Approximating the Number of Network Motifs

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• Approximating the Number of Network Motifs
• Mira Gonen
• gonenmir_at_post.tau.ac.il
• Joint Work with Yuval Shavitt

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Talk Outline

• Background and Motivation
• Our Algorithm for counting Network Motifs
• The Color-Coding Technique
• High level description of the algorithm
• Summery of main results
• An example counting the number of cycles with a
  chord.
• Conclusions and Future Work

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Background and Motivation

• World Wide Web, Internet, coupled biological and
  chemical systems, neural networks, and social
  interacting species, are only a few examples of
  systems composed by a large number of highly
  interconnected dynamical units.
• The first approach to capture the properties of
  such systems is to model them as graphs whose
  nodes represent the dynamical units, and whose
  links stand for the interactions between them.

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Background and Motivation

• Such networks have been extensively studied by
  exploring their global topologies such as
  power-law degree distribution and the existence
  of dense-core.
• However two networks which have similar global
  features can have significant differences in
  structure.
• ? Local structures must be examined.
• These networks contain characteristic patterns,
  termed network motifs, which occur far more often
  than in randomized networks with the same degree
  sequence.
• Different motifs were found in different
  networks. The motifs reflect the underlying
  processes that generate each type of network.

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Background and Motivation
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Background and Motivation

• Milo, Shen-Orr, Itzkovitz, Kashtan, Chklovskii
  and Alon found motifs in the World Wide Web, and
  networks from biochemestry and neurobiology.
• Graphlet distribution of a vertex - a new
  systematic measure of a network local topology
  that was suggested by Przulj. They count for each
  vertex the number of all motifs of size at most
  five that are adjacent to the vertex.
• Gordon, Livneh, Pinter, and Rubin discussed
  counting the number of motifs a node is part of
  as a method to classify nodes in the network.
• Hales and Arteconi presented results from a motif
  analysis of networks produced by peer-to-peer
  protocols. They showed that the motif profiles of
  such networks closely match protein structure
  networks.

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Background and Motivation
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(?,?) - Approximation

• An algorithm for a counting problem f is an
  (?,?)-approximation if it takes an input instance
  and two real values ?, ? and produces an output y
  such that
• Pr(1-?)?f?y?(1?)?f?1-2?

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The Color Coding Technique

• Combinatorial approach that was introduced by
  Alon, Yuster, and Zwick to detect simple paths,
  trees and bounded treewidth subgraphs in
  unlabeled graphs.
• It is based on
• assigning random colors to the vertices of an
  input graph
• considering only subgraphs for which each vertex
  has a unique color.

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Main Contribution

• Using the color coding technique to approximate
  the number of network motifs a node is part of
  for k-length cycles, k-length cycles with a
  chord, (k-1)-length paths, where kO(log
  V), and for all motifs of size at most 4.
• The time complexity of our algorithm is
  O(e2k?E?V2?log(1/?)/?2)

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Counting the Number of Motifs v is part of
v
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Counting the Number of Motifs v is part of
v
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Counting the Number of Motifs v is part of
v
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Counting the Number of Motifs v is part of
v
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Counting the Number of Motifs v is part of
v
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Counting the Number of Motifs v is part of
v
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Counting the Number of Motifs v is part of
v
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4-Nodes Motifs
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1
2
4
5
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3. Results
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O(log(V)-Nodes Motifs
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8
9
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Our Main Results
Motif Time Complexity 1
O(E?V) 2
O(E?log(1/?)/?2) 3
O(E) 4
O(E2V?E?log(1/?)/?2) 5
O(E?V?logVE2) 6
O(E?V)
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Our Main Results
Motif Time Complexity 7
O(e2k?E?log(1/?)/?2
) 8
O(e2k?E?V2?log(1/?)/?2) 9
O(e2k?E?V2?log(1/?)/?2)
kO(log V)
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An Example Result

• Approximation Algorithm for Counting the
  Number of Cycles with a Chord

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Mathematical Notations

• C(v,u,S) the number of colorful paths from v to
  u in a specific coloring, using the colors in S.
• P(v,u,w,S) the number of colorful paths from u
  to w that are adjacent to v in a specific
  coloring, using the colors in S.

S
v
u
w
v
u
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The Algorithm

• Algorithms Input
• A graph G(V,E), a vertex v, fault-tolerance ?,
  error probability ?
• Notation let AV,z,b(S) be the set of all pairs
  (S1,S2) such that the following hold
• S1 z1,
• S2b-z1,
• S1?S2S,
• S1\col(u)u?V?S2\col(u)u?V ?

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The Algorithm
Repeat tlog(1/?) times

• 1. Color each vertex of G independently and
  uniformly at random with one of the k
  colors.
• 2. Compute the number of cycles with a chord in
  the coloring there are two cases
• 2.1 Compute X1,v the number of
  k-length cycles with a chord in case 1.
• 2.2 Compute X2,v the number of
  k-length cycles with a chord in case 2.
• 3. Let Yv the average of all the s X1,vX2,v.
• 4. Return the median of all the t Yv multiplied
  by kk/k!.

Case 2
Case 1
Repeat s4kk/?2k! times
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Computing the number of paths between v,w, for
every color-set S
w
v

• For all S?? k s.t Sl C(v,w,S)1 if
  col(v)col(w)l, and 0 otherwise.
• For q2 to k, for all S?? k s.t Sq
• C(v,w,S) ? C(u,w,S\col(v))

u?N(v)
w
v
u?N(v)
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Case 1
1?z?l-1

• P(v,u,w,S) ?1?z?l-1? C(v,w,S1)?C(v,u,S2)

l-z-length colorful paths between v and u using
colors in S2
z-length colorful paths between u and w using
colors in S1
The sum is over all (S1,S2) in Av,z,l(S)
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Case 1

• of cycles with a chord in case 1 ? ? ?
  P(v,u,w,S3)?C(u,w,S4)

1?z?l-1
l
(u,w)?E
l-length colorful paths between u and w that
are adjacent to v using colors in S3
k-l-length colorful paths between u and w using
colors in S4
The sum is over all (S3,S4) in Au,wl,k(k) and
over all (S3,S4) in Au,w,k-l,k(k)
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Case 2

• of cycles with a chord in case 2 ? ? ?
  C(v,w,S1)?C(v,w,S2)

l
w?N(v)
k-l-length colorful paths between v and w using
colors in S2
l-length colorful paths between v and w using
colors in S1
The sum is over all (S1,S2) in Av,w,l,k(k)
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Time Complexity
Case 1
The time complexity of computing C(v,w,S) for
every color-set S and every pair of vertices v,w
in a given coloring is O(2k?E?V2). The time
complexity of computing P(v,u,w,S) for every
color-set S and fixed v,u,w (assuming C(v,w,S) is
known) is
Choosing the colors of the path between u and w
that is going through v
Choosing the colors of the path between v and w
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Time Complexity
Case 1
?The time complexity of the first case, for every
edge (u,w), every vertex v, and every color-set
S, in a given coloring is
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Time Complexity
Case 2
The time complexity of the second case, for every
vertex v, every neighbor of v, and every
color-set S, in a given coloring is
Choosing the colors of the path between v and w
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Time Complexity
?The total time complexity, for every vertex v
and every color-set S, in a given coloring, is
O(ek?E?V2).
Time complexity O(e2k?E?V2?log(1/?)/?2)
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Conclusions and Future Work

• We have presented a fast algorithm for
  approximating the number of Network Motifs a node
  a part of.
• We also presented algorithms for counting the
  total number of occurrences of these Network
  Motifs when no efficient algorithm exists.
• Can we find sublinear algorithms for counting
  these motifs?

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Thank You!

• gonenmir_at_post.tau.ac.il