Network Motifs: Simple Building Blocks of Complex Network

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Network Motifs Simple Building Blocks of Complex
Network

• Lecturer Jian Li

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Introduction

• Recently, it was found that biochemical and
  neuronal network share a similar property they
  contain recurring circuit elements which occur
  more often far more than that in randomized
  networks.
• We call such simple building blocks network
  motifs.

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Introduction

• In the case of biological regulation networks, it
  has been suggested that network motifs play key
  information processing roles.

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Introduction

• Some examples

Three major network mortifs were found in the
transcription network of bacteria and yeast.
One of these the feed-forward loop, has been
shown theoretically to perform information
processing tasks such as sign-sensitive
filtering, response acceleration and
pulse-generation.
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Introduction

• Some examples

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Introduction

• Schematic Illustration

Red dashed line indicate edges that participate
in the feedforward loop motif, which occur five
times in the real network.
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Introduction

• Applications in other network
• Ecology (food web)
• Neurobiology (neuron connectivity)
• Engineering (electronic circuit, WWW)

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Introduction

• Some remarks
• The solution we get is closely related to the
  randomized network model. So a reasonable select
  of randomized network model is very important.
• Some functional-important but less-frequent
  building block will be missed no matter how we
  select our model. To find this type of things
  need specific knowledge and information which are
  beyond the sweep of graph theory approach.

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Related Problems

• Theoretical Perspective
• efficiently counting cycle.
• counting spanning trees.
• number of nonisomorphic graphs
• testing isomorphism
• approximating perfect matching.
• approximating frequent subgraphs based on the
  regularity lemma.

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Related Problems

• Data mining perspective.
• Mining frequent subgraphs.
• Mining a given subgraph.
• Mining subgraphs in sparse network.
• Graph-based substructure pattern
  mining(gSpan)

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Related Problems

• Random network.
• Generating randomized network with prescribed
  degree sequence.
• Estimating subgraphs in random networks.

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Related Problems

• Random network.
• Erdos model
• -the distribution of the number of edges per node
  exhibit a Poissonian distribution.
• Scale-free model
• -the distribution of the number of edges per node
  exhibit a exponential distribution.

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Randomized Network

• Generating randomized network
• Here we only give a simple algorithm.
• We employed a Markov-chain algorithm, based on
  starting with the real network and repeatedly
  swapping randomly chosen pairs of connections
  (X1-gtY1, X2 -gtY2 is replaced by X1-gtY2, X2-gtY1)
  until the network is well randomized.
• Switching is prohibited if the either of the
  connections X1-gtY2 or X2-gtY1 already exist.

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Randomized Network

• Controlling for Appearances of (n 1)-Node
  Motifs
• We generate a series of randomized network
  ensembles, each of which has the same (n
  1)-node subgraph count as the real network, as a
  null hypothesis for detecting n-node motifs.
• This is done to avoid assigning high significance
  to a structure only because of the fact that it
  includes a highly significant substructure.

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Randomized Network

• Controlling for Appearances of (n 1)-Node
  Motifs
• Metropolis Monte-Carlo approach
• Vreal,k be the number of appearances of each of
  the kth (n-1)-node subgraphs in the real network
  and Vrand,k be the corresponding vector in the
  randomized network.
• We define an energy
• E ?k(Vreal,k Vrand,k/(Vreal,k
  Vrand,k)).
• The energy E is zero only when all the three-node
  subgraph counts of the real and randomized graphs
  are equal.

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Randomized Network

• Controlling for Appearances of (n 1)-Node
  Motifs
• start by fully randomizing the network according
  to first algorithm.
• Then, we generate a random switch (X1-gtY1, X2-gt
• Y2 to (X1-gtY2, X2-gtY1), and similarly for double
  edges, as described above).
• If this switch lowers E, it is accepted.
• Otherwise, it is accepted with probability exp(M
  E/T), where ME is the difference in energy before
  and after the switch and T is an effective
  temperature.

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Graph Theoretical Results

• Controlling for Appearances of (n 1)-Node
  Motifs
• This process is repeated, with a simulated
  annealing regiment to lower T slowly until a
  solution with E 0 is obtained.
• This can be readily generalized to form (n
  1)-node null-hypothesis networks

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Algorithm Counting

• Goal find all n-node network motif
• Method
• Do the following for both real network and
  randomized network
• Simply enumerate all the possible n node
  subgraphs, classify them into non-isomorphic
  class.
• Count the number of subgraphs in each class.see
  all types of 3,4node nonisomorphic graphs

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Algorithm Counting

• Efficiently count all connected n-node subgraphs
  in a connectivity matrix M
• main
• for all rows i
• for each nonzero element (i, j)
• search (i,j)
• search(i,j)
• for each k such that Mik 1 and k!j
• if an n-node subgraph is obtained then
  record it and return
• else search (i,k)
• do similar things for each Mki 1, Mkj 1, Mjk
  1

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Algorithm Counting

• A table is formed that counts the number of
  appearances of each type of subgraph in the
  network,
• This process is repeated for each of the
  randomized networks. The number of appearances of
  each type of subgraph in the random ensemble is
  recorded, to assess its statistical significance.

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Algorithm Counting

• Criteria for Network Motif Selection
• (i) The probability that it appears in a
  randomized network an equal or greater number of
  times than in the real network is smaller than P
  0.01.
• (ii) The number of times it appears in the real
  network with distinct sets of nodes is at least
  4.
• (iii) The number of appearances in the real
  network is significantly larger than in the
  randomized networks Nreal Nrand gt 0.1Nrand.
  This is done to avoid detecting as motifs some
  common subgraphs that have only a slight
  difference between Nrand and Nreal but have a
  narrow distribution in the randomized networks.

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Algorithm Counting

• Result

CiNi/?i Ni Z-scores Z (Creal
Crand)/Varrand (note the inequality P(X-E(x))
gtZVar(x)lt1/Z2 ) High Z-scores indicate the
event is quit unlikely.
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Algorithm Sampling

• A clever trade-off between accuracy and
  efficiency.
• The counting algorithm can exactly enumerate the
  number of subgraph, but to detect network motifs,
  we only need to know which type of subgraph occur
  more frequently in real network than in
  randomized network.

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Algorithm Sampling

• Using random sampling method can do pretty good
  estimation.
• Random sampling has many applications.
• -approximating dense subset
• -approximating P-complete problem
• -mechine learning

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Algorithm Sampling

• This algorithm does not enumerate subgraphs
  exhaustively but instead samples subgraphs in
  order to estimate their relative frequency.
• The runtime of the algorithm asymptotically does
  not depend on the network size.
• Surprisingly, few samples are needed to detect
  network motifs reliably.
• The sampling method is useful for analyzing very
  large networks or for detection of high-order
  motifs, which are beyond the reach of exhaustive
  enumeration algorithms.

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Algorithm Sampling

• DefinitionEs is the set of picked edges
• Vs is the set of all node that are touch be the
  edges in Es
• ALGORITHM Sampling
• Initiate Vs? and Es ?
• 1.Pick a random edge e1(vi,vj),update
  Ese1,Vsvi,vj
• 2.Make a list L of all neighboring edges of Es,
  omit all edges between Vs.if L? return to 1
• 3.pick a random edge e(vk,vl)from L. Update
  EsEs U e, VsVs U vk,vl
• 4.Repeat steps 2-3 until completing n-node
  subgraph S.
• 5.Calculate the probability P to sample S.

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Algorithm Sampling

• The probability of sampling the subgraph is the
  sum of the probabilities of all such possible
  ordered sets of n-1 edges
• Where Sm is a set of all (n-1)-permutations of
  the edges from the specific subgraph edges that
  could lead to a sample of the subgraph. Ej is the
  j -th edge in a specific (n-1)-permutation (s).

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Algorithm Sampling
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Algorithm Sampling

• Add score W 1/P to the accumulated score, Si ,
  of the relevant subgraph type i Si Si W.
  After ST samples, assuming we sampled L different
  subgraph types, we calculate the estimated
  subgraph concentrations
• Ci Si/?k1L Sk

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Algorithm Sampling

• Z-scores is calculated as before.
• Z (Creal ltCrandgt)/Varrand
• where Creal is the concentration in the real
  network, ltCrandgt and Varrand are the mean and SD
  in the randomized networks.

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Algorithm Sampling
Sampling method versus exhaustive enumeration,
Highlighted subgraphs were found to be network
motifs.
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Algorithm Sampling

• Algorithm convergence
• The subgraph concentrations calculated by the
  sampling algorithm converged to the fully
  enumerated concentrations. Different numbers of
  samples were required for achieving good
  estimations for different subgraphs and in
  different networks.
• All of the simulations we performed, on a variety
  of networks, showed that the results converge
  toward the real values within ST 105 samples or
  less.

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Algorithm Sampling

• Algorithm convergence
• It is seen that even with a small number of
  samples one can estimate reliably concentrations
  as low as C 10-5.
• It is possible to use convergence studies in
  order to decide the required number of
  samples.(adaptive sampling method,using
  instantaneous convergence rate to decide how many
  samples are enough)

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Algorithm Sampling

• The sampling method allows accurate counting of
  rare, high-order subgraphs and motifs

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Some discuss and Future attempt

• We focus on comparing between the real network
  and the randomized network with prescribed degree
  sequence. So our question is whether some real
  frequent building block are caused by the degree
  sequence.
• If so, so what we have done will miss this type
  of building block. Some other randomized network
  model (rather than the ones with prescribed
  degree sequence) could be introduced to deal with
  such case.

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Some discuss and Future attempt

• Embedding the graph to euclidean space, and
  considering the subgraph with no only topological
  properties but also geometric properties.

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THANKS