3rd Quarter FY07 Progress Report
Task Highlights for the Quarter Q3’07 [Period: 4/1/07 - 6/30/07]
Subtask 1 Scalable MANET Design and Analysis (Telcordia: Kant, Krishnan & Gopalakrishnan)

(a couple of bullet items)
Subtask 2: General Optimization Framework for Multi-User Flow Control (UMinn: Giannakis)

(a couple of bullet items)
Subtask 3: Co-operative Communications and Network Coding (UMd: Ephremedies)

(a couple of bullet items)
Subtask 4: Fundamental Properties and Cross-Layer Design of Wireless Sensor Networks: A
Stochastic Optimization Approach (UCDavis: Zhao)

(a couple of bullet items)
Subtask 5: Non-Local Models for MANETs (UMd: La, Makowski, Martins, UMd)
Subtask 5.1: Non-local models for MANETs

We continued studying how the distribution of nodes affects the critical transmission range in
wireless networks which can be modeled as geometric random graphs on intervals. In
Q3FY2007 the emphasis was given to understanding the nature of associated phase
transitions.

We continued investigating the structural properties of the random connection model, a model
recently introduced by Hekmat and van Mieghem in the context of wireless networks. This
class of models offers the possibility of incorporating pathloss and fading which are more
realistic than in the usual disk model.
Subtask 5.2: Overhead in MANETs

We continued our investigation of the minimum routing overhead in static MANETs. Using a
family of strategies, we identified a trade-off between propagation of topology information in
a local neighborhood (maintenance of routing information in a local neighborhood of
destinations) and flooding of data packets outside the neighborhood with topology
information.
Subtask 5.3: Impact of MAC on network connectivity
1
3rd Quarter FY07 Progress Report

We refined a model we developed in the previous quarter for studying the impact of medium
access control on network connectivity in MANETs. Our model captures the inherent
spreading of packet transmissions in space and in time.
Task ID: SWMN-07-04
Task Name: Network Science
Principal Investigators:
Associate Investigators:
Industry PI: Dr. Latha Kant
Name: Dr. K.R.Krishnan, Telcordia Technologies
Affiliation: Telcordia Technologies
Citizenship: USA
Address: RRC 1A 235, One Telcordia Drive, Task#: 1
Piscataway, NJ 08854
Email: lkant@telcordia.com
Phone: 732.699.2428; Fax: 732.336.7026
Country of Citizenship: USA
ARL PI: Dr. Ananthram Swami
Name: Praveen Gopalakrishnan, Telcordia
Affiliation: ARL
Technologies
Address:
Citizenship: India
Email: aswami@arl.army.mil
Task#: 1
Phone: 301.394.2486
Country of Citizenship: USA
Name: Dr. Georgios B. Giannakis, University of
Minnesota
Citizenship: USA
Task #: 2
Name: Dr. Anthony Ephremides, University of
Maryland
Citizenship: USA
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3rd Quarter FY07 Progress Report
Task#: 3
Name: Dr. Qing Zhao
Citizenship: P.R. China
Task#: 4
Name: Dr. Richard J. La, University of Maryland
Citizenship: USA
Task#: 5
Name: Dr. Armand M. Makowski, University of
Maryland
Citizenship: Belgium
Task#: 5
Name: Dr. Nuno Martins, University of Maryland
Citizenship: Portugal
Task#: 5
Research Issues Addressed
Broadly speaking, we view “Network Science” as the “science of interconnections”. We then observe
that when an “interconnected structure” or “network” exhibits properties that are not mere extrapolations
of the properties of the individual elements of the structure, we are in the realm of network science. The
objective of this task to identify and address issues that are fundamental to the realm of network science.
Before we expand on the research issues identified, we observe that since the term “network” is very
broad, for example, it can correspond to electronic communications networks, social networks, biological
networks, etc., the focus of this technical area will be on electronic communications networks. In
particular, since mobile ad hoc networks (MANETs) are becoming the basis of the future Army networks
and the network centric warfare (NCW) paradigm due to their enormous flexibility and dynamism, the
focus of this technical area will be on mobile ad hoc networking and MANET-related issues.
The flexibility and dynamism of MANETs that render them very powerful also introduce fundamental
research questions and render analyses very challenging. We identify below, basic questions and
associated research issues that will be addressed by Tasks 1 through 5, under this technical area of
Network Science.

When a network of mobile ad hoc nodes need to communicate, what is the underlying network
capacity? Note network capacity differs from an individual node’s capacity in wireless networks.
Does an analogue of “Shannon’s Limit” exist for the MANET world and if so what is it? Recall,
MANET capacity is a time varying quantity – a stochastic process whose quantification depends
on a variety of factors.

What specific information control techniques based on a systematic set of reasoning principles
and algorithms vs. heuristic techniques need to be researched and developed in order to reliably
deliver applications based on their Quality of Service (QoS) requirements over a collection of
mobile ad hoc nodes?

What is common to both Cooperative Communications and Network Coding and how can we
exploit them to achieve more efficient communications amongst nodes that have random
distribution (mobility)? Basic research in the area of Network coding for MANETs will be
conducted to provide insights to the above problem.
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3rd Quarter FY07 Progress Report

How do the PHY and MAC layers interact when designing large scale sensor networks? The
cross-layer nature of this topic combined with the uncertainty of the underlying network,
fundamental research using stochastic optimization techniques will be employed to answer the
above question.

How is connectivity defined and determined amongst mobile ad hoc nodes? For example, edge
connectivity should be determined by conditions involving several random variates such as node
location and fading intensity, to name a few. Connectivity graph models should incorporate the
important observation that a communication link between two nodes is determined by all active
users, and not just by the pair of users, giving rise to non-local connectivity in these models. An
additional question that arises is: “are there formal techniques to quantify MANET routing
overheads and understand their scaling properties, and if so, what are they based on?”
Subtasks 1 through 5 address each of the above-mentioned issues and more. In what follows below, we
provide a status update on progress-to-date for each of the five tasks.
Progress Summary
SUBTASK 1: Scalable MANET Design and Analysis (Kant, Krishnan & Gopalakrishnan,
Telcordia Technologies)

(a few bullet items)
SUBTASK 2: General Optimization Framework for Multi-User Flow Control (Giannakis,
UMinn)

(a few bullet items)
SUBTASK 3: Cooperative communications and Network Coding (Ephremides, UMd)

(a few bullet items)
SUBTASK4: Fundamental Properties and Cross-Layer Design of Wireless Sensor
Networks: A Stochastic Optimization Approach (Zhao, UC Davis)

(a few bullet items)
SUBTASK5: Non-Local Models for MANETs (La, Makowski, Martins, UMd)
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3rd Quarter FY07 Progress Report
Subtask 5.1: Non-local models for MANETs

As a way to better understand the impact of mobility on connectivity, we have continued
investigating how the distribution of nodes affects the critical transmission range for connectivity
in geometric random graphs on intervals. In Q3FY2007 we further developed an asymptotic
theory for the critical transmission range when the node placement density does not vanish. This
was achieved by establishing a limit result for the maximal spacing associated with i.i.d. samples
drawn from a distribution with compact support and non-vanishing density. This result
complements the very strong zero-one law for connectivity (established in Q3FY2007) when the
node placement distribution admits a non-vanishing density, and leads to exact asymptotics of
the transition width for network connectivity. We have continued exploring the implications of
these results for resource dimensioning.

We have developed zero-one laws for the property of node isolation in the random connection
model in two dimensions under uniform node placement.
Subtask 5.2: Overhead in MANETs

We continued our investigation of the minimum routing overhead in static MANETs: We
considered a family of routing strategies (which can be viewed as a combination of on-demand
and table-driven routing protocols) and identified a trade-off between propagation of topology
information in a local neighborhood (maintenance of up-to-date routing information to
destinations in local neighborhoods) and flooding of data packets outside the neighborhood
around the intended destination with topology information.
Subtask 5.3: Impact of MAC on network connectivity

We refined the new model we developed in the second quarter for studying the impact of MAC
protocol on network connectivity of a MANET with interference. The model captures (i) the time
sharing aspect of MAC and (ii) the resulting interference as a function of the local density, i.e.,
the number of neighbors, in a neighborhood.
Deliverables
Due
Milestone / Deliverable
Status
1Q’07
Quarterly Report; conference papers (Telcordia,
SubTask 1)
Telcordia – Task 1: Complete (Quarterly Report).
Conference papers in preparation.
Report, conference, and journal papers on optimal
congestion control in contention-based wireless
tactical networks (UMinn – SubTask 2)
Extension of the simple relay idea for MAC-level
cooperation through Network Coding. (UMd –
SubTask 3)
5
UMin – Task 2: Completed. Drafts + Matlab
codes available upon request. See publication list
below.
UMd- Task 3: Our first milestone to convert
throughputs to spectral efficiencies has been
fully met. Next set of milestones with
regards to examining cooperative relay
techniques by means of Network Coding and
3rd Quarter FY07 Progress Report
identifying similarities
networks have begun.
Quarterly report;
SubTask 4)
conference
papers
(UCD-
to
biological
UCD – Task 4: Complete (Quarterly report); See
publication list below.
UMd – Task 5: One paper being prepared for
submission to the IEEE Transactions on
Information Theory.
Development of a model for studying the minimum
expected routing overhead in static MANETs
(UMd - SubTask 5)
One paper submitted to ISIT 2007.
Complete
2Q’07
Telcordia – Quarterly Report Completed.
Quarterly Report (Telcordia, Task 1)
Preliminary Design Document on
Capacity Analysis and Scalable
(Telcordia, SubTask 1)
MANET
Routing
Report , conference, and journal papers on joint
congestion, routing and contention control
algorithms (UMinn – SubTask 2)
Use of Fountain Codes for wireless multicasting
for both transmitter-based as well as receiver-based
throughput measures. (UMd – SubTask 3)
Quarterly report; conference and journal papers
(UCD – SubTask 4)
Telcordia - Completed and Delivered document
on “Preliminary design document on MANET
Capacity Analysis” along with this quarterly
report.
UMinn- Quarterly report completed. See
references for conference and journal paper
submission.
UMd (Ephremedies) – Quarterly report complete.
UCD – Task 4: Complete (Quarterly report); See
also “Publications” for papers submitted.
Study of SINR graphs with fading (onedimensional case); zero-one laws for node
isolation. (UMd – SubTask 5)
UMd – Task 5: Quarterly Report completed.
routing overhead in static MANETs. (UMd –
SubTask 5)
One paper is being prepared for submission to the
IEEE Transactions on Information Theory
Development of a model for studying the effects of
MAC protocols. (UMd – SubTask 5)
Paper was accepted for inclusion in the program
of ISIT 2007 (Nice, France, June 2007)
Quarterly Report and
(Telcordia, SubTask 1)
.
One paper is being prepared for submission
to the IEEE Transactions on Information
Initial investigation of the minimum expected Theory
3Q’07
Paper
Submissions
Report, conference, and journal papers on joint
flow congestion control and multi-user scheduling
in hybrid wireline-wireless tactical MANETs
(UMinn – SubTask 2)
6
3rd Quarter FY07 Progress Report
Conversion of throughput gains to Spectral
Efficiencies for Cross-Layer Integration (UMd–
SubTask3)
Quarterly report; conference and journal papers
(UCD – SubTask 4)
Quarterly Report and Paper Submissions (UMd,
SubTask 5)
UMd – Task 5: Quarterly Report completed
One paper was submitted to the IEEE
Transactions on Information Theory
One paper is being prepared for submission
to the IEEE Transactions on Information
Theory
Study of SINR graphs with fading (twodimensional case); zero-one laws for node isolation
(UMd – SubTask 5)
Zero-one laws for one-dimensional geometric
random graphs – non-uniform case (UMd SubTask 5)
Exact asymptotics for width of phase transition
have been derived
Continuing investigation of the minimum expected
routing overhead in static MANETs with a focus
on identifying a lower bound on the routing
overhead (as a function of the delay constraint T).
(UMd – SubTask 5)
A formula was obtained for quantifying a tradeoff between latency and transmission rate, due to
routing overhead.
Integration of carrier sensing in the model and
initial study of the effects of carrier sensing on
network connectivity. (UMd – SubTask 5)
4Q’07
Delayed
Quarterly Report (Telcordia, SubTask 1)
Final Design Document on MANET Capacity
Analysis and Scalable Routing, and Paper
Submissions (Telcordia, SubTask 1)
Report, conference, and journal papers on
distributed version of our joint congestion control
and scheduling design (UMinn – SubTask 2)
Exploration of Similarities to Biological Networks.
(UMd – SubTask 3)
Comparison of Cooperative vs Competitive
Multiple Access through Network Coding and
other diversity techniques and Connection to the
Back-pressure algorithm ( UMd – SubTask 3)
Quarterly report and Summary of GFY07 Progress;
7
A manuscript, for submission in the IEEE
Transactions on Information Theory is under
preparation.
Delayed
3rd Quarter FY07 Progress Report
conference and journal papers (UCD – SubTask 4)
Study of SINR graphs with fading (onedimensional case); zero-one laws for connectivity.
(UMd – SubTask 5)
Initial investigation of routing overhead in mobile
networks with a focus on identifying a lower bound
on the overhead due to node mobility. (UMd –
SubTask 5)
Continuing study of the effects of carrier sensing
on network connectivity and integration of the
power control (with fixed modulation and coding
schemes) in the model. (UMd – SubTask 5)
Plans for Next Quarter
SUBTASK 1: Scalable MANET Design and Analysis (Kant, Krishnan & Gopalakrishnan,
Telcordia Technologies)

(a couple of bullet items)
SUBTASK 2: General Optimization Framework for Multi-User Flow Control (Giannakis,
UMinn)

(a few bullet items)
SUBTASK 3: Cooperative communications and Network Coding (Ephremides, UMd)

(a few bullet items)
SUBTASK4: Fundamental Properties and Cross-Layer Design of Wireless Sensor
Networks: A Stochastic Optimization Approach (Zhao, UC Davis)

(a few bullet items)
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3rd Quarter FY07 Progress Report
SUBTASK5: Non-Local Models for MANETs (La, Makowski, Martins, UMd)
Subtask 5.1: Non-Local Models for MANETs

We shall continue to investigate the structural properties of the random connection model in two
dimensions. In particular, building on progress made in Q3FY2007 we will develop zero-one laws
for graph connectivity when the connection function has bounded support.

We plan to begin a study of how the distribution of nodes affects the critical transmission range in
wireless networks which can be modeled as geometric random graphs in two dimensions, say the
original disk model of Gupta and Kumar. In Q4FY2007, as a preliminary step, emphasis will on
understanding the existence of zero-one laws for the property that no isolated nodes exist.
Subtask 5.2: Overhead in MANETs

We will continue the investigation of the minimum routing overhead in static MANETs. In
particular, we will focus on identifying a lower bound on the minimum expected routing overhead
subject to a delay constraint with increasing network size. We will also start formulating the
problem of studying routing overheads with mobile nodes.
Subtask 5.3: Impact of MAC on network connectivity

Using the model developed in 3QFY07, we will investigate the impact of the MAC protocol on
network connectivity with a physical layer model (with interference). We will also extend the model
to the case with power control by the nodes.
Detailed Progress
SUBTASK 1: Scalable MANET Design and Analysis (Kant, Krishnan & Gopalakrishnan,
Telcordia Technologies)
(detailed write-up goes in here)
SUBTASK 2: General Optimization Framework for Multi-User Flow Control (Giannakis,
UMinn)
(detailed write-up goes in here)
SUBTASK 3: Cooperative communications and Network Coding (Ephremides, UMd)
(detailed write-up goes in here)
9
3rd Quarter FY07 Progress Report
SUBTASK4: Fundamental Properties and Cross-Layer Design of Wireless Sensor
Networks: A Stochastic Optimization Approach (Zhao, UC Davis)
(detailed write-up goes in here)
SUBTASK5: Non-Local Models for MANETs (La, Makowski, Martins, UMd)
Subtask 5.1: Non-local models for MANETs

We consider the random network where n points are placed independently on the unit interval
[0,1] according to some probability distribution function F. Two nodes communicate with each
other if their distance is less than some transmission range  > 0. When F admits a continuous
density f with f *  inf( f ( x), x  [0,1]) > 0, we have shown [HanMakowski2007a] by direct
probabilistic methods that the property of graph connectivity for the underlying random graph
admits a strong critical threshold given by
 n* 
1 log( n)
, n  2,3,...
f* n
(1)
This amounts to the following statement: Let P(n;) denote the probability that the random graph
with n nodes is connected when the communication range is . Then, we have
lim n

0 if 0  c  1
P(n; c )  
1 if 1  c

*
n
for any range function  : N 0  R
[HanMakowski2007a] for details.
such
that
 n ~ c n*
for
some
c>0.
See
In Q2FY2007 we also showed that the property of graph connectivity admits a very strong
threshold, and we identified the corresponding (very strong) threshold: Under mild conditions on
the density function f, we showed that
lim n

0 if lim n  n  
P(n;  )  
1 if lim
n   n  

*
n
when the range function  : N 0  R is determined through
1
log( n)  log(log( n))   n
r
n 
, n  1,2,...
n
with deviation function  : N 0  R where r > 0 is some parameter indicating how “flat” this
density is near its minimum; see [HanMakowski2007b] for additional details.
10
3rd Quarter FY07 Progress Report
Very strong thresholds are forerunners of sharp phase transitions: For each n=2,3,…, P(n;) is
monotone increasing in  with a transition from P(n;  )  0 to P(n;  )  1 as  varies across
some critical range, and the sensitivity to “small” deviations implied by the last equation suggests
that the width of this transition interval decreases rapidly with large n. It is therefore natural to
seek estimates of how quickly this transition takes place. This issue is by now well understood
under uniform node placement, e.g., see [HanMakowski2006]. In Q3FY2007 we have made
some progress on this issue in the non-uniform case: Under the assumptions leading to the very
strong zero-one law, we have derived a limit result on the maximal spacing associated with n i.i.d
samples drawn from the distribution with density f. This result generalizes a well-known result of
P. Levy for uniform samples, and leads very naturally to the conclusion that the transition width
is of the form
 n a  
1 C a   1 
 o 
f* n
n
 log a 
 . Here  n a    n 1  a    n a  where for
 log 1  a  
for all 0  a  1 / 2 with C a   log 
each b in 0,1 ,  n b  is the unique solution to the equation
Pn,    b, 0    1 .
See [HanMakowski2007c] for details. From a network engineering viewpoint, a small transition
width would imply that if the transmission range of the nodes is set slightly larger than the
critical threshold in a large network, then the probability that the network remains connected will
be high (i.e., close to one). On the other hand, if the transition width is large, then the
transmission range may need to be set considerably larger than the critical threshold in order to
ensure network connectivity for most of the time. It is noteworthy that the density function enters
the critical thresholds through the inverse of its minimum, a quantity which is notoriously hard to
estimate. Therefore, the lack of accurate estimates or simply ignorance of the true density
function, may force rather large assignments, in fact much larger than the ones associated with
the (very) strong zero-one laws.
The work on the random connection model under uniform placement is still in a preliminary
phase and will be developed further in Q4FY2007. However, some progress was made
concerning the derivation of very strong zero-one laws for the property that there exists no
isolated nodes under the assumption that the connection function has bounded support.
Subtask 5.2: Overhead in MANETs

The model we use to study the minimum routing overhead in static MANETs is as follows:
Consider a network of n nodes that are just deployed over some domain D in R2 or R3. These
nodes are assumed to be unaware of the locations of other nodes. Further, we assume that the
nodes are placed independently of each other according to a common distribution and once
deployed, the locations of the nodes are fixed, i.e., a static network. There is a fixed set of
source-destination pairs that need to communicate with each other. The source and destination
nodes are randomly selected and unaware of the location of each other. Each source needs to
send a packet to the corresponding destination without any prior routing information. We impose
a delay constraint that the packet must reach the destination within T seconds for all source-
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3rd Quarter FY07 Progress Report
destination pairs. One can view this delay as the initial delay incurred before the network can
begin to deliver packets between the source and the destination.
The distance between a pair of source and destination nodes will depend on both the locations of
the nodes. Hence, the minimum overhead required to route the packet will be random. Instead of
attempting to compute the minimum required overhead for every realization, we focus on the
expected minimum overhead. The questions we are interested in investigating are: (i) For a fixed
T, how does the expected minimum overhead in bits  meter/second scale with increasing
network size, especially in relation to network throughput? (ii) How does the expected minimum
overhead behave as a function of the deadline T? Note that not every node needs to be involved
in delivering packets, and a subset of nodes may suffice. Related to these questions is the
following issue: How long will it take on the average before a network becomes operational in
the sense that it can deliver packets between sources and destinations, as the network size
increases?
Our preliminary results for static networks show that, for any fixed time horizon T after the
initial deployment, the fraction of the nodes that can, at least, communicate their label (or ID) to
one neighbor converges to zero as the number of nodes grows. This fact clearly proves the
importance of studying such an initial transitory behavior. At a larger timescale, there is also a
trade-off between allocating the network resources for exchanging/propagating routing
information or for transmitting information. First, consider the following two extreme cases: In
the first case the network topology information is broadcast throughout the network to all nodes.
This allows the nodes to construct a network map and to use an efficient routing algorithm, e.g.,
Dijkstra's algorithm. In the second case, on the other hand, most of the network resource is
allocated for the communication between nodes, and only little resource is assigned to
propagating network topology information. The first scheme may demand that most of the
network resource be dedicated to propagation of network topology information, leaving very
little or no resource for the transmission of information between nodes. In the latter scheme, the
lack of information about the network topology will preclude the use of efficient routing
algorithms, but instead force the use of flood strategies, leading to low throughput.
Our initial studies show that, in general, this problem persists even if we allow strategies inbetween these two extreme cases. In particular, we can show that there exists a universal lower
bound on the expected minimum delay after deployment, before which no communication can
take place. In addition, for any given positive throughput, there is a minimum delay before which
such a throughput is not feasible. This finding can be summarized by the following equation:
 

n  min
A
latency


Tn
 min
Rn log 2  n   Dn     
min 2  
Rnmin  C 
n
  Rn  
 

where Tnlatency is the required latency, Rnmin is the minimum radius around the destinations where
the routing information to the destinations is available, Cn is the network transport capacity, A is
the area of the domain, and  ,  , are positive constants. When Gupta and Kumar’s results are
used, we obtain
latency
Tn

12
n  log 2 n

3rd Quarter FY07 Progress Report
Subtask 5.3: Impact of MAC on network connectivity

When distributed packet scheduling is employed in MANET, the interference at a receiver is
unlikely to be known in advance. This is because the set of simultaneous transmitters is in
general unknown to the receivers, and is hard to control without a centralized controller. This
difficulty is further complicated by two additional factors: (i) Oftentimes there are constraints on
the set of transmitters T when Medium Access Control (MAC) schemes are employed, and such
constraints are difficult to model accurately, and (ii) the set of transmitters changes with time
(i.e., T(n), n = 0, 1, …, in discrete time model), making it virtually impossible to predict the
interference at a receiver accurately.
First, many of MAC protocols (e.g., IEEE 802.11) employ carrier sensing; a node first listens to
the channel to make sure that it does not hear any other transmissions taking place over the same
channel (in its neighborhood). Carrier sensing tends to limit the number of transmissions in a
local neighborhood and spread them out both in space and in time. Therefore, it imposes a
constraint on the set of transmitters that access the channel simultaneously.
Second, as the set of transmitters changes with time and is not known in general, it may be too
restrictive, if not impossible, to insist that the realized SINR exceeds a certain threshold for all
transmissions in order for two nodes to be neighbors (i.e., can communicate with each other). In
practice, an automatic repeat request (ARQ) protocol will retry unsuccessful transmissions. Thus,
it is not necessary for every transmission to be successful. Thus, we assume that there exists a
link between two nodes (i.e., the two nodes are neighbors) if a transmission between them is
successful with a high probability.
This requirement can be written as




Pi Gij
P
 SINR   
 N j   Pk Gkj

kT \{ i , j }


where 0 < < 1, N j is the noise power at receiver j, Pk and G kj are the transmit power of
node k and channel gain from node k to node j, respectively, SINR is the threshold on signal to
interference plus noise ratio (SINR) for successful transmission, and T is a random set of
transmitters. Note that this condition can be rewritten as




Pi Gij
P
 SINR   
 N j   I k Pk Gkj

k i , j


where I k  1 if k  T and I k  0 otherwise.
In general these indicator functions will not be mutually independent due to carrier sensing. Let
T *  2 N be the set of all subsets of the set of nodes N. Now the question is what suitable
probability distribution on T * one may assume.
From the viewpoint of studying network connectivity, the problem is more interesting when we
assume the worst scenario where the interference is as strong as it is allowed to be. One can
13
3rd Quarter FY07 Progress Report
argue that the interference experienced at the receivers will be the largest when the network is
congested and every node has packets to transmit and attempts to access the channel at all times.
This is implicitly assumed throughout the rest.
1. Static model: First order approximation without dynamics: When every node always has
packets to transmit, the fraction of time a node gets to access the channel will be (roughly
speaking) inversely proportional to the number of its neighbors whose transmissions it can sense.
One may be able to approximate the number of such neighbors within a given carrier sensing
range using the neighbors that lie within a fixed radius  from the node (similar to the disk
model). The variable  denotes the sensing range of a node. Let N i denote the number of
neighbors within radius 
 
of node i. One can now assume that the fraction of time
i 1
node i is active is proportional to N 
, i.e., the fraction of time node i transmits is  / N i _for
some 0<< 1. We now replace the previous condition with






Pi Gij
P
 SINR   
Pk Gkj
N 


j
k


k i , j N 


This is equivalent to replacing the indicator functions in the previous condition with their
expected values. This model attempts to capture the effects of carrier sensing of MAC schemes:
A node in a dense neighborhood may experience stronger interference from its neighbors that are
active. However, in such a dense neighborhood, the number of simultaneously active nodes will
be limited by carrier sensing.
Refinement: The previous model we focused on in the previous quarter attempts to model the
first-order effect of scheduling in that it limits the fraction of time a node accesses the channel as
a function of the number of neighboring nodes whose transmission it can sense. However, it
suffers from an obvious shortcoming that it tends to overestimate the interference from the
neighbors within the sensing range of a receiver. This is because the unconditional distribution of
interference at the receiver j is likely to be considerably different from that conditional on the
event that it is receiving a packet. In other words, when the receiver broadcasts a control packet,
e.g., CTS, to reserve the channel, most (if not all) of its neighbors that can detect the control
packet will remain silent.
We model this by introducing a function f(d) in the previous condition as follows:






Pi Gij
P
 SINR   
f (d kj ) Pk Gkj
N 


j
k


N
k i , j


where d kj
is the distance between nodes k and j, and f(d) is an increasing function with
f(0) = 0 and f(d) = 1 for all d > . The function f(d) models the probability that
another node within the sensing range of the receiver will access the channel at the same time as
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3rd Quarter FY07 Progress Report
a function of their distance. If carrier sensing is perfect, we should have f(d) = 0 if 0<d <
and f(d) = 1 otherwise.
2. Probabilistic model: As mentioned earlier, a source of difficulty is that the set of
simultaneous transmitters that causes interference at a receiver is random. There has been some
recent work on trying to understand the distribution over the sets of simultaneous transmitters
T * (e..g, work by Durvy, Dousse and Thiran, IEEE Infocom 2007). In this quarter, we started
developing a probabilistic model that first identifies the distribution over T * and allows us to use
the first criterion for determining the one-hop connectivity.
Publications & Presentations
Journal Papers:
[HanMakowski2007a] G. Han and A.M. Makowski, “A strong zero-one law for connectivity in onedimensional geometric random graphs with non-vanishing densities," submitted to IEEE Transactions on
Information Theory (2007).
[HanMakowski2007b] G. Han and A.M. Makowski, “A very strong zero-one law for connectivity in onedimensional geometric random graphs with non-vanishing densities," submitted to IEEE Transactions on
Information Theory (2007).
[HanMakowski2007c] G. Han and A.M. Makowski, “On the width of phase transition for connectivity in
one-dimensional geometric random graphs with non-vanishing densities," in preparation and to be
submitted to IEEE Transactions on Information Theory (2007).
Conference Papers:
[HanMakowski2006] G. Han and A.M. Makowski, “Very sharp transitions in one-dimensional
MANETs," in Proceedings of the International Conference on Communications (ICC 2006), Istanbul
(Turkey), June 2006.
Presentations:
A.M. Makowski,
“On the critical communication range in one-dimensional geometric random graphs: Sensitivity to node
placement distribution,"
Workshop on Mathematical Modeling and Analysis of Computer Networks, Ecole Normale
Superieure, Paris (France), June 2007.
“On the critical communication range in one-dimensional random networks under non-uniform node
placement,"
Seminaire Reseaux, INRIA Sophia-Antipolis, Valbonne (France), June 2007.
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3rd Quarter FY07 Progress Report
Summer Research Institute, School of Computer and Communication Sciences, Ecole
Polytechnique Fed\'erale de Lausanne, Lausanne (Switzerland), July 2007.
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