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What is the best way for a group of individuals to cooperate? This is a longstanding question with roots in game theory, a branch of science which uses mathematical models of how individuals should best strategize for the optimal result.

A simple example is the prisoner’s dilemma: Two people are arrested for an alleged bank robbery. The police take them downtown and place them in individual, isolated interrogation rooms.

The police admit they don’t have enough evidence to convict them both, and give each the same option: if he confesses and his partner does not, they will release the confessor and convict the other of the serious charge of bank robbery. But if one does not confess and the other does, the first will get a lengthy prison sentence and the other will be released. If both confess, they will both be put away for many years. If neither confesses, they will be arraigned on a lesser charge of gun possession.

What should each do to minimize their time in jail? Does an individual stay silent, trusting his partner to do the same and accept a shorter prison sentence? Or does he confess, hoping the other stays silent. But what if the other confesses too? It is an unenviable position.

There is no correct solution to the prisoner’s dilemma. Other similar problems are the game of chicken, where each driver races towards the other, risking a head-on crash, or swerving away at the last minute and risking humiliation—being called “chicken” for a lack of courage. Many other simple games exist.

Now imagine a group—they may be people, or they may be cellular organisms of some sort. What kind of cooperation gives the optimal result, when each individual is connected to some others and pays a cost (money, energy, time) to create a result that benefits all? It’s a given that individuals are selfish and act in their own best interests, but we also know that cooperation can result in a better outcome for all. Will any take the risk, or look out only for themselves?

A long-standing result is that, in a homogeneous network where all individuals have the same number of neighbours, cooperation is favoured if the ratio between the benefit provided by a cooperator and their associated cost paid exceeds the average number of neighbours.

But people are not homogeneous, they’re heterogeneous, and they don’t usually have the same number of links to neighbours as does everyone else or change their strategy at the same rates.

It is also known that allowing each individual to update their strategy at exactly the same time, such as immediately mimicking their neighbour, significantly alters the evolution of cooperation. Previous investigations have reported that pervasive heterogeneous individual connections hinder cooperation when it’s assumed that individuals update their strategies at identical rates.

Now a group of researchers located in China, Canada and the US have found a surprising result: when individuals’ strategy update rates vary inversely with their number of connections, heterogeneous connections outperform homogeneous ones in promoting cooperation. The study is published in the journal Nature Communications.

“How to analyse the quantitative impact of the prevalent heterogeneous network structures on the emergence of group optimal strategies is a long-standing open question that has attracted much attention,” said Aming Li, a co-author and Assistant Professor in Dynamics and Control at Peking University.

Li’s team solved the problem by analytical calculations backed up by computer simulations, to find the fundamental rule for maintaining collective cooperation: “The nodes with substantial connections within the complex system should update their strategies infrequently,” he says. That is, individual strategy update rates should vary inversely with the number of connections they have in the network. In this way, a network with heterogeneous connections between individuals outperforms a network with homogeneous connections in promoting cooperation.

The team has also developed an algorithm that most efficiently finds the optimal strategy update rates that brings about the group’s optimal strategies, which they call OptUpRat. This algorithm helps collective utility in groups and, Li says, “is also essential in developing robotic collaborative systems.” The finding will be useful to researchers in such multidisciplinary fields as cybernetics, artificial intelligence, systems science, game theory and network science.

“We believe that utilizing AI-related techniques to optimize individual decisions and drive collective intelligence will be the next research hotspot.”

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Credit of the article given to David Appell , Phys.org

 

 

Calculating probabilities can be complicated, as this classic “what’s behind the doors” problem shows, says Peter Rowlett.

Calculating probabilities can be tricky, with subtle changes in context giving quite different results. I was reminded of this recently after setting BrainTwister #10 for New Scientist readers, which was about the odds of seating two pairs of people adjacently in a row of 22 chairs.

Several readers wrote to say my solution was wrong. I had figured out all the possible seating arrangements and counted the ones that had the two groups adjacent. The readers, meanwhile, seated one pair first and then counted the ways of seating the second pair adjacently. Neither approach was wrong, depending on how you read the question.

This subtlety with probability is illustrated nicely by the Monty Hall problem, which is based on the long-running US game show Let’s Make a Deal. A contestant tries to guess which of three doors conceals a big prize. They guess at random, with ⅓ probability of finding the prize. In the puzzle, host Monty Hall doesn’t open the chosen door. Instead, he opens one of the other doors to reveal a “zonk”, an item of little value. He then offers the contestant the opportunity to switch to the remaining door or stick with their first choice.

Hall said in 1991 that the game is designed so contestants make the mistaken assumption that, since there are now two choices, their ⅓ probability has increased to ½. This, combined with a psychological preference to avoid giving up a prize already won, means people tend to stick

Marilyn vos Savant published the problem in her column in Parade magazine in 1990 along with the answer that you are much more likely to win if you switch. She received thousands of letters, many from mathematicians and scientists, telling her she was wrong.

Imagine the host opened one of the unchosen doors at random: one-third of the time, they would reveal the prize. But in the remaining cases, the prize would be behind the chosen door half the time, for a probability of ½.

But that isn’t really the problem being solved. The missing piece of information is that the host knows where the prize is, and of course the show must go on. There is a ⅓ probability that the prize is behind the chosen door, and therefore a ⅔ probability that it is behind one of the other two. Being shown a zonk behind one of the other two hasn’t changed this set-up – the door chosen still has a probability of ⅓, so the other door carries a ⅔ probability. You should switch.

Probability problems depend on the precise question more than people realise. This is why it might seem surprising when you run into a friend, because you aren’t considering the number of people you walked past and how many friends you might see. And for scientists, it is why they have to be very careful about what their evidence is really telling them.

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*Credit for article given to Peter Rowlett*

Whether it’s physical phenomena, share prices or climate models—many dynamic processes in our world can be described mathematically with the aid of partial differential equations. Thanks to stochastics—an area of mathematics which deals with probabilities—this is even possible when randomness plays a role in these processes.

Something researchers have been working on for some decades now are so-called stochastic partial differential equations. Working together with other researchers, Dr. Markus Tempelmayr at the Cluster of Excellence Mathematics Münster at the University of Münster has found a method which helps to solve a certain class of such equations.

The results have been published in the journal Inventiones mathematicae.

The basis for their work is a theory by Prof. Martin Hairer, recipient of the Fields Medal, developed in 2014 with international colleagues. It is seen as a great breakthrough in the research field of singular stochastic partial differential equations. “Up to then,” Tempelmayr explains, “it was something of a mystery how to solve these equations. The new theory has provided a complete ‘toolbox,’ so to speak, on how such equations can be tackled.”

The problem, Tempelmayr continues, is that the theory is relatively complex, with the result that applying the ‘toolbox’ and adapting it to other situations is sometimes difficult.

“So, in our work, we looked at aspects of the ‘toolbox’ from a different perspective and found and proved a method which can be used more easily and flexibly.”

The study, in which Tempelmayr was involved as a doctoral student under Prof. Felix Otto at the Max Planck Institute for Mathematics in the Sciences, published in 2021 as a pre-print. Since then, several research groups have successfully applied this alternative approach in their research work.

Stochastic partial differential equations can be used to model a wide range of dynamic processes, for example, the surface growth of bacteria, the evolution of thin liquid films, or interacting particle models in magnetism. However, these concrete areas of application play no role in basic research in mathematics as, irrespective of them, it is always the same class of equations which is involved.

The mathematicians are concentrating on solving the equations in spite of the stochastic terms and the resulting challenges such as overlapping frequencies which lead to resonances.

Various techniques are used for this purpose. In Hairer’s theory, methods are used which result in illustrative tree diagrams. “Here, tools are applied from the fields of stochastic analysis, algebra and combinatorics,” explains Tempelmayr. He and his colleagues selected, rather, an analytical approach. What interests them in particular is the question of how the solution of the equation changes if the underlying stochastic process is changed slightly.

The approach they took was not to tackle the solution of complicated stochastic partial differential equations directly, but, instead, to solve many different simpler equations and prove certain statements about them.

“The solutions of the simple equations can then be combined—simply added up, so to speak—to arrive at a solution for the complicated equation which we’re actually interested in.” This knowledge is something which is used by other research groups who themselves work with other methods.

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Credit of the article given to Kathrin Kottke, University of Münster

The first commercial gambling operations emerged, coincidentally or not, at the same time as the study of mathematical probability in the mid-1600s.

By the early 1700s, commercial gambling operations were widespread in European cities such as London and Paris. But in many of the games that were offered, players faced steep odds.

Then, in 1713, the brothers Johann and Jacob Bernoulli proved their “Golden Theorem,” known now as the law of large numbers or long averages.

But gambling entrepreneurs were slow to embrace this theorem, which showed how it could actually be an advantage for the house to have a smaller edge over a larger one.

The book “The Gambling Century: Commercial Gaming in Britain from Restoration to Regency,” WEexplain how it took government efforts to ban and regulate betting for gambling operators to finally understand just how much money could be made off a miniscule house edge.

The illusion of even odds in games that were the ancestors of roulette and blackjack proved immensely profitable, sparking a “probability revolution” that transformed gambling in Britain and beyond.

A new theorem points to sneaky big profits

The law of large numbers refers to events governed by chance.

When you flip a coin, for example, you have a 50% – or “even money” – chance of getting heads or tails. Were you to flip a coin 10 times, it’s quite possible that heads will turn up seven times and tails three times. But after 100, or 1000, or 10,000 flips, the ratio of “heads” to “tails” will be closer and closer to the mathematical “mean of probability” – that is, half heads and half tails.

Mathematicians Johann and Jacob Bernoulli developed what’s known today as the law of large numbers. Oxford Science Archive/Print Collector via Getty Images

This principle was popularized by writers such as Abraham De Moivre, who applied them to games of chance.

De Moivre explained how, over time, someone with even the smallest statistical “edge” would eventually win almost all of the money that was staked.

This is what happens in roulette. The game has 36 numbers, 18 of which are red and 18 of which are black. However, there are also two green house numbers – “0” and “00” – which, if the ball lands on them, means that the house can take everyone’s wager. This gives the house a small edge.

Imagine 10 players with $100 apiece. Half of them bet $10 on red and the other half bet $10 on black. Assuming that the wheel strictly aligns with the mean of probability, the house will break even for 18 of 19 spins. But on the 19th spin, the ball will land on one of the green “house numbers,” allowing the house to collect all the money staked from all bettors.

After 100 spins, the house will have won half of the players’ money. After 200 spins, they’ll have won all of it.

Even with a single house number – the single 0 on the roulette wheels introduced in Monte Carlo by the casino entrepreneur Louis Blanc – the house would win everything after 400 spins.

This eventuality, as De Moivre put it, “will seem almost incredible given the smallness of the odds.”

Hesitating to test the math

As De Moivre anticipated, gamblers and gambling operators were slow to adopt these findings.

De Moivre’s complex mathematical equations were over the heads of gamblers who hadn’t mastered simple arithmetic.

Gambling operators didn’t initially buy into the Golden Theorem, either, seeing it as unproven and therefore risky.

Instead, they played it safe by promoting games with long odds.

One was the Royal Oak Lottery, a game played with a polyhedral die with 32 faces, like a soccer ball. Players could bet on individual numbers or combinations of two or four numbers, giving them, at best, 7-to-1 odds of winning.

Faro was another popular game of chance in which the house, or “bank” as it was then known, gave players the opportunity to defer collecting their winnings for chances at larger payouts at increasingly steep odds.

Faro was a popular game of chance in which players could delay collecting their winnings for the chance to win even bigger sums. Boston Public Library

These games – and others played against a bank – were highly profitable to gambling entrepreneurs, who operated out of taverns, coffeehouses and other similar venues. “Keeping a common gaming house” was illegal, but with the law riddled with loopholes, enforcement was lax and uneven.

Public outcry against the Royal Oak Lottery was such that the Lottery Act of 1699 banned it. A series of laws enacted in the 1730s and 1740s classified faro and other games as illegal lotteries, on the grounds that the odds of winning or losing were not readily apparent to players.

The law of averages put into practice

Early writers on probability had asserted that the “house advantage” did not have to be very large for a gambling operation to profit enormously. The government’s effort to ban games of chance now obliged gaming operators to put the law of long averages into practice.

Further statutes outlawed games of chance played with dice, cards, wheels or any other device featuring “numbers or figures.”

None of these measures deterred gambling operators from the pursuit of profit.

Since this language did not explicitly include letters, the game of EO, standing for “even odd,” was introduced in the mid 1740s, after the last of these gambling statutes was enacted. It was played on a wheel with 40 slots, all but two of which were marked either “E” or “O.” As in roulette, an ivory ball was rolled along the edge of the wheel as it was spun. If the ball landed in one of the two blank “bar holes,” the house would automatically win, similar to the “0” and “00” in roulette.

EO’s defenders could argue that it was not an unlawful lottery because the odds of winning or losing were now readily apparent to players and appeared to be virtually equal. The key, of course, is that the bar holes ensured they weren’t truly equal.

Although this logic might not stand up in court, overburdened law enforcement was happy for a reason to look the other way. EO proliferated; legislation to outlaw it was proposed in 1782 but failed.

In the 19th century, roulette became a big draw at Monte Carlo’s casinos.Hulton Archive/Getty Images

The allure of ‘even money’

Gambling operators may have even realized that evening the odds drew more players, who, in turn, staked more.

After EO appeared in Britain, gambling operations both there and on the continent of Europe introduced “even money” betting options into both new and established games.

For example, the game of biribi, which was popular in France throughout the 18th century, involved players betting on numbers from 1 to 72, which were shown on a betting cloth. Numbered beads would then be drawn from a bag to determine the win.

In one iteration from around 1720, players could bet on individual numbers, on vertical columns of six numbers, or other options that promised large payouts against steeper odds.

By the end of the 18th century, however, one biribi cloth featured even money options: Players could bet on any number between 36 and 70 being drawn, or on any number between 1 and 35. Players could also select red or black numbers, making it a likely inspiration for roulette.

In Britain, the Victorian ethos of morality and respectabilityeventually won out. Parliament outlawed games of chance played for money in public or private in 1845, restrictions that were not lifted until 1960.

By 1845, however, British gamblers could travel by steamship and train to one of the many European resorts cropping up across the continent, where the probability revolution had transformed casino gambling into the formidable business enterprise it is today.

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Credit of the article given to The Conversation

 

Australian teenagers have more disruptive maths classrooms and experience bullying at greater levels than the OECD average, a new report shows.

But in better news, Australian students report high levels of curiosity, which is important for both enjoyment and achievement at school.

The report, by the Australian Council for Educational Research (ACER) analysed questionnaire responses from more than 13,430 Australian students and 743 principals, to understand how their school experiences impact on maths performance.

What is the research?

This is the second report exploring Australian data from the 2022 Programme for International Student Assessment (PISA).

Australian teenagers have more disruptive maths classrooms and experience bullying at greater levels than the OECD average, a new report shows.

But in better news, Australian students report high levels of curiosity, which is important for both enjoyment and achievement at school.

The report, by the Australian Council for Educational Research (ACER) analysed questionnaire responses from more than 13,430 Australian students and 743 principals, to understand how their school experiences impact on maths performance.

 

What is the research?

This is the second report exploring Australian data from the 2022 Programme for International Student Assessment (PISA).

Author provided (no reuse)

The advantage gap

ACER’s first PISA 2022 report showed students from disadvantaged socioeconomic backgrounds were six times more likely to be low performers in maths than advantaged students.

It also showed the achievement gap between these two groups had grown by 19 points (or about one year of learning) since 2018.

This second report provides more insight into the challenges faced by disadvantaged students.

It shows a greater proportion of this group report learning in a less favourable disciplinary climate, experience lower levels of teacher support and feel less safe at school than their more advantaged peers.

Girls are more worried than boys

In last year’s report, the mean score for maths performance across OECD countries was nine points lower for girls than it was for boys. In Australia, the difference was 12 points.

The new report also showed differences in wellbeing. In 2022, a greater number of girls reported they panicked easily (58% compared to 23% of boys), got nervous easily (71% compared to 39%) and felt nervous about approaching exams (75% compared 49%).

Almost double the percentage of girls reported feeling anxious when they didn’t have their “digital device” near them (20% compared to 11%). Whether this was a phone, tablet or computer was not specified.

Overall, students who reported feeling anxious when they did not have their device near them scored 37 points lower on the maths test than those who reported never feeling this way or feeling it “half the time”.

Author provided (no reuse)

Curiosity a strong marker for performance

Curiosity was measured for the first time in PISA 2022. This included student behaviours such as asking questions, developing hypotheses, knowing how things work, learning new things and boredom.

Students in Singapore, the highest performing country in PISA 2022, showed the greatest levels of curiosity, followed by Korea and Canada. These were the only comparison countries to have a significantly higher curiosity score than Australia, with the Netherlands showing the lowest curiosity score overall.

As ACER researchers note: “curiosity is associated with greater psychological wellbeing” and “leads to more enjoyment and participation in school and higher academic achievement”.

They found Australia’s foreign-born students reported being more curious than Australian-born students, with 74% compared to 66% reporting that they liked learning new things.

What next?

Their findings highlight concerns for Australian education, such as persistently poor outcomes for disadvantaged students and higher stress levels experienced by girls. We need to better understand why this is happening.

But they also identify behaviours and conditions – such as high levels of curiosity – that contribute to a good maths performance and can be used by schools and policymakers to plan for better outcomes.

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Credit of the article given to The Conversation

Complex projects are made up of many activities, the duration of which vary according to a power law; this model can be used to predict overall project duration and delay.

We have all been frustrated when a project is delayed because one sub-task cannot begin before another ends. It is less well known that the process of scheduling projects efficiently can be described in mathematical terms.

Now, Alexei Vazquez, of technology company Nodes & Links and based in Cambridge, U.K., has shown that the distribution of activity lengths in a project follows the mathematical relationship of power law scaling. He has published his findings in The European Physical Journal B.

Any relationship in which one quantity varies as a power of another (such as squared or cubed) is known as a power law. These can be applied to a wide range of physical (e.g., cloud sizes or solar flares), biological (e.g. species frequencies in a habitat) and man-made (e.g. income distribution) phenomena.

In Vazquez’ analysis of projects, the quantities that depend on power laws were the duration of each of the activities that make up the project and the slack times between each activity, or “floats.”

Vazquez analysed data on 118 construction projects, together comprising more than 1,000 activities, that was stored in a database belonging to his company. The activity durations in a given project fitted a power law with a negative exponent (i.e., there were more short-duration activities, and a “tail” of small numbers of longer ones); the value of the exponent varied from project to project. The distribution of float times for the activities in a project can be expressed in a similar but independent power law.

He explained that these power law scalings arise from different processes: in the case of the activities, from a historical process in which a generic activity fragments over time into a number of more specialized ones. Furthermore, he showed that estimation of delays associated with a whole project depends on the power law scaling of the activities but not of the floats. This analysis has the potential to forecast delays in planned projects accurately, minimizing the annoyance caused by those long waits.

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Credit of the article given to Clare Sansom, SciencePOD

 

Gossip often has a negative connotation, but imagine you are part of a group deciding on a job candidate to hire or a local political candidate to back. Candidates who get a good reputation by helping others may be more likely to receive help in the form of a job offer or endorsement, a feedback loop known as indirect reciprocity. Gossip can facilitate

Previous research has shown that people tend to cooperate more when they think their peers are gossiping about their behaviour, gossipallows people to avoid potential cheaters, and gossip can punish freeloaders. Yet little was understood about how much gossip is required to foster cooperation and how incorrect information impacts the effects of gossip.

Researchers in the Plotkin Research Group in Mathematical Biology in the School of Arts & Sciences studied this issue by creating a model that incorporates two sources of gossip: randomly selected people versus a single source. They show that there is a mathematical relationship between these forms of gossip—meaning that understanding gossip with a single source also allows them to understand gossip with peers—and developed an analytical expression for the amount of gossip required to reach sufficient consensus and sustain cooperation.

Their findings are published in Proceedings of the National Academy of Sciences.

“The study of the spread of social information and the study of the evolution of cooperative behaviour are very mature fields, but there hasn’t been as much work done to combine those,” says first author Mari Kawakatsu, a postdoctoral researcher in the lab of biology professor Joshua B. Plotkin, the paper’s senior author.

“By merging ideas from the two fields, we were able to develop a mechanistic model of how information spread can help cooperative behaviour.”

Co-author Taylor A. Kessinger, also a postdoctoral researcher with a background in physics, says this analysis bridges the critical gap in past work on no gossip, where everyone’s opinion is private and independent, and infinitely fast gossip with total agreement about reputations. Kessinger has also seen the central role that indirect reciprocity plays on X, formerly known as Twitter, and how disagreement about reputations and ingroup-outgroup dynamics can incentivize bad behaviour.

“Systems of morality and reputation help ensure that good actors get rewarded and bad actors get punished. That way, good behaviour spreads and bad behaviour doesn’t,” Kessinger says. “If you punish a bad actor, you need to be sure that other people agree they’re guilty of wrongdoing. Otherwise, they might see you as the bad actor. Gossip can be one way to accomplish this.”

Plotkin says while past work has taken the basic model of indirect reciprocity and added various complications, such as stereotyping, this paper goes back and fills a gap in the theory. The paper provides a quantitative model that explains how many rounds of gossip are sufficient for people to change their cooperative or noncooperative behaviours, he says.

The paper involves a game-theoretical model where an interaction takes the form of a donation game, with each “donor” choosing whether to cooperate with each “recipient” by paying a cost to provide a benefit. All individuals serve once each as donor and recipient. Each then privately assesses the reputation of every donor by assessing their action toward a randomly selected participant, and a period of gossip about reputations follows. Private assessments and gossip continue until reputations equilibrate.

The authors note that behavioural strategies vary. Some always cooperate, some always defect, and some discriminate, meaning they cooperate when the recipient has a good reputation and defect when the recipient has a bad one. The researchers found that both forms of gossip tend to increase agreement about reputations, which in turn improves the equilibrium reputations of discriminators.

So, if gossip runs long enough, discriminators can eventually outcompete cooperators and defectors, which is a good outcome because discriminators are highly cooperative with one another and stable against noncooperative behaviours.

The researchers further found that biased gossip, meaning the spread of false information, can either facilitate or hinder cooperation, depending on the magnitude of gossip and whether the bias is positive or negative. But as gossip becomes more prone to unbiased “noise,” the population must gossip for longer to stabilize the equilibrium.

Kawakatsu next wants to think about how information flow interacts with altruism. The paper also notes that future research could explore how the number of gossip sources impacts cooperation, the conditions that would cause a rift in how an individual is viewed, and how bias may be applied differently for in-group and out-group members.

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Credit of the article given to University of Pennsylvania

 

Most people have heard the famous phrase “the enemy of my enemy is my friend.” Now, Northwestern University researchers have used statistical physics to confirm the theory that underlies this famous axiom. The study, “Proper network randomization is key to assessing social balance,” is published in the journal Science Advances.

In the 1940s, Austrian psychologist Fritz Heider introduced social balance theory, which explains how humans innately strive to find harmony in their social circles. According to the theory, four rules—an enemy of an enemy is a friend, a friend of a friend is a friend, a friend of an enemy is an enemy and, finally, an enemy of a friend is an enemy—lead to balanced relationships.

Although countless studies have tried to confirm this theory using network science and mathematics, their efforts have fallen short, as networks deviate from perfectly balanced relationships. Hence, the real question is whether social networks are more balanced than expected according to an adequate network model.

Most network models were too simplified to fully capture the complexities within human relationships that affect social balance, yielding inconsistent results on whether deviations observed from the network model expectations are in line with the theory of social balance.

The Northwestern team, however, successfully integrated the two key pieces that make Heider’s social framework work. In real life, not everyone knows each other, and some people are more positive than others. Researchers have long known that each factor influences social ties, but existing models could only account for one factor at a time.

By simultaneously incorporating both constraints, the researchers’ resulting network model finally confirmed the famous theory some 80 years after Heider first proposed it.

The useful new framework could help researchers better understand social dynamics, including political polarization and international relations, as well as any system that comprises a mixture of positive and negative interactions, such as neural networks or drug combinations.

“We have always thought this social intuition works, but we didn’t know why it worked,” said Northwestern’s István Kovács, the study’s senior author.

“All we needed was to figure out the math. If you look through the literature, there are many studies on the theory, but there’s no agreement among them. For decades, we kept getting it wrong. The reason is because real life is complicated. We realized that we needed to take into account both constraints simultaneously: who knows whom and that some people are just friendlier than others.”

“We can finally conclude that social networks align with expectations that were formed 80 years ago,” added Bingjie Hao, the study’s first author. “Our findings also have broad applications for future use. Our mathematics allows us to incorporate constraints on the connections and the preference of different entities in the system. That will be useful for modeling other systems beyond social networks.”

Kovács is an assistant professor of Physics and Astronomy at Northwestern’s Weinberg College of Arts and Sciences. Hao is a postdoctoral researcher in his laboratory.

What is social balance theory?

Using groups of three people, Heider’s social balance theory maintains the assumption that humans strive for comfortable, harmonious relationships.

In balanced relationships, all people like each other. Or, if one person dislikes two people, those two are friends. Imbalanced relationships exist when all three people dislike each other, or one person likes two people who dislike each other, leading to anxiety and tension.

Studying such frustrated systems led to the 2021 Nobel Prize in physics to Italian theoretical physicist Giorgio Parisi, who shared the prize with climate modelers Syukuro Manabe and Klaus Hasselmann.

“It seems very aligned with social intuition,” Kovács said. “You can see how this would lead to extreme polarization, which we do see today in terms of political polarization. If everyone you like also dislikes all the people you don’t like, then that results in two parties that hate each other.”

However, it has been challenging to collect large-scale data where not only friends but also enemies are listed. With the onset of Big Data in the early 2000s, researchers tried to see if such signed data from social networks could confirm Heider’s theory. When generating networks to test Heider’s rules, individual people serve as nodes. The edges connecting nodes represent the relationships among individuals.

If the nodes are not friends, then the edge between them is assigned a negative (or hostile) value. If the nodes are friends, then the edge is marked with a positive (or friendly) value. In previous models, edges were assigned positive or negative values at random, without respecting both constraints. None of those studies accurately captured the realities of social networks.

Finding success in constraints

To explore the problem, Kovács and Hao turned to four large-scale, publicly available signed network datasets previously curated by social scientists, including data from 1) user-rated comments on social news site Slashdot; 2) exchanges among Congressional members on the House floor; 3) interactions among Bitcoin traders; and 4) product reviews from consumer review site Epinions.

In their network model, Kovács and Hao did not assign truly random negative or positive values to the edges. For every interaction to be random, every node would need to have an equal chance of encountering one another. In real life, however, not everyone actually knows everyone else within a social network. For example, a person might not ever encounter their friend’s friend, who lives on the other side of the world.

To make their model more realistic, Kovács and Hao distributed positive or negative values based on a statistical model that describes the probability of assigning positive or negative signs to the interactions that exist. That kept the values random—but random within limits given by constraints of the network topology. In addition to who knows whom, the team took into account that some people in life are just friendlier than others. Friendly people are more likely to have more positive—and fewer hostile—interactions.

By introducing these two constraints, the resulting model showed that large-scale social networks consistently align with Heider’s social balance theory. The model also highlighted patterns beyond three nodes. It shows that social balance theory applies to larger graphlets, which involve four and possibly even more nodes.

“We know now that you need to take into account these two constraints,” Kovács said. “Without those, you cannot come up with the right mechanisms. It looks complicated, but it’s actually fairly simple mathematics.”

Insights into polarization and beyond

Kovács and Hao currently are exploring several future directions for this work. In one potential direction, the new model could be used to explore interventions aimed at reducing political polarization. But the researchers say the model could help better understand systems beyond social groups and connections among friends.

“We could look at excitatory and inhibitory connections between neurons in the brain or interactions representing different combinations of drugs to treat disease,” Kovács said. “The social network study was an ideal playground to explore, but our main interest is to go beyond investigating interactions among friends and look at other complex networks.”

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Credit of the article given to Northwestern University

Malawi introduced free primary education in 1994. This has significantly improved access to schooling. However, the country—which is one of the poorest in the world—still faces a high learning poverty rate of 87%. Learning poverty is a measure of a child’s inability to meet minimum proficiency in reading, numeracy and other skills at the primary school level. Malawi’s rate means that 87% of children in standard 4, at age 10, are unable to read. Only 19% of children aged between 7 and 14 have foundational reading skills and 13% have foundational numeracy skills. This leads to social and financial dependency. It also limits the extent to which individuals can actively participate in society. Children become especially vulnerable to pernicious social issues such as forced marriage, female genital mutilation, and child labor.

The primary education sector also has many challenges. These include overcrowded classrooms, limited learning materials, and a shortage of trained teachers.

There is a pressing need for innovative, transformative approaches to providing foundational education to meet the goals envisioned in Malawi 2063, the country’s long-term national plan. To accomplish this, the government of Malawi is using scientific evidence to enable meaningful and effective learning happen at scale.

This evidence has been generated in parallel by researchers from the University of Nottingham in the UK and the NGO Imagine Worldwide in the US and Africa. We have been testing the efficacy of an interactive educational technology (EdTech) developed by UK-based non-profit onebillion to raise foundational education by different groups of learners in Malawi.

The EdTech delivers personalized, adaptive software that enables each child to learn reading, writing and numeracy at the right level. Children work on tablets through a carefully structured course made up of thousands of engaging activities, games and stories. Over the past 11 years, we have built a complementary and robust evidence base focusing on different aspects of the software and program.

In 2013, I conducted the first pupil-level randomized control trial at a state primary school in Malawi’s capital city, Lilongwe. Randomized controlled trials are prospective studies that measure the effectiveness of a new intervention compared to standard practice. They are considered the gold standard in effectiveness research. We wanted to test whether the EdTech could raise young children’s numeracy skills. The study showed that after eight weeks of using the EdTech for 30 minutes a day, learners in grades 1–3 (aged 6 to 9) made significant improvements in basic numeracy compared to standard classroom practice. Teachers were also able to put the EdTech to use with ease.

Now, after many studies, Malawi’s government, in collaboration with Imagine Worldwide, is embedding the EdTech program in all state primary schools nationwide. This will serve 3.8 million children per year in grades 1–4 across all 6,000 state primary schools in Malawi.

Rigorous testing

After our initial 2013 study, we kept testing the EdTech through rigorous studies. Oneshowed that the EdTech program significantly raised foundational numeracy and literacy skills of early grade learners. Our results showed similar learning gains for girls and boys with the EdTech. This equalizes foundational education across gender.

Another study showed that children with special educational needs and disabilities could interact and learn with the EdTech, albeit at a slower pace than mainstream peers.

The EdTech wasn’t just tested in Malawi. We wanted to see if it could address learning poverty in different contexts, thus equalizing all children’s opportunities, no matter where they live.

Research in the UK demonstrated that the same EdTech raised the basic numeracy skills of children in the early years of primary schools compared to standard classroom instruction. It was also found to support numeracy acquisition by developmentally young children, including those with Down syndrome.

It was also shown to be effective in a bilingual setting. Brazilian children’s basic numeracy skills improved compared to standard practice after instruction with the EdTech delivered in either English, their language of instruction, or their home language, Brazilian-Portuguese.

Alongside the research from the University of Nottingham, Imagine Worldwide undertook a series of studies in Malawi and other countries to investigate how this EdTech could raise foundational skills over longer periods of time and in different languages and contexts, including refugee camps.

Imagine Worldwide conducted six randomized control trials, including two of the longest over eight months and two years. They showed robust learning gains in literacy and numeracy. They also found that children’s excitement about school, their attendance, and their confidence as learners improved.

The EdTech program also mitigated against learning loss during school closures. During Imagine’s 2-year randomized control trial in Malawi, program delivery was interrupted for seven months by COVID-related closures. Yet, results showed that children who had participated in the EdTech program prior to schools closing returned to school with higher achievement levels than their peers who had received standard instruction only.

Applying the evidence to policy

Malawi’s government was pleased with the early results and the program was expanded to about 150 schools, with the help of UK non-profit Voluntary Service Overseas. A national steering committee was established by Malawi’s government to monitor the program and review additional emerging research. In 2022 the Education Ministry formally launched the program through which the EdTech will be rolled out; it was introduced in 500 new schools at the start of the 2023/2024 school year, in September 2023.

To achieve the promise of the early research, ongoing implementation research and monitoring is helping to ensure program quality and impacts are sustained as it rolls out nationwide.

Strong evidence

Basic literacy and numeracy are the keys to unlocking a child’s potential—improving their health, wealth and social outcomes. Our combined research has shown that child-directed EdTech can deliver high-quality education for millions of marginalized children worldwide. The evidence is strong, diverse and replicable. Now governments need to follow the lead of Malawi to abolish learning poverty and make foundational education a reality for all children, everywhere.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to Nicola Pitchford and Dr. Karen Levesque, The Conversation

 

French director Justine Triet’s “Anatomy of a Fall,” winner of the 2023 Oscar for best original script, reconstructs a fatal fall in order to dissect the collapse of the romantic relationship between the film’s leading couple, Sandra Voyter and Samuel Maleski.

Far from an exception, breakups of the sort depicted in the film are commonplace: global data shows high levels of marriage failure, with a marked increase towards the end of the last century.

In some Western countries, as many as 50% of marriages do not make it past 25 years, giving rise to the popular maxim “half of all marriages end in divorce.”

According to Triet, “the strange thing is for a relationship to work. The majority are hellish, and the film aims to go deep into that hell.”

Importantly, divorce statistics do not account for the number of relationships that are unhappy. Perhaps the majority are indeed hellish, but some marriages today are long-lasting, and seem stronger and more loving than any that came before. This dichotomy—widespread failure or exceptional success– seems to summarize the current state of marriage in the West. This has been dubbed the “all or nothing” marriage.

Supplying relationship energy

Scientific studies have established that romantic relationships tend to drop off, meaning that, on average, satisfaction levels reduce over time. Successful couples are able to arrest this fall, finding a satisfying level that can last indefinitely. Many others, however, gradually decline to the point where breaking up is only a matter of time.

Relationship psychology shows that love alone is not enough to keep a couple together –it requires effort. Relationship scientist John Gottman likens this to the second law of thermodynamics, whereby a closed system—such as a marriage– degenerates unless energy is supplied. As he puts it, “if you do nothing to make things get better in your marriage but do not do anything wrong, the marriage will still tend to get worse over time.”

The “all or nothing” theory therefore suggests that successful relationships require a significant investment of time and energy. Couples who make this commitment will be rewarded with a high level of satisfaction, while those who fail to do so, like Samuel and Sandra in Triet’s film, are destined to fail.

But why do some couples manage to stop this fall and stay happy? Like Samuel and Sandra, all couples start out in love, and want to be happy together forever. If we assume that they are compatible and willing to make the effort together, they form what some call an “Adam and Eve” relationship—the Biblical archetype of a harmonious, lasting union.

Analysing the ‘Adam and Eve’ relationship

Using dynamic systems to analyse this relationship model confirms the “all or nothing” theory.

Dynamic systems are a mathematical tool for understanding the evolution of a variable over time. In the case of romantic relationships, we are interested in the “feeling” of love in a couple. Because effort is needed to sustain the relationship, it becomes a dynamic system controlled by effort: effort regulates “feeling,” with the objective of making the “feeling” last forever.

By applying this effort control theory, our research has found that a successful relationship requires effort beyond the partners’ preferred level, and that this effort gap is difficult to sustain over time.

The mathematical anatomy of a fall

As Sandra Voyter says in Triet’s film, there are times when a relationship is chaotic, others when you fight alone, sometimes alongside your partner, and sometimes against your partner.

Samuel and Sandra’s relationship has elements in common with any other couple’s relationship. The starting point is very high: “feeling” is at its peak, and there is a shared belief that it will never end. Both are willing to contribute to the happiness of the relationship by making their own individual efforts, and both know that some kind of shock or external event will eventually alter this state.

Generally speaking, couples with the same socioeconomic, cultural, or religious background—known as homogamous couples— are more stable. Many couples, however, are heterogamous, meaning they differ in one or more of these regards.

Heterogamy can extend beyond an individual’s circumstances: on its most elemental level, it can boil down to a mismatch or imbalance in how efficient one member of a couple is in transforming effort into “feeling” or happiness. Such a disparity may lead to asymmetrical levels of effort being dedicated to making the relationship successful, which are already higher than those both members would prefer to make.

This is the case in Samuel and Sandra’s relationship: at one point in the film Samuel highlights this imbalance, and Sandra replies that she does not believe a couple should make equal efforts, saying she finds the idea depressing.

Who contributes more?

Our latest computational models for assessing the dynamics of imbalanced effort levels in couples allow us to simulate the evolution of happiness in a relationship, both in predictable environments and with varying levels of uncertainty. Our simulations suggest that Sandra is right: each partner does not have to make the same level of effort.

One of the film’s scenes—where Sandra and Samuel reproach each other for the efforts made or not made to sustain the relationship—displays typical negative couple dynamics, where each has a bone to pick. The film also implies that Samuel has made or is making more effort than Sandra in their relationship. Our analysis shows, perhaps surprisingly, that the more emotionally efficient partner has to make a greater effort to sustain the relationship. In the film, it appears that this is Samuel.

External events play a big part

Our analysis also shows that when the couple is subjected to a stressful episode, both partners need to increase their effort levels if the relationship is to survive. However, the more efficient partner’s effort level has to increase more. In the film, Sandra and Samuel’s relationship is subjected to a tremendous misfortune, which has a prolonged and pronounced effect on its narrative arc. This is why Samuel feels much more stressed than Sandra.

Mathematics offers an outcome in line with the film’s plot: the continuous overexertion of the most emotionally efficient partner—amplified by a prolonged period of crisis—leads to the relationship falling apart. In the case of the film, this also leads to Samuel’s fall.

For more such insights, log into our website https://international-maths-challenge.com

Credit of the article given to José-Manuel Rey and Jorge Herrera de la Cruz