Self-avoiding pruning random walk on signed network

Abstract

A signed network represents how a set of nodes are connected by two logically contradictory types of links: positive and negative links. In a signed products network, two products can be complementary (purchased together) or substitutable (purchased instead of each other). Such contradictory types of links may play dramatically different roles in the spreading process of information, opinion, behaviour etc. In this work, we propose a self-avoiding pruning (SAP) random walk on a signed network to model e.g. a user's purchase activity on a signed products network. A SAP walk starts at a random node. At each step, the walker moves to a positive neighbour that is randomly selected, the previously visited node is removed and each of its negative neighbours are removed independently with a pruning probability r. We explored both analytically and numerically how signed network topological features influence the key performance of a SAP walk: the evolution of the pruned network resulted from the node removals, the length of a SAP walk and the visiting probability of each node. These findings in signed network models are further verified in two real-world signed networks. Our findings may inspire the design of recommender systems regarding how recommendations and competitions may influence consumers' purchases and products' popularity.products where two products are connected if when a user is purchasing one product the other product is recommended 3 by the online retail platform like Amazon [24][25][26]. However, these models have not considered the substitute relations between products and the fact that once a product has been purchased, its substitutable products will be unlikely to be purchased afterwards. Hence, we propose in this work a self-avoiding pruning (SAP) walk on a signed network to model, e.g. a user's purchase behaviour on a signed network of products. As shown in figure 1, a SAP walk starts at a random node in a signed network at t=0. At each step, the walker moves from its current location node i to a positive neighbour 4 j that is randomly selected, its previous location, i.e. node i is removed from the signed network 5 and each of node iʼs negative neighbours is removed independently with probability r. The walker repeats such steps until there is no new location to move to. Since each node pair can be connected by either a positive or negative link, but not both, the walker could equivalently, at each step, remove each negative neighbour of its current visiting node i independently with probability r, then move to a random positive neighbour j and afterwards remove the previously visited node i.In the context of a signed product network, a SAP walk may model the purchase trajectory of a user on the network of products: initially, the user purchases a random product and afterwards buys a random complementary product of his/her previous purchase; however, the user will not buy the same product repetitively and unlikely buy the substitutable products of what he/she has bought. If the negative layer is...

Non-Markovian random walks characterize network robustness to nonlocal cascades

“…We name this stochastic process the self-avoiding teleporting random walk (SATRW). Notice that our model interpolates between percolation, which can be seen as a purely nonlocal phenomenon, if α = 1 and φ = 1 − t/N 0 , where N 0 1 is the initial network size, and the purely local process of a growing SARW when α = 0 [37][38][39].…”

Efficient network exploration by means of resetting self-avoiding random walkers

“…Yet, during the last years, some theoretical foothold has been gained, by addressing, for example, questions such as the connectivity constant µ and the mean length of the walks computed in several types of networks [30][31][32][33][34][35], as well as the degree distribution of the network not visited by the walker [36][37][38]. From an application viewpoint, SARWs have been used, for instance, to detect communities [39], to model purchase activity on signed network products [40] or to model overflow cascade spreading [38].…”

Optimal networks revealed by global mean first return time

“…Random walks are popular models with wide applications [1,2], which include target search [3][4][5], reaction kinetics [6][7][8], descriptions of financial markets [9][10][11] and polymer chains [12,13]. Random walks can also be used to model epidemic or opinion spreading [14][15][16], help predict the arrival time of diseases (or opinion) spreading on networks [17] and estimate the occurrence (or recurrence) of extreme events on the networks [18][19][20][21], and etc.…”