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University of Michigan
SI 614
Network subgraphs (motifs)
Biological networks
Lecture 11
Instructor: Lada Adamic
Outline
 motifs
 motif detection (software & Pajek)
 review of network characteristics
 used to compare model with real-world network
 one more: degree assortativity
 biological networks
 types
 characteristics
 hierarchical modularity model
Schematic view of network motif detection
Motifs can overlap in the network
graph
motif matches in the target graph
http://mavisto.ipk-gatersleben.de/frequency_concepts.html
motif to be found
Examples of network motifs (3 nodes)
 Feed forward loop
 Found in neural networks
 Seems to be used to neutralize
“biological noise”
 Single-Input Module
 e.g. gene control networks
All 3 node motifs
Examples of network motifs (4 nodes)
 Parallel paths
W
 Found in neural networks
 Food webs
X
Y
Z
4 node subgraphs (computational expense increases with
the size of the graph!)
Network motif detection
 Some motifs will occur more often in real world networks
than random networks
 Technique:
 construct many random graphs with the same number of nodes
and edges (same node degree distribution?)
 count the number of motifs in those graphs
 calculate the Z score: the probability that the given number of
motifs in the real world network could have occurred by chance
 Software available:
 http://www.weizmann.ac.il/mcb/UriAlon/
What the Z score means
m = mean number of times the motif
appeared in the random graph
s standard deviation
the probability observing a Z
score of 2 is 0.02275
In the context of motifs:
Z > 0, motif occurs more often
than for random graphs
Z < 0, motif occurs less often
than in random graphs
# of times motif
appeared in random graph
zx =
x - mx
sx
|Z| > 1.65, only a 5% chance of
random occurence
Finding classes on graphs based on their motif “profiles”
Finding motifs (cliques and subgraphs) in Pajek
 Create a second network that is the subgraph you are
looking for
e.g. an undirected triad
*Vertices
3
1 "v1"
2 "v2"
3 "v3"
*Arcs
*Edges
2
3
1
1
2
1
1
3
1
finding motifs with Pajek
 Use the two drop down menus in the ‘networks’ list to
specify two networks:
 Then run Nets>Fragment (1 in 2)>Find
 under Net>Fragment (1 in 2)>Options
 can select ‘induced’ subnetwork containing only overlapping
fragments
in
finding motifs with Pajek (cont’d)
 Now we have just the triads:
 Creates a hierarchy object with
the membership of each triad
listed
Comparing network models with the real thing
 check for structural similarity between the artificial
network (the model) and the real world network
 degree distribution
 assortativity
 do high degree nodes connect to other high degree nodes?
 average shortest path
 dependence on size of network
 clustering coefficient
 compare to a randomized version conserving node degree
 dependence on node degree
 dependence on size of network
 motif profile
How can we randomize a network while
preserving the degree distribution?
 Stub reconnection algorithm (M. E. Newman, et al, 2001, also known in
mathematical literature since 1960s)
 Break every edge in two “edge stubs”
AB to A
B
 Randomly reconnect stubs
 Problems:
 Leads to multiple edges
 Cannot be modified to preserve additional topological
properties
Local rewiring algorithm
 Randomly select and rewire two edges (Maslov, Sneppen, 2002, also
known in mathematical literature since 1960s)
 Repeat many times
 Preserves both the number of upstream and downstream
neighbors of each node
Conserving additional low-level topological
properties
 In addition to ki one may also conserve:
 The exact numbers of loops or other motifs
 The size and numbers of components: Internet – all nodes have
to be connected to each other
 Metropolis algorithm: two edges are rewired based on
E=(Nactual-Ndesired)2/Ndesired
 If E0 rewiring step is always accepted
 If E>0 rewiring step is accepted with p=exp(-E/T)
Assortativity
 Social networks are assortative:
 the gregarious people associate with other gregarious people
 the loners associate with other loners
 The Internet is disassortative:
Assortative:
hubs connect to hubs
Random
Disassortative:
hubs are in the
periphery
Correlation profile of a network
 Detects preferences in linking of nodes to each other
based on their connectivity
 Measure N(k0,k1) – the number of edges between nodes
with connectivities k0 and k1
 Compare it to Nr(k0,k1) – the same property in a properly
randomized network
 Very noise-tolerant with respect to both false positives
and negatives
Correlation profiles give complex networks
unique identities
2D picture
Protein interactions
slide by Sergei Maslov
Internet
Correlation profiles give complex networks
unique identities
Sergei Maslov: 2D histogram
Protein interactions
Internet
Correlation profiles -cont’d
 Pastor-Satorras and Vespignani: 2D plot
average degree
of the node’s neighbors
degree of node
Correlation profiles -cont’d
 Newman: single number
-0.189
internet degree correlation coefficient
The Pearson correlation coefficient of nodes on each
side on an edge
Other examples of assortative mixing
 Assortativity is not limited to degree-degree correlations
other attributes
 social networks: race, income, gender, age
 food webs: herbivores, carnivores
 internet: high level connectivity providers, ISPs, consumers
 Tendency of like individuals to associate: ‘homophily’
 Scott Feld paper
Biological networks
 In biological systems nodes and edges can represent
different things
 nodes
 protein, gene, chemical
 edges
 mass transfer, regulation
 Can construct bipartite or tripartite networks:
 e.g. genes and proteins
GENOME
protein-gene interactions
PROTEOME
protein-protein interactions
METABOLISM
bio-chemical reactions
slide after Reka Albert
Cellular processes form networks on many levels
 metabolic reaction networks (tri-partite)
 Node types:
 metabolites (substrates or products), open rectangles
 metabolite-enzyme complexes (black rectangles)
 enzymes (open ovals)
 Edges
 substrate to complex or complex to product
 symmetrical edges
slide after Reka Albert
regulatory networks
nodes: genes, proteins
edges: translation
regulation: activating
inhibiting
slide after Reka Albert
the yeast two-hybrid method
 Activation and binding
domains are separated
and each attached to a
different protein
 If the proteins interact,
the two domains will be
brought together and
activate the
transcription of a
reporter gene
 Can do simultaneous
genome-wide
experiments
slide after Reka Albert
Resulting interaction network
slide after Reka Albert
Properties and problems of resulting networks
 Properties
 giant component exists
 power law distribution with an
exponential cutoff
 longer path length than
randomized
 higher incidence of short loops
than randomized
 Problems
 false positives
 false negatives
 only 20% overlap between
different studies
Implications
 Robustness
 resilient to random breakdowns
 mutations in hubs can be
deadly
 Evolution
 most connected hubs
conserved across organisms
(important)
 gene duplication hypothesis
 new gene still has same output
protein, but no selection
pressure because the original
gene is still present. So some
interactions can be added or
dropped
 leads to scale free topology
Metabolic networks: how to represent them
 Can consider the one-mode
projection of substrate
interactions (undirected)
slide after Reka Albert
Metabolic networks are scale-free
 In the bi-partite
graph:
 the probability that
a given substrate
participates in k
reactions is
k-a
 indegree:
a = 2.2
 outdegree:
a = 2.2
(a) A. fulgidus (Archae) (b) E. coli
(Bacterium) (c) C. elegans (Eukaryote), (d)
averaged over 43 organisms
Modularity
 No modularity
 Modularity
 Hierarchical modularity
E. Ravasz et al., Science 297, 1551 -1555 (2002)
(Pajek!)
How do we know that metabolic networks are modular?
clustering
decreases with
degree as
C(k)~ k-1
randomized
networks (which
preserve the
power law degree
distribution) have
a clustering
coefficient
independent of
degree
How do we know that metabolic networks are modular?
 clustering coefficient is the same across metabolic networks in
different species with the same substrate
 corresponding randomized scale free network:
C(N) ~ N-0.75 (simulation, no analytical result)
bacteria
archaea (extreme-environment
single cell organisms)
eukaryotes (plants, animals,
fungi, protists)
scale free network of the same
size
review: what would the clustering coefficient of a
random network be
 assume average degree of node is k
 probability of one neighbor linking to another is ~ k/N
 scales as N-1
Constructing a hierarchically modular network
RSMOB model
 Start from a fully
connected cluster of
nodes
 Create 4 identical replicas
of the cluster, linking the
outside nodes of the
replicas to the center
node of the original (N =
25 nodes)
 This process can
repeated indefinitely
 (initial number of nodes
can be different than 5)
Properties of the hierarchically modular model
RSMOB model
 Power law exponent g = 2.26 (in agreement with real
world metabolic networks)
 C ≈ 0.6, independent of network size (also
comparable with observed real-world values)
 C(k) ≈ k-1, as in real world network
 How to test for hierarchically arranged modules in
real world networks
 perform hierarchical clustering on the topological overlap
map (we’ll cover hierarchical clustering in a few weeks…)
 can be done with Pajek
Topological overlap
 A: Network consisting of nested modules
 B: Topological overlap matrix
hierarchical
clustering
Hubs may act within a module, or connect modules
 Party hub:
 simultaneous interactions
 tends to be within the same
module
 Date hub:
 sequential interactions
 connect different modules
Han et al, Nature 443, 88 (2004)
slide after Reka Albert
 some matching
motifs frequently
overlap (e.g. feed
forward loop)
Zhang et al, J. Biol 4, 6 (2005)