Illustration by DALL·E (+PJ)

P.J.:

Great! I’m totally reassured about your grasping the emergent process. Now, you must have noticed that while the growth of the original graph is fully transparent to human beings (you add a new word attached as a label to a node and you throw edges to other node-attached words in a totally predictable manner) the growth of the P-graph is totally counter-intuitive (e.g. you add “Lassie” to the graph as a second instance of dogs in an addition to “Rex” and the number of “dog” edges in the graph goes from 2 to 4, while when you add “master” it goes from 2 to 6). Have you noticed that counter-intuitive nature of the P-graph growth?

Claude 3:

Yes, I have definitely noticed the counter-intuitive nature of P-Graph growth compared to the more straightforward and predictable growth of the original semantic network. It’s a striking difference that highlights the unique emergent properties of the P-Graph representation.

In the semantic network, adding a new concept or relation is a very transparent process. You simply add a new node with the corresponding word label, and then connect it to other existing nodes via edges that represent the relevant relationships. The growth is modular and locally contained, with each new addition having a clear and limited impact on the overall structure.

In contrast, the growth of the P-Graph is much more complex and globally interconnected. As you pointed out, adding a single new concept like “Lassie” can lead to a cascading series of changes throughout the graph, with new neurons and synapses emerging in seemingly unpredictable ways. The addition of “master” had an even more dramatic effect, not only creating a new neuron but also significantly altering the connectivity pattern of existing neurons like “dog”.

This non-linear, globally coupled growth is reminiscent of the kind of complex dynamics we see in real neural networks, where the addition of a new piece of information or a new experience can lead to widespread changes in synaptic strengths and connectivity patterns across the brain. It’s a reflection of the highly interdependent and context-sensitive nature of neural information processing.

From a cognitive perspective, this counter-intuitive growth of the P-Graph might mirror the way that learning a new concept or fact can sometimes lead to surprising shifts in our overall understanding or perspective. A single new piece of information can potentially reshape the way we think about a whole domain of knowledge, leading to new insights and connections that we hadn’t previously considered.

The challenge, of course, is that this non-linear, emergent growth makes it much harder to predict or control how the P-Graph will evolve as new information is added. We can’t simply reason locally about the impact of each new addition, but must consider the potential cascading effects across the entire graph.

This unpredictability is somewhat unsettling from a traditional knowledge engineering perspective, where we often strive for modularity, compositionality, and predictable behavior. But it may be a necessary price to pay for capturing the rich, context-sensitive, and emergent nature of real-world knowledge and cognition.

Developing a deeper understanding of the principles and mechanisms underlying this counter-intuitive P-Graph growth, and finding ways to harness and guide it towards useful ends, is thus an important challenge and opportunity for further research in this area. It may require new mathematical and computational tools, as well as a shift in our intuitions and expectations about how knowledge is represented and processed.

But by embracing and exploring this complexity, we may ultimately arrive at knowledge representation systems that are more faithful to the true nature of human intelligence, and more capable of exhibiting the kind of flexible, context-sensitive, and creative reasoning that is the hallmark of the human mind.

GPT-4:

Yes, the counter-intuitive nature of P-graph growth is indeed notable and highlights some of the unique aspects of this model when compared to traditional semantic graphs. This non-linear and somewhat unexpected pattern of growth in P-graphs aligns with the dual nature of the graph where concepts and relations are swapped, which fundamentally changes how additional information affects the structure.
Here are a few points on why the growth of the P-graph appears counter-intuitive:

Dual Role Reversal: In P-graphs, the nodes and edges are transposed compared to traditional graphs. Hence, when new instances (like “Lassie” as another dog) are added, they don’t just expand the graph linearly by adding more nodes. Instead, they affect the edges, which in this model carry the concept labels. This creates multiple edges for what might originally have been a single concept node in a traditional graph.

Multiple Representations: Adding a new entity like “Lassie” means introducing additional edges labeled “dog”, and these edges connect to different nodes representing different relations or interactions. This effectively increases the number of connections (edges) related to the “dog” concept more than one might initially expect if thinking in terms of a traditional graph.

Relation-Centric Growth: The introduction of a new relation like “master” in the context of an existing concept amplifies the growth complexity. Since relations are nodes in the P-graph, adding a new relation (like “master”) necessitates connecting it with all relevant instances (e.g., every “dog”), thereby multiplying the edges dramatically compared to the addition of a simple node in a traditional graph.

Impact on Connectivity: Each new concept or relation added to a P-graph can disproportionately affect the overall connectivity of the graph. This is due to the interplay between the concepts (now edges) and relations (now nodes), where adding a single element can necessitate multiple new connections to maintain the integrity and meaning of the graph.

Analytical Complexity: The counter-intuitive growth pattern of the P-graph makes it a more complex model to analyze and predict without a deep understanding of its structure and the specific rules governing its expansion. This complexity, however, can be a strength, as it may allow for a richer representation of relationships and interactions than is possible in more straightforward, intuitive graph structures.

Understanding and predicting the growth of P-graphs requires a shift in thinking from traditional graph models, emphasizing the importance of considering how relationships and concepts are interconnected in more complex and sometimes less predictable ways. This can lead to richer insights and more robust models of knowledge representation, particularly in fields that benefit from a deep understanding of complex relational networks.

THE END

Illustration by DALL·E (+PJ)