Complexity of Social Networks with Applications in Wellness
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Complexity of Social Networks with Applications in Wellness
ความซับซ้อนของเครือข่ายสังคมและการประยุกต์กับสุขภาวะที่ดี
Nopadon Juneam 1
Joint work with
Assoc. Prof. Dr. Sanpawat Kantabutra 2
Prof. Dr. Raymond Greenlaw 3
1Department of Computer Science, Chiang Mai University
2Department of Computer Engineering, Chiang Mai University
3Center of Cyber Security Studies, United States Naval Academy
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 1 / 18
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Background Wellness Profile Model (WPM)
Model Definition
A Wellness Profile Model (WPM) W is an eight-tuple
(M, C, P, I, A, T , S, V), where
1. The set M = {m1, m2, . . . , mk} is a finite collection of members,
where k ∈ N.
2. The set C = {c1, c2, . . . , cl} is a finite collection of characteristics,
where l ∈ N. Corresponding to each ci, for 1 ≤ i ≤ l, is a set Γ(ci)
representing the set of all possible values for each ci. There is a total
order ti on Γ(ci), for 1 ≤ i ≤ l.
3. The set P = {P1, P2, . . . , Pk} is a finite collection of preferences such
that, for 1 ≤ i ≤ k, Pi = (pi1, . . . , pil), where for 1 ≤ j ≤ l, pij ∈ Γ(cj)
and pij is the preference value for member mi on characteristic cj.
4. The set I = {I1, I2, . . . , Ik} is a finite collection of intervals such
that, for 1 ≤ i ≤ k, Ii = [nia, nib] with nia, nib ∈ N and nia ≤ nib, and
Ii is the interval representing the number of desired partners for
member mi not including himself.
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 4 / 18
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Background Wellness Profile Model (WPM)
Model Definition (cont.)
5. The set A = {a1, a2, . . . , ao} is a finite collection of activities, where
o ∈ N. For 1 ≤ i ≤ k, Ai ⊆ A is the set of desired activities for
member mi.
6. The set T = {T1, T2, . . . , Tk} is a finite collection of available times
such that, for 1 ≤ i ≤ k and some fixed r ∈ N, Ti = {[ti,s1 , ti,e1 ],
[ti,s2 , ti,e2 ], . . . , [ti,sr , ti,er ]}, where for 1 ≤ p ≤ r, sp, ep ∈ N and
sp < ep. We call [ti,sx , ti,ex ] the xth-interval of availability for member
mi and ti,ex − ti,sx the duration of such an interval.
7. The set S = {s1, s2, . . . , sq} is a finite collection of vital statistics,
where q ∈ N. Corresponding to each si, for 1 ≤ i ≤ q, is a set ζ(si)
representing the set of all possible values for each si.
8. The set V = {V1, V2, . . . , Vk} is a finite collection of members’ vital
statistics such that, for 1 ≤ i ≤ k, Vi = (vi1, . . . , viq), where for
1 ≤ j ≤ q, vij ∈ ζ(si) and vij is the statistic value for member mi on
vital statistic sj.
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 5 / 18
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Recent Works Compatible Wellness Group in WPM
Originality
Basically, a compatible wellness group is a group consisting of those
who have similar interests and have their physical abilities close to one
another.
Finding a largest compatible wellness group of members is considered
a desirable task, especially in social networking, as groups of users can
be formed by friend suggestions or group recommendations.
Here in the wellness application such groups can lead to wellness
communities where people in the communities can help improve one
another’s health.
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 7 / 18
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Recent Works Compatible Wellness Group in WPM
Preliminaries
Definition (Vital Statistics’s Classification)
For q = |S|, the vital statistic’s classification is given by a set λ = {λ1, λ2,
. . ., λq}, where for 1 ≤ i ≤ q, each λi is a partition of ζ(si). We call λi the
classification for vital statistic si and an element in λi is called range.
Definition (Within Range Members’s Vital Statistics)
For q = |S|, the vital statistics of two distinct members Vi, Vj ∈ V are said
to be within range if, for 1 ≤ p ≤ q, the values of vip, vjp fall in the same
range in partition λp.
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 8 / 18
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Recent Works Compatible Wellness Group in WPM
Preliminaries (cont.)
Definition (Compatible Wellness Group)
A set G ⊆ M is called a group of members, where |G| denotes its size. G is
called a compatible wellness group if G has the five following conditions:
1. The preferences of all members are the same.
2. |G| − 1 is in the interval of the number of desired partners for each
member.
3. There is some activity common to all members.
4. There is some time period common to all members.
5. For all members in G, the vital statistics of each pair of members are
within range.
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 9 / 18
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Recent Works Compatible Wellness Group in WPM
Problem Definitions
Definition (Compatible Wellness Group with Target Member
Problem or CWGTMP)
Instance: A WPM W = (M, C, P, I, A, T , S, V), a set of
classifications λ, target member my ∈ M, and integer r.
Question: Does W contain a compatible wellness group G ⊆ M such
that my ∈ G and |G| ≥ r ?
Definition (Compatible Wellness Group Problem or CWGP)
Instance: A WPM W = (M, C, P, I, A, T , S, V), a set of
classifications λ, and an integer r.
Question: Does W contain a compatible wellness group G ⊆ M of size r
or more?
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 10 / 18
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Recent Works Compatible Wellness Group in WPM
Time Complexity of The CWGTMP
Theorem (CWGTMP’s time complexity)
Let n = |M|. The CWGTMP is solvable in O(n)2 time.
Proof.
An exhaustive search for a largest group G such that my ∈ G takes
1. O(n) time to seek my’s exact match on P and C.
2. O(n) time to seek my’s exact match on S and V.
3. O(n2) time to try a possible match for my with maximum number of
partners on each a ∈ Ay and each t ∈ Ty. ⇒ O(|Ay||Ty|n2) time in
total.
Since |Ay| and |Ty| are constant in general, the running time of the search
algorithm takes O(n) + O(n) + O(n2) = O(n2) time.
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 11 / 18
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Recent Works Compatible Wellness Group in WPM
Polynomial-time Algorithm for The CWGTMP
Algorithm Match on Component Time Complexity
Group-By-Preferences P, C O(n)
Group-By-Statistics S, V O(n)
Group-By-Availability a ∈ Ay O(n)
Group-By-Activity t ∈ Ty O(n)
Max-Partners I O(n2)
CWGTM-Algorithm largest G ⊆ M, my ∈ G O(n2)
Table: Characterization of the CWGTM-Algorithm.
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 12 / 18
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Recent Works Compatible Wellness Group in WPM
Time Complexity of The CWGP
Lemma (CWGP’s subcase)
Let W = (M, C, P, I, T , A, S, V) be an instance of the WPM, λ be a set
of classifications, and r be an integer. (W, λ, r) is a YES-instance of the
CWGP if and only if (W, λ, m, r) is a YES-instance of the CWGTMP,
for some target member m ∈ M.
Proof.
(⇒) Obviously, a group G ⊆ M such that |G| ≥ r for the CWGP is a
group of size r or more in W that includes m as a member for some m ∈ G.
(⇐) Clearly, a group G ⊆ M such that |G| ≥ r and m ∈ G for the
CWGTMP is also a group of size r or more in W.
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 13 / 18
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Recent Works Compatible Wellness Group in WPM
Time Complexity of The CWGP (cont.)
Theorem (CWGP’s time complexity)
Let n = |M|. The CWGP is solvable in O(n)3 time.
Proof.
The previous lemma implies that an exhaustive search algorithm for the
CWGTMP can be used to solve the CWGP. In particular, instead of
computing a largest group G for one particular member, we expand the
search space through every member in M. Therefore, our new algorithm
for the CWGP runs in time O(n) × O(n2) = O(n3).
Nopadon Juneam (Chiang Mai University) Semester Progress Report April 30, 2014 14 / 18
