1) The document discusses an algebraic approach to measuring conflict in social networks that contain both positive (friendship) and negative (enmity) relationships. 2) It describes using a signed graph Laplacian matrix L to represent the network and formulate the measurement of conflict (frustration) as the minimization of a quadratic form related to L. 3) Minimizing this quadratic form is equivalent to computing the minimum eigenvalue of L, which provides a measure of relative relaxed frustration F* that indicates the proportion of edges that would need to be removed to make the network balanced.

1. naXys – Namur Centre for Complex Networks – Univ. of Namur
Measuring the Conflict in a Social
Network with Friends and Foes:
A Recent Algebraic Approach
Jérôme KUNEGIS
Based on work performed in collaboration with Christian Bauckhage, Andreas Lommatzsch,
Stephan Spiegel, Jürgen Lerner, Fariba Karimi and Christoph Carl Kling

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Social Network

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http://slashdot.org/

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The Slashdot Zoo
Slashdot Zoo: Tag users as friends and foes
You are the fan of your friends and the freak of your foes.
Graph has two types of edges: friendship and enmity
me
FriendFoe
FanFreak
The resulting graph is sparse,
square, asymmetric and has
signed edge weights.

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Wikipedia Edit Wars

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Signed Bipartite Networks

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Other Signed Networks
konect.uni-koblenz.de/networks

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Polarization
 Slashdot: 23.9% of edges are negative
Polarization,
No conflict

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Dyadic Conflict (Reciprocity of Valences)
Balance Conflict

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Measuring Dyadic Conflict
C =₂
#conflictTriads
#totalTriads
C = 0₂

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Tryadic Conflict (Balance Theory)
Balance
Conflict
(Harary 1953)

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Measuring Tryadic Conflict
 Definition:
C =₃
#negTriangles
#totalTriangles
(cf. Signed relative clustering coefficient, Kunegis et al. 2009)
C = 0₃

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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

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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

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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

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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

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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
∪

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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
*

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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

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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
theoremf* = λmin
[L]8
|V|
f* = λmin
[L]
|V|
8

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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|

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Properties of L (Unsigned Graphs)
 L is positive-semidefinite (all λ[L] ≥ 0)
 Multiplicity of λ = 0 equals number of connected
components

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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

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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)⇐

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What Unsigned L can Do

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What Signed L Can Do

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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');

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Temporal Analysis of C₂
Epinions ratings
Wikipedia elections

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Temporal Analysis of C₃
Epinions ratings
Wikipedia elections

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F* over Time
MovieLens 100k
Wikipedia elections

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Cross-Dataset Comparison of F*
For larger networks, a smaller
proportion of edges must be removed
to make it balanced