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MX4023 Project: Random Graphs\
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Author:  Marcell Veiner\t\
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Imagine organizing a party for guests who initially do not know each other, \
while offering them wine and cheese and you will soon see them chatting in \
groups of two to three. Now mention to Mary, one of your guests, that the red \
wine in the unlabeled dark green bottles is a rare vintage, much better than \
the other ones. If she shares this information only with her acquaintances, \
you may think that your expensive wine is safe, as she only had time to meet \
a few others so far. \
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Yet, you would be wrong. As the guests continue to mingle, subtle paths form \
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while John has not yet met Mary, they have both met Mike, so there is an \
invisible path from John to Mary through Mike. As time goes on, the guests \
will be increasingly interwoven by such elusive links.\
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Random network theory tells us that we do not have to wait until all \
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Rather, soon after each person meets at least one other guest, an invisible \
network will emerge that will allow the information to reach all of them.\
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We tell Mary at the beginning of the party about the rare wine.\
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Each person at the party talks to 3 other for 30 minutes. This can be someone \
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We will create an empty adjacency matrix for a 100 node graph. We will also \
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In mathematics, random graph is the general term to refer to a probability \
space over graphs, thus a random graph is described by a probability \
distribution or by a random process, which generates it. From a mathematical \
perspective, random graphs are used to answer questions about the properties \
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Key ideas already appeared in sociology and biophysics [7; 8]\
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Mathematical foundation by Paul Erd\[ODoubleAcute]s and  Alfr\[EAcute]d  R\
\[EAcute]nyi  \"On  random  Graphs\",  published  in  1959 [3] and then a \
seminal series of papers\
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Duncan J. Watts and Steven Strogatz bringing random graphs closer to \
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Systematical study of random networks and general network science by Albert \
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A more local property of random graphs than hubs are so called, clusters.  \
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We have seen how the number of edges or the degree of vertices are likely to \
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Using the Lemma above, the Poisson Summation Formula and the Generalized Mean \
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Deriving the formula would be too long to be contained in this presentation. \
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Not specifying the theorem used in showing the convergence of the series in \
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The small-world phenomenon, also known as six degrees of separation, has long \
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Many real-world networks posses the small-world property, which our random \
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Cell["The Small-World Property", "Section",
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Cell["\<\
Although the simplicity of Gilbert random graphs has made the model an \
attractive choice for many applications, their degree distributions and \
clustering coefficients do not resemble networks observed in the real world. \
To address these differences an alternative model was proposed in [9], called \
the Watts-Strogatz model.\
\>", "Text",
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Cell["Definition: Watts-Strogatz Random Graph", "Subsection",
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Cell[TextData[{
 "Let \[ScriptN] denote the number of vertices and let \[ScriptK] be an even \
integer, denoting the average degree of a vertex. Furthermore, assume that \
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chaos.  However, by studying random graphs, we were able to quantify some of \
these differences, and even explore ideas that tried to patch the \
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In fact the theory of random graphs serves more as a reference point when \
studying real-world networks.  Upon discovering a new property of a network, \
a question random graphs can answer is: if it could have emerged by chance? \
If the property cannot be found in random graphs, it may represent a \
signature of order, \[OpenCurlyDoubleQuote]requiring a deeper explanation\
\[CloseCurlyDoubleQuote] [2].  This is the relevance of random graphs to the \
study of networks.\
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[1]  R\[EAcute]ka Albert and Albert-L\[AAcute]szl\[OAcute] Barab\[AAcute]si.  \
Statistical mechanics of complex networks. Reviews of modern physics, \
74(1):47, 2002.\
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[3]  Paul Erd\[ODoubleAcute]s and Alfr\[EAcute]d R\[EAcute]nyi.  On the \
evolution of random graphs. Publ.Math. Inst. Hung. Acad. Sci, \
5(1):17\[Dash]60, 1960.\
\>", "Item",
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[4]  Agata Fronczak, Piotr Fronczak, and Janusz A Holyst. Average path length \
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\>", "Item",
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[5]  Edgar N Gilbert.  Random graphs. The Annals of Mathematical Statistics, \
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[6]  Stanley  Milgram.   Six  degrees  of  separation.Psychology Today,  2:60\
\[Dash]64,1967.\
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[7]  Jacob L Moreno and Helen H Jennings.  Statistics of social \
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[8]  Ray Solomonoff and Anatol Rapoport.  Connectivity of random nets.The \
bulletin of mathematical biophysics, 13(2):107\[Dash]117, 1951.\
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[9]  Duncan  J  Watts  and  Steven  H  Strogatz.   Collective  dynamics  of  \
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