In many real situations, networks grow only via local interactions. New nodes are added to the growing network with only information pertaining to a small subset of existing nodes. Multilevel marketing, social networks, and diseases models can all be depicted as growing networks based on local distance information. In these examples, all nodes whose distance from a chosen center is less than d form a subgraph. We thence grow networks with information only from these subgraphs. Moreover, we use a likelihood-based method, where at each step we modify the networks in the way that will make the graph closer to the expected degree distribution. Combining local information and the likelihood method, we grow networks that exhibit novel features. We discover that the likelihood method, at certain parameter ranges, can generate networks with highly modulated communities, even when global information is not available. Communities and clusters are abundant in real-life networks, and the method proposed here provides a natural mechanism for the emergence of communities in scale-free networks. In addition, the algorithmic implementation of network growth via local information is substantially faster than global methods and allows for the exploration of much larger networks. Introduction. -Many systems in the physical and 1 biological realms exhibit behaviours governed by local 2 rather than global interaction. Growth of such systems 3 is often governed by local knowledge -when a new 4 node is added to the network, it is not provided with 5 perfect global information. Examples abound in different 6 fields [1, 2]. Social relationship, computers under virus 7 attack, and insects affected by infectious diseases all form 8 networks based on local evolving rules. 9 10 After Barabási and Albert's investigation into scale-free 11 networks [3], there has been a huge amount of work de-12 voted to the growth, properties and applications of these 13 networks. The mechanism of the growth, the hub, and 14 the size effect has been discussed relating to the scale-free 15 properties [4-7]. In contrast to growing networks from 16 a global view, local-world scale-free networks has the 17 advantage that it is, in many situations, closer to reality 18 [8]. Moreover, most of these networks prove to be 19 scale-free [9]. 20 21 47 defined way of choosing between candidate nodes. The 48 idea of likelihood is proposed and elucidated in [16] and 49 [17]. By referring to likelihood, we mean the probability 50 of a graph being selected P (G N ) among all the graphs 51 G N of N nodes. It is computed based on an underlying 52 probability distribution (power-law distribution) and the 53 degree sequence of G N . Here we explore the problem 54 with local information: at each step, we select a center 55 node and nodes within distance d from the center form a 56 subgraph. Then we introduce a mechanism to add nodes 57 or edges within the subgraph as long as the operation 58 increases P (G N ), the likelihood of the new graph -59 where likelihoods are only comput...
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