Max-Plus Opinion Dynamics With Temporal Confidence
MMax-Plus Opinion Dynamics With Temporal Confidence
Daniel Feinstein and Ebrahim Patel University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG [email protected]
Often in the setting of human-based interactions, the existence of a temporal hierarchyof information plays an important role in diffusion and opinion dynamics within com-munities [1]. For example at the individual agent level, more recently acquired infor-mation may exert greater influence during decision-making processes [2]. To facilitatefurther exploration of this effect, we introduce an efficient method for modelling tempo-rally asynchronous opinion updates, where the timing of updates depends on the timingof incoming opinion states received from neighbours. The framework enables the in-troduction of information arrival-time lag by means of lag-vectors . These are used toweight the relevance of information received by agents, based on the delay betweenits receipt and the subsequent opinion update. The temporal dynamics (i.e. the timesat which information is transmitted) are governed by an underlying algebraic structurecalled max-plus algebra ([3], [4]). We investigate the resulting continuous opinion dy-namics under the max-plus regime using a modified Hegselmann-Krause model [5],replacing the conventional confidence-interval based on the distance between opinionswith one based instead on the recency of information received by agents.Our model works as follows: at time-step k =
0, each agent (represented by a nodein a network) is assigned an initial opinion from the interval [ , ] uniformly at randomand transmits this value to all neighbours in the network. If an edge from agent i to j exists, the information leaving agent i arrives at j after A ji time units (e.g. minutes),where A is the max-plus adjacency matrix (which is nothing more than the transposeof the conventional adjacency matrix). After sending their current opinion, each agententers a dormant period where it waits to receive all incoming opinion values. Oncereceived, agents update their opinions before immediately re-sending their new valuesto all neighbours (possibly at different times), and this process continues for a desiredduration.The event-times (the times at which opinion updates occur) can be conveniently mod-elled using the max-plus algebra which we denote R ∞ . Let (cid:126) x ( k ) ∈ R ( n × ) ∞ denote thevector of the ( k + ) st time agents communicate their opinions. Then (cid:126) x i ( k ) is the timeof the ( k + ) st transmission of agent i and is defined, in line with the above description,by: (cid:126) x i ( k ) = max { (cid:126) x j ( k − ) + A i j : j = , . . . n } , for all k ≥ . (1)In the notation of the max-plus algebra this becomes, (cid:126) x ( k ) = A ⊗ (cid:126) x ( k − ) , for all k ≥ . (2) a r X i v : . [ phy s i c s . s o c - ph ] J a n e also conveniently model the number of time units agent i has been sitting on theopinion value received from agent j before its next update: (cid:126) ξ ( k , i ) = (cid:126) x i ( k ) (cid:126) I − (cid:126) x ( k − ) − A Ti (3)where (cid:126) I is the ( n × ) unit column vector and A Ti is the transposed i th row of A . We referto the vector above as the lag-vector for opinions arriving at agent i , having been sentneighbours at time-step ( k − ) .To simulate the resulting opinion dynamics, we modify the Hegselmann-Krause model[5] (which from here, we refer to just as the HK model) to incorporate the lag-vectors.At each time-step ( k + ) , every agent i updates their opinion according the followingupdate-rule: o i ( k + ) = (cid:12)(cid:12) N (cid:0) i , k (cid:1)(cid:12)(cid:12) − ∑ j ∈ N (cid:0) i , k (cid:1) o j ( k ) , (4)where N ( i , k ) = { ≤ j ≤ n (cid:12)(cid:12) ≤ (cid:126) ξ j ( i , k ) ≤ ε } , i.e. the set of i (cid:48) s neighbours whoseopinion values are sat on by i for at most ε time units. Note the standard, unmodi-fied HK model update-rule is given by replacing N ( i , k ) with M (cid:0) i , k (cid:1) = { ≤ j ≤ n (cid:12)(cid:12) | o i ( k ) − o j ( k ) | ≤ ε } .To summarize the entire process for each time-step: each agent waits until it has re-ceived all incoming information (modelled by equation 2). On receipt of the final in-coming opinion value for the current time-step, agents update their opinion using themodified HK update-rule (equation 4) and send this to all neighbours. We show via extensive computational simulations that the updated HK model (usingthe temporally bounded confidence-interval in equation 4) supports multi-opinion con-sensus clusters despite the absence of the conventional confidence-interval based on thedistance between neighbouring opinions (Fig 1). This is significant because it demon-strates that opinion fragmentation is possible even with seemingly innocent sorting ofcontent based only on recency considerations. Simulations are carried out on randomweighted strongly-connected and directed Barab´asi–Albert, Erd˝os–R´enyi and Watts-Strogatz graphs consisting in each case of 100 nodes. Furthermore, we examine typicalbehaviours emerging from varying the threshold beyond which agents fail to take opin-ions from their neighbours into account, i.e. from varying ε , while keeping all otherinitial conditions fixed. Two noticeably distinct regimes emerge. The first is a gradualtransition from multiple consensus clusters to a single global consensus. As epsilon isincreased, the number of consensus clusters grows slowly, until ε becomes large enoughfor a single global consensus to form. The second regime is of a more abrupt and dis-crete change. The number of consensus clusters remains fixed before a sudden transitionoccurs beyond some critical value of ε <
1, where multiple opinion clusters suddenlycollapse into global consensus. ig. 1.
Multi-opinion consensus clus-ters emerging using the modified HKupdate-rule with ε =
1. The sim-ulation was carried on a randomweighted strongly-connected directedBarab´asi–Albert graph with 100 nodes,Barab´asi–Albert parameter of 2, andedge-weights drawn uniformly at ran-dom from { ,..., } . Fig. 2.
Oscillatory behaviour of opin-ions emerging using the modified HKupdate-rule with ε =