This is a supplemental blog for a course which will cover how the social, technological, and natural worlds are connected, and how the study of networks sheds light on these connections.


If you could buy a network, how much would you pay?

http://www.bloomberg.com/apps/news?pid=newsarchive&sid=a0jloTW0QDe8

With the recent FCC auction of the 700 MHz band fetching bids that total to more than $19 billion, one has to ask exactly how much is a nationwide wireless network worth, and who can afford it? In rules specified by the FCC before the auction’s start bidders are to submit confidential bids in series of rounds. At the end of each round the highest bid is revealed but the bidder remains confidential. In this highly public yet somewhat secretive auction the stakes are huge with many technology behemoths such as AT&T, Verizon, and even Google vying to win a coveted piece of the wireless spectrum. How have these companies gone about deciding how much to bid and what drove the prices to exceed the government’s original predictions by as much as $7 billion?

In this type of silent auction theory might suggest that a bidder’s best strategy should be to simply bid their true value of the goods up for auction. In this case the true value of a wireless spectrum can vary greatly from company to company. Verizon and AT&T, two natural bidders that they are seeking to extend their already national wireless networks, can obviously derive a lot of value out of the 700 MHz band. However, another closely watched bidder, Google, does not have as transparent goals in this auction. Google publicly announced before the auction that they would bid on the coveted C-block of the spectrum until the bidding surpassed the $4.6 billion threshold that the FCC set for triggering open-access provisions. However, outside of this stated goal, the potential value of a wireless network for Google may be limited and as such Google may not be truly interested in bidding to win but rather bidding to raise the price.

Google’s presence in the auction probably affected the actions of all the other bidders. The significance of this impact however, will not be known until all the bids are set and the winner is announced in the following weeks.

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The Power of Plastic: A New Corporate Gatekeeper?

The power of gatekeepers on the corporate level is undeniable. In the software industry Microsoft holds most of the power. The reason; the Windows operating system is by far the most widely used platform. If software companies want to reach the consumers they will do well to do so through a Windows compatible program. Users then need Windows to run the software, and Microsoft grows. Google’s role as an online advertising giant is also strengthened through its gatekeeper functions. With the advent of Google Analytics and Google AdSense internet users have data sent through one of many Google servers every time they leave one page and go to another. The more information that passes through Google the more they can tailor their advertising; the more effective their advertising the more websites will use their services; and the more websites that use their services the more data they are able to collect. Hence the advantage of being a gatekeeper on the corporate level.

Now a set of new corporations are set to take on a gatekeeping role. Credit card companies have been pushing hard to make “plastic” the number one form of payment over cash and checks, and with Visa about to go public with the largest expected IPO in history they will be well positioned to pursue this goal. Evidence of trends toward using”plastic” in place of cash is everywhere these days.

In Visa’s recent line of commercials for their Check Card they suggest that cash and checks are a slow and outdated way to pay for goods.

http://www.youtube.com/watch?v=ddwa3YlkMlU&feature=related

Even the classic board game Monopoly is making adjustments to keep up with the times. In recent editions of the game, Monopoly Money has been replaced by a Monopoly Visa Debit Card.

http://www.geekologie.com/2006/07/monopoly_replaces_cash_with_vi.php

So what implications does this have on Visa and other credit card companies (Mastercard also has similar products) as a Gatekeepers in the corporate world? With the elimination of cash as a form of payment an increasing number of purchases are being handled through these companies. As gatekeepers in these buyer/seller transactions Visa gains access to a large amount of market data as well as bargaining power both with the buyers who carry their cards and sellers who accept them. How they will use this to their advantages is a tale that time will tell, but we can be certain that their power will increase as their gatekeeping role becomes more prominent.

Additional information about the Visa IPO can be found here:

http://news.yahoo.com/s/afp/20080225/ts_alt_afp/usipocompanyfinancevisastocks_080225160547

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Haggling: Realizing How Networks are Connected

Most shoppers don’t think to bargain. If an item is marked at a certain price, it’s stressful to walk up to a salesperson and ask if the price can be lowered. People aren’t used to haggling, so the idea may seem embarrassing, or just too out of the ordinary to try. How often does it work? According to Getting the Best Deal: Better Bargaining, by John Przybys on ReviewJournal.com, those who bargain end up saving money 90% of the time. Not only do these people save money, “Asked for their biggest score, most said they saved $50 or more while, in the case of cell phone plans and medical fees, more than 25 percent of respondents saved at least $100.” It may seem intimidating to ask for a lower price, but really, all  sellers want is a deal that leaves them with a profit. The fact that they seem more powerful than that only helps them end up selling at higher prices. While it’s common to bargain for better prices on houses and cars, the article mentions that bargaining very often works in electronics stores, large department stores, independently owned stores, and even grocery stores. People don’t try this because they don’t realize the position of power they are in as a buyer.

Where does this power come from? In class we discussed graphs with buyers, sellers, and traders, and how prices can be affected based on who is able to trade with whom. The issue of the buyer’s actual power can be roughly approximated as that type of graph. A customer enters a store and sees a price on a piece of furniture that is slightly high. The furniture isn’t hand crafted or particularly special, and similar furniture could easily be found in other stores. The customer however, sees himself as only able to buy from the one trades he is dealing with. He is imagining a graph where many traders buy from similar sellers, but where he is in a weak position, where he can not really trade with other traders, and he must either accept the asked price or leave. However, the store’s manager realizes that when the customer turns down a price, he can easily look for similar items at another store. Once the customer realizes that he is actually in a position of power, bargaining is easy. If both the buyer and the trader get a good deal by haggling, there’s very little reason why a store owner would not want to turn down a bargainer.

While this example isn’t exactly what we’ve covered in class (bid and ask prices first, then buyers and sellers picking traders) it definitely reflects elements from this graph, and from all discussions of which nodes have more or less powerful placements on a graph. A node that is only connected to a single other node generally has less power than a node with multiple neighbors. The idea that haggling is uncommon and uncomfortable stops the buyer from realizing that he’s in a powerful position in a network.

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Botnets: Exploiting Network Phenomena for Personal Gain

A botnet is a collection of computers, or bots, that have been compromised by some sort of malicious software. This software gives the attacker the ability to control the bot remotely. Botnets can vary greatly in size, from a handful of computers to hundreds of thousands. One such example of a very large botnet is the Storm botnet. The malicious software spreads like a normal computer virus, in which each infected machine attempts to infect more machines using methods such as sending infected files over peer-to-peer networks or sending spam emails with infected attachments. Once a computer has been compromised, an IRC bot is installed. IRC stand for Internet Relay Chat, which is a form of real time chat, usually used for large group conversations. Typically, the IRC bot connects to a specific IRC channel, where it waits for instructions. The attacker can then issue commands to the bots through that IRC channel.

The ability to control so many internet-connect computers at once allows the controller of the botnet to gain unfair advantages in situations that can be explained by network theory. One could only dream of having such an advantage in other situations: imagine being able to command 50,000 people to vote for a particular person, or to invest in a particular stock! One example of an unfair advantage is Google AdSense abuse. Companies that use AdSense can display ads on their website that earn the company money every time that they are clicked. An attacker could abuse AdSense by creating a website that displays these advertisements, and then instructing a botnet to repeatedly click on the ads, generating unfairly earned income. One could imagine this scenario using our knowledge of market interaction: Imagine an auction where there is one seller (Google) with unlimited supply of a good that they value at zero. The seller is connected to a single trader (the AdSense user), who is connected to a very large number of buyers. The trader varies his offer to the buyers by changing the contents of his website. The ads displayed on the website adapt to its content, so they would change also. The buyer’s value could be how interesting the ad looks, and their cost of clicking the ad could be the time spent. In this case, the trader isn’t just strictly a trader, he has an affect on how the buyers value the good. So, assuming the trader is rational, it would be in his best interest to maintain an interesting website so that more people will value the ads enough to click on them.

But what about the case of the trader being able to control a botnet? In this case, the botnet controller simply tells a large amount of buyers to buy his good (clicking on the ads). In this situation, the Trader has no incentive to create an interesting website. So in this simple auction model, the buyer ends up losing, since they end up buying the good at a price greater than they value it. This interpretation does not translate exactly back into the real life scenario, because there isn’t a person that is actually forced to click on the ads. Instead, one could say that the buyer loses because the computer user experiences a slowdown while their computer is clicking on ads by itself. There is another entity that loses due to a botnet. A company paid Google to create an ad in order in order to bring people to the company’s own website, possibly to purchase things. However, if all of their ad clicks come from a botnet, they won’t make any sales. In the end, a botnet controller has the ability to “cheat” in the AdSense market, profiting at the expense of others.

Sources:

http://en.wikipedia.org/wiki/Botnet

http://www.honeynet.org/papers/bots/

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Leaving Facebook.com?

Article link: http://www.nytimes.com/2008/02/13/technology/13face.html?_r=1&scp=3&sq=facebook&st=nyt&oref=slogin 

This article talks about the difficulties of quiting Facebook.com. Many people who have joined Facebook and no longer want an account there find it difficlt to completely get rid of it and only finds a solution to “deactivate” their account. By deactiviating the account, the information is still stored by Facebook, and your name can still be searched by Facebook users. The logic behind this according to Facebook is because of the numerous users that think they want to shut down their account only to restart at a later date. A Facebook spokesperson claimed that on a given day, the number of people who reactivate their account is roughly half of those deactivating their account. Facebook users have a reason to be angered; as one user puts it, “I did not want to be on it after junior associates at work whom I have to manage saw my Facebook page.” Currently, Facebook users who wished to close their accounts had been unable to do so, even after contacting Facebook’s customer service representatives. Many have attempted to erase their accounts for weeks to months without success.

 Why has Facebook with over 64 million worldwide (MySpace has an estimated 110 million monthly active users) not complied with requests by users to make deletion of accounts easier? Would losing a small fraction of users be a big deal? In the first week of class, we briefly talked about the network dynamics of coordination. This claims that the value in social networking sites is in the potential for interacting with others and the site becomes more and more valuable as others join it. Perhaps by making it difficult to leave Facebook, the company can still claim that their network is one of the largest social networking sites and quote a larger number of users than really exist to lure more users; this can be also be a selling point to companies who wish to advertise on Facebook pages and led to believe that there are more users than there really are. Another reason to keep these pages maybe to attract users who wishes to see their boss’s old Facebook page that he cannot delete. There can be a multitude of reasons of why Facebook is being particularly difficult in this. Perhaps game theory (loosely) comes in to play here. What is the best strategy for Facebook.com when faced with a group that is determined to leave its network? If these users are determined to leave the site, then Facebook’s best response to this action is obvious. Facebook cannot try to individually persuade these members to stay if they are determined to leave. So Facebook’s best response is to prevent them from permanently leaving by not allowing them to delete their information.

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Game Theory - Not Applicable In All Situations

        In this article, the author describes the entrance of a new competitor into the mobile market.  In particular, in Great Britain, a new company, Hutchison 3G has attempted to break into the ultra-competitive wireless communications industry.  However, as expected, they have had a considerable amount of difficulty persuading customers to leave their current wireless providers and join the new company.  This is reflected in the fact that Hutchison 3G has not yet been able to even gain 100,000 subscribers to its network.  As a result, Hutchison 3G has launched a new marketing plan that offers considerably lower billing fees per month for the same number of minutes.  Their new billing structure is up to 50% cheaper than their competitor’s prices.  Understandably, this has created a shockwave in the mobile market as Hutchison 3G’s offer is much better than those currently offered by Hutchison 3G’s competitors (the dominant market forces).  In “prisoner’s dilemma” terms, by lowering prices a significant amount, Hutchison 3G has just entered a situation where it has “confessed” in the sense that they chose an optimal strategy to gain a greater “deal” in this given situation.  By “confessing” (lowering prices), they are forcing their competitors to somehow react to this shock to the mobile market.  According to game theory, Hutchison 3G’s competitors (the mobile companies that currently have a large market share in the mobile industry) should also “confess” by lowering prices for their phone plans in order to compete with Hutchison 3G because consumers are most likely to take the more cost-efficient option (which at this point is offered only by 3G).  Thus, as Hutchison 3G’s competitors lose market share to Hutchison 3G, they will be forced to “confess” because not doing so will result in a greater cost in terms of profitability.

However, the author states in this article that in this case, game theory does not hold because if Hutchison 3G’s competitors were to lower prices, it would not result in optimal profitability because the author believes that all mobile companies will lower prices in reaction to this shockwave and it will just result in a total loss in profitability for the mobile industry while the market share stays roughly the same due to the lower monthly phone plans.  The author goes on to state that these competitors should rather defy game theory by offering a micro-level alternative in which it comes up with innovative solutions to maintain their high level of profitability even though it may lose some market share.  Such an alternative does not require these companies to lower their monthly phone plans, but for example, improve efficiency or customer satisfaction.

For example, in Ithaca, the majority of students at Cornell choose Verizon Wireless over AT&T due to the fact that Verizon’s coverage in this area is significantly better than AT&T’s even though AT&T offers lower monthly phone plans.  Furthermore, Verizon’s INNetwork, in which Verizon customers can place free phone calls to other Verizon customers, is an example of an innovative solution to the problem because Verizon is not “confessing” while companies such as AT&T are “confessing” by lowering the price of monthly phone plans.

After reading this article and another article, which is linked, states that game theory does not always apply in every single situation, I am convinced that game theory is not always the best option for every situation.  Although it may apply for most situations, there are instances, such as the one cited in this article, where using game theory is not always the best strategy for the long run.  This example shows that game theory is limited in the sense that in this innovative and competitive society in which products and services can always be offered for lower prices by competitors who are looking to gain an upper hand in a given situation, new ways of thinking and strategizing are very crucial to the success of any business or industry.

http://www.lexisnexis.com/us/lnacademic/results/docview/docview.do?risb=21_T3177246735&format=GNBFI&sort=BOOLEAN&startDocNo=26&resultsUrlKey=29_T3177245650&cisb=22_T3177245649&treeMax=true&treeWidth=0&csi=293847&docNo=40

http://www.lexisnexis.com/us/lnacademic/results/docview/docview.do?risb=21_T3177246735&format=GNBFI&sort=BOOLEAN&startDocNo=51&resultsUrlKey=29_T3177245650&cisb=22_T3177245649&treeMax=true&treeWidth=0&csi=293847&docNo=55

Posted in Topics: Education, General, Mathematics, Technology, social studies

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“A Balance More Replete”: Structural Balance and Dénouements

It is a truth universally acknowledged, that a play or novel or film nearing its conclusion must be in want of a dénouement.

Well, perhaps not universally acknowledged, but whether or not we know the term for it we expect the play we see or the story we read to come to a certain completion: plots untangled, tensions played out, couples joined in marriage or enemies joined in death. We expect a dénouement, a moment where the work resolves the conflicts and events that have been propelling the action and arrives at a certain final harmony. (It doesn’t matter if this is a necessarily positive harmony. A work in which everyone ends up killing each other still arrives at sort of harmonious conclusion: there’s no tension if everyone is dead.)

This sense of harmony, while by no means reducible to by any sort of network theory, nevertheless proves deeply linked to what is essentially a sort of structural balance. Like any group of “people,” the characters of a play or novel or other work can be looked at as a social network, one which changes over time as the work progresses and which should follow all the tendencies of a real social network, including that of structural balance. Tensions between characters along the different edges warp and wane as the plot progresses, motivate the action, and otherwise drive things forward. Resolving the plot necessitates also resolving these tensions, and so we should expect a dénouement to be as much about achieving structural balance among characters as the resolving the events stemming from it.

More importantly however, this final structural balance would seem to lean much more heavily towards one-group rather than two-group solution. Either by altering relationships between characters or eliminating nodes all together (either by death or just dropping out of the story) dénouement tend to result in a single unified group instead of two mutual opposed ones. The Importance of Being Earnest, like most comedies ends with everyone getting along and getting married, bringing everyone together to a single balanced group. Elizabethan revenge dramas end with almost everyone implicated in a negative edge dead, eliminating everyone impeding overall balance. Dickens, Sophocles contemporary romantic comedies, all seem to follow this general trend.

This holds for many works–in fact I would argue it describes the vast majority of conventional Western comedy and drama–but for clarity and simplicity sake, I’d like to switch to a more specific, representative example. It will be much clear and more manageable if we do.

Let’s take a moment then to look a one, relatively simple case in point: William Shakespeare’s A Midsummer Night’s Dream. Perhaps not the most exciting of choices, but the main plot’s small cast and the abruptness with which relationships between its characters change make it a much more manageable choice than most.

If you’ve forgotten, the main plot of A Midsummer Night’s Dream is pretty straightfoward: Enter two pairs of young lovers. Two of them, Lysander and Hermia, are a couple. A third, Demetrius loves Hermia, who hates him. The fourth, Helena, loves Demetrius, who hates her. They all end up in the woods at night where a fairies drug them, making them fall in love with different people. They fight, until they fall asleep. The fairies drug them again making them all fall in love with the right people. Everyone gets married and lives happily ever after. This gives a pretty simple, three-stage social network in which the tension inherent in the original setup is resolved as the play reaches its conclusion. (See the associated diagram below). You could chart the development of almost any work, but Dream’s smallish central cast and clear divisions make it an unusually easy task

.The Dynamic Social Network of A Midsummer Night’s Dream

This observation is by no means universal. There are plenty of works which do not resolve themselves, either deliberately or simply by happenchance, and there are others which do resolve themselves but in completely different manners, but this does describe a certain type of plot, most obviously, perhaps, in traditional Western comic (and to a lesser extent tragic) theatre, a type of plot which often serves as a point of comparison for most literate Westerners, extremely well.

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What does it mean to “win” on eBay?

In class, we’ve talked about auctions and specifically about Second Price Auctions.  As it says in our textbook, “The auction form use on eBay is essentially a second-price auction” and this makes sense.  On eBay, the winner at the end pays only the 2nd highest bid since eBay will incrementally increase your bid until it just beats the 2nd highest bidder.  Even if you bid higher (your valuation is higher) eBay will not make you pay that higher bid.  Therefore, the discussion of the bidder’s dominant strategy being to just bid the bidder’s valuation for the item outright is very pertinent to eBay.  In fact, eBay even tells us this strategy outright in it’s Tips to Bidding saying, “When you place a bid, we suggest that you enter the maximum amount that you’re willing to pay for the item.”

In this article written by Chris Gaylord from the Christian Science Monitor,  it seems like there’s a bit more psychologically than just winning an item and receiving that item: it literally is “winning” that item.  This can easily be seen through a feature that some sellers can use called “Buy Now”.  This feature allows the seller to sell immediately to a buyer who is willing to pay the “Buy Now” price that the seller has set.  For the buyer, this means the buyer’s valuation is higher than the seller’s and therefore would not have to go through the entire auctioning process to win the item.  There are occasions where the bidding will actually surpass the “Buy Now” price (the feature disappears after a set number of bids or price) but if everyone followed this dominant strategy, this outcome should never happen.  Beyond the mechanic nature of the dominant strategy though is the idea of actually winning the bidding.  As it says in the article, this is called calor licitantis or bidder’s heat.  Essentially, the bidders get so wrapped up in bidding against each other, that the price is driven up because the valuation of the item to each bidder is now compared to other bidders.  In fact, about 45% to 50% of iPod auctions on eBay with a “Buy Now” price had those prices surpassed in the actual bidding.  Fun eh?

Finally, there’s also an interesting notion about when we should bid.  This portion never seemed to be mentioned in class, but I expect it was because it makes the auctioning more complicated.  I believe part of the reason for this increase in prices is partly due to the fact that there isn’t only one submission for a bid, but that we can continually raise our bids until we feel like stopping or the bidding is over.  This means, according to game theory, we’re not only playing one game and submitting bids without having knowledge of the other bids and bidders, but instead we get the opportunity to see what the current highest bid is, and then recalculate our valuation on the item at hand.  This means that the dominant strategy only really holds if everyone bid at the same time, and not in this countdown fashion.  Interestingly enough, the more bids early on an item, the more attention is drawn to that item and the higher likelyhood that the ending price will be higher.  Therefore the idea is to bid as late as possible at the valuation that you decided upon initially.  Win or lose, at least you won’t pay more than you originally intended to.

Sources:

http://www.csmonitor.com/2007/0716/p13s02-wmgn.htm

http://pages.ebay.com/help/buy/bidding-ov.html#about

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Independent Thinkers Cowork

http://www.nytimes.com/2008/02/20/business/businessspecial2/20cowork.html?pagewanted=1&sq=networking&st=nyt&scp=19

 

Coworking (no hyphen) has become a new social phenomenon that are emerging across the country. It is where someone creates an office and rents out desks to professionals who have different jobs but who share the same values.  Coworking sites are not only meant to be a solutions for professionals who work independently like entrepreneurs, contractors, or designers, it is also meant as an intellectual commune for networking.  Coworking sites are usually equipped with internet access, conference rooms, and work stations.  While sites are popping up all over the world, there is a common ideal that these sites are for collaboration and not necessarily profit from desk rents.

This type of social networking is unique in that it brings together many interesting relationships. Since coworking brings individuals who have their own networks together into one working space, local bridges are formed. However, when it is clear that someone inhibits the openness the space is intended for like “aggressively recruit employees for [their] own startup” or professionals fail to “respect their space and leave it clean” then they are asked to leave. The discussion of the theory of structural holes is particularly apparent in this type of environment because someone can be introduced to an entirely different network that he would not have encountered otherwise. The idea of coworking relies on the flourishing of individuals with different structural advantages.

Ronald S. Burt’s paper Structural Holes and Good Ideas  stresses that one advantage of a social network like that in coworking is the creativity that can occur from standing at one end of these local bridges. He suggests that innovations often occur of the synergy of ideas that are distinct and unrelated. It can be noted that coworking can bring success to its participants despite how embedded these individuals are if you compare them to their original situation. It also caters to the goal of all participants as well: to spur the success of their businesses by sharing ideas with similarly-minded strangers.

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Structural Imbalance in Apartment 319

Problems always arise when people live in close proximity to one another. Even the closest of friends often find fault with each other’s habits and routines if they’re around one another enough. In my own apartment, this is certainly the case, though the number of conflicts has increased substantially this semester. I attribute this conflict explosion to the one of our roommates, Raymond, going abroad to Australia. There are five of us that currently reside in Apartment 319: Casey, Alex, Ryan, Brian, and Dave (some names were changed for identity preservation). Dave moved in this semester to fill Raymond’s spot while he is abroad. In speaking to Ray on the phone last week, I blamed him for the increased level of conflict in our apartment. He responded by saying “It’s impossible. Dave gets along with more people in the apartment than I do, things have to be easier than before.” While it may be true that Dave meshes better with more roommates, I believe that his presence actually causes more conflict.

In delving further into the logic behind structural balance, I came across Frank Harary’s 1953 paper “Of the Notion of Balance of a Signed Graph.” It is from this paper that the idea of structural balance originally arose. Harary claims that everyone in a social network either must be friends, or the network is split up into two factions. I believe that our apartment satisfied the latter of these two conditions with Raymond present, and that the balance has been disrupted by the roommate switch. Perhaps graph theory can shed some light on why this is the case, and prove why my insight is correct.

Our apartment this semesterThe graph to the right is a depiction of our apartment at the present. An edge with a (+) means that two roommates have little or no conflict. Conversely, a (-) indicates that two roommates argue frequently about apartment-related things. Given Harary’s rules, it is clear to see that the (+) edge between Ryan and Dave causes a structural imbalance, and is perhaps the root of the increased conflict in Apartment 319. According to the rules set forth by Frank Harary in 1953, this edge causes two (+) (+) (-) triangles, and hence puts latent stress on Dave and the network as a whole. It’s seems strange that a good relationship between two roommates causes more conflict than a bad one, but perhaps an example of a recent conflict can elucidate why this is the case.

Casey and Alex prefer the apartment temperature to be 72 degrees, and turn the thermostat up to 72 whenever Brian and Ryan aren’t around. Brian and Ryan, however, change it back to 68 whenever they get the chance. Neither group addresses the problem with the other, and the game continues indefinitely without conflict. However, Dave sees that two sets of his friends are not getting along, and takes in upon himself to correct the problem. His addressing the problem causes the two groups to argue, and perpetual conflict ensues. If Dave simply chose one group (Alex and Casey, or Ryan and Brian), then the two groups could get along by rationalizing that the members of the other group are just jerks.

Our apartment next semesterWhen Dave graduates, and Raymond (node Y) returns in the fall, I would expect the level of conflict to diminish. Raymond doesn’t get along with Ryan in the way that Dave does. The graph below illustrates this new scenario. Here, the graph can clearly be divided in two groups, with Alex/Casey/Raymond in one group and Brian/Ryan in the other. Since there are two distinct groups, and each group gets along with everyone within it and argues with everyone in the other group, the graph achieves structural balance, just as Harary would predict. It is worth noting though, that it is still early in the semester, and relationships may change. There is pressure on Dave to choose one group or the other, and so in time, he may sever the positive relationship with Ryan, since he does not argue with either of the members of the other group. However, this hasn’t happened yet. The rules for structural balance seem to make logical sense, but it is interesting to see that they hold true under empirical observation.

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