Subsuming Complex Networks by Node Walks

TL;DR

This work introduces node_walks as a dynamic process where a walking node subsumes adjacent nodes, progressively transforming the network topology and yielding a single-node limit. Dynamics are characterized by the walking-node strength $s_t$ and the accumulated strength $S=\sum_{t=0}^{N-1} s_t$, studied across Erdős–Rényi, Barabási–Albert, and geometric networks under uniformly random, largest-degree, and smallest-degree neighbor selection. Key findings show that the subsumption signatures are highly sensitive to both topology and walk rule, with BA networks and largest-degree preferences often producing larger $S$, while GEO networks exhibit greater dispersion due to borders; correlations between initial topology (e.g., degree $k$) and $S$ are present but not universally predictive. These results demonstrate a topology-dynamics coupling via node_walks and suggest avenues for modeling mergers, hierarchical growth, and dynamic-topological co-evolution in complex networks.

Abstract

The concept of node walk in graphs and complex networks has been addressed, consisting of one or more nodes that move into adjacent nodes, henceforth incorporating the respective connections. This type of dynamics is then applied to subsume complex networks. Three types of networks (Erdós- Rény, Barabási-Albert, as well as a geometric model) are considered, while three node walks heuristics (uniformly random, largest degree, and smallest degree) are taken into account. Several interesting results are obtained and described, including the identification that the subsuming dynamics depend strongly on both the specific topology of the networks as well as the criteria controlling the node walks. The use of node walks as a model for studying the relationship between network topology and dynamics is motivated by this result. In addition, relatively high correlations between the initial node degree and the accumulated strength of the walking node were observed for some combinations of network types and dynamic rules, allowing some of the properties of the subsumption to be roughly predicted from the initial topology around the waking node which has been found, however, not to be enough for full determination of the subsumption dynamics. Another interesting result regards the quite distinct signatures (along the iterations) of walking node strengths obtained for the several considered combinations of network type and subsumption rules.

Figures (7)

• Figure 1: The subsumption of a graph performed along iterations $t=0, 1, 2, 3, 4$ respectively to walking node 2 (shown with red border) and choosing the next node (in green) in a uniformly random manner. At each iteration, one of the neighbors of the walking node is chosen with uniform probability and incorporated into the walking node, which inherits the respective links. Subsumed links have their links superimposed. For instance, the incorporation of node 3 into the walking node 2 leads to the links to nodes 4 and 5 to have increased from 1 to 2. The strength $s$ of the walking node, corresponding to the sum of the weights of the respective links, is also shown for each iteration.
• Figure 2: The strengths of the walking node in the example presented in Fig. \ref{['fig:walk']} are shown in terms of the iteration $t$. The accumulated strength $S=16$, corresponding to the sum of the strength of the walking node for $t=0, 1, 2, 3, 4$, is also presented. The value of $S$ indicates the interaction between the walking node and its first neighbors accumulated along the complete subsumption dynamics.
• Figure 3: The set of signatures $s_t$ obtained for each of the nodes in ER, BA, and GEO networks under the three considered types of dynamics. The whole set of signatures in case are shown in random colors. The average $\pm$ standard deviation signatures in each case are shown in red. Mostly distinct shapes and dispersions have been obtained.
• Figure 4: The larger dispersion of the accumulated strength $S$ obtained in the case of the considered GEO networks is a consequence of the marked variation of node accessibility travenccolo2009bordertravenccolo2008accessibilityviana2010characterizingbenatti2022complex and the presence of borders characterizing this type of network. The plate (a) shows a geographical network with $N=100$ nodes and its border nodes (in red) identified as the 20 nodes with the smallest node accessibility considering $h=3$ hierarchical levels. The signatures of accumulated walking node strengths obtained for uniform random choice of neighbors are shown in (b), with the colors being respective to that of the nodes in (a). The signatures obtained for the border nodes tend to present the smallest values of $S$.
• Figure 5: Scatterplots showing the relationship between the initial degree $k$ of the walking node and its total strength $S$. The Pearson correlation coefficients are respectively indicated. Three cases -- namely (c), (e), (f), and (h) -- resulted in larger correlation values. Observe, however, that the largest correlation does not correspond to a well-defined linear relationship.