On the Impact of Disorder on Dynamic Network Navigation
Anh-Dung Nguyen, Patrick Senac, Michel Diaz
anh-dung.nguyen@isae.fr, patrick.senac@isae.fr, michel.diaz@laas.fr
INFOCOM 2013, Turin, Italy
Problem
Dynamic networks, e.g. DTNs, exhibit some degree of
regularity on their temporal contact patterns [1]. We show that this temporal dimension, if correctly exploited,
can improve significantly routing perfor- mances in these networks.
time
Basic Concepts
In our work, dynamic networks are modeled as a tem- poral graph [2] G(V
t, E
t) where E
tand V
tare respec- tively the set of edges and vertexes of the network at time t.
A dynamic path from node i to node j at time t is
a time ordered sequence of temporal links
which allows a message to go from i to j according to the store-move-and-forward communication mechanism [3]. The shortest dynamic path from node i to
node j at time t can be based on two metrics
• Delay d
ji(t): the delay of node i with respect to node j at time t is the elapsed time from the last moment when there was a dynamic path from j to i.
d
ji(t) = {inf(t − t
0)|t
0≤ t and
∃ a dynamic path from j to i at time t
0}
• Hops h
ji(t): the number of nodes on the path
Temporal Structure
If nodes in a dynamic network keep track of the
couple of metrics [delay,hops] with respect to other nodes, the metric values will maintain an order rela- tion between all the nodes in the network which rep- resents a temporal and spatial proximity be-
tween nodes. In consequence, this order relation forms a
gradient field from a node towards the others. This
temporal structure is inherent to any dynamic network. To build such structure, we need to maintain two scalars d
ki(t) and h
ki(t) for each destination k at each node i.
1
while t do
2
if contact with a node j then
3
for all known destination k do
4
d
ki(t + 1) = max(d
ki(t), d
kj(t));
5
if d
ki(t) < d
kj(t) then
6
h
ki(t + 1) = h
kj(t) + 1;
hops = 2
hops = 2
hops = 2 hops = 1
hops = 1 hops = 1
t0
t1
t2 gradient (delay, hops) target
node
References
[1] A. Clauset, N. Eagle Persistence and Periodicity in a Dynamic Proximity Network, DIMACS/DyDAn Workshop on Computational Methods for Dynamic Interaction Networks, 2007
[2] P. Holme, J. Saramaki: Temporal Networks. Physics Reports, vol 519, no. 3, pp. 97-125, 2012
[3] A.D. Nguyen, P. Senac, M. Diaz: Understanding and Modeling the Small-world Phenomenon in Dynamic Networks, ACM MSWIM, Paphos, Cyprus, 2012
Disorder Degree of Dynamic Networks
Network model: Let a network of N nodes n
0, . . . , n
N−1that evolves over time with a periodic contact pattern as follows. At time 0, n
0is connected to n
1; at time 1, n
1is connected to n
2; at time 2, n
2is connected to n
3and so on. This contact pattern repeats until time T
maxbeyond which we no longer observe the network.
We introduce disorder into this network by rewiring its links as follows. For each contact, with a probability p we replace it by a contact between another pair of nodes. With probability 1 − p we let the contact unchanged. We process the rewiring by advancing in time until time T
max(i.e. we rewire the first contact then the second contact and so on until the contact at T
max).
Estimation of the disorder degree of real dynamic networks: To estimate the rewiring probability p of real networks, we apply the following algorithm to detect the contact patterns and estimate the value of p . For each pair of nodes (i, j ), their contact times are collected and a time series c ij (t) in which c
ij(t) = 1 if i encounters j at time t and 0 otherwise is generated. The auto-correlation coefficients of these series
Corr
ij(k) =
Tmax−k
X
i=1
c
ij(i)c
ij(i + k)
give us the number of contacts that are separated by k time units. We then reorder the auto-correlation coefficients in descending order and select the second mode for each pair of nodes (the first mode corresponds to k = 0).
The lag k that gives this mode is the most frequent contact period. For each pair of nodes, we count the number of k-periodic contacts then sum up globally. Consequently, an estimate of p is given by
p real = contacts − P
ij
k-periodic contacts between i,j contacts
. . . . . .
. . .
. . .
. . .
time
. . .
Dataset Size (nodes/days) Environment p
real(%)
MIT 104/246 campus,city 57.13
St-Andrews 25/79 city 58.63
Cambridge06 36/54 city 71.14
Cambridge05 12/6 laboratory 76.24
Intel 9/6 laboratory 70.51
Infocom06 98/4 conference 74.75
Infocom05 41/4 conference 78.3
Navigation in Dynamic Networks Using Temporal Structure
One-message Routing Algorithms Class: Routing in dynamic networks is finding an efficient dynamic path from a node to another which minimizes some cost functions . In this work we consider the basic and worst case solution where only one copy of each message is kept. We introduce here two greedy algorithms in which the messages follow either upward or downward the gradient slope
of the temporal structure to reach the destination.
1
while t do
2
if contact with another node k then
3
if d
jk(t) < d
ji(t) then
4
forward(m);
5
else if d
jk(t) = d
ji(t) then
6
if h
jk(t) < h
ji(t) then
7
forward(m);
Algorithm 1: GRAD-DOWN
1
while t do
2
if contact with another node k then
3
if d
jk(t) > d
ji(t) then
4
forward(m);
5
else if d
jk(t) = d
ji(t) then
6
if h
jk(t) > h
ji(t) then
7
forward(m);
Algorithm 2: GRAD-UP
Results: We consider as cost functions the delivery delay, the delivery rate and the overhead .
We measure the performances of GRAD-DOWN, GRAD-UP, PROPHET, DirectDelivery and FirstContact as a function of the disorder degree. The results show that routing performances depend strongly on the disorder degree and GRAD-DOWN outperforms other algorithms in almost all contexts by minimizing the delivery delay while maximizing the delivery rate and keeping a low overhead. At p = 20%,
GRAD-DOWN achieves its optimum performances.
GRAD-DOWN GRAD-UP FirstContact PROPHET DirectDelivery
p
DeliveryDelay(timeunits)
0 0.2 0.4 0.6 0.8 1
0 500 1000 1500 2000 2500 3000 3500
p
DeliveryProbability(%)
0 0.2 0.4 0.6 0.8 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 p
Overhead
0 0.2 0.4 0.6 0.8 1
0 20 40 60 80 100 120 140 160 180 200
Explanation: By increasing the disorder degree in the network, we introduce shortcuts that shorten the con- tact cycle. GRAD-DOWN takes advantage of these short- cuts. Too many shortcuts seems to suppress the cue to find efficient paths because the message may be for- warded backwardly in the contact cycle.
rewiring start
target
start
target target
start
We back up this intuition by collecting the traces of messages which successfully arrived at their destinations. These traces contain the chains of nodes by which the message passed through and also the time when it was for- warded to these nodes. At p = 20%, the mes- sage passing chains have the shortest av- erage chain length.
2000 10000
0 100
1000 8000
0 100
Node"sID
1000 5000
0 100
2000 10000
0 100
Time (unit)
1000 7000
0 100
p
0 0.2 0.4 0.6 0.8 1 18
20 22 24 26 28 30 32 34
