“Modularity” of Social Networks
Image: Ramki (www.wildventures.com)
Image : R K Pan
Sitabhra Sinha
The Institute of Mathematical Sciences, Chennai
Outline or Why ? What ? How ?
 Why networks ?
 What structures underlie real life networks ?
R K Pan & SS, EPL (2009)
 How do such structures affect network dynamics ?
In particular,
 How does it affect ordering dynamics in spin models ?
S Dasgupta, R K Pan & SS, PRE Rapid (2009)
Significant for understanding social coordination processes
such as consensus formation & adoption of innovations
Nodes
Why Networks ?
• In a network of interacting components
• Emergence
• of qualitatively different behavior from individual
components.
E.g., component = neuron, system = brain
Interactions add a new layer of complexity!
The aim of studying networks:
how interactions → complexity at systems-level
Links
Ubiquity of Networks
Networks appear at all scales
Proteins
Neuronal
Intra-cellular
communication
signalling
Social
contact
Food webs
10-9 m
10-6 m
10-3 m
1m
103 m
106 m
Molecules
Cells
Organisms
Populations
Ecologies
Theoretical understanding of networks
• Regular lattice or grid (Physics)
• average path length ~ N (no. of nodes)
• clustering high
• delta function distribution of degree (links/node)
•Random networks (Graph theory)
• average path length ~ log N
• clustering low
• Poisson distribution of degree
Empirical networks are not random –
many have certain structural patterns
Example: small-world networks
Regular Network “Small-world” Network Random Network
p=1
0<p<1
p=0
Increasing Randomness
p: fraction of random, long-range connections
Watts and Strogatz (1998): Many biological, technological and social
networks have connection topologies that lie between the two
extremes of completely regular and completely random.
Small-world networks can be highly clustered (like regular networks ),
yet have small characteristic path lengths (as in random networks ).
Characteristic path length ( ℓ ( p ) ) and clustering coefficient
(C( p ) ) as network randomness increases.
A prominent example of small-world networks:
Social networks
Nodes : individuals (N = 3 – 109)
Links : social interactions (< 150/node)
• Represent the “underlying” social
network (e.g. a knows b)
• Approximated by real interactions
between a & b (e.g. through phone calls,
emails, trades, …)
• Weight represents interaction
strength
Example of social interaction network
Collaboration Networks
Authors
1
2
A
Papers
IMSc Physics co-authorship network (2000-08)
3
4
5
B
C
D
Bipartite Paper-Author network
1 Co-authorship network
2
3
4
5
Re-constructing social networks
Example: Macaque troupe social network
UAS-GKVK Campus, Bangalore
Data: Anindya Sinha (NIAS,B’lore)
The Bonnet Macaque (Macaca radiata) seen widely in southern India
Usually live in large (~ 40)
multi-male, multi-female
troops where the adult
individuals (~ 10) develop
strong affiliative relationships
Image: Ramki (www.wildventures.com)
Image: Arunkumar
Macaque Social Networks can be defined in terms of
grooming frequency
total grooming time
approach frequency
Numbers refer to rank
among the adult females
from 11 (most dominant)
to 1 (least dominant)
Data: 1993-1997
grooming frequency
grooming time
approach frequency
Network analysis can predict group dynamics !
Female bonnet macaques
•usually remain in the group throughout their life
•as adults, form strong linear matrilineal dominance hierarchies that are
stable over time
Male bonnet macaques
•as adults, form unstable dominance hierarchies
•occupy low ranks when young, high when mature and at peak of health
Community detection generates consistent partitions for females, not for males
Predictive power: Observation in 1998 showed the group had split into two
(11,10,9,8,7,2) and (6,5,4,3) [1 had died]
Importance of SW social network structure
Contagion propagation in society through contact
Spread of SARS
from Taiwan, 2003
“The small-worlds of
public health”
(CDC director)
Chen et al, Lect Notes Comp Sci 4506 (2009) 23
Worldwide spread of SARS through international airline network
Why small-world pattern in
complex networks at all ?
We have shown that structurally
Watts-Strogatz
network
≡
R K Pan and SS, EPL 2009
Modular network
All the classic “small-world” structural properties of
Watts-Strogatz small world networks
e.g., high clustering, short average distance, etc.
are also seen in Modular networks
Modular Networks: dense connections within certain subnetworks (modules) & relatively few connections between
modules
Modules: A mesoscopic organizational
principle of networks
Going beyond motifs but more detailed than global description (L, C etc.)
Micro
Kim & Park, WIREs Syst Biol & Med, 2010
Meso
Macro
Ubiquity of modular networks
Modular Biology (Hartwell et al, Nature 1999)
Functional modules as a critical level of biological
organization
Modules in biological networks are
often associated with specific functions
Metabolic network of E coli
(Guimera & Amaral)
Chesapeake Bay foodweb
(Ulanowicz et al)
How about social small-world networks ?
Let’s look at real re-constructed social contact
patterns: A mobile phone interaction
network
Onnela et al. PNAS,104,7332 (2007)
Data:
• A mobile phone operator in an European country, 20% coverage
• Aggregated from a period of 18 weeks
• > 7 million private mobile phone subscriptions
• Voice calls within the operator
• Require reciprocity of calls for a link
• Quantify tie strength (link weight)
7 min
5 min
15 min
(3 calls)
Aggregate call
duration
Total number
of calls
3 min
J-P Onnela
Reconstructed network
Onnela et al. PNAS,104,7332 (2007)
Modularity of social networks
4.6 106 nodes
7.0  106 links
Modularity: Cohesive groups
communities with dense
internal & sparse external
connections
Other examples of modular
social networks
– Scientific collaborators
– e-mail communication
– PGP encryption ”web-oftrust”
– non-human animals
Onnela et al. PNAS,104,7332 (2007)
Q. What structure ?
Ans. Modular
Next…
How do such structures affect dynamics ?
Over social networks, such dynamics can be of
 information or epidemic spreading
 consensus or opinion formation
 adoption of innovations
A simple model of modular networks
Module ≡ random network
Adjacency matrix
Model parameter r : Ratio of inter- to intra-modular connectivity
The modularity of the network is changed keeping avg degree constant
Comparison with Watts-Strogatz model
Structural measures used:
Communication
efficiency
Clustering
coefficient
E = [avg path length, ℓ ]-1 = 2 /N(N-1) i>jdij
C = fraction of observed to potential triads
= (1 /N) i2ni / ki (ki - 1)
WS and Modular networks
behave similarly as function of p
or r
(Also for between-ness centrality,
edge clustering, etc)
In fact, for same N and <k>, we
can find p and r such that the WS
and Modular networks have the
same “modularity” Q
Then how can you tell them apart ?
Dynamics on Watts-Strogatz network different from that
on Modular networks
Network topology
Consider ordering or alignment of orientation on such networks
e.g., Ising spin model: dynamics minimizes H= -  Jij Si (t) Sj (t)
2 distinct time scales in Modular networks: t modular & t global
Global order
Modular order
Time required for
global ordering
diverges as r →0
Universality
Almost identical two time scale behavior is seen for oscillator
synchronization (viz., nonlinear relaxation oscillators)
Network topology
Consider synchronization on modular networks
e.g., Kuramoto oscillators: di /dt = w + (1/ki)Kij sin (j -  i)
2 distinct time scales in Modular networks: t modular & t global
Eigenvalue spectra of the Laplacian
Shows the existence of spectral gap  distinct time scales
Modular network Laplacian spectra
gap
Spectral gap in modular
networks diverges with
decreasing r
No gap
WS network Laplacian spectra
Existence of distinct time-scales in Modular networks
No such distinction in Watts-Strogatz small-world networks
Diffusion process on modular networks
Random walker moving from one node to randomly chosen nghbring node
Also shows the existence of 2 distinct time scales:
• fast intra-modular diffusion
Relevant for diffusion of
• slower inter-modular diffusion
innovation or epidemics
Distrn of first passage times for rnd walks
to reach a target node starting from a
source node
Localization of eigenmodes of
transition matrix
ER
WS
Modular
Within modules
diffusion
Between modules
diffusion
How about real small-world networks ?
The networks of cortical
connections in
mammalian brain have
been shown to have
small-world structural
properties
gap
gap
Our analysis reveals
their dynamical
properties to be
consistent with modular
“small-world” networks
We again turn attention back to consensus
formation dynamics
Q. How does individual behavior at micro-level
relate to social phenomena at macro level ?
Order-disorder transitions in Social Coordination
The emergence of novel phase of
collective behavior
Spin models of statistical physics: simple models of
coordination or consensus formation
•Spin orientation: mutually exclusive choices
•Choice dynamics: decision based on
information about choice of majority in
local neighborhood
Simplest case: 2 possible choices
Ising model with FM interactions: each agent can
only be in one of 2 states (Yes/No or +/-)
Types of possible order in modular
network of Ising spins
Modular order
Global order
N spins, nm modules
FM interactions: J > 0
Avg magnetic moment / module
Total or global magnetic moment
But how do the different ordered phases occur as a
function of modularity parameter r and temperature T ?
Long transient to global order can be mistaken as modular order
Possible Phase Diagrams
Tgc
Tgc
T gc
T
T
T
0
r
1
0
r
1
0
Can modular order be seen as a phase at all ?
r
1
fraction of “up” spins in module
Magnetic moment of a single module
At eqlbm, for strong modularity (r << 1)
fraction of modules with +
Total magnetic moment
Minimizing free energy w.r.t. f+
Continuous transition to “modular order” phase below
“Modular” critical temp
# links within module
As T is lowered,
Another continuous transition to “global order” phase below
“Global” critical temp
conn prob betn modules
Phase diagram: two transitions
r = 0.002
T
There will be a phase corresponding to modular but no
global order (coexistence of contrary opinions) even when all
mutual interactions are FM (favor consensus) !
Even when global order is possible …
Divergence of relaxation time with modularity
To switch from +
to -
, module crosses free energy barrier:
Relaxation time to global order:
-
At low T:
+
Thus, even when
strongly modular network takes very long to show global
order
Time required to achieve consensus increases
rapidly for a strongly modular social organization
But then …
How do certain innovations get adopted rapidly ?
Possible modifications to the dynamics:
Positive feedback
Different strengths for inter/intra couplings
I. The effect of Positive Feedback
Brian Arthur
The case of “counter-clockwise” clocks
Microsoft & the 7 Dwarves
The effect of Positive Feedback
Introduce a field H = hM (proportional to magnetization)
 Effectively increases intermodular interactions
 Drives system away from
critical line by increasing Tgc
 reduces
II.Varying Strength of
Coupling Between &
Within Modules
Marc Granovetter
J0 : strength of inter-modular connections
Ji : strength of intra-modular connections
increasing J0 / Ji  increasing r
Non-monotonic behavior of
relaxation time vs ratio of strengths
of short- & long-range couplings
Not seen in Watts-Strogatz SW networks
(Jeong et al, PRE 2005)
Ongoing and Future work
Effect of modular contact structure in formal games Is it easier for cooperation to emerge for Prisoners Dilemma on
modular networks ?
Could modularity in social networks have arisen as a result
of modules promoting cooperation – necessary for building
social organization ?
Role of multiple (>2) choices : q-state Potts model
Having different types of interactions : spin glass behavior on
modular networks
Question:
Why Modular Networks ?
Suggestion: modularity imparts robustness
Not quite !
Consider stability of a
random network with a
modular structure.
E.g.,Variano et al, PRL 2004
N=256
As random networks
are divided into more
modules (m) they
become more
unstable…
… however, we see modular networks all around us.
Why ?
Question:
Why Modular Networks ?
Clue:
Many of these modular networks also possess multiple hubs !
Most real networks have non-trivial degree distribution
Degree: total number of connections for a node
Hubs: nodes having high degree relative to other nodes
Why Modular Networks ?
R K Pan & SS, PRE 76 045103(R) (2007)
Hypothesis: Real networks optimize between several
constraints,
•Minimizing link cost, i.e., total # links L
•Minimizing average path length ℓ
•Minimizing instability max
Minimizing link cost and avg path length yields …
… a star-shaped network with single hub
But unstable !
Instability measured by max ~ max degree (i.e.,
degree of the hub)
In fact, for star network, max ~ N
How to satisfy all three constraints ?
Answer: go modular !
As star-shaped networks are
divided into more modules they
become more stable …
 = 0.4
Increasing stability, average path length
 = 0.78
=1
…as stability increases by
decreasing the degree of hub
nodes max ~ [N/m]
Shown explicitly by
Network Optimization
Fix link cost to min (L=N-1) and
minimize the energy fn
E () =  ℓ + (1- ) max
 [0,1] : relative importance of path
length constraint over stability
constraint
N = 64, L = N-1
Transition to star configuration
The robustness of modular structures
We have considered dynamical instability criterion for network
robustness – how about stability against structural perturbations ?
E.g., w.r.t. random or targeted removal of nodes
Scale-free network: robust w.r.t. random removal
Random network: robust w.r.t. targeted removal
Is the modular network still optimal ?
Surprisingly YES !
At the limit of extremely small L, optimal
modular networks ≡ networks with
bimodal degree distribution…
…has been shown to be robust w.r.t.
targeted as well as random removal of
nodes
Tanizawa et al, PRE 2005
Similar mechanisms explain the emergence
of hierarchical structures in society
Many complex networks that we see in society (and also
nature) have hierarchical structures. Why ?
Emergence of hierarchy
Consider social networks as solutions to multi-constraint
problems:
• Increase communication efficiency  Decrease average path
length l in a network
• Minimize information load at each node  Decrease max
degree kmax
• Minimize overall communication needs  Decrease total
number of links L
Emergence of hierarchy
Minimize the energy
function,
E = a l + g L + (1-a-g) kmax
• For a = 0, g =1 optimal
network is a chain
• For a = 1, g = 0, optimal
network is a clique
• For a = g =0.5, optimal
network is a star
• Over large range of values
of a and g, hierarchical
networks emerge –
hierarchy measured by HQ
Cycles
Hierarchical network
Conclusions:
 Modular networks are small-world networks: indistinguishable
from WS model generated networks by using structural
measures
 Dynamics on modular networks (but not in WS networks)
show distinct, separate time-scales; manifested as Laplacian
spectral gap seen in real networks (cortico-cortical brain
networks)
 Collective behavior such as ordering in spin models show a
novel phase corresponding to modular order, in addition to
global order and no order – even when mutual interactions favor
consensus
Thanks:
Raj Kumar Pan
Sumithra Surendralal
Abhishek Dasgupta
Subinay Dasgupta
Anindya Sinha