From the perspective of network attackers, finding attack sequences that can make significant damage on network controllability is an important task, which also helps defenders improve the robustness during network constructions. Therefore, developing effective attack strategies is a key aspect of research on network controllability and its robustness. In this paper, we propose a Leaf Node Neighbor-based Attack (LNNA) strategy that can effectively disrupt the controllability of undirected networks. The LNNA strategy targets the neighbors of leaf nodes, and when there are no leaf nodes in the network, the strategy attacks the neighbors of nodes with a higher degree to produce the leaf nodes. Results from simulations on synthetic and real-world networks demonstrate the effectiveness of the proposed method. In particular, our findings suggest that removing neighbors of low-degree nodes (i.e., nodes with degree 1 or 2) can significantly reduce the controllability robustness of networks. Thus, protecting such low-degree nodes and their neighbors during network construction can lead to networks with improved controllability robustness.

It has recently been shown that, contrary to the wide belief that a shift-enabled condition (necessary for any shift-invariant filter to be representable by a graph shift matrix) can be ignored because any non-shift-enabled matrix can be converted to a shift-enabled matrix, such a conversion in general may not hold for a directed graph with non-symmetric shift matrix. This paper extends this prior work, focusing on undirected graphs where the shift matrix is generally symmetric. We show that while, in this case, the shift matrix can be converted to satisfy the original shift-enabled condition, the converted matrix is not associated with the original graph, that is, it does not capture anymore the structure of the graph signal. We show via a counterexample, that a non-shift-enabled matrix cannot be converted to a shift-enabled one and still maintain the topological structure of the underlying graph, which is necessary to facilitate localized signal processing.

This article describes the transition of an undirected active network, used for numerical and logical analysis, and made up of communicating operators and nodes, and being self-extensible, into a network of semantic structures, and the extensions necessary to handle language. The network, called Active Structure, was infused with knowledge in the form of tens of thousands of definitions taken from a human-readable dictionary. Where the dictionary fell short, from circularity, lack of a hierarchical form for its senses and sub senses, or inadequate detail around verbs, the entries were extensively curated. The result is a machine which has access to every meaning of every word in its vocabulary, and needs to manage the right meaning for every use of a word, a task that our unconscious minds now handle. To demonstrate that such a network can aspire to human-level tasks, we have begun loading an Anti-Money Laundering Act so that the machine can devise and test strategies against money launderers. There is a higher goal being aimed at, the breaking of the Four Pieces Limit. That is, the network is not intended to save money on human cognitive labor, but to raise an absurdly low limit on it.