Space-Time Network Computation

Abstract

This paper proposes a novel network computation model termed "Space-Time Network Computation" (STNC). The core claim is that integrating spatial and temporal dimensions offers a more efficient approach to processing data exhibiting spatial and temporal dependencies. STNC leverages an extension of Graph Neural Networks (GNNs) by introducing the concepts of time streams and spatial streams. These streams are utilized within a graph structure to simulate and compute spatial-temporal interactions. The fundamental innovation lies in overcoming the dimensional limitations of traditional network computations, providing a new paradigm for handling complex spatial-temporal data. This approach utilizes graph structures to represent relationships, incorporating temporal evolution through time streams and spatial relationships through spatial streams. The key mathematical formulation is presented as follows: Let *G* = (*V*, *E*) be a graph, where *V* represents the set of nodes (vertices) and *E* represents the set of edges. Each node *vi* ∈ *V* is associated with a state vector *si* ∈ ℝd, where *d* is the dimension of the state vector. Each edge *eij* ∈ *E* is characterized by a weight *wij* ≥ 0. A *time stream* τ is defined as a sequence of state vectors: τ = (*si(0)*, *si(1)*, *si(2)*, ..., *si(T)*) , where *si(t)* is the state vector of node *i* at time *t*. A *spatial stream* σ is defined as a set of state vectors representing the spatial neighborhood of a node: σ(*i*) = {*sj(t)* | (*j*, *i*) ∈ *E*} for each node *i*. The core computation within STNC involves updating node states based on their time streams and spatial streams. The update rule can be expressed as: *si(t+1)* = f(*si(t)*, τi, σ(*i*)) where *f* is an update function, τi is the time stream associated with node *i*, and σ(*i*) is the spatial stream for node *i*. This update function can be a neural network or a more traditional rule-based system. The goal is to iteratively evolve the state vectors of nodes over time, capturing the influence of their neighbors and the temporal dynamics. The effectiveness of STNC is predicated on the efficient representation and manipulation of these time streams and spatial streams within the graph structure. The choice of *f* is crucial and will be explored in subsequent sections. The proposed model offers a flexible framework capable of accommodating diverse spatial-temporal data and associated dynamics. Further research will focus on optimizing the update function *f* and exploring its applicability to various domains.