Contagion threshold and contagion probability in arbitrary finite networks / Keng Ying Ying

Keng , Ying Ying (2024) Contagion threshold and contagion probability in arbitrary finite networks / Keng Ying Ying. PhD thesis, Universiti Malaya.

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Abstract

Given a simple undirected connected network ⌧ = (+, ⇢), we study how likely a given seed set (will cause full contagion throughout the network, considering both heterogeneous and homogeneous distributions of adoption thresholds among nodes. We assume that the nodes in ( stay active throughout the spreading dynamics, while each non-seed E is active if and only if the fraction of its currently active neighbors is at least its adoption threshold q(E). The contagion probability ?2 (() of ( is the probability that full contagion occurs from ( when each node chooses its adoption threshold independently and uniformly at random from [0, 1]. By mapping each threshold function q : + ! [0, 1] to a point in the =-cube [0, 1]=, where = = |+|, we call q a contagious point of ( if q can induce full contagion from (. Thus ?2 (() is naturally the Lebesgue measure or the volume of the set of contagious points of ( in [0, 1]=. We derive a formula for ?2 ((), applicable to any seed set (. The formula leads to the observation that ?2 (() is the sum of the probabilities that influence from ( can spread to the entire network through each spanning tree of the quotient graph ⌧( of ⌧ in which the set ( is represented as a single node. As a specific corollary, the contagion probability of each node is directly proportional to its degree. In the homogeneous case, all nodes share the same adoption threshold that is chosen uniformly at random from [0, 1] and the likelihood of ( causing full contagion is quantified using the contagion threshold @2 (() of (, which is the relative length of the set of contagious points of ( along the main diagonal of [0, 1]=. In contrast to contagion probability, contagion threshold of individual nodes does not always preserve the vicinal preorder. More specifically, the inclusion of the closed (resp. open) neighborhood of a node D in that of another node E guarantees that @2 (D)  @2 (E) if the degree of D is at most 6 (resp. 3); however, this does not extend to higher degrees of D in some networks. Besides, we characterize the seed sets that have a sufficiently high contagion threshold. In addition, we make some comparisons between contagion probability and contagion threshold. We show that the presence of a cycle in ⌧( is a necessary but not sufficient condition for ?2 (() > @2 (().

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