Transcript Modular network

Networks, Modules & Persistence
Dynamics of agent interactions on complex
networks with mesoscopic organization
Sitabhra Sinha
in collaboration with
R K Pan and T Jesan (IMSc Chennai)
S Dasgupta (Calcutta Univ)
Primate
Networks:
Genesis of
hierarchy
Epidemics or
contagion
spreading
Opinion or
Consensus
formation
Social Network
Research
@ IMSc
Popularity:
Diffusion of
products or
ideas
Games on
Graphs
Market
Dynamics:
Bubble &
Crashes
Themes
• Are there common structural mesoscopic features in
social networks ?  Statics: modularity
• What role do such network features play in governing
network function ?  Response & dynamics: polarization
• How can such structures arise in general ?  Evolution:
multi-constraint optimization
• Are there specific processes in social networks
promoting such motifs ?  Social games: Emergence of
co-operation
Importance of 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
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)
Are social networks modular ?
Let’s look at real re-constructed human social
contact network: 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 contagion spreading
 consensus or opinion formation
 adoption of innovations
Social Network of a Karnataka Village
Data: microfinance institution Bharatha Swamukti Samsthe
Nodes: Individuals
Links: Social relations
Described in Banerjee et al, Science (2013)
Village “52”
Population:1525 individuals
Largest connected component
1497 individuals
43 modules
Largest module contains
15.5% of nodes
32 modules contain less than
1% of nodes
Node colors represent the community to which they belong
Measuring modularity
How to quantify the degree of modularity ?
One suggested measure:
(Newman, EPJB, 2004)
A: Adjacency matrix
L : Total number of links
ki : degree of i-th node
ci : label of module to which i-th node belongs
A
For directed & weighted networks:
(
W: Weight matrix
si : strength of i-th node
Modules determined through a generalization of the
spectral method (Leicht & Newman, 2008)
C
B
)
D
To model the transmission of contagion on the network
we use
SIRS dynamics
The “infectiousness” of a contagion
To measure the force of infection of a contagion (how quickly it can
spread) we use the empirically measurable quantity
Basic reproduction number R0
Mean number of new infections caused by a single infectious
individual in a wholly susceptible population (as in the beginning of
an epidemic): If each infected person on average infects more than
one other individual, R0 > 1  Epidemic
For a contagion spreading on a contact network
• if the probability of infection is , and,
• if k is the average degree of the contact network
Then R0  kg
where g : mean generation interval
We investigate the long-term transmission of contagia in
modular contact networks
In particular,
Persistence
i.e., existence (circulation) of contagion in the
population for an indefinite period, i.e., I(t) > 0 as t
A contagion that starts out as an
epidemic, eventually either dies out
[I(t)=0] or becomes endemic
Hollingsworth, J Pub Health Policy, 2009
Contagia in empirical
network comprising
communities are
surprisingly persistent
in contrast to those in
degree-preserved
randomized networks
which do not have
modular organization
Difference even
more pronounced if
modularity is
enhanced by
decreasing the intermodular
connectivity
Can modular organization of contact network have significant
impact on the long-term dynamics of contagia spreading ?
A simple model of modular networks
Model parameter r : Ratio of inter- to intra-modular connection density
Module ≡ random network
Comparison with Watts-Strogatz model
Structural measures used:
Communication
efficiency
E = 1 /avg path length, ℓ = 2 /N(N-1) i>jdij
Clustering
coefficient
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)
How can you tell them apart ?
Dynamics on Modular networks different from that on
other small-world (Watts-Strogatz) 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
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
Signal propagation on contact networks with community
organization can take extremely long times
Diffusion process on modular networks
E.g., Random walker moving from one node to randomly chosen neighboring node
Pan & Sinha, EPL 2009
Distrn of first passage times
for random walks
Shows the existence of two
distinct time scales:
• fast intra-modular diffusion
• slower inter-modular diffusion
while random networks show a
continuous range of time scales
In modular networks, signal spreads slowly from module to module !
SIRS Dynamics: Existence of communities can
make highly infectious contagia persistent
m=64 modules of size n=16
avg degree <k> = 12
Avgd over 100 rlzns
r = 0.0002: rapid extinction
For isolated
modules (r=0) and
homogeneous
networks (r=1)
epidemic with high
R0 dies out quickly
r = 0.002: persistence
r = 0.02: rapid extinction
However, for
a critical range of
modularity for
contact network
(r ~ 10-3), highly
infectious contagia
are persistent.
Persistence in a critical range of modularity
Distribution of
persistence time 
shows bimodal
character for large r –
with the upper branch
diverging for a critical
range of modularity…
Effect of increasing the
number of modules
…while for lower r
the distribution is
unimodal with avg 
decreasing rapidly as
the modules are
effectively isolated
(R0 = 6)
The mechanism of enhanced persistence
Time-scale for global
diffusion (inverse of
smallest finite
Laplacian eigenvalue)
decreases with r…
…. So does the timescale separation
between inter- and
intra-modular events
(the Laplacian spectral
gap)
However – the ratio
of the two show nonmonotonic
dependence on
contagion spreads slowly from module to module, allowing
parts of the network to recover before return of infection!
modularity
Order-disorder transitions in Social Coordination
How does existence of different time-scales affect consensus
formation dynamics on networks with communities
Q. How does individual behavior at micro-level
relate to social phenomena at macro level ?
E.g., can social polarization occur as a result of modular
network structure even when all relations are “friendly” ?
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
•Temperature: noise or degree of
uncertainty among agents
Simplest case: 2 possible choices
Ising model with FM interactions: each agent can
only be in one of 2 states (Yes/No or +/-)
possible order in modular network of
“spins” No consensus
Consensus
Modular order
Global order
N spins, nm modules
FM interactions: J > 0 (“Friendly” interactions)
Avg modular order parameter
System or global order parameter
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) !
Divergence of relaxation time with modularity
Even when global order is possible, i.e.,
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 order
parameter)
 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)
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
Modularity via games between
selfish rational agents
So far, we looked at rather general processes for evolving
modularity
Our agents are relatively “naïve”, whose rules for making choices do
not explicitly take into account how the other agents choose
What happens if our agents are capable of “strategic” decision
making (i.e., taking into account how other agents will behave) ?
Can there be specific processes at work in social
networks that promote the emergence of community
structure ?
Social games: Emergence of co-operation
Prisoners Dilemma
originally framed by Merrill Flood and Melvin
Dresher at RAND (1950)
In the iterative setting, an ideal
model for analyzing the
conditions for the emergence of
cooperation
Payoff Matrix
Cooperate
Defect
Cooperate
R,R
S,T
Defect
T,S
P,P
T: Temptation to defect
R: Reward for cooperation
P: Punishment for mutual defection
S: Sucker’s payoff
In general, T > R > P > S
Usually, R=1, P=0, S=0 and 1<T<2
Prisoners Dilemma in a network
Spatial Prisoners 1.75< T <1.8
Dilemma:
Nowak & May, Nature 1992
1.8< T <2
Agents play with
neighbors on a
lattice, adopting
strategy of neighbor
with max payoff
Network Prisoners Dilemma: agents play with nbrs in the network,
compare payoff with randomly chosen nbr and adopt strategy of neighbor having
higher payoff with a probability
Stability w.r.t. defection  Robustness w.r.t. removal of nodes
Role of modules:
Fragmenting the network into several sparsely connected communities,
each with star configuration, allows cooperation to be robust – modularity
beneficial if degree distribution is bimodal
The evolution of cooperation driving the
emergence of modular structures
C
C
C
D
D
C
C
D
C
D
Link
adaptation
dynamics
C
C
C
C
D
C
D
Could modularity in social networks have arisen
as a result of modules promoting cooperation –
necessary for building social organization ?
D
To Summarize
• Are there common structural mesoscopic features in
social networks ?  Statics: modularity
• What role do such network features play in governing
network function ?  Response & dynamics: polarization
• How can such structures arise in general ?  Evolution:
multi-constraint optimization
• Are there specific processes in social networks
promoting such motifs ?  Social games: Emergence of
co-operation
Thanks