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Dominating Sets, Multiple Egocentric Networksand Modularity Maximizing Clustering
of Social Networks
Moses A. Boudourides1 & Sergios T. Lenis2
Department of Mathematics
University of Patras, Greece
[email protected]@gmail.com
1st European Conference on Social NetworksBarcelona, July 1-4, 2014
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Community Partitions in a GraphDefinition (Newman & Girvan, 2004)
Let G = (V ,E ) be a graph.A clustering of vertices of G is a partition of the set of verticies Vinto a (finite) family C ⊂ 2V of subsets of vertices, often calledmodules, such that
⋃C∈C C = V and C ∩ C ′ = ∅, for each
C ,C ′ ∈ C,C 6= C ′.
A clustering C may be assessed by a quality function Q = Q(C),called modularity, which is defined as:
Q =∑C∈C
[|E (C )||E |
−(
2|E (C )|+∑
C ′∈C,C 6=C ′ |E (C ,C ′)|2|E |
)2]
=∑C∈C
[|E (C )||E |
−(∑
v∈C degree(v)
2|E |
)2],
where E(C) is the number of edges inside module C and E(C ,C ′) is thenumber of edges among modules C and C ′. Essentially, modularity comparesthe number of edges inside a given module with the expected value for arandomized graph of the same size and same degree sequence.
A clustering that maximizes modularity is usually called communitypartition and the corresponding modules are called communities.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Theorem (Brandes, Delling, Gaertler, Gorke, Hoefer, Nikoloski &Wagner, 2008)
Modularity is strongly NP–complete.
Approximate Algorithmic Computation of Community Detection
Here, we are using the community detection algorithm (throughmodularity maximization) of the Louvain method (Blondel,Guillaume, Lambiotte & Lefebvre, 2008) as implemented in Pythonby Thomas Aynaud.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Communities in the Network of Florentine Families
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Dominating Sets in a Graph
Definition
Let G = (V ,E ) be a (simple undirected) graph. A set S ⊆ V ofvertices is called a dominating set (or externally stable) if everyvertex v ∈ V is either an element of S or is adjacent to an elementof S .
RemarkA set S ⊆ V is a dominating set if and only if:
for every v ∈ V r S , |N(v) ∩ S | ≥ 1, i.e., V r S is enclaveless;
N[S] = V ;
for every v ∈ V r S , d(v , S) ≤ 1.Above N(v) is the open neighborhood of vertex v , i.e., N(v) = {w ∈ V : (u,w) ∈ E}, N[v ] is the closedneighborhood of vertex v , i.e., N[v ] = N(v) ∪ {v}, d(v, x) is the geodesic distance between vertices v and x andd(v, S) = min{d(v, s) : s ∈ S} is the distance between vertex v and the set of vertices S .
Standard Reference
Haynes, Teresa W., Hedetniemi, Stephen T., & Slater, Peter J.(1998). Fundamentals of Domination in Graphs. New York:Marcel Dekker.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Definition
Let S be a set of vertices in graph G .
S is called independent (or internally stable) if no twovertices in S are adjacent.
If S is a dominating set, then
S is called minimal dominating set if no proper subsetS ′ ⊂ S is a dominating set;S is called a minimum dominating set if the cardinality of Sis minimum among the cardinalities of any other dominatingset;The cardinality of a minimum dominating set is called thedomination number of graph G and is denoted by γ(G );S is called independent dominating set if S is both anindependent and a dominating set.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Example
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The sets {1, 3, 5}, {3, 6, 7, 8} and {2, 4, 6, 7, 8} (red circles) are allminimal dominating sets. However, only the first is a minimumdominating set. Apparently γ = 3 for this graph.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Theorem (Ore, 1962)
A dominating set S is a minimal dominating set if and only if, foreach vertex u ∈ S , one of the following two conditions holds:
1 u is such that N(u) ⊆ V r S , i.e., u is an isolate of S ,
2 there exists a vertex v ∈ V r S such that N(v) ∩ S = {u},i.e., v is a private neighbor of u.
Theorem (Ore, 1962)
Every connected graph G of order n ≥ 2 has a dominating set S .
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Complexity of the Dominating Set Problem
Theorem (Johnson, 1974)
The dominating set problem is NP–complete.
Approximate Algorithmic Computation of Community Detection
As the dominating set problem is NP–complete, there arecertain efficient algorithms (typically based on linear andinteger programming) for its approximate solution.
Here, we are algorithms that have been already implementedin Python’s Sage. However, these implementations return onlyone of the minimum or independing (minimimum) dominatingsets.
To be able to obtain all possible (minimum and independent)dominating sets for rather small graphs, we have written aPython script implementing the brute force algorithm fordominating sets.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Egocentric Subgraphs Induced by a Dominating Set in a Graph
Definition
Let G = (V ,E ) be a (simple undirected) graph.
Let U ⊂ V be a set of vertices in a graph G and let u ∈ U.
The subgraph induced by U (in G ) is called an egocentric(sub)graph (in G ) if
U = N[u],
i.e., if all vertices of U are dominated by u;vertex u is the ego of the egocentric (sub)graph N[u];vertices w ∈ N(u) are called alters of ego u.
Let S ⊂ V be a dominating set in G and let v ∈ S .
The subgraph induced by N[v ] (in G ) is called a dominatingegocentric (sub)graph;vertex v is the dominating ego;vertices w ∈ N(v) are the dominated alters by v .Graph G is called multiple egocentric or |S |-egocentricgraph corresponding to the dominating set S .
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Remark
Given a (simple undirected) graph, typically, there is amultiplicity of dominating sets for that graph.
Therefore, any egocentric decomposition of a graph dependson the dominating set that was considered in the generationof the constituent egocentric subgraphs.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Private and Public Alters
Definition
Let S be a dominating set in graph G = (V ,E ) (|S | ≥ 2) and letv ∈ S an ego.
An alter w ∈ N(v) (w /∈ S) is called private alter ifN(w) ⊂ N[v ].
An alter w ∈ N(v) (w /∈ S) is called public alter ifN(w) r N[v ] 6= ∅.
Remark
Two adjacent egos u, v are not considered alters to eachother! Nevertheless, they define an ego–to–ego edge (bridge).
A private alter is always adjacent to a single ego.
A public alter is adjacent either to at least two egos or to asingle ego and to another alter which is adjacent to a differentego.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Figure : Five private alters (blue circles) adjacent to an ego (red square).
Figure : Left: one public alter (green circle) shared by two egos (redsquares). Right: two public alters (green circles), each one adjacent to adifferent ego (red square).
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Dominating and Dominated Bridges
Definition
Let S be a dominating set in a graph G = (V ,E ) (|S | ≥ 2).
If two egos v1, v2 ∈ S are adjacent, then edge (v1, v2) ∈ E iscalled dominating edge or bridge of egos.
If two private alters w1 ∈ N(v1),w2 ∈ N(v2) are adjacent(possibly v1 = v2), then edge (w1,w2) ∈ E is calleddominated edge among private alters or bridge of privatealters.
If two public alters w1 ∈ N(v1),w2 ∈ N(v2) are adjacent(possibly v1 = v2), then edge (w1,w2) ∈ E is calleddominated edge among public alters or bridge of publicalters.
If a private alter w1 ∈ N(v1) and a public alter w2 ∈ N(v2)are adjacent (possibly v1 = v2), then edge (w1,w2) is calleddominated edge among private–public alters or bridge ofprivate–to–public alters.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Figure : One edge (bridge) among private alters (yellow line), two edges(bridges) among public alters (magenta lines) and one edge (bridge)among private–to–public alters (brown lines).
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Communities in 1–Egocentric Networks (γ = 1)
Proposition
Let G = (V ,E ) be 1-egocentric network, i.e., G is a graph in which thereis a single ego a and all other vertices are alters of a. In other wordes,S = {a} is a (minimum) dominating set or γ = 1. We denote by
Vleaves, the set of private alters having degree 1 (leaves) and, by
Gjoint alter, the subgraph induced by Vjoint alter = V r (Vleaves ∪ {a}).
Then, if C is a (modularity maximizing) community partition of G , one ofthe following cases holds:
1 G is a single community and a necessary and sufficient condition forthis is that the alter subgraph Galter = G r {a} is a proper clique;
2 {a} ∪ Vleaves is a community in C;
3 there is a collection Hjoint alter of connected components of Gjoint alter
such that {a} ∪ Vleaves ∪Hjoint alter is a community in C;
4 there is a subset Ujoint alter ⊂ Vjoint alter such that{a} ∪ Vjoint leaves ∪ Ujoint alter is a community in C.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
The four types of communities in a 1–egocentric network.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Ego–Free Communities
Definition
Let S be a domination set in a graph G and C be a communitypartition of G . A community C ∈ C is called ego–free ifC ∩ S = ∅, i.e., if no ego is included among the vertices whichbelong to that community.
Corollary
Any community in a 1–egocentric network is either an ego–freecommunity or a (proper) clique (i.e., a structurally equivalentclass).
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Communities in a Barabasi–Albert Network(n = 100,m = 3)
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Definition
Let C be a community partition in a graph G ald let C ∈ C be acommunity.
The set of boundary vertices of C , denoted by ∂C , is the setof vertices of C which are incident to links among C and anyother community in C.
The set of interior vertices of C , denoted by intC , is the setof vertices of C such that any edge incident to them remainsinside C .
Thus,C = ∂C ∪ intC .
Proposition
If C is an ego–free community, then intC = ∅.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Relationship between Number of Communities andDomination Number in a Graph
Conjecture
For every graph G with community partition C and dominationnumber γ,
|C| ≤ γ.
Proposition
If in a graph G with community partition C, for every C ∈ C,intC 6= ∅, then |C| ≤ γ.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Six Empirical Networks: Minimum Dominating Egocentric Subgraphs and their
Community Partitions
Network |V | |E | |Vprivate| |Vpublic| |Eprivate| |Epublic| |Eprivate− public| |Eego| γ |C|Florentine Families 15 20 5 5 0 2 2 3 5 4
Zachary’sKarate Club
34 78 15 15 3 19 17 1 4 4
Les Miserables 77 254 38 29 28 70 40 10 10 6
AmericanCollege Football
115 613 0 103 0 487 0 3 12 10
Food–Web 161 592 30 116 0 230 1 13 15 4
C. elegans 297 2148 41 240 28 1471 132 12 16 5
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
The Network of Florentine Families: Egos in Communities
Five egos distributed inside four communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Zachary’s Karate Club Network: Egos in Communities
Four egos distributed inside four communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Network of Les Miserables: Egos in Communities
Ten egos distributed inside six communities with one of them being ego–free.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
American College Football Network: Egos in Communities
Twelve egos distributed inside ten communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
Food–Web Network: Egos in Communities
Fifteen egos distributed inside four communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities
C. elegans Network: Egos in Communities
Sixteen egos distributed inside five communities.Egos are colored red, private alters blue and public alters green.Dotted rectangles embrace vertices lying inside the same communities.
M.A. Boudourides & S.T. Lenis Dominating Egocentric Networks & Communities