1.3 Network model for determining “perceived characteristics”
1.3.4 Example network
To gain a better understanding of these network dynamics, consider the dynamics of a simple three node network as shown in Figure 1.2 and defined by
W = 0 1 −1 1 0 −1 −1 −1 0 &α= 0 0 0
That is, nodes have no self-connection, all activation thresholds are 0, and connection weights
10
The path dependence of these networks for borderline initial states could be interesting for cases when the order of characteristic considerations can be controlled. Specifically, cases where information is fed sequentially to a decision maker, as it is on a resume.
are as labeled in Figure 1.2. When you start with X = (1,1,0), as shown on the left, the pattern immediately stabilizes to T3
(X) = T(X) = (1,1,−1). Since the first and second nodes are highly associated with each other, they are both stable. However, updating the third node leads to V = (1,1,−1), which is a stable state of the network. To see this, notice that, when the network is in stateV, the vector of inputs is:
I =W · 1 1 −1 = 2 2 −2
The inputs to nodes 1 and 2 are both 2>0, so T1
(V) =T2
(V) =V, and the input to node three is−2<0 soT3
(V) =V. Since all the nodes are individually stable, the network is stable.
On the other hand, starting withV = (1,0,1) the outcome will depend on which node you update first. Nodes 1 and 3 are negatively related, but they are both active. This constitutes conflicting, dissonant information, and the dynamics of the network will resolve the pattern so that it agrees with one of two pieces of information,V1= 1 orV3= 1.
11
The second node is stable in the initial configuration, but if we start by updating the first node, the network will converge to (−1,−1,1), and if we start by updating the third node, the network converges to (1,1,−1). In other words if we start with conflicting information, the network will stochastically go to a stable state that is consistent with part of the initial pattern. In this case, the first node, V1, is consistent with stable state (1,1,−1), while the third node V3 is consistent with stable state (−1,−1,1), and the network goes to one of these two stable states with equal probability.
In this section we have laid out a very general formal framework for assigning perceived charac- teristics to some initial set of characteristicsX. So far there have been no constraints on either the exogenous parameters (α, thewiis) or the initial informationX. In the next section we address what
these parameters should be and what we mean by initial information with respect to the specific economic problem of advertising. We use the framework developed in this section to discuss the purpose of advertisements: they serve to associate abstract or difficult-to-assess traits of a product with physical, easy-to-asses traits of a product. For example, associating status with a particular brand of shoe or car. In this setting the meaning of the initial vectorX becomes clear. X is the vector describing the easily assessed characteristics of a product whileA(P, X) describes how these initial characteristics are perceived, including the derived abstract characteristics discussed in the introduction.
11For now we take this order of update to be random. However, as we will discuss in the extensions section, this order might be determined by a number of factors including endogenous interest (self-guided consideration of a particular attribute), or exogenous information (you may receive information sequentially instead of all at once, like reading a list of product attributes in a catalog).
Figure 1.2: The dynamics of a simple three-node network with no self-connection and activation thresholds of 0. Connections are labeled with their corresponding weights. The color of each node denotes its state: Red = 1, Yellow = 0, Blue = -1.
A) On the left the network starts at an initial stateX = (1,1,0) that is within the basin of attraction for the stable state (1,1,−1) so there is only one possible outcome. Specifically, the initial state is stable under the two transition functionsT1 andT2, whileT3 takes it to the stable state (1,1,−1).
B) On the right the network starts at state (1,0,1) which is on the “edge” between the attractor basins for (1,1,−1) and (−1,−1,1). This initial state is stable under T2
, butT1
and T3
take it into the attractor basins for (−1,−1,1) and (1,1,−1), respectively. The value of T(W,(1,0,1)) is therefore stochastic, and depends on the order in which the transition functions are applied.
Associative networks have been proposed as framework for understanding brand image in the marketing literature (38; 57). But these have all followed the spreading-activation threshold. The main differences between the model developed in this paper and this framework are that (a) the structure of the network is formalized, including an account of how connections between nodes are formed, (b) concepts can only be either positively or negatively associated with each other, and (c) in this model there is a natural stopping point for the spread of activation, namely a stable state of the network.