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The spectral properties of traditional (dyadic) graphs, where an edge connects exactly two vertices, are widely studied in different applications. These spectral properties are closely connected to the structural properties of dyadic graphs. We generalize such connections and characterize higher-order networks by their spectral information. We first split the higher-order graphs by their “edge orders” into several uniform hypergraphs. For each uniform hypergraph, we extract the corresponding spectral information from the transition matrices of carefully designed random walks. From each spectrum, we compute the first few spectral moments and use all such spectral moments across different “edge orders” as the higher-order graph representation. We show that these moments not only clearly indicate the return probabilities of random walks but are also closely related to various higher-order network properties such as degree distribution and clustering coefficient. Extensive experiments show the utility of this new representation in various settings. For instance, graph classification on higher-order graphs shows that this representation significantly outperforms other techniques.
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Analytical results for the distribution of first return times of non-backtracking random walks on configuration model networks
Abstract We present analytical results for the distribution of first return (FR) times of non-backtracking random walks (NBWs) on undirected configuration model networks consisting of $$N$$ nodes with degree distribution $P(k)$. We focus on the case in which the network consists of a single connected component. Starting from a random initial node $$i$$ at time $t=0$, an NBW hops into a random neighbor of $$i$$ at time $t=1$ and at each subsequent step it continues to hop into a random neighbor of its current node, excluding the previous node. We calculate the tail distribution $$P ( T_{\rm FR} > t )$$ of first return times from a random initial node to itself. It is found that $$P ( T_{\rm FR} > t )$$ is given by a discrete Laplace transform of the degree distribution $P(k)$. This result exemplifies the relation between structural properties of a network, captured by the degree distribution, and properties of dynamical processes taking place on the network. Using the tail-sum formula, we calculate the mean first return time $${\mathbb E}[ T_{\rm FR} ]$$. Surprisingly, $${\mathbb E}[ T_{\rm FR} ]$$ coincides with the result obtained from Kac's lemma that applies to simple random walks (RWs). We also calculate the variance $${\rm Var}(T_{\rm FR})$$, which accounts for the variability of first return times between different NBW trajectories. We apply this formalism to Erd{\H o}s-R\'enyi networks, random regular graphs and configuration model networks with exponential and power-law degree distributions and obtain closed-form expressions for $$P( T_{\rm FR} > t )$$ as well as its mean and variance. These results provide useful insight on the advantages of NBWs over simple RWs in network exploration, sampling and search processes.
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- Award ID(s):
- 2102832
- PAR ID:
- 10649999
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Journal of Physics A: Mathematical and Theoretical
- ISSN:
- 1751-8113
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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