Transcript Modeling and Optimization for Multi

Maryland Hybrid Networks Center
(HyNet)
Wireless Network Design:
Implicit Loss Models, PHY Feedback,
Performance Sensitivities
John S. Baras
Institute for Systems Research
Department of Electrical and Computer Engineering
Department bof Computer Science
Fischell Department of Bioengineering
University of Maryland College Park
USC Workshop on Wireless Networks
Los Angeles, May 20-21, 2008
Copyright © John S. Baras 2008
1
Network Science - 1
• What is a Network?
– In several fields or contexts: social, economic, communication,
sensor, biological, physics and materials
• A collection of nodes, agents, … that collaborate to accomplish
actions, gains, … that cannot be accomplished without collaboration
– Network nodes gain from collaborating
– To collaborate they need to communicate (cost)
Constrained Coalitional Games
• Trade-off: gain from vs cost of collaboration (vector metrics)
• Many problems in networks
fields), can be traced to this fundamental
Phase (all
transitions
trade off
• Most significant principle for autonomic networks
n=20
 Example 1: Network Formation -- Effects on Topology
 Richer than percolation,
finite time dynamics
 Example 2: Collaborative communications
 Example 3: Web-based social networks and services
Copyright © John S. Baras 2008
2
Network Science - 2
 How to synthesize networks to specifications, resilient, robust,
adaptive to mission requirements?
– Current network models and network software are monolithic, brittle and nonfault tolerant, expensive to change, intractable to verification and
performance analysis except via simulations
Component-Based Network Analysis & Synthesis (CBN)




Components result in: modularity, implementation cost reduction ,
re - usability, adaptability to mission, facilitate technology insertion
Executable
Formalmakes a set of
With the right (powerful)
interfaces, programmability
Models
components much Models
more integrated – intelligent
networks
Components facilitate validation and verification
Theory and Practice of Component-Based Networks
–
–
–
–
Heterogeneous components and their composition
Constructivity theory including results on compositionality
Performance of components and of their compositions
Performance
Must be able to move back and
forth form performance - optimization domain to
correctness and timing analysis Models
domain and have composition theory preserving
the properties of components as you try to satisfy specifications in both domains
– Create new tools for performance and functionality analysis
Copyright © John S. Baras 2008
3
Communication Networks
•
•
•
•
•
•
•
•
Broadband wireless: multiple MBps to the mobile user (WiMax & beyond)
and Baras 04,
AD/IPA/CE
Multiple networks-multiple
routes optimized for different [Liu
loads
Subgradients
Baras et al 07]
Performance
Dynamic spectrum management
and allocation
deterministic
Metrics andbased on probing feedback
stochastic
sensitivities
Fixed
Integrated dynamic coordination
of or
interference management via on-line feedback
from
PHY (MIMO, directionalhybrid
antennas,
new PHY concepts)
Multi-objective
Topology -Mobility
MANET
broadband
Performance
designer/optimizer
Traffic patterns
Integrated
design of adaptive
MAC and
dynamic hierarchical
routing
design
parameters
Models
Net Utility Max
and matrix
and
multiple
dynamic
interfaces
Separate
control-signaling
plane
fromand
traffic
(multimedia) plane
Network Conditions
Extensions
architecture
QoS
Robustness and resiliency
CBN and -- Monitoring, probing, evaluation of situation,
Self-configurable networks
extensions
environment and demand
to create best topology and paths
Component-based network protocol design – a more
sophisticated version of cross-layer design
Mobile wireless communication networks as distributed,
asynchronous, feedback (many loops), hybrid automata
(dynamical systems) -- emergent behavior
Performance models linked to sensitivity computations and
multi-criteria constrained optimization (constrained NUM,
optimization - constraint based reasoning, semi-rings)
Copyright © John S. Baras 2008
QoS Driven Design
•
Performance Models ( Rates – Throughput, packet losses, delays, etc. )
Sensitivity Computation and Trade offs ( Automatic Differentiation /
Perturbation
Analysis
/ Cross networks
Entropy ) to meet specifications?
HowInfinitesimal
to design
mobile
wireless
CONSOL -OPTCAD, ILOG CPLEX -SOLVER
4
Social and Cognitive Networks
• Social Network models: multiple directed graphs
– Nodes: agents, organizations; Links: ties, relationships
– Weights on links : value (strength, significance) of tie
– Weights on nodes : importance of node (agent)
• Real-life problems: Dynamic, time varying graphs, relations, weights
• Organizational needs
Network architecture and operation
• Distributed trust management systems in autonomic networks
ˆ
si (k trust
 1) management
 f J ji , s j (k ) | j  Ni
– Trust document distribution, trust metrics,
– Trust (and Mistrust) spreading, dynamics, voting, policies
– Distributed, high integrity reputation and recommender systems

• Effects of topology on convergence (efficiency -- small world graphs)
1 a 2 b
3
• Trust as incentive for collaboration (lubricant for collaboration
– Arrow)
• Trust evaluation: direct and indirect ways; reputations, profiles
– Pairwise games on graphs with players of many types
a
– Trust computation via linear iterations in ordered semirings
• Spin glasses (from statistical physics), phase transitions
b observations
• Infer organizational structure from partial communications
• Learning on graphs and network games: behavior, adversaries, network
Copyright © John S. Baras 2008
5
Implicit Loss Models and AD
• Simple approximate loss network model
that couples the physical, MAC and
routing layers to estimate the total network
throughput
• Optimization framework using Automatic
Differentiation (AD) for sensitivity analysis;
applications to robustness, to maximizing
throughput w.r.t. routing parameters, etc.
• Substantial extensions to include multiple
paths, non-saturated flows, hidden nodes,
scheduling, MAC/PHY failures
Copyright © John S. Baras 2008
6
Motivation
• Discrete packet simulation tools (e.g. ns2,
OPNET) take too much time even for simple
network configurations and light traffic
• Our objective: to develop models that estimate
the performance of a MANET fast. Design for
predicatble performance bounds (specifications)
• Inputs: network topology (could be time
varying), neighborhood relations (channel
conditions), traffic demand (source-destination
pairs, data rates, number of paths).
Copyright © John S. Baras 2008
7
Approach
• We define two sets of equations:
– The first set describes the specific MAC and PHY models we
consider and expresses the loss parameters and the outgoing
rate of a link in terms of the incoming rates and the interference
from neighboring links
– The second set describes the relations between flows and the
scheduling parameters at each node
• Two sets are coupled iteratively, on the entire network, in
a fixed point setting till they converge to a consistent
solution
• Then, we evaluate the robustness of the solution using
AD:
– AD provides the partial derivatives of the performance metric
(throughput) w.r.t. design parameters (e.g. routing probabilities)
– Using these partial derivatives with the gradient projection
method we solve for the design parameters that maximize the
total network throughput
Copyright © John S. Baras 2008
8
Preliminaries
• We consider the DCF function of IEEE 802.11with the
RTS/CTS mechanism
• The network consists of N nodes, set of paths P
• The unit of time is one slot equal to the backoff slot of
IEEE 802.11
• We assume packet collisions occur only during the
RTS/CTS exchange
• Notation:
– Pi is the set of paths through node i
– Ci is the set of nodes within range from node i
– Ci+ is Ci plus node i
_
– Ci is the set of nodes not in Ci
– hi,p is the next hop of node i in path p
Copyright © John S. Baras 2008
9
Prior Related Work
•
•
•
•
•
•
Bianchi [2000]
Kumar, Altman, et al [2005]
Garetto, Salonidis, Knightly [2006]
Medepalli, Tobagi [2006]
Hira, et al [2006]
Liu, Baras [2004]
Copyright © John S. Baras 2008
10
PHY/MAC Parameters
•
•
•
•
•
Stage 1 : RTS/CTS sent
Stage 2 : ACK and data packets sent
i,p : Prob. of PHY or MAC failure (stages 1, 2)
i,p : Prob. of PHY failure (stage 2)
li,j : Prob. of PHY failure (stage 1 or 2)
• i,p : arrival rate of path p packets at node i
• Ti,p : service time of path p packets at node i
(from the time scheduler selects it)
 ki,p : average serving rate of path p packets at
node i
Copyright © John S. Baras 2008
11
Scheduler Coefficients
 i, p '
  i, p
if 
E(Ti , p ' )  1
m
 (1   m ) ,
p 'Pi (1   i , p ' )
i, p


 i, p
k i, p  
m
(1


i, p )

, otherwise

 i, p '
E( Ti , p ' )

m
 p ' Pi (1   i , p ' )
m : max No of packet transmission retries
Fraction of time node i is serving path p packets:
i :
i, p  ki, p E(Ti, p )
total average throughput of node i
i   ki , p E( Ti , p )
p Pi
Copyright © John S. Baras 2008
12
Collision and Hidden
Nodes Modeling (1)
Channel access probability [Bianchi 2000], L back-off
stages, min window size W:
''
i, p
a

2(1  i , p )
W (1  2i , p )  i , p (W  1)(1  (2i , p ) L )
Average transmission time of node i during Ti,p : vi,p
Average time successful + average time in failed transmissions
Probability of collision:
i , p  1  (1  li ,h )(1  h ,i )
i,p
i,p

jChi , p
(1 
Ci
 a j , p ',hi , p )
p 'Pj

jChi , p Ci
(1 
i,p
a
)
 j , p ',hi , p
V
p 'Pj
''
a


(1


)
a
Probability of receiving a packet: i , p, j
i, p
i, j
i, p
Copyright © John S. Baras 2008
13
Collision and Hidden
Nodes Modeling (2)
Probability of transmissions from hidden nodes:
i , j

vn, p ' 
 1   1    n , p '
)



E(
T
)
p
'

P
nCi C j 
n, p '
n

Average transmission time:
i, p
i, p
i, p
vi , p  1  im, p TRTS
 TCTS
 TACK
 TPi , p  3  SIFS

1  im, p
1  i , p
  i, p i, p

  i, p  i, p
i, p
i, p
i , p 
TRTS  TCTS  TP  3  SIFS  1 
 TRTS  SIFS 


 i , p

 i , p 


Copyright © John S. Baras 2008
14
Time Components (1)
• di,p : time for successful transmissions of path p
packets at i
• ui,p : average time for successful transmissions
of i neighbors
• bi,p : average back-off time for path p packets at i
• ci,p : average time for failed transmissions
Copyright © John S. Baras 2008
15
Time Components (2)
Average service time:
E(Ti , p )  1  im, p  di , p  bi , p  ui , p  ci , p
Successful transmission time:
di, p  T
i, p
RTS
T
i, p
CTS
T
i, p
ACK
T
i, p
P
 3 SIFS
m
Average backoff time:
bi , p  Wn in, p
n 0
Average time spent in failed transmissions (in neighborhood of i ):
ci , p 
zi , p  ri , p
Copyright © John S. Baras 2008
qi , p
wi , p
16
Time Components (3)
qi,p: Prob. of successful transmission of i when scheduled
to transmit path p packets
ri,p: Prob. of successful transmission in the neighborhood of i
zi,p: Prob. of at least one transmission in the neighborhood of i ,
when scheduled to transmit a path p packet
wi,p: average time for failed transmissions in neighborhood of i



zi , p  1  1  ai'', p   1  1   j ,i     j , p ' a ''j , p '  
 p 'P


jCi
 j


ri , p



 1  1  qi , p   1  1   j ,i     j , p ' q j , p '  
 p 'P


jCi
 j


qi , p  ai'', p 1  i , p 
wi , p
  i, p i, p


  i, p
''
i, p
i, p
a


1


T

T

T

3

SIFS


 RTS CTS P
 1  
  p
j, p' j, p' j, p'  
j ,i 



jCi  'Pj
i, p


 i, p



''
a j , p '  j , p '  j , p '  1   j ,i 
  p

jCi  'Pj

Copyright © John S. Baras 2008

 i, p
T

SIFS
  RTS
 


17
Time Components (4)
Average time of successful transmissions of
neighbors of node i :
ui , p 
ri , p  qi , p
qi , p
q
where:
g j, i, p 
p 'Pj
jCi
j, p'
j ,i , p
dj
 j , p ' 1   j ,i 
ri , p  qi , p

dj 
g
p 'Pj
k j , p ' d j , p ' 1   jm, p ' 

p 'Pj
k j , p ' 1   jm, p ' 
Copyright © John S. Baras 2008
18
Fixed Point Network Model
• Combine the first set of equations given in previous
slides with the second set of equations that describe
the relation between the incoming and outgoing flows at
each node:
h
i,p ,P
 ki , p 1  
m
i, p

• The fixed point algorithm starts from the source node
of each path at each iteration where the arrival rate is
fixed and given.
• Next, the algorithm computes the scheduling
coefficients.
• Then we use the second set of equations to compute the
next hop incoming traffic rate.
• We repeat the same procedure for the next hop. The
algorithm terminates when a fixed point is reached.
Copyright © John S. Baras 2008
19
Design Methodology (1)
• The fixed point algorithm provides a performance
analysis tool.
• Using this tool we develop a methodology for network
configuration and optimization.
• Example: optimal probabilistic routing.
– For each source-destination pair k paths are discovered using
the Dreyfus k-shortest path algorithm.
– We use Automatic Differentiation (AD) to compute the partial
derivatives of the total network throughput w.r.t. the routing
probabilities for each path in each source-destination pair.
– AD is a necessary tool since the dependence of the throughput
on the routing probabilities is not given explicitly, but rather
implicitly through the set of equations that constitute the fixed
point algorithm.
– Using the gradient projection method we solve for the optimal
routing probabilities for each path in each source-destination pair
that maximize the total network throughput.
Copyright © John S. Baras 2008
20
Design Methodology (2)
Copyright © John S. Baras 2008
21
Automatic Differentiation (AD)
• Goal: maximize network throughput w.r.t. the routing
probabilities.
• There is no analytic expression that gives the throughput
as a function of the routing probabilities; instead, the
code that computes the fixed point for the set of
equations presented above provides such description
(implicitly).
• Thus, AD is necessary: it repeatedly applies the chain
rule to combine the local partial derivatives of each
executed operator in the code.
• The optimization method used is Gradient Projection.
Copyright © John S. Baras 2008
22
Comparison Between
Fixed Point Method and OPNET
Algorithm finds shortest paths between nodes 3 and 7 : 3-0-1-5-7
Copyright © John S. Baras 2008
23
Sensitivity Analysis
Three active connections : 3-5, 16-21, 17-22
Three strategies (algorithms): one path, equal probabilities,
optimization-based
Performance of optimization-based algorithm improves as
No of available paths increases
Copyright © John S. Baras 2008
24
Comparison with 3 Connections
Copyright © John S. Baras 2008
25
Computation Time (in secs)
Number of
connections
1
3
5
7
9
C code
0.51
2.86
4.37
5.90
10.38
Opnet
190
309
352
466
476
Copyright © John S. Baras 2008
26
CBN -- Beyond Modularity
• With the right interfaces, programmability
makes the set of components much more
integrated – intelligent networks
• Like in networked embedded systems
• Insisting on orthogonallity of concerns for
components is not correct
• Components will interact – We need to control
the interaction through integration – possible
only with new more powerful interfaces
27
CBR -- Performance Metrics for
Routing Components
• Component selection: how to evaluate and
compare different components under different
environments (Network topology, Traffic scenario,
Mobility profile, Link states)?
• Meaningful Component Metrics are crucial for
components performance evaluation, comparison,
selection
• Finer metrics than System Performance Metrics
(Latency, Throughput, Packet Loss Ratio)
• Statistics can be collected during network
activities
5/30/2007
28
CBR -- Routing Protocol Metrics
vs Component Derivative Metrics
• Goal: Evaluate the components against relevant metrics
that will not only differentiate the various components but
will also relate the performance of the component with the
routing protocol performance.
Also link to other layer metrics (e.g. MAC)
• Derivative Metrics:
•
•
•
•
5/30/2007
Route selection latency (sec)
Route selection overhead (packets/sec)
Number of routes found and ranking
Quality of the routes (stability, E2E rate delay loss)
29
Modeling and Performance of
MANET Routing (and beyond)
1: Dynamic computation and maintenance of
neighborhoods (CBR)
– Mathematically: maintaining up to date information and
characterization of the Adjacency Matrix : A(t )
– Here
 1, if j "can hear" i
Aij (t )  
0, if j "cannot hear" i
2: Algorithms for computing paths between designated
sets of origin and destinations nodes. Floyd-Warshall,
Dreyfus, and variations with an eye towards faster
distributed execution
– Input is A(t)
– Output is multiple-paths from each origin to each
destination node
30
Modeling and Performance (cont.)
3: Assignment of QoS to paths and algorithms to
compute all pairs shortest paths, all pairs bottleneck
paths, all pairs k-shortest paths. Approximations for
speed and scalability
– Mathematically: Weighted Adjacency Matrix : A(t )
– Examples: distance, hops, bottleneck, MAC and
interference
– Can handle IERs effectively as constraints
4: Multiple-path Routing Algorithms (CBR) for optimizing
performance metrics or trade-offs between metrics
– Gallager minimum delay and extensions
– Fast approximations
– Tradeoffs between delay, throughput etc
31
Modeling and Performance (cont.)
5: Estimation of Overhead. Use Information Dissemination
Component (CBR and CBN). Trade-offs between
performance and Overhead.
– Use our recent results on linking interference models,
throughput, routing via fixed point methods and MVA.
– Use our recent results on linking the above with Automatic
differentiation: Thus for the first time enabling a “gradient like”
approach to design even when models are in the form of
programs or numerical algorithms or simulations.
6: Dynamic scheduling for joint MAC and multiple-path
Routing design for better performance. Estimation of OH.
– Estimate delay for all O-D pairs
– Trade-offs via extensions of (5)
– Linkage with much improved version of “back pressure”
algorithm and cross-layer designs.
32
Modularity of Routing Protocols
• All MANET routing protocols studied (AODV, DSR,
OLSR, TORA, …) can be modularized into five
functional components:
• Neighborhood Discovery Component
• Selector of Topology Information to Disseminate
Component
• Topology Information Dissemination Component
• Route Selection Component
• Packet Forwarding Component
5/30/2007
33
CBR – Details
• A modular and structural method for design and
analysis of ad-hoc routing protocols
• Specification of the main components of a routing
protocol
• Specification of the inputs, outputs, metrics,
exogenous inputs, parameters of each component
• Specification of the performance metrics and
bounds for each component.
• We have implemented these models for OLSR and
OLSR-like routing algorithms
The Model
Mapping 1
Neighborhood discovery
Link Probability
pij
Mapping 2
Selector topology
Info. Dissemination
MPR Probability
 ij , P(Ci )
Mapping 3
Route Selection
Next hop Probability
Neighborhood discovery
overhead rate H
eij , Tij
Mapping 5
MAC & PHY layer models
Packet Failure and
Service Time
Topology information
Dissemination
overhead rate T

ij
(k )
Mapping 4
Packet forwarding
Scheduling Rate
ij (k )
Neighborhood Discovery
•
•
•
•
•
•
Periodic transmission of HELLO messages to detect bidirectional links and
identify second order neighbors.
We model this block as a finite state Markov Chain.
The control parameters are U and D.
The input is transmission success probability
The output is probability of link detection, which can be computed from
probability of being in the blue states.
Average delay in detection of an operational link and a broken link can also be
computed.
sij
sij
s
ij
1
fij
2
sij
U-1
fij
fij
sij
fij
U+D
U+D
-1
fij
U
sij
sij
U+2
fij
U+1
fij
Performance Metric for NDC
• Link detection probability or entropy curve
Selector of Topology Information to Disseminate
(MPR selection):
•
•
•
•
State of a node represents its selected MPRs, which is a random
process.
We use Monte Carlo simulation to estimate the state probabilities for
each node.
For the state probability computation, we have to consider links of a
node and its neighbors in the simulation.
Hence, the simulation time for each node is scalable and is not
affected by the network size.
Route Selection Component
(reverse network)
• Input is the link detection and state probabilities
• Output is next-hop probabilities
• In the reverse network links are from nodes to their
MPRs
• We compute the average hop count using a
probabilistic set of Bellman-Ford (DP) equations on
the reverse network:
h(i, Ci , k )  1  min (h( j, k ), h(i, k )),
jMPR ( Ci )
h(i, k )   P(Ci )h(i, Ci , k ).
Ci
• We also take into account that for the last hop,
nodes do not need to use their MPR
Route Selection Component
(forward network)
• We use the average hop count on the reverse
network to compute the average hop count on the
forward network
• After computing the average distance, we estimate
the hop distance using Maximum Entropy Estimator
• For each node i at state Ci, probability of selecting
an MPR j as the next hop is probability of j having
minimum distance among neighbors
• We use this principle together with the estimated
distance distributions to estimate the next hop
probabilities on the forward network
Estimation of the hop count pdf
• We use the B-F iterative computation on the reverse
network to estimate the average hop count.
• The minimum hop count can be computed directly.
• The maximum hop count is assumed to be:
•
M (i, k )  max  2D(i, k )  m(i, k ),2m(i, k ) 
The maximum entropy pdf is determined p j   exp( j )
M ( i ,k )
min

j  m ( i ,k )

p j log p j ,

j  m ( i ,k )
p j  1,

e  j  1
j  m ( i ,k )
s.t.
M ( i ,k )
M ( i ,k )
M ( i ,k )

j  m ( i ,k )
jp j  D (i, k )

M ( i ,k )

j  m ( i ,k )
je   j  D (i, k )
Next hop Probability Computation
• Every node always select the neighbor with minimum
distance
• For each node compute the probability of each
combination of the detected neighbors

2
– For a node with  neighbors, there are
possible
combinations
– We assume that the link detections are independent events
• For each combination, if X and Y are hop counts of two
detected neighbors, we compute probability of selecting
X over Y
– We always select the neighbor with smaller distance.
– If distances are equal, we select the one with lower index.
– The selection probability of a node is proportional to probability
of that node being selected over all detected neighbors
A Simple Example
• 5x5 grid topology with 0.6 link detection probability.
• Routing probabilities for source node 13 to
destination node 1.
• Consider the combination that 4 neighbors
(8,12,14,19) are detected with probability 0.64
• The average distances using B-F are: (4.70, 4.74,
7.42, 7.39)
• The maximum entropy pdf distribution for distance
of neighbors are
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Hop Distance 3
4
5
6
7
8
9
10
Node 8
0.23
0.22
0.27
0.27
0.18
0.18
0.31
0.33
0.17
0.17
0.17
0.17
0.16
0.16
0.16
0.16
0.16
0.16
Node 12
Node 14
Node 18
0.19
0.18
A Simple Example (cont)
• The probability of selection of neighbor i
over neighbor j
x
0.64
0.94
0.94
0.637
0.36
x
0.94
0.94
0.359
0.06
0.06
x
0.58
0.002
0.06
0.06
0.42
X
0.002
The selection probability of
node i over node j
The selection
probability of node i
End-to-End Success Probability
• Comparison of OLSR neighborhood
discovery with our proposed design
Effects of Graph Topology on
Convergence of Network Protocols
• Distributed algorithms frequently arise in
networked systems
– Group of agents with simple/complex abilities
– Agents sense their “local” neighborhood
– Communicate with neighbors and process the
information
– Perform a local action
– Emergence of a global behavior.
Example: iterative actions leading to convergence to
an agreement about ”coordination variables” in
consensus problems
– Group topology affects group performance
Copyright © John S. Baras 2007
46
Effects of Graph Topology on
Convergence of Network Protocols
• Distributed algorithms frequently arise in networked
systems
– Emergence of a global behavior from local behaviors
• Effectiveness of these algorithms depends on:
– The speed of convergence
– Robustness to agent/connection failures
– Energy/ communication efficiency
• Design problem: Favorable tradeoff between
performance improvement (benefit) of collaborative
behaviors vs. costs of collaboration
– Small world graphs achieve such tradeoff
– Two-level hierarchy to provide efficient communication
• Many applications: communication and sensor networks,
networked control, biology, sociology, economics
Copyright © John S. Baras 2007
47
The Importance of Being
Well-Connected
• Local majority voting (Peleg ’96)
– Each of n citizens has an opinion about voting
Yes or No
– Rule: Each citizen’s vote is based on the
majority of its neighbors, including itself
– What is the minimum number of No-voters
that can guarantee a No result?
– A few number of well connected nodes can
determine the outcome of the process!
Copyright © John S. Baras 2007
48
The Importance of Being
Well-Connected (cont.)
White circles: NO voters
Black circles: YES voters
Order of voting matters!
Iterative polling : Oscillation or
If NO voters do not follow the
protocol, then 2 NO voters, are
sufficient to change the other n-2
YES voters’ opinion.
Even if NO voters follow the protocol
a negligible minority of 2 n can result
in one step convergence to NO
Copyright © John S. Baras 2007
49
The Importance of Being
Well-Connected (cont.)
Simple Lattice
C(n,k)
Small world model:
rewiring a portion 
Conjecture: Short paths
between distant parts of
the network will result in
fast information spreading
which helps global
coordination
Small world: Slight
variation adding nk 
Copyright © John S. Baras 2007
50
Spectral Gap gain
vs
C(500,3)
C(1000,5)
Simulations support conjecture
Copyright © John S. Baras 2007
51
Mean Field Explanation for Fast
Convergence in Small World Networks
• Simulation results show that adding a small number of
well chosen links to ring structured graphs results in high
convergence rate
• Eigenvalue analysis difficult due to non-symmetric
matrices
• Use a slightly different model to explain
– Start from ring structure
G0=C(n,1)
F0
K
– Perturb zero elements in the positive direction by  
for fixed
n
and K  0 ,   1.
– Perturb the formerly nonzero elements equally, such that the
stochastic structure of the F matrix is preserved
Fε
– Analyze the SLEM as a function of the perturbation as α varies

Copyright © John S. Baras 2007
52
Mean Field Explanation for Fast
Convergence in Small World Networks
• Refer to the perturbations as ε-shortcuts
in the limit n  
– For   3 the effect of ε-shortcuts on convergence
rate is negligible
– For   2 the shortcuts dominantly decrease SLEM
– For   1almost all of the nodes communicate
effectively and thus SLEM is very small
• ε-shortcuts are loosely analogous to the
shortcuts in Small World networks
• α=2 is the onset of small world effect
Copyright © John S. Baras 2007
53
Thank you!
[email protected]
301- 405- 6606
http://www.isr.umd.edu/~baras
Questions?
Copyright © John S. Baras 2008
54