The document proposes a distributed algorithm for network size estimation. Each node in the network runs simple first-order dynamics that exchanges information only with neighbors. The dynamics are designed such that the individual solutions of all nodes will converge to the total number of nodes N in the network. The algorithm provides a deterministic estimate of N and does not require initialization, making it "plug-and-play ready" for dynamic networks where nodes can join or leave over time. It is proven that if the gain k is larger than N^3, the estimates will converge to the true value N within a finite settling time.

1. Distributed Algorithm
for Network Size Estimation
Donggil Lee, Seungjoon Lee, Taekyoo Kim, and Hyungbo Shim
Control & Dynamic Systems Lab. Seoul National University
57th IEEE Conference on Decision and Control
December 18, 2018

2. We propose a Distributed Algorithm for Network Size Estimation
Network size: total number N of nodes in a given network
Goal: design a node dynamics whose individual solutions all
converges to N. We pursue
decentralized design: design of node dynamics doesn't use
much information about the network
distributed algorithm: each node exchanges information only
with its neighbors
2 / 20

3. We propose a Distributed Algorithm for Network Size Estimation
Network size: total number N of nodes in a given network
Goal: design a node dynamics whose individual solutions all
converges to N. We pursue
decentralized design: design of node dynamics doesn't use
much information about the network
distributed algorithm: each node exchanges information only
with its neighbors
2 / 20

4. Distributed estimation of N is useful in many applications
For example,
Distributed Optimization [Nedic, Ozdaglar (2009)]
1 requires
N to obtain the convergence rate.
Distributed Kalman Filter [Kim, Shim, Wu (2016)]
2 requires
N is known to all nodes.
1
Nedic, Ozdaglar, Distributed subgradient methods for multi-agent
optimization, IEEE TAC, 2009
2
Kim, Shim, Wu, On distributed optimal Kalman-Bucy ltering by
averaging dynamics of heterogeneous agents, IEEE CDC, 2016
3 / 20

5. Distributed estimation of N is not trivial
N is a property of a network and so is a global parameter.
Each node is able to see only its neighbor
4 / 20

6. Previous results for network size estimation
(Baquero, et al., IEEE Trans. Parallel and Distrib. Sys., 2012)
(Lucchese, Varagnolo, ACC, 2015)
obtains the estimate ˆN in statistical manner by exchanging M
pieces of information with neighbors
⇒ the estimate is not deterministic
E[ ˆN] = N with Var[ ˆN] =
N2
M − 2
(Kempe et al., IEEE Symp. Foundations of Comp. Sci., 2003)
(Shames et al., ACC, 2012)
obtains 1/N asymptotically by average consensus
They require initialization which should be done over the network
⇒not ready for plug-and-play.
5 / 20

7. Previous results for network size estimation
(Baquero, et al., IEEE Trans. Parallel and Distrib. Sys., 2012)
(Lucchese, Varagnolo, ACC, 2015)
obtains the estimate ˆN in statistical manner by exchanging M
pieces of information with neighbors
⇒ the estimate is not deterministic
E[ ˆN] = N with Var[ ˆN] =
N2
M − 2
(Kempe et al., IEEE Symp. Foundations of Comp. Sci., 2003)
(Shames et al., ACC, 2012)
obtains 1/N asymptotically by average consensus
They require initialization which should be done over the network
⇒not ready for plug-and-play.
5 / 20

8. The proposed algorithm
Assumptions
1. Communication graph is undirected and connected with unit
weight.
2. ∃ one special node always belonging to network; say node 1.
node 1: ˙x1(t) = 1−x1(t) + k
j∈N1
xj(t) − x1(t)
all other nodes: ˙xi(t) = 1 + k
j∈Ni
xj(t) − xi(t)
gain k will be designed
algorithm is simple, only scalar xi(t) ∈ R is exchanged
initial condition xi(0) is arbitrary
6 / 20

9. The proposed algorithm
Assumptions
1. Communication graph is undirected and connected with unit
weight.
2. ∃ one special node always belonging to network; say node 1.
node 1: ˙x1(t) = 1−x1(t) + k
j∈N1
xj(t) − x1(t)
all other nodes: ˙xi(t) = 1 + k
j∈Ni
xj(t) − xi(t)
gain k will be designed
algorithm is simple, only scalar xi(t) ∈ R is exchanged
initial condition xi(0) is arbitrary
6 / 20

10. How the proposed algorithm works?
Overall dynamics:





˙x1
˙x2
.
.
.
˙xN





= −





k





l11 l12 · · · l1N
l21 l22 · · · l2N
.
.
.
.
.
.
.
.
.
.
.
.
lN1 lN2 · · · lNN





+





1 0 · · · 0
0 0 · · · 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0 · · · 0










x +





1
1
.
.
.
1





=: −(kL + J11
)x + 1N (L is Laplacian matrix)
Lemma: If k 0, then the matrix −(kL + J11) is Hurwitz.
Therefore, x(t) converges to the equilibrium
x∗
= x∗
(k) = (kL + J11
)−1
1N .
Lemma: x∗
1(k) = N, ∀k 0, and x∗
i (k) → N as k → ∞ for i ≥ 2.
Therefore, if k is large enough such that |x∗
i (k) − N| 0.5, ∀i,
lim
t→∞
round(xi(t)) = lim
t→∞
xi(t) = N.
7 / 20

11. How the proposed algorithm works?
Overall dynamics:





˙x1
˙x2
.
.
.
˙xN





= −





k





l11 l12 · · · l1N
l21 l22 · · · l2N
.
.
.
.
.
.
.
.
.
.
.
.
lN1 lN2 · · · lNN





+





1 0 · · · 0
0 0 · · · 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0 · · · 0










x +





1
1
.
.
.
1





=: −(kL + J11
)x + 1N (L is Laplacian matrix)
Lemma: If k 0, then the matrix −(kL + J11) is Hurwitz.
Therefore, x(t) converges to the equilibrium
x∗
= x∗
(k) = (kL + J11
)−1
1N .
Lemma: x∗
1(k) = N, ∀k 0, and x∗
i (k) → N as k → ∞ for i ≥ 2.
Therefore, if k is large enough such that |x∗
i (k) − N| 0.5, ∀i,
lim
t→∞
round(xi(t)) = lim
t→∞
xi(t) = N.
7 / 20

12. How the proposed algorithm works?
Overall dynamics:





˙x1
˙x2
.
.
.
˙xN





= −





k





l11 l12 · · · l1N
l21 l22 · · · l2N
.
.
.
.
.
.
.
.
.
.
.
.
lN1 lN2 · · · lNN





+





1 0 · · · 0
0 0 · · · 0
.
.
.
.
.
.
.
.
.
.
.
.
0 0 · · · 0










x +





1
1
.
.
.
1





=: −(kL + J11
)x + 1N (L is Laplacian matrix)
Lemma: If k 0, then the matrix −(kL + J11) is Hurwitz.
Therefore, x(t) converges to the equilibrium
x∗
= x∗
(k) = (kL + J11
)−1
1N .
Lemma: x∗
1(k) = N, ∀k 0, and x∗
i (k) → N as k → ∞ for i ≥ 2.
Therefore, if k is large enough such that |x∗
i (k) − N| 0.5, ∀i,
lim
t→∞
round(xi(t)) = lim
t→∞
xi(t) = N.
7 / 20

13. How large k should be?
To compute the minimal value of k, we impose
One more assumption
3. ∃ upper bound of network size ¯N ( ¯N N), and ¯N is known
to every node (this is the only global information needed.)
Theorem: If
k ¯N3
then the proposed algorithm
node 1: ˙x1 = 1 − x1 + k
j∈N1
(xj − x1)
all other nodes: ˙xi = 1 + k
j∈Ni
(xj − xi)
with arbitrary initial conditions yields estimation of N because
lim
t→∞
|xi(t) − N| 0.5, ∀i ∈ N.
8 / 20

14. How large k should be?
To compute the minimal value of k, we impose
One more assumption
3. ∃ upper bound of network size ¯N ( ¯N N), and ¯N is known
to every node (this is the only global information needed.)
Theorem: If
k ¯N3
then the proposed algorithm
node 1: ˙x1 = 1 − x1 + k
j∈N1
(xj − x1)
all other nodes: ˙xi = 1 + k
j∈Ni
(xj − xi)
with arbitrary initial conditions yields estimation of N because
lim
t→∞
|xi(t) − N| 0.5, ∀i ∈ N.
8 / 20

15. How to obtain N in nite time?
The problem is to nd minimal value T such that
|xi(t) − N| 0.5, ∀t T
To nd the time T, we need
convergence rate
bounded initial condition
9 / 20

16. How to obtain N in nite time?
The problem is to nd minimal value T such that
|xi(t) − N| 0.5, ∀t T
To nd the time T, we need
convergence rate
bounded initial condition
9 / 20

17. Convergence rate of proposed algorithm
Recall overall dynamics:





˙x1
˙x2
.
.
.
˙xN





= −





k





l11 l12 · · · l1N
l21 l22 · · · l2N
.
.
.
.
.
.
. . .
.
.
.
lN1 lN2 · · · lNN





+





1 0 · · · 0
0 0 · · · 0
.
.
.
.
.
.
. . .
.
.
.
0 0 · · · 0










x +





1
1
.
.
.
1





=: −(kL + J11
)x + 1N
Lemma: If k ¯N3, then
λmax −(kL + J11
) ≤ −
1
4 ¯N
.
convergence rate is decreasing function w.r.t. ¯N
10 / 20

18. Main Result
Our last assumption (Bounded initial condition)
Suppose xi(0) ∈ [0, ¯N] for all i.
reasonable initial guess ( N ¯N)
Theorem (Finite-time Estimation of N)
Under all the assumptions, if k ¯N3, then the proposed algorithm
guarantees
xi(t) = N, ∀t T(k), ∀i
where the settling time T(k) is given by
T(k) = 4 ¯N ln
2 ¯N1.5k
k − ¯N3
.
11 / 20

19. Advantages of the proposed algorithm
node 1: ˙x1 = 1 − x1 + k
j∈N1
(xj − x1)
all other nodes: ˙xi = 1 + k
j∈Ni
(xj − xi)
1. simple rst-order dynamics
2. exchanges single variable with neighbors
3. obtains N directly within nite time
4. independent of initialization
→ while the algorithm is running, new node can join or some
node can leave the network
This property is often called
`plug-and-play ready' or
`open MAS (multi-agent system)' or
`initialization-free algorithm'
12 / 20

20. A Remark for Practical Application
1. To obtain correct estimate of N, it takes T(k) time from the
network change
2. However, not every node can detect the changes.
3. Possible solution: allow the changes only at specied time,
i.e., some nodes can join or leave the network at t = j · T
where T T(k), assuming every node has same clock.
Example scenario:
there is a unit length of time T T(k)
1. two nodes 1 and 2 belong to the network
from time T0 = 0
2. node 3 joins the network at T1 = T
3. node 3 leaves the network at T2 = 2T
13 / 20

21. A Remark for Practical Application
1. To obtain correct estimate of N, it takes T(k) time from the
network change
2. However, not every node can detect the changes.
3. Possible solution: allow the changes only at specied time,
i.e., some nodes can join or leave the network at t = j · T
where T T(k), assuming every node has same clock.
Example scenario:
there is a unit length of time T T(k)
1. two nodes 1 and 2 belong to the network
from time T0 = 0
2. node 3 joins the network at T1 = T
3. node 3 leaves the network at T2 = 2T
13 / 20

22. (two nodes belong to the network from T0 = 0)
Every node initializes its state within [0, ¯N]
Estimation is guaranteed for t T(k)
14 / 20

23. (node 3 joins the network at T1 = T )
x3(T1) is initialized within [0, ¯N]
both x1(T1) and x2(T1) are within [0, ¯N]
15 / 20

24. (node 3 leaves the network at T2 = 2T )
all x1(T2) and x2(T2) are within [0, ¯N]
correct estimation is always available for t Tj + T(k)
16 / 20

25. Behind the scene
How we came up with the proposed algorithm?
17 / 20

26. Blended dynamics approach
A tool for analysis of heterogeneous multi-agent system
Node's dynamics:
˙xi = fi(xi) + k
j∈Ni
xj − xi , i ∈ {1, 2, · · · , N}
Blended dynamics (average of vector elds fi):
˙s =
1
N
N
i=1
fi(s) with s(0) =
1
N
N
i=1
xi(0)
Theorem3
Suppose blended dynamics is stable. Then, ∀ 0, ∃k∗ such that
for all k ≥ k∗,
lim sup
t→∞
|xi(t) − s(t)| , ∀i.
3
Kim, Yang, Shim, Kim, Seo, Robustness of synchronization of
heterogeneous agents by strong coupling and a large number of agents, IEEE
TAC, 2016
18 / 20

27. We designed node dynamics so that their blended dynamics
has desired property.
The proposed node dynamics:
˙x1 = 1 − x1 + k
j∈N1
xj − x1
˙xi = 1 + k
j∈Ni
xj − xi , ∀j ∈ {2, . . . , N}
Their blended dynamics:
˙s =
1
N
N
i=1
fi(s) =
1
N
(N − s) = −
1
N
s + 1
Therefore, with suciently large k, we have
lim sup
t→∞
|xi(t) − s(t)| = lim sup
t→∞
|xi(t) − N| , ∀i ∈ N
information about N is embedded in the vector elds (not in
the initial conditions) → key to the `plug-and-play'.
19 / 20

28. Summary
the design of proposed algorithm is based on blended dynamics
˙s = −
1
N
s + 1
each node obtains network size exactly with arbitrary initial
condition
⇒ the algorithm supports plug-and-play operation
the estimation is guaranteed within nite time
Thank you!
Donggil Lee (dglee@cdsl.kr)
20 / 20

29. Simulation for 30 nodes
Black dotted line: N(t) ± 0.5