Soka University

On composing a resilient tree in acommunication network with intermittentconnections based on stress centrality

Genya IshigakiNorihiko Shinomiya

ISCC 2016

June 29, 2016

1. Introduction: Assumed Environment

Disruption Tolerant Networking (DTN)

! communications in a challenging environment (Wei et al., 2014)! communications under huge disasters (vehicle, drone networks)! networks in communication challenged regions

! no guarantee of end-to-end paths between communication nodes(intermittent connections) (Fall, 2003)

(Internet.org, NTT docomo)2

1. Introduction: Assumed Environment

Disruption Tolerant Networking (DTN)

! communications in a challenging environment (Wei et al., 2014)! communications under huge disasters (vehicle, drone networks)! networks in communication challenged regions

! no guarantee of end-to-end paths between communication nodes(intermittent connections) (Fall, 2003)

(Internet.org, NTT docomo)3

2. Related Works: Topology Design

Maximum Reliability Tree Problem (Yoshida et al., 2010)

composing a tree maximizing the weakest path reliability

Path Reliability := the product of weights on edgesTree Reliablity (TA) := reliability of the weakest path

✓ ✏consideration ONLY to the total evaluation value✒ ✑

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3

Works on Graph Reliability

Survivable Network Design Problem (Korte, 2007) Input: weighted graph G, required connectivity c Problem: find a subgraph H of G that satisfies c and minimizes total weight

Minimum Spanning Tree problem (c = 1, H is spanning.)

consideration to relationship between location and edge reliability

Maximum Availability Tree Problem (Yoshida et al., 2010) composing a tree maximizing the weakest path reliabilityTree Availability (TA) := product of weights on edges in the weakest path

TA

0.7

0.6

0.6

0.9

0.7

0.6

0.9

0.7

0.6

0.6

0.7

0.60.9

0.6⇥ 0.7⇥ 0.9 0.6⇥ 0.6⇥ 0.7 0.6⇥ 0.7⇥ 0.9

4

3. Big Picture of this Study

The Purpose of this Study

composition of a resilient topology in a network with intermittent linksconsidering the difference on relative locations of links

1-connected (tree) 2-connected

5

3. Big Picture of this Study

The Purpose of this Study

composition of a resilient topology in a network with intermittent linksconsidering the difference on relative locations of links

What is more “resilient”?! difference among existence possibility of communication links! relative location of communication links in composed topology

6

4. Modeling: Probabilistic Aggregated Contact Graph

Contact Graph (CG) Gti

representation of connectionsbetween communication nodesat time ti

Probabilistic ACG (PACG) G′RACG assigned a probability ofconnection existence to eachedge GR = (V, ER), where

ER =!

ti∈REti

(R = {t0, t1, ..., tr})

v2 v4

v1 v3

e3

e1

e2 e4

CG at t0.

v2 v4

v1 v3

e3

e1

e2 e4

CG at t1.

v2 v4

v1 v3

e3

e1

e2 e4

CG at t2.

v2 v4

v1 v3

e3

e1

e2 e4

0.33

0.33

1.0 0.67

7

4. Modeling: Metrics on an Edge

1. Availability A(ei)

probability assigned to each edgethat shows reliability of the edge

2. Commonality comG(ei)

the number of paths sharing theedge, which indicates importance ofthe edge in a graph

comT (e1) = 3.comT (e2) = 4.comT (e3) = 3.

* vTu: the path from a vertex v to uon a tree T

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7

1. Commonality of an edgethe number of paths sharing the edge, which indicates importance of the edge in a graph

Especially, in tree,

comT (ei)

e.g.

2. Availability of an edgeprobability assigned to each edge that shows reliability of the edge

A(ei)

Metrics

topological characteristics, i.e., relative location of the edge

in a tree.

2. 1 Survivable Network Design Problem

Survivable Network Design Problem (SNDP) is extrac-

tion of subgraph H = (V (H), E(H)) from weighted graph

G = (V, E) with weight function w : E −→ R+ such that the

number of edge disjoint paths between vi and vj is greater

than and equal to a designated value pij for every pair of

vertices. Note that two paths are edge disjoint when the

paths do not share any edge in themselves. When pij = 1

for all combinations of vi, vj , the subgraph H is tree. Then,

SNDP is converted to Steiner Tree Problem, so the opti-

mum solution for SNDP is equal to minimum spanning tree

if V (H) = V .

2. 2 Maximum Availability Tree Problem

Maximum Availability Tree Problem (MATP) is pro-

pounded to construct the most reliable network in terms of

path reliability [5]. The study defines reliability of a path as

the product of weights of edges on the path. The objective

of the problem is to find a tree whose weakest path in terms

of the path reliability is the most reliable of the other trees

in a given graph. This paper also states that the desired op-

timum tree is obtained by minimum diameter spanning tree

algorithm.

3. Definitions

3. 1 Tree and Path

Tree T = (V (T ), E(T )) is a connected acyclic graph in

graph G = (V,E). When a tree covers all vertices in G

(V (T ) = V ), the tree is said spanning. Because a tree does

not contain any cycles from its definition, it is minimally

connected. In other words, only one path exists between ar-

bitrarily chosen two vertices vi, vj . The path between vi and

vj in T is denoted as viTvj .

A graph G contains different spanning trees as its sub-

graphs, so a set of all possible spanning trees in G is denoted

as TG = {Ti}.A leaf vertex li is a vertex whose degree is 1 in T , so it is

possible that a tree has many leaf vertices. One of the ver-

tices in V (T ) can be designated as root vertex r of T . Then,

the tree T is called a rooted tree and represented as T (r) to

declare the root vertex.

In a rooted tree, edges can be labeled based on the distance

from the root vertex r. The label of each edge ei is denoted

as level(ei). Figure 1 describes the level of each edge in a

rooted tree T (v1).

Any tree is classified into either centered or bicentered

trees. The (bi)center vertex γ (γ1, γ2 in the case of a bi-

centered tree) of a tree is found by continuous remove of leaf

vertices until the number of left vertices becomes less than

3. For instance, v2 is the center vertex of T (v1) in Figure 1.

3. 2 Ordered Set

A set M is said strictly ordered under binary relation <

and denoted as (M,<) when x, y, z ∈ M satisfy the follow-

ing properties.

∀x ∈ M, ¬ (x < x) (irreflexivity) (1)

x < y ⇒ ¬ (x > y) (asymmetry) (2)

x < y, y < z ⇒ x < z (transitivity) (3)

Relation < is total in the case that any pair of elements

x, y ∈ M can be compared under <. Thus, a strictly ordered

set (M,<) is called totally strictly ordered set or strict chain

when < is total.

4. Metrics

4. 1 Commonality of an Edge

Commonality of an edge comG(ej) is the number of paths

that contain edge ej in graph G. In Figure 1, comT (v1)(e4) =

5 because e4 is on path v6Tv4, v6Tv2, v6Tv5, v6Tv1, v6Tv3.

In contrast, comT (v1)(e3) = 8.

Especially, commonality of edge ej in spanning tree Ti =

(V, E(Ti)) is obtained by product of the size of two compo-

nents separated by ej . Because a tree is an acyclic graph,

deletion of any edge divides the tree into two connected

components, namely, C1(ej), C2(ej) (|C1(ej)| <= |C2(ej)| =|V |− |C1(ej)|). Note that Cp(ej) = (V (Cp(ej)), E(Cp(ej)))

is a subgraph of Ti, and its size is defined as |V (Cp(ej)|.Then, the commonality of ej is calculated as follows.

comTi(ej) := |C1(ej)|× |C2(ej)|. (4)

Equation (4) implies that commonality marks the mini-

mum value when |C1(ej)| = 1. Additionally, commonality

increases in accordance with the augmentation of |C1(ej)|and reaches the maximum value when |C1(ej)| = |C2(ej)|.This is because the two components satisfy C1(ej)∪C2(ej) =

V ∧ C1(ej)∩C2(ej) = ∅. Due to this property, topology of

a tree and locations of edges in the tree seriously influence

commonality of each edge.

4. 2 Edge Availability

In a practical network, reliability of a communications link

demonstrates the stability of the link to properly operate.

Modeling a network with a graph, each edge is assigned a

weight showing its probability not to fail. This weight is

named edge availability, A(en). Because edge availability in-

dicates a probability, edge availability in graph G = (V, E)

is formally denoted as follows.

weight function A : E −→ (0.0, 1.0] . (5)

Figure 2 is a graph G1 whose edges are assigned their avail-

ability.

— 2 —

topological characteristics, i.e., relative location of the edge

in a tree.

2. 1 Survivable Network Design Problem

Survivable Network Design Problem (SNDP) is extrac-

tion of subgraph H = (V (H), E(H)) from weighted graph

G = (V, E) with weight function w : E −→ R+ such that the

number of edge disjoint paths between vi and vj is greater

than and equal to a designated value pij for every pair of

vertices. Note that two paths are edge disjoint when the

paths do not share any edge in themselves. When pij = 1

for all combinations of vi, vj , the subgraph H is tree. Then,

SNDP is converted to Steiner Tree Problem, so the opti-

mum solution for SNDP is equal to minimum spanning tree

if V (H) = V .

2. 2 Maximum Availability Tree Problem

Maximum Availability Tree Problem (MATP) is pro-

pounded to construct the most reliable network in terms of

path reliability [5]. The study defines reliability of a path as

the product of weights of edges on the path. The objective

of the problem is to find a tree whose weakest path in terms

of the path reliability is the most reliable of the other trees

in a given graph. This paper also states that the desired op-

timum tree is obtained by minimum diameter spanning tree

algorithm.

3. Definitions

3. 1 Tree and Path

Tree T = (V (T ), E(T )) is a connected acyclic graph in

graph G = (V,E). When a tree covers all vertices in G

(V (T ) = V ), the tree is said spanning. Because a tree does

not contain any cycles from its definition, it is minimally

connected. In other words, only one path exists between ar-

bitrarily chosen two vertices vi, vj . The path between vi and

vj in T is denoted as viTvj .

A graph G contains different spanning trees as its sub-

graphs, so a set of all possible spanning trees in G is denoted

as TG = {Ti}.A leaf vertex li is a vertex whose degree is 1 in T , so it is

possible that a tree has many leaf vertices. One of the ver-

tices in V (T ) can be designated as root vertex r of T . Then,

the tree T is called a rooted tree and represented as T (r) to

declare the root vertex.

In a rooted tree, edges can be labeled based on the distance

from the root vertex r. The label of each edge ei is denoted

as level(ei). Figure 1 describes the level of each edge in a

rooted tree T (v1).

Any tree is classified into either centered or bicentered

trees. The (bi)center vertex γ (γ1, γ2 in the case of a bi-

centered tree) of a tree is found by continuous remove of leaf

vertices until the number of left vertices becomes less than

3. For instance, v2 is the center vertex of T (v1) in Figure 1.

3. 2 Ordered Set

A set M is said strictly ordered under binary relation <

and denoted as (M,<) when x, y, z ∈ M satisfy the follow-

ing properties.

∀x ∈ M, ¬ (x < x) (irreflexivity) (1)

x < y ⇒ ¬ (x > y) (asymmetry) (2)

x < y, y < z ⇒ x < z (transitivity) (3)

Relation < is total in the case that any pair of elements

x, y ∈ M can be compared under <. Thus, a strictly ordered

set (M,<) is called totally strictly ordered set or strict chain

when < is total.

4. Metrics

4. 1 Commonality of an Edge

Commonality of an edge comG(ej) is the number of paths

that contain edge ej in graph G. In Figure 1, comT (v1)(e4) =

5 because e4 is on path v6Tv4, v6Tv2, v6Tv5, v6Tv1, v6Tv3.

In contrast, comT (v1)(e3) = 8.

Especially, commonality of edge ej in spanning tree Ti =

(V, E(Ti)) is obtained by product of the size of two compo-

nents separated by ej . Because a tree is an acyclic graph,

deletion of any edge divides the tree into two connected

components, namely, C1(ej), C2(ej) (|C1(ej)| <= |C2(ej)| =|V |− |C1(ej)|). Note that Cp(ej) = (V (Cp(ej)), E(Cp(ej)))

is a subgraph of Ti, and its size is defined as |V (Cp(ej)|.Then, the commonality of ej is calculated as follows.

comTi(ej) := |C1(ej)|× |C2(ej)|. (4)

Equation (4) implies that commonality marks the mini-

mum value when |C1(ej)| = 1. Additionally, commonality

increases in accordance with the augmentation of |C1(ej)|and reaches the maximum value when |C1(ej)| = |C2(ej)|.This is because the two components satisfy C1(ej)∪C2(ej) =

V ∧ C1(ej)∩C2(ej) = ∅. Due to this property, topology of

a tree and locations of edges in the tree seriously influence

commonality of each edge.

4. 2 Edge Availability

In a practical network, reliability of a communications link

demonstrates the stability of the link to properly operate.

Modeling a network with a graph, each edge is assigned a

weight showing its probability not to fail. This weight is

named edge availability, A(en). Because edge availability in-

dicates a probability, edge availability in graph G = (V, E)

is formally denoted as follows.

weight function A : E −→ (0.0, 1.0] . (5)

Figure 2 is a graph G1 whose edges are assigned their avail-

ability.

— 2 —

v1 v2 v3 v4e2e1 e30.82 0.92 0.85

v1 v2 v3 v4

Tree T

v1Tv3

v2Tv3

v2Tv4

v1Tv4

e2e1 e3

v1 v2 v3 v4e2e1 e30.82 0.92 0.85

v1 v2 v3 v4

Tree T

v1Tv3

v2Tv3

v2Tv4

v1Tv4

e2e1 e3

comT (e2) = 4

comT (e1) = 3

paths including e2

Availability on edges.

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7

1. Commonality of an edgethe number of paths sharing the edge, which indicates importance of the edge in a graph

Especially, in tree,

comT (ei)

e.g.

2. Availability of an edgeprobability assigned to each edge that shows reliability of the edge

A(ei)

Metrics

topological characteristics, i.e., relative location of the edge

in a tree.

2. 1 Survivable Network Design Problem

Survivable Network Design Problem (SNDP) is extrac-

tion of subgraph H = (V (H), E(H)) from weighted graph

G = (V, E) with weight function w : E −→ R+ such that the

number of edge disjoint paths between vi and vj is greater

than and equal to a designated value pij for every pair of

vertices. Note that two paths are edge disjoint when the

paths do not share any edge in themselves. When pij = 1

for all combinations of vi, vj , the subgraph H is tree. Then,

SNDP is converted to Steiner Tree Problem, so the opti-

mum solution for SNDP is equal to minimum spanning tree

if V (H) = V .

2. 2 Maximum Availability Tree Problem

Maximum Availability Tree Problem (MATP) is pro-

pounded to construct the most reliable network in terms of

path reliability [5]. The study defines reliability of a path as

the product of weights of edges on the path. The objective

of the problem is to find a tree whose weakest path in terms

of the path reliability is the most reliable of the other trees

in a given graph. This paper also states that the desired op-

timum tree is obtained by minimum diameter spanning tree

algorithm.

3. Definitions

3. 1 Tree and Path

Tree T = (V (T ), E(T )) is a connected acyclic graph in

graph G = (V,E). When a tree covers all vertices in G

(V (T ) = V ), the tree is said spanning. Because a tree does

not contain any cycles from its definition, it is minimally

connected. In other words, only one path exists between ar-

bitrarily chosen two vertices vi, vj . The path between vi and

vj in T is denoted as viTvj .

A graph G contains different spanning trees as its sub-

graphs, so a set of all possible spanning trees in G is denoted

as TG = {Ti}.A leaf vertex li is a vertex whose degree is 1 in T , so it is

possible that a tree has many leaf vertices. One of the ver-

tices in V (T ) can be designated as root vertex r of T . Then,

the tree T is called a rooted tree and represented as T (r) to

declare the root vertex.

In a rooted tree, edges can be labeled based on the distance

from the root vertex r. The label of each edge ei is denoted

as level(ei). Figure 1 describes the level of each edge in a

rooted tree T (v1).

Any tree is classified into either centered or bicentered

trees. The (bi)center vertex γ (γ1, γ2 in the case of a bi-

centered tree) of a tree is found by continuous remove of leaf

vertices until the number of left vertices becomes less than

3. For instance, v2 is the center vertex of T (v1) in Figure 1.

3. 2 Ordered Set

A set M is said strictly ordered under binary relation <

and denoted as (M,<) when x, y, z ∈ M satisfy the follow-

ing properties.

∀x ∈ M, ¬ (x < x) (irreflexivity) (1)

x < y ⇒ ¬ (x > y) (asymmetry) (2)

x < y, y < z ⇒ x < z (transitivity) (3)

Relation < is total in the case that any pair of elements

x, y ∈ M can be compared under <. Thus, a strictly ordered

set (M,<) is called totally strictly ordered set or strict chain

when < is total.

4. Metrics

4. 1 Commonality of an Edge

Commonality of an edge comG(ej) is the number of paths

that contain edge ej in graph G. In Figure 1, comT (v1)(e4) =

5 because e4 is on path v6Tv4, v6Tv2, v6Tv5, v6Tv1, v6Tv3.

In contrast, comT (v1)(e3) = 8.

Especially, commonality of edge ej in spanning tree Ti =

(V, E(Ti)) is obtained by product of the size of two compo-

nents separated by ej . Because a tree is an acyclic graph,

deletion of any edge divides the tree into two connected

components, namely, C1(ej), C2(ej) (|C1(ej)| <= |C2(ej)| =|V |− |C1(ej)|). Note that Cp(ej) = (V (Cp(ej)), E(Cp(ej)))

is a subgraph of Ti, and its size is defined as |V (Cp(ej)|.Then, the commonality of ej is calculated as follows.

comTi(ej) := |C1(ej)|× |C2(ej)|. (4)

Equation (4) implies that commonality marks the mini-

mum value when |C1(ej)| = 1. Additionally, commonality

increases in accordance with the augmentation of |C1(ej)|and reaches the maximum value when |C1(ej)| = |C2(ej)|.This is because the two components satisfy C1(ej)∪C2(ej) =

V ∧ C1(ej)∩C2(ej) = ∅. Due to this property, topology of

a tree and locations of edges in the tree seriously influence

commonality of each edge.

4. 2 Edge Availability

In a practical network, reliability of a communications link

demonstrates the stability of the link to properly operate.

Modeling a network with a graph, each edge is assigned a

weight showing its probability not to fail. This weight is

named edge availability, A(en). Because edge availability in-

dicates a probability, edge availability in graph G = (V, E)

is formally denoted as follows.

weight function A : E −→ (0.0, 1.0] . (5)

Figure 2 is a graph G1 whose edges are assigned their avail-

ability.

— 2 —

topological characteristics, i.e., relative location of the edge

in a tree.

2. 1 Survivable Network Design Problem

Survivable Network Design Problem (SNDP) is extrac-

tion of subgraph H = (V (H), E(H)) from weighted graph

G = (V, E) with weight function w : E −→ R+ such that the

number of edge disjoint paths between vi and vj is greater

than and equal to a designated value pij for every pair of

vertices. Note that two paths are edge disjoint when the

paths do not share any edge in themselves. When pij = 1

for all combinations of vi, vj , the subgraph H is tree. Then,

SNDP is converted to Steiner Tree Problem, so the opti-

mum solution for SNDP is equal to minimum spanning tree

if V (H) = V .

2. 2 Maximum Availability Tree Problem

Maximum Availability Tree Problem (MATP) is pro-

pounded to construct the most reliable network in terms of

path reliability [5]. The study defines reliability of a path as

the product of weights of edges on the path. The objective

of the problem is to find a tree whose weakest path in terms

of the path reliability is the most reliable of the other trees

in a given graph. This paper also states that the desired op-

timum tree is obtained by minimum diameter spanning tree

algorithm.

3. Definitions

3. 1 Tree and Path

Tree T = (V (T ), E(T )) is a connected acyclic graph in

graph G = (V,E). When a tree covers all vertices in G

(V (T ) = V ), the tree is said spanning. Because a tree does

not contain any cycles from its definition, it is minimally

connected. In other words, only one path exists between ar-

bitrarily chosen two vertices vi, vj . The path between vi and

vj in T is denoted as viTvj .

A graph G contains different spanning trees as its sub-

graphs, so a set of all possible spanning trees in G is denoted

as TG = {Ti}.A leaf vertex li is a vertex whose degree is 1 in T , so it is

possible that a tree has many leaf vertices. One of the ver-

tices in V (T ) can be designated as root vertex r of T . Then,

the tree T is called a rooted tree and represented as T (r) to

declare the root vertex.

In a rooted tree, edges can be labeled based on the distance

from the root vertex r. The label of each edge ei is denoted

as level(ei). Figure 1 describes the level of each edge in a

rooted tree T (v1).

Any tree is classified into either centered or bicentered

trees. The (bi)center vertex γ (γ1, γ2 in the case of a bi-

centered tree) of a tree is found by continuous remove of leaf

vertices until the number of left vertices becomes less than

3. For instance, v2 is the center vertex of T (v1) in Figure 1.

3. 2 Ordered Set

A set M is said strictly ordered under binary relation <

and denoted as (M,<) when x, y, z ∈ M satisfy the follow-

ing properties.

∀x ∈ M, ¬ (x < x) (irreflexivity) (1)

x < y ⇒ ¬ (x > y) (asymmetry) (2)

x < y, y < z ⇒ x < z (transitivity) (3)

Relation < is total in the case that any pair of elements

x, y ∈ M can be compared under <. Thus, a strictly ordered

set (M,<) is called totally strictly ordered set or strict chain

when < is total.

4. Metrics

4. 1 Commonality of an Edge

Commonality of an edge comG(ej) is the number of paths

that contain edge ej in graph G. In Figure 1, comT (v1)(e4) =

5 because e4 is on path v6Tv4, v6Tv2, v6Tv5, v6Tv1, v6Tv3.

In contrast, comT (v1)(e3) = 8.

Especially, commonality of edge ej in spanning tree Ti =

(V, E(Ti)) is obtained by product of the size of two compo-

nents separated by ej . Because a tree is an acyclic graph,

deletion of any edge divides the tree into two connected

components, namely, C1(ej), C2(ej) (|C1(ej)| <= |C2(ej)| =|V |− |C1(ej)|). Note that Cp(ej) = (V (Cp(ej)), E(Cp(ej)))

is a subgraph of Ti, and its size is defined as |V (Cp(ej)|.Then, the commonality of ej is calculated as follows.

comTi(ej) := |C1(ej)|× |C2(ej)|. (4)

Equation (4) implies that commonality marks the mini-

mum value when |C1(ej)| = 1. Additionally, commonality

increases in accordance with the augmentation of |C1(ej)|and reaches the maximum value when |C1(ej)| = |C2(ej)|.This is because the two components satisfy C1(ej)∪C2(ej) =

V ∧ C1(ej)∩C2(ej) = ∅. Due to this property, topology of

a tree and locations of edges in the tree seriously influence

commonality of each edge.

4. 2 Edge Availability

In a practical network, reliability of a communications link

demonstrates the stability of the link to properly operate.

Modeling a network with a graph, each edge is assigned a

weight showing its probability not to fail. This weight is

named edge availability, A(en). Because edge availability in-

dicates a probability, edge availability in graph G = (V, E)

is formally denoted as follows.

weight function A : E −→ (0.0, 1.0] . (5)

Figure 2 is a graph G1 whose edges are assigned their avail-

ability.

— 2 —

v1 v2 v3 v4e2e1 e30.82 0.92 0.85

v1 v2 v3 v4

Tree T

v1Tv3

v2Tv3

v2Tv4

v1Tv4

e2e1 e3

v1 v2 v3 v4e2e1 e30.82 0.92 0.85

v1 v2 v3 v4

Tree T

v1Tv3

v2Tv3

v2Tv4

v1Tv4

e2e1 e3

comT (e2) = 4

comT (e1) = 3

paths including e2Paths including edge e2.

8

5. Problem Formulation

Focus of this Study

consideration to probabilistic stabilityand location importance

Methodology

! Modeling a com. network with aPACG

! 2 kinds of edge weights

3

3

4

v2 v4

v1 v3

e3

e1

e2 e4

0.33

0.33

1.0 0.67

Tree 1

4

3 3

v2 v4

v1 v3

e3

e1

e2 e4

0.33

0.33

1.0 0.67

Tree 2

Com

mon

ality

Availability1.0

e1

0.33

4

3

e2 e1

e2

1.00.33

4

3

Availability

Com

mon

ality

Tree 1 (Left) Tree 2 (Right)

9

5. Problem Formulation

Focus of this Study

consideration to probabilistic stabilityand location importance

Methodology

! Modeling a com. network with aPACG

! 2 kinds of edge weights

3

3

4

v2 v4

v1 v3

e3

e1

e2 e4

0.33

0.33

1.0 0.67

tree 1

4

3 3

v2 v4

v1 v3

e3

e1

e2 e4

0.33

0.33

1.0 0.67

tree 2

Com

mon

ality

Availability1.0

e1

0.33

4

3

e2 e1

e2

1.00.33

4

3

Availability

Com

mon

ality

Tree 1 (Left) Tree 2 (Right)

10

6. Problem Formulation

“resilient” in our Study! more stable edge located at more important location! more paths sharing edges with higher availability in a tree

Maximize"

vi∈V

"

v j∈Vmax

ek∈viTv jA(ek)

3

3

4

v2 v4

v1 v3

e3

e1

e2 e4

0.33

0.33

1.0 0.67

tree 1

4

3 3

v2 v4

v1 v3

e3

e1

e2 e4

0.33

0.33

1.0 0.67

tree 2

† the real numbers assigned to the edges are availability A(ei). 11

7. Tree Composition Approach

Order for an Edge set based on the Metrics

Exploiting the pair set on availability A(e) and commonality comG(e), theorder ≤ of an edge set is defined as follows.

e j ≤ ek ⇔ A(e j) ≤ A(ek) ∧ comG(e j) ≤ comG(ek) .

Tree 1

Tree 2

Com

mon

ality

Availability1.0

e1

0.33

4

3

e2 e1

e2

1.00.33

4

3

Availability

Com

mon

ality

Tree 1 (resilient) Tree 212

7. Tree Composition Approach

Paths Totally Ordered Trees (PTOTs)! decrease in the edge commonality along with increase in the

distance from a center γ! ordering of a path from a center to a leaf E(γTlk)! for each leaf lk, totally ordered path E(γTlk)⇒ PTOT

Procedure to obtain Paths Totally Ordered Trees (PTOTs)

Input: r Contact Graphs (CGs) over time range R = {t0, t1, ..., tr}Task: Check whether each tree on PACG GR is PTOT

Soka University

6

Big Picture of this Study

composition of a tree that alleviates the influence of single link failure on the whole network considering significance of location in the tree

each edge assigned 2 types of weights: 1. commonality : significance of location in a given tree 2. availability : reliability of the edge

all spanning trees

resilient treescompose a subset of trees whose paths are totally ordered under reliability relation on edges

13

7. Tree Composition Approach: Example

Check whether a tree on PACG GR is PTOT

leaf v1, v5, v6, v7center v3

v1

leaf l1

v2 v3

center γ

v4 v5

leaf l2

v6

leaf l3

v7

leaf l4

14

7. Tree Composition Approach: Example

Check whether a tree on PACG GR is PTOT

leaf v1, v5, v6, v7center v3paths v3Tv1, v3Tv5, v3Tv6, v3Tv7

v1

leaf l1100.8

60.5

v2 v3

center γ

v3

center γ

v4 v5

leaf l2

v3

center γ

v6

leaf l3

v3

center γ

v4

v7

leaf l415

7. Tree Composition Approach: Example

Check whether a tree on PACG GR is PTOT

leaf v1, v5, v6, v7center v3paths v3Tv1, v3Tv5, v3Tv6, v3Tv7

v1

leaf l1100.2

60.5

v2 v3

center γ

v3

center γ

v4 v5

leaf l2

v3

center γ

v6

leaf l3

v3

center γ

v4

v7

leaf l416

8. Simulation: Overview

Resiliency of every path in a tree! Does “good-thing” (high availability edges are shared much)

happen?

1|P|"

P in T

maxe∈P

A(e).

! Does NOT “bad-thing” (low availability edges are not shared much)happen?

1|P|"

P in T

mine∈P

A(e).

† P is a set of all paths in a tree T .

17

8. Simulation

Graph Settings! |V | = 50, 75, 100

125, 150! 10 sparse NWS

random graphs(k = 2, p = 0.1)

! BFS, DFS trees

Eval. index1|P|#

P in T maxe∈P A(e)1|P|#

P in T mine∈P A(e)0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

50 75 100 125 1500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1 |P|!

P⊂E(T

)max

e∈PA(e)

1 |P|!

P⊂E(T

)min

e∈PA(e)

The number of vertices

MAX TOMAX Others

MIN TOMIN Others

Results! more paths sharing edges with higher availability in a tree! less paths sharing edges with lower availability in a tree

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9. Conclusion and Future Work

Summary

! Modeling of a network with a Probabilistic Aggregated ContactGraph (PACG)

! Edge Metrics! availability : probability of an edge to operate properly! commonality : importance of the location of an edge

! Formulation of the composition problem of a resilient tree on PACG! Composition method: Totally Ordered Paths Trees with relation ≤ on

edges

Future Tasks! some variant of this resilient topology design problem

(degree-constrained, connectivity requirements)! the proofs of the intractability of this problem

19