In this talk, I present recent results on the measurement of conflict in signed social network. A signed social network is a network in which both positive and negative ties are present, for instance representing friendship and enmity, or trust and distrust. Such networks have long been studied under the aspect of balance theory, i.e., considering whether the individuals can be grouped into two groups, such that the sign of all ties reflects the partition, or equivalently in terms of the individual configuration of tryads. The failure of such a structure to be present in a signed social network is usually designated as conflict, and measuring the amount of conflict in a given signed social network is an open problem. In this work, I present a novel measure of conflict, based on algebraic graph theory, and considering a previously known variant of the Laplacian matrix for signed graphs. The talk will show both theoretical motivations for the measure, as well as an evaluation of its utility at signed social network analysis using multiple real-world signed social networks.

1. Institute for Web Science and Technologies · University of Koblenz-Landau, Germany
Algebraic Graph-theoretic
Measures of Conflict
Jérôme Kunegis
Based on work performed in collaboration with Christian Bauckhage, Andreas
Lommatzsch, Stephan Spiegel, Jürgen Lerner, Fariba Karimi and Christoph Carl Kling
Journée Graphes et Systèmes Sociaux (JGSS), March 18, 2016, Avignon

2. Algebraic Graph-theoretic Measures of Conflict 2Jérôme Kunegis
Social Network

3. Algebraic Graph-theoretic Measures of Conflict 3Jérôme Kunegis
http://slashdot.org/

4. Algebraic Graph-theoretic Measures of Conflict 4Jérôme Kunegis
The Slashdot Zoo
Slashdot Zoo: Tag users as friends and foes
Graph has two types of edges: friendship and enmity

5. Algebraic Graph-theoretic Measures of Conflict 5Jérôme Kunegis
Signed Directed Social Network
me
FriendFoe
FanFreak
The resulting graph is sparse,
square, asymmetric and has
signed edge weights.
You are the fan of your friends and the freak of your foes.

6. Algebraic Graph-theoretic Measures of Conflict 6Jérôme Kunegis
Wikipedia Edit Wars

7. Algebraic Graph-theoretic Measures of Conflict 7Jérôme Kunegis
Signed Bipartite Networks

8. Algebraic Graph-theoretic Measures of Conflict 8Jérôme Kunegis
Other Signed Networks
konect.uni-koblenz.de/networks

9. Algebraic Graph-theoretic Measures of Conflict 9Jérôme Kunegis
Polarization
 Slashdot: 23.9% of edges are negative
Polarization,
No conflict

10. Algebraic Graph-theoretic Measures of Conflict 10Jérôme Kunegis
Dyadic Conflict (Reciprocity of Valences)
Balance Conflict

11. Algebraic Graph-theoretic Measures of Conflict 11Jérôme Kunegis
Measuring Dyadic Conflict
C =₂
#conflictDyads
#totalDyads
C = 0₂

12. Algebraic Graph-theoretic Measures of Conflict 12Jérôme Kunegis
Tryadic Conflict (Balance Theory)
Balance
Conflict
(Harary 1953)

13. Algebraic Graph-theoretic Measures of Conflict 13Jérôme Kunegis
Measuring Tryadic Conflict
 Definition:
C =₃
#negTriangles
#totalTriangles
(cf. Signed relative clustering coefficient, Kunegis et al. 2009)
C = 0₃

14. Algebraic Graph-theoretic Measures of Conflict 14Jérôme Kunegis
Balance on Longer Cycles
 Equivalent definitions: a graph is balanced when
(a) all cycles contain an even number of negative edges
(b) its nodes can be partitioned into two groups such
that all positive edges are within each group, and all
negative edges connect the two groups

15. Algebraic Graph-theoretic Measures of Conflict 15Jérôme Kunegis
Frustration
 Definition: The minimum number of edges f that
have to be removed from a signed graph to make
the graph balanced.
Example: f = 1

16. Algebraic Graph-theoretic Measures of Conflict 16Jérôme Kunegis
Frustration (partitioning view)
 Definition: minimum number of edges that are
frustrated (i.e., inconsistent with balance) given any
partition of the graph's nodes into two groups.
Example:
f = 1

17. Algebraic Graph-theoretic Measures of Conflict 17Jérôme Kunegis
Frustration: Computation
 Computation of frustration is equivalent to
MAX-2-XORSAT
max (a XOR b) + (¬a XOR c) + (b XOR ¬d) ⋯
 MAX-2-XORSAT is NP complete
 Solution: Relax the problem
see overview in (Facchetti & al. 2011)
a
b
c
d

18. Algebraic Graph-theoretic Measures of Conflict 18Jérôme Kunegis
Algebraic Formulation
 Let G = (V, E, σ) be a signed graph. σuv = ±1 is the
sign of edge (uv).
 Given a partition V = S T, let x be the characteristic
node-vector:
 Number of frustrated edges: Σ (xu − σuv xv)²
xu
=
+1/2 when u ∈ S
−1/2 when u ∈ T
uv∈E
∪

19. Algebraic Graph-theoretic Measures of Conflict 19Jérôme Kunegis
Frustration as Minimization
 f is given by the solution to:
f = min Σ (xu − σuv xv)²
s.t. x ∈ {±1/2}V
Σ xu² = |V| / 4
⇔ ||x|| = |V| / 2
uv∈Ex
u
Relaxation
*

20. Algebraic Graph-theoretic Measures of Conflict 20Jérôme Kunegis
Using Matrices
 The quadratic form can be expressed using matrices
Σ (xu − σuv xv)² = xT L x
where L ∈ ℝV × V is the matrix given by
Luv = −σuv when (uv) is an edge
Luu = d(u) is the degree of node u
 L = D − A is the signed graph Laplacian
uv∈E
1
2

21. Algebraic Graph-theoretic Measures of Conflict 21Jérôme Kunegis
Minimizing Quadratic Forms
f* = min xT L x
s.t. ||x|| = |V| / 2
xT L x
xT x
x
f* = min
x
1
2
8
|V|
Rayleigh
quotient
min-max
theorem
f* = λmin
[L]8
|V|
f* = λmin
[L]
|V|
8

22. Algebraic Graph-theoretic Measures of Conflict 22Jérôme Kunegis
Relative Relaxed Frustration
 Definition: Proportion of edges that have to be removed
to make the graph balanced
F* =
F* =
0 ≤ F* ≤ ≤ 1
λmin[L] ≤
f*
|E|
|V|
8 |E|
λmin
[L]
f
|E|
8 |E|
|V|

23. Algebraic Graph-theoretic Measures of Conflict 23Jérôme Kunegis
Properties of L (Unsigned Graphs)
 L is positive-semidefinite (all λ[L] ≥ 0)
 Multiplicity of λ = 0 equals number of connected
components
 Smallest eigenvalue measures conflict
 Second-smallest eigenvalue measures connectivity
(“algebraic connectivity”)

24. Algebraic Graph-theoretic Measures of Conflict 24Jérôme Kunegis
Properties of L (Signed Graphs)
L = Σ L(uv)
L(uv)
= [ ]when σuv = +1
L(uv)
= [ ]when σuv = –1
uv∈E
1 –1
–1 1
1 1
1 1

25. Algebraic Graph-theoretic Measures of Conflict 25Jérôme Kunegis
Minimal Eigenvalue of L (Signed Graphs)
 L is positive-semidefinite (all eigenvalues ≥ 0)
 L is positive-definite (all eigenvalues > 0) iff all
connected components are unlabanced
– Proof “ ”: by equivalence by all-positive graph⇒
– Proof “ ”: by contradiction (eigenvector would be all-zero)⇐

26. Algebraic Graph-theoretic Measures of Conflict 26Jérôme Kunegis
What Unsigned L can Do

27. Algebraic Graph-theoretic Measures of Conflict 27Jérôme Kunegis
What Signed L Can Do

28. Algebraic Graph-theoretic Measures of Conflict 28Jérôme Kunegis
Computing λmin
[L]
 Sparse LU decomposition + inverse power iteration:
O(|V|²) memory, but then very fast
% Matlab pseudocode
[ X Y ] = sparse_lu(L);
[ U D ] = eigs(@(x)(Y X x), k, 'sm');

29. Algebraic Graph-theoretic Measures of Conflict 29Jérôme Kunegis
Temporal Analysis of C₂
Epinions ratings
Wikipedia elections

30. Algebraic Graph-theoretic Measures of Conflict 30Jérôme Kunegis
Temporal Analysis of C₃
Epinions ratings
Wikipedia elections

31. Algebraic Graph-theoretic Measures of Conflict 31Jérôme Kunegis
F* over Time
MovieLens 100k
Wikipedia elections

32. Algebraic Graph-theoretic Measures of Conflict 32Jérôme Kunegis
Cross-Dataset Comparison of F*
For larger networks, a smaller
proportion of edges must be removed
to make it balanced

33. Algebraic Graph-theoretic Measures of Conflict 33Jérôme Kunegis
Merci
konect.uni-koblenz.de
Contact: Jérôme Kunegis
Université Koblenz-Landau
<kunegis@uni-koblenz.de> @kunegis
We want more datasets with negative edges, and
timestamps.
 In particular: with changing and/or dissapearing edges