[…] a Walk can be imagined as an actual walk of a Traveler along the edges in a diagrammatic representation of the Graph under consideration. The traveler always walks along an Edge from one end-vertex to the other. Suppose now that we allow the traveler to change his mind when coming to the midpoint of an edge: instead of continuing along the edge towards the other end-vertex, he could return to the initial end-vertex and continue as he wishes. Then the basic constituent of a walk is no longer an edge; rather we could speak of a Walk as a Sequence of Semi-Edges. Such walks could be called semi-edge walks. A semi-edge in a walk could be followed by the other semi-edge of the same edge (thus completing the edge) or by the same semi-edge in which case the traveler returns to the vertex at which he started. A formal definition of a semi-edge walk is obtained from the above definition of a walk by deleting the word “distinct" from the description of end-vertices. Hence we have the following definition.
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Both standard and semi-edge walks can be considered as Random Walks. […]
Based on these observations, we could suggest to try to create models for Virus Propagation and the Spread of Knowledge in which the adjacency matrix would be replaced by the signless Laplacian. In such models desirable graphs for anti-virus protection would be those with small q1 and for R&D networks those with large q1. We believe that there are situations in which viruses or Knowledge move along lazy random walks rather than along standard random walks. This can be expected in situations when the vertices when receiving something from their Neighbors are likely to respond back with some action.
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CVETKOVIĆ, Dragoš, ROWLINSON, Peter and SIMIĆ, Slobodan, 2009. An Introduction to the Theory of Graph Spectra. Cambridge University Press. ISBN 978-0-511-80151-8. [Accessed 5 March 2024].