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Archive for October, 2010


ABSTRACT – Applications of graph theory are primarily, but not exclusively, concerned with labeled graphs and various specializations of these.Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be represented by graphs. The link structure of a relationship between a boy and a girl could be represented by a directed graph: the vertices are the boys and girls having some degree of inclination for each other, available for an open or closed system and a directed edge from a boy B[i] to a girl G[j] exists if and only if both B[i] and G[i] have an equal or almost equal inclination toward each other.

 
In the mathematical discipline of graph theory, a matching or independent edge set in a graph is a set of edges without common vertices. It may also be an entire graph consisting of edges without common vertices. Given a graph G = (B[i],G[j] for all 0

 

A vertex(a Boy or a Girl) is matched (or saturated) if it is incident to an edge in the matching. Otherwise the vertex is unmatched.A maximal matching is a matching M of a graph G with the property that if any edge not in M is added to M, it is no longer a matching and the system becomes unstable. In other words, a matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. The whole system has got maximum stability when each vertex(Boys and Girls) are part of one and only one Matching M, with no intersection at all with other Matchings. In simpler words, if each Boy B[i] has an inclination for only one Girl G[j] and vice-versa. For maximum stability, i should be equal to j, for all i,j=Z+.

 

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PROPOSAL – Modern Game Theory has always been used to predict the stability of a system having multiple forces, taking into account the behaviour of other points in the system. A Simple Boy Girl Relationship can be expressed as a directed graph, with each vertex representing a Boy and a Girl, inclined towards each other. Such a Directed Edge, in the absence of any other third isolated node or any other unmatched graph, is the most stable system possible.

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AXIOM 1 – A single Node(A single Boy B[i], or a Single Girl G[j]), is stable for the time interval in which it is isolated. In the presence of other single nodes, such an isolated node will eventually form a directed graph with some other node, which may be isolated or part of a matching graph.

AXIOM 2 – As stated earlier, a directed graph consisting of only two nodes, one of which is a boy B1, and the other girl G1, with equal inclination towards each other, is the most stable system possible. All complex systems whether stable or unstable, strive to attain this most stable system by various means.

Corollary – If this directed edge between B1 and G1 is a part of a set of B[i] and G[j] as shown below, and B1 and G1 do not have equal inclination towards each other, then there is a possibility of B1 and G1 forming the maximum stable system with rest of G[j] and B[i] respectively. In case of even number of total nodes, the whole system still remains stable because of smaller stable sub-systems. In case of odd number of total nodes, we have a highly unstable super-system with constantly shifting stable sub-systems and one always isolated node.

 

 

AXIOM 3 – A system with two B[i]s and one G[j] or two G[j]s and one B[i], is a highly unstable system, also known as the Classic L Triangle. The system constantly tries to attain stability by B[i]-G[j] pairing, but there always exists a single unstable B[i] or G[j] at any point of time, resulting in an overall unstable system.

THEOREM : The Degree of instability of a system, is defined as the number of isolated B[i] or G[j] nodes , divided by the total number of directed graphs in the system. Consequently, a system with higher U-score tends to be more stable than a system with lower U-score.

The U-score for fully stable system is Zero.

The U-score for the classic L-triangle is 1.

AXIOM 4 – For systems with 3 or more nodes, as stated earlier, the system tends to form stable sub-systems consisting of directed graphs between B[i]s and G[j]s. The U-score of the system decreases with the number of directed graphs in the system and increases with the number of isolated nodes in the overall system.

A system consisting of only G[j]s or only B[i]s , has a U-score of infinity and is highly unstable.

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CONCLUSION: Thus we see how Graph theory can be applied for Modern Day relationships to get some logical analysis. The systems considered above are but ideal, and the following assumptions have been made, which may deviate from a real case scenario.

1) Each B[i] and G[j] is assumed to have the same inclinations towards all other G[j]s and B[i]s. In Reality, the L-Bond between no two B[i]-G[j] pair is equal in strength,

2) It has been found after extensive study, that B[i]s and G[j]s also interact among themselves and have a tendency to form directed graphs amongst themselves, often in the absence of nodes of the other kind. However for the above study, such interactions have not been considered.

3) As per Classical Game theory, each B[i] and each G[j] has multiple choices at any point of time. The overall stability of the system, ultimately depends on the choices made by these nodes. Hence a detailed study of the application of game theory also needs to be considered for the above problem.

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With these assumptions in place, it can be safely said, that Mathematics can be used with proper care, to explain something as random and chaotic as Modern Day Relationships. As it’s always said, “There is always some madness in love. But there is also always some reason in madness.”

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It’s a sunny morning and the weather outside is generally hot. A guy comes to office and settles down in front of his desktop. Switches on the monitor and begins surfing through his mails, almost drowsing in the process. Suddenly a chat box comes up on his window and he opens his half closed eyes to look at the chat window….which said “Dude…”

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Guy1 : 10:30:45 AM : Dude

Guy2 : 10:30:49 AM : Hoy

Guy1 : 10:31:05 AM : Smoke?

Guy2 : 10:32:00 AM : Yups (almost sleepily)

Guy1 : 10:32:38 AM : Meet at Reception, 2 mins

Guy2 : 10:33:00 AM : OK

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And then two guys, in two different corners of the office get up from their seats, to take a break from their much busy work schedule , and to get a smoke outside office. On the way, they call and pick up any other guy who is interested to join. Most of the guys they ask, do join them, except the guys who have been coming from there already. Sometimes those join too for another round of cigarettes. A group of normally 4 and at least 2, then proceeds down the fire escape to go out for a round of smoke. A typical sunny morning it is.

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Thats how a typical smoker’s club looks like and functions in any modern office these days. A small group of people, who know each other’s weaknesses for the addiction and who are willing to smoke any time , any moment. And in today’s much stressful loves of people, such clubs are but a norm and an increasing one too.

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So the guys head down the fire escape, slowly, almost as if in a trance. Most of them are bachelors and slept lat last night , so had to rush to office in the morning with a lot of leftover sleep in their eyes. One of them ventures out a question in the open, almost meaning it for the others, “So, did you drink last night too?”.  One of them picks up the question and says, “Yeah, a half a bottle whiskey before sleeping off at 5am. I was talking to this girl , you know over yahoo and just kept on drinking. In between , had to get up and go get myself a pack of cigarettes at 1am. Man, am so screwed…need to quit doing all this”. Others don’t really give their opinion. They don’t offer advises when sober and never so early in the mornings.

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One of the other guys says however, “Last night we and went to this pub man, awesome place, excellent rock music and good food. The liquor was a bit expensive but it was so totally worth it man. You guys should have seen this hot reception chick. And those girls singing karaoke man. Incredible. You guys should have been there. And those idiots charged me 250 rupees for a 100 rupee cigarette pack. What about you ?”

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The first guy who has been mostly listening till now, says, “Me? Well i had some work to finish off, so i went home late. Drank a bit at home only. I just had 2 cigarettes and found no shop open at 2am. I smoked two as soon as i got up this morning man. Do you have cigarettes or do we need to go out?”

“Nah….” the others say out loud.

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So the group of half-asleep-half-awake people, make it to the cigarette shop opposite the office building on the other side of the road and ask the guy selling cigarettes in the hot sun, under an umbrella, “4 cigarettes, 3 milds, 2 gold….and do u guys want tea??…..4 cups of tea too”.  And then they go to the one side and smoke up peacefully. There’s the usual chit-chat , about how each one’s bosses suck, how they never get a promotion in spite of working so hard, how the foreign trip got cancelled due to visa reasons, how the new girl in office was, which pub played what and served what….and other such issues of local importance. They smoke for 15 minutes, a couple of them smoke multiple cigarettes in the same time while the other just refrain in an effort to save their failing healths. 20 minutes later, they pay up the cigarette guy and move back to offices. It’s almost 11:10 am now. Meetings to be attended. Important work to be done. And lunch time coming up.

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And that’s how such clubs meet. They meet almost everyday, almost after every 2-3 hours , as and when their schedules and health permit. Often they meet more than 5-6 times a day, and spend up a lot of time in these club meetings. Nothing useful ever comes out of these meetings. Once the group for the club is formed, there isn’t much networking opportunities either as a lot of people believe. Yet, they are so much the norm these days. Evey corporate thrives on many such small internal clubs and if you have one small supplier close by , it’s a flourishing business.

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For men will smoke for no apparent reason. Addiction is how they term it. But addiction itself is such a vague term as most people would say. It’s just something, which you do so much, that it becomes a habit and then you just can’t get it out of your mind. For men, they’ll smoke anytime. You just need to ask these smokers…

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…Care for a smoke??…

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