Opinion Formation under Bounded Confidence via Gossip Algorithms and Applications

Transcription

1 Opinion Formation under Bounded Confidence via Gossip Algorithms and Applications Nguyen Thi Hoai Linh 1, Takayuki Wada 1, Izumi Masubuchi 2, Toru Asai 3, Yasumasa Fujisaki 1 1 Osaka University, 2 Kobe University, 3 Nagoya University This work is supported by CREST, JST 1 Motivation & Purpose 2 Constant Bounded Confidence Algorithm 3 Clustering Convergence Analysis 4 Numerical simulations 5 Increasing bounded confidence algorithm 6 Applications

2 1. Motivation & Purpose Motivation: Public opinion formation Purpose: Study dynamics of opinion forming in social networks Considering Problem We consider a network consisting of n agents. At each time instant agents hold continuous opinions about some issue: x i (k), i = 1, 2,..., n, k N. How agents adjust their opinions while interacting with each other? Which dynamics can results in different formation: Consensus, partial consensus, polarization, fragmentation?

3 Literature review Variety of pattern formation Stubborn Agents Bounded confidence algorithms Hegselmann-Krause model Deffuant-Weisbuch model 1 x i (k + 1) = N i (k) x j(k), i V, N i (k, d) = {j V : x i (k) x j (k) d} Chose (i, j) at k, x i (k + 1) = αx i (k) + (1 α)x j (k) x j (k + 1) = αx j (k) + (1 α)x i (k) if x i (k) x j (k) d.

4 2. Constant Bounded Confidence Algorithm n-agent network,v={1,2,...,n} x i (0) R, i V : initial opinion, d: confidence threshold k = 0 k = k + 1 Choose (i, j) with p = 2/(n(n 1)) x i (k + 1) = x j (k + 1) = x i (k) + x j (k) 2 x l (k + 1) = x l (k), l V \{i, j} yes x i (k) x j (k) d? no x l (k + 1) = x l (k), l V

5 2. Constant Bounded Confidence Algorithm Algorithm in compact form Here x(k + 1) = W (i(k), j(k), x(k))x(k). (1) x(k) = [x 1 (k) x 2 (k) x n (k)] T { W ij, if x i (k) x j (k) d, W (i, j, x(k)) = I otherwise. W ij = I 1 2 (e i e j )(e i e j ) T.

6 3. Clustering Convergence Lemma 1 (Equilibrium point) A vector x is an equilibrium point of (1), i.e., x = W (i, j, x )x, i, j V (2) if and only if it has the form xi = xj or xi xj > d i, j V. (3) Lemma 3 Given ε 0 = d/(n 1), P(Ω) = 1, Ω = { k k 0 s. t. i, j V, k 0 N x i ( k) x j ( k) ε 0 or x i ( k) x j ( k) > d}.

7 3. Clustering Convergence For each x(k), we equip the network with a graph G(x(k)) = (V, E(k)), where (i, j) E(k) x i (k) x j (k) ε 0. Definition 1. (d-clusters) The clusters V 1 (k), V 2 (k),..., V Gd (k)(k)(k) induced from connected components of G(x(k)) are d-clusters if (i) dia V p (k) := (ii) dist(v p (k) V q (k)) := d p q. max x i(k) x j (k) d i,j V p(k) p, and min x i(k) x j (k) > i V p(k),j V q(k)

8 3. Clustering Convergence Theorem 4. For any initail opinion profile and a given confidence threshold, with probability one, the opinion profile is partitioned into d-clusters within finite step. Theorem 5. Consider d-cluster V 1 (0) with V 1 (0) d. The constant bounded confidence algorithm drives the opinions of all agents to their average (1 T x(0))/m almost surely.

9 3. Clustering Convergence Definition 2. (Clustering convergence) Given a parameter d > 0. An algorithm given by x(k + 1) = f (x(k)) for k = 0, 1, 2,... is said to achieve d-clustering convergence if starting from any initial state, the system eventually converges to some state x of form (3), i.e., lim x(k) = x. k Theorem 6 Given a confidence threshold d. For any initial opinion profile, Algorithm1 achieves d-clustering convergence almost surely. Theorem 8. (Upper bound of clusters) Given x(0) [0, 1] n, c {1, 2,..., n}, if d 1/c, then for any sample path of choosing interaction pairs, the number of clusters formed is at most c.

10 4. Numerical Simulation n = 100, x = i(0) (0, 1) uniformly Opinion 0.5 Opinion k k 10 3 d = 0.14 d = 1

11 5. Increasing Bounded Confidence Algorithm n-agent network, V = {1, 2,..., n} c : desired number of clusters, x = x(0), d = 0, δd: threshold s increment, κ: maximum number of steps Superviser calculate c(x ): number of clusters in x c(x ) > c? no Stop yes x = x(κ) Constant bounded confident algorithm with x(0) = x, d = d + δd, κ

12 5. Increasing Bounded Confidence Algorithm Theorem 10. c(l), which is the number of clusters in x (l), is a non-increasing function of l with high probability. Example δd = 0.001, c = 5. Final confidence threshold d = Opinion k x 10 5

13 6. Applications We consider the applications of the algorithms in the problem of clustering multi-dimensional data and intrusion dectection. Clustering Problems Given a set of data points, usually in high-dimensional space, partition them into clusters so that: Points within each cluster a similar to each other. 1 Points from different clusters 0 are dissimilar

14 6. Applications n data objects each of which has d attributes Multi-agent system V = {1, 2,..., n}, x i (0) R d X (k) := [x 1 (k) x 2 (k) x n (k)] T R n,d { I 1 W (i, j, X (k)) := 2 (e i e j )(e i e j ) T, if X i (k) X j (k) δ, I otherwise.

15 6. Applications Constant confidence threshold Increasing confidence threshold Core for clustering-based distributed outlier/anomaly detection in power distrubution systems.

16 7. Conclusions We have studied two algorithms for the dynamics of opinion formation and given a rigorous convergence analysis. The algorithms are distributed asynchronous gossip The algorithms express the veriety of opinion formation: consensus, clustering, fragmentation which are observed frequently in social network. have potential to be used as cores of clustering-based distributed outlier/anomaly detection protocol in networks or computer systems, such as power distribution systems.