimensions, we split each group into S
smaller groups of the same size. In this paper S 2 f1; 2; :::; 5g, therefore we
conducted experiments for numbers of 7, 14, 21, ..., 35 agents in the model.
Evolutionary algorithm The main parameters that need to be adjusted in
social mood model to re ect the real-world data are the network topology, 0,
' and parameters of the agents. In our approach, we start with a random
graph or scale-free graph network topology (with the same probability), random
values of 0 vector generated independently from [
We start with population of P = 100 randomly generated models of social mood3. For each model we conduct the simulation and calculate the mean absolute percentage error (MAPE). M AP E = [M AP E1; M AP E2; :::; M AP E7] is also a vector, because there are 7 mood dimensions, so we calculate the mean value of this vector coordinates obtaining our nal error Er.
We then sort our models ascending by the value of Er and build next generation of models in the following way. The m ttest models (where m is equal to 50% in our simulations) are retained in the next generation and the others are discarded. A single mutated copy is made of each remaining model so that the size of the population always remains constant. Mutations are applied to r agents from a particular model (where r is equal to 10%) and can take four forms with equal probability: 1. The agent receives new values of inf luence ' and impressionability parameters. They are generated independently and randomly from [0; 1] interval. 2. A new link in the network is added between the agent and di erent, randomly chosen agent. 3. An existing, randomly chosen link of the agent is removed from the network. 4. The agent receives new values of 0 vector. They are generated independently and randomly from [ 1; 1] interval.
After G generations, we obtain the model with the least error Er which is the best t to the data. 3 Some parameters of the evolutionary algorithm are constrained (eg. P = 100, m = 50%, r = 10%). The values were handpicked to optimize the performance.
In order to evaluate the framework, the experiments are conducted to see how well can it approximate the social mood changes and predict the future values of social mood. To evaluate the quality of the framework's approximations of social mood changes, for each value of the splitting parameter S and for the number of generations G = 300, we test it on 60 one-hour-long time intervals. They span across 10 days in July 2014, from 7th till 11th and from 14th till 18th. The periods of time lay between 9:45 and 15:45 EST, as this is the time when New York Stock Exchange is opened (actually it is 9:30 - 16:00, but rst and last quarters are the most unstable, that is why we do not want to include them). Much of the research on social mood and electronic sentiment is focused on nding nancial applications, therefore we wanted to follow that trend. Each time period starts at [H:45; H:50), which is the starting point, and then there are 11 time intervals that are approximated [H:50; H:55), [H:55; H + 1:00], ..., [H + 1:40; H + 1:45). Thus, we test the approximation on 10d 6h 11 = 660 time slots. The results can be seen in the Table 1. They were obtained against a benchmark of G = 300 generations in the evolutionary algorithm. These outcomes can be further improved if the computations are longer (e.g. for S = 1 we can achieve 2 percentage point better results if we set G to 600). In order to evaluate the predictive power of the models, for each value of the splitting parameter S and for each model computed in previous subsection between 9:45 and 14:45 EST, we predict twelve following ve-minutes-long time intervals. Therefore, on each of ten days, for ve di erent starting hours, we predict twelve time intervals. Thus, in our experiment we predict 10d 5h 12 = 600 time slots. The results can be seen in the Table 1.
The comparison of social mood approximations for di erent values of the splitting parameter S con rms the intuitive anticipation that the larger the value is, the better are the approximations (in a matter of fact concerning all comparison indicators: mean, median and standard deviation). One could suspect this fact. In larger graphs there are more agents and more connections among them. Thus, it can be easier for the model to tune to the data. On the other hand, the person studying the graphical model would like to have as small network as possible. They are then easier to analyze and to understand. As the experiments show, the approximations of the models with smaller splitting parameters are worse by around one percentage point. In most cases, this should still be a satisfactory level of error, which one can accept for the sake of the clarity of the graphical model.
As far as the predictive power of models is concerned, the problem with social mood assessed by the means of Twitter is that this kind of system is not closed. In other words, external factors a ect the social mood on Twitter, and not only users in uence each other. Therefore, prediction power of a particular model is limited by the way of how the next time interval is similar to the previous one in terms of mood changes dynamics.
During our experiments the predictive power of models with the splitting parameter S = 1 turns out to be the best, even though they are not the best t to the data. Most probably it is due to the over tting of models with greater splitting parameter. In addition to this, in case of S = 1 there is no noise created by the interactions between the representatives of the same mood. In the Figure 1, MAPEs of ten predicted time intervals for di erent values of S are presented. One can notice said smaller amounts of noise for S = 1.
Topologies of the evolved networks are something that could be a topic of a separate investigation. From the models that we obtained during our experiments, we can state that they are di erent for di erent moments of time, concerning not only their shapes but also their parameters. These facts are not surprising and are probably due to the fact that in di erent moments of time people were exposed to di erent external factors. The question of what kind of social situation causes which kind of network topology would be an interesting issue for the future research.
From our conclusions about topologies, rst notable fact is that, during the evolution, bigger networks lose their "scale-free property" (understood as a degree distribution following the power law in a graph which is not in nite). Some models' degree distributions look quite similar to the distributions following the power law, but still are disturbed. Rest of the networks turn into more random graphs.
Another fact concerning bigger networks produced by the algorithm (with S > 2) is that they are not really easy to read and analyze. Each mood dimension have a few representatives, but usually only some of them are connected to others. It is not clear how someone should interpret such a graph. We can say that only part of the people with particular dominant mood dimension is engaged in interactions, but still conclusions are chaotic. Another issue is the interaction between representatives of the same mood dimension. The aim of the framework is to translate the complexity to something simple. It is not obvious if bigger networks can make that much of a simpli cation.
This brings our attention to smaller networks. Small graphs, with only one representative for each mood dimension (S = 1), do not possess the problems stated above. They also have bigger predictive power of social mood changes (only the approximation is a little bit worse, but as it was stated earlier - it is satisfactory). Thus, we may recommend them as a better source of information and a better tool to investigate social mood. We can see an example of such a network in Figure 2. 5
In this paper, we investigate whether a complex network of individual emotions in uencing each other can be approximated by a small graph with similar properties. Our experiments show that small networks can indeed approximate social mood with reasonable mean absolute percentage errors ranging from 9.19% to 10.69%. These results can be further improved using longer computations. Our studies show also that if the following period of time is similar to the previous one, meaning that it is not a ected by big amount of external factors, models can be even predictive of the future social mood states. The models with the splitting parameter S = 1 can predict following twelve intervals of time with mean MAPE = 18.63%.
In the literature, there are hardly any data-driven models of social mood dynamics, as well as data-driven models of social in uence (there are models of these phenomena, but they are not data-driven). Thus, our attempt proposes quite complete framework of assessing and analyzing social mood based on the real-world data. One could argue, that other techniques could be used to approximate/predict social mood like Markov Models, Conditional Random Fields etc. Although, the bene t of our approach is that it creates "the map" of interactions between the mood dimensions. We are able to notice which dimension is connected to which one. We can therefore infer where the in uences are and how strong they are (looking at the values of in uence and impressionability parameters).
Presented framework may be useful in situations where quick information about emotion dynamics is needed. For instance, for people responsible for communication or public relations. Di erent situations may include, for example, nancial intraday trading: algorithmic as well as conducted by human. On the other hand, when the temporal resolution of mood measurements is changed to longer periods of time, like days for instance, the framework can also be useful for long-term analysis of social mood dynamics. Such investigations might be of high interest for variety of institutions monitoring societies.
As far as future work in concerned, one can notice, that during the evolution most of the networks lose their scale-free property. Therefore, it is not clear if this is a good starting point of evolutionary algorithm. One could consider di erent initial topologies. Another interesting question is what types of graphs are being produced by the framework. Is it a single class or a group of them?
Apart from further work on the framework itself or studying types of networks that it produces, there are also issues more of a sociological nature. One could investigate whether di erent network topologies and model parameters are somehow related to the nature of real-world social events. We hypothesize that deeper studies of graphs' "shapes" and distributions of in uence/impressionability measures can give interesting conclusions about society dynamics, as well as re ect its qualitative properties. Such correlations between social reality and topologies of small graphs could be of a great interest for social scientists. 6
The paper is cofounded by the European Union from resources of the European Social Fund. Project PO KL "Information technologies: Research and their interdisciplinary applications".