Random Graph Models of large networks

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Random Graph Models of large networks

• Alan Frieze
• with special thanks to
• Abraham Flaxman

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Graphs

• A graph G is defined by a vertex set V and an
  edge set E.

It may look like this
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Real Graphs

• Large graphs are all around us in the world
  today, for example, the World Wide Web.

The vertex set V consists of all web pages, and
the edge set E consists of all hyperlinks.
V ¼ 109, average 7 links per page.
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More Examples

• Internet
• Metabolic Networks
• Social Networks
• Neural Networks
• Peer to Peer Networks
• Tunnels of an Ant Colony

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Modeling Graphs

• We assume that they arise via some random
  process.
• Why not model them as random graphs
• Erdos and Renyi model
• Vertex set 1,2,,n,
• Each of the graphs with
• edges equally likely.

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Problem

• Suppose
• For small k the number of vertices degree k is
  whp
• In many real world cases the number of vertices
  of degree k
• for some A,? (close to 3 in WWW graph)
• e.g. Faloutsos,Faloutsos and Faloutsos.

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Fixed Degree Sequence Models Choose a degree
sequence d1,,dn and then choose a graph
uniformly from graphs with this degree sequence.
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Molloy and Reed

• Let
• ?lt0 implies all components of G are small whp.
• ?gt0 implies that G contains a giant component
  (size ?(n)) whp.

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Explanation represent a vertex of degree k by a
set of size k. Randomly pair up points of sets.
Vertex of Degree 3
Expected increase in free dots is
?idi(di-2)/?idi
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Aiello, Chung, Lu
Given ?,? they considered random graphs where the
number of vertices y of degree x satisfies
For various ranges for ?,?.
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• Flavour of results existence of giant
• component whp
• ?gt?03.478.. implies there is no giant component
• ?lt?0 implies there is a unique giant
• component.
• Bounds given on size of second largest
• component too.

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Cooper and Frieze Random Digraphs
They consider random digraphs D with n vertices
where the number of vertices with in-degree i
and out-degree j is li,j . Let ? n?i,jli,j be
the number of arcs in D. Let
d?i,jijli,j/(? n).
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• Flavour of results size of largest strong
• component (size ?(n)) whp
• dlt1 implies there is no giant strong component.
• dgt1 implies there is a giant strong component S.

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• More on dgt1
• Let L be the set of vertices with giant
• fan-out and L- be the set of vertices
• with giant fan-in. Then whp
• SLÅ L-

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Papadimitriou and Mihail

• Model Fix a1,a2,an and then add an
• edge between vertices i and j with
• probability aiaj/An where An? ai.
• Let ?1 ?2 be the largest eigenvalues
• of the adjacency matrix of the random
• graph produced.

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• Suppose ½lt?lt1. Suppose that aia1i-?
• for small i. Then for small i we have
• ?i ai where ?i denotes the ith
• largest degree, and

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Dynamic modelsPreferential Attachment Model (PAM)

• We build the graph dynamically

1. At time t (a) add vt (b) connect vt to u
chosen randomly
2. Every m steps contract the most recently added
m vertices into a single vertex.
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• 1. (b) Connect vt to u chosen randomly

Randomly how?
The rich get richer
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Preferential Attachment Model

• History

Yule, 1925 - Zoology
Simon, 1955 - Word frequencies, academic papers,
cities, income, more zoology
Barabasi and Albert, 1999 - WWW
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Heuristic Analysis of Degrees
Let dv(t) denote the degree of v at time t.
Then,
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• Suppose that v is added at time s. Then
• we get
• Thus the number of vertices of degree
• exceeding k at time t is

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• And the number of vertices of degree
• exactly k is

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• A rigorous proof of the following is given
• in Bollobas, Riordan, Spencer, Tusnady
• With probablity 1-o(1), as ,
• the number of vertices of degree k is

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More Work
B. Bollobas, O. Riordan Diameter Robustness
Suppose we delete the first ct vertices for some
clt1.
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• Researchers have developed similar, but
• more complex models which are
• mixtures of preferential and random
• attachment. These give arbitrary
• exponents for the power law.

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Copying Model

• Communities A large dense bipartite
• sub-graph of the WWW indicates a
• community. Experiments indicate a
• larger number of these than you would
• get from say the simple model PAM.
• The next model does give many though it is
• due to Kumar,Raghavan,Rajakopalan,Sivakumar
• and Upfal.

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As in PAM at each stage we add a new vertex vt
and we give it m incident edges. Its
construction rests on a parameter ?. Then a
vertex u is chosen uniformly at random from
Vtv1,v2,,vt-1 and then for i1,2,,m we 1.
With probability ? we create edge (vt,x) where x
is chosen randomly from Vt. 2. With probability
1-? we create edge (vt,y) where y was the ith
choice of u.
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• 1. Whp the degree sequence has a power
• law with exponent .
• 2. Whp the number of copies of Ki,i, i m is
  ?(te-i).
• This contrasts with the simple preferential
• model where the expected number is
• O(1).

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More on PAM

• Let Di ith max degree at time t
• Fenner,Flaxman,Frieze Fix k independent
• of t.
• Whp for any f(t) with f(t)!1 as t!1, and iltk
• t1/2/f(t) ?i t1/2f(t)
• and
• ?i1 ?i - t1/2/f(t).

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• Furthermore, if ?1,?k
• are the kth largest eigenvalues then whp

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Crawling on web graphs

• Cooper and Frieze considered the
• following scenario We have the model
• PAM. Plus there is a spider S which does
• a random walk as the graph is growing.
• Let ?m(t) denote the expected number of
• vertices not visited at least once by the
• spider up to time t.

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• Let
• then

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Heuristically Optimized Trade-Offs Fabrikant,
Koutsoupias and Papadimitriou

• This a random graph model of the growth
• of the internet that exhibits a power law
• in its degree sequence.

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• The model builds a tree on n random
• points X1,X2,,Xn in the unit square 0,12
• ?n is a parameter of the model.
• Suppose that we have built a tree T on
• X1,X2,,Xi-1 and we wish to connect
• Xi up a close point on T.

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• We connect Xi to Xj where j minimises
• ?ndi,jhj
• Here di,j is Euclidean distance and
• hj is the tree distance (in edge count)
• from Xj to the root X1.

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Results

• ?nlt2-1/2 implies that T is a star with root X1
• ?n?(n1/2) implies that the degree
• distribution of T is exponential.
• 4 ?no(n1/2) gives a power law
• for the degree distribution of T.

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